Figure 26 shows a typical example of a normalized power spectral density computed for an ensemble of 161 accelerograms recorded on alluvium. Studies (217, 218) have shown that strong motion segment of accelerograms constitutes a locally stationary random process and that the power spectral density can be presented as a timedependent function G(t,ω) in the form:
(217)
where S(t) is a slowly varying timescale factor which accounts for the local variation of the mean square acceleration with time. Power spectral density is useful not only as a measure of the frequency content of ground motion but also in estimating its statistical properties. Among such properties are the rms acceleration ψ, the central frequency ωc, and the shape factor δ defined as
ψ = λ0
(218)
ω c = λ2 / λ0
(219)
δ = 1 − (λ12 / λ0 λ2 )
(220)
2. Earthquake Ground Motion and Response Spectra
59
Figure 26. Normalized power spectral density of an ensemble of 161 horizontal components of accelerograms recorded on alluvium. [After Elghadamsi et al. (218).]
where λr is the rth spectral moment defined as
λr = ∫
ω0
ω r G (ω )dω
(221)
Smooth power spectral density of the ground acceleration has been commonly presented in the form proposed by Kanai (219) and Tajimi (220) as a filtered white noise ground excitation of spectral density G0 in the form
G (ω ) =
1 + 4ξg2 (ω / ω g ) 2
[1 − (ω / ω ) ] 2
g
2
+ (2ξg ω / ω g )
2
G0
(222) The KanaiTajimi parameters ξg, ωg, and G0 represent ground damping, ground frequency, and ground shaking intensity. These parameters are computed by equating the rms acceleration, the central frequency, and the shape factor,
Equations 218 to 220, of the smooth and the raw (unsmooth) power spectral densities (218, 221) . Table 23 gives the values of ξg, ωg, and G0 for the normalized power spectral densities on different soil conditions. Also shown are the central frequency ωc and the shape factor δ. Using the KanaiTajimi parameters in Table 23, normalized power spectral densities for horizontal and vertical motion on various soil conditions were computed and are presented in Figures 27 and 28. The figures indicate that as the site becomes stiffer, the predominant frequency increases and the power spectral densities spread over a wider frequency range. This observation underscores the influence of site conditions on the frequency content of seismic excitations. The figures also show that the power spectral densities for horizontal motion have a sharper peak and span over a narrower frequency region than the corresponding ones for vertical motion.
60
Chapter 2
Table 23. Central Frequency, Shape Factor, Ground Frequency, Ground Damping, and Ground Intensity for Different Soil Conditions. [After Elghadamsi et al. (218)] Site Category
No. of Records
Central Frequency fc (Hz)
Shape Factor δ
Ground Frequency fg (Hz)
Ground Damping ξg
Ground Intensity G0 (1/Hz)
Horizontal Alluvium Alluvium on rock Rock
161 60 26
4.10 4.58 5.41
0.65 0.59 0.59
2.92 3.64 4.30
0.34 0.30 0.34
0.102 0.078 0.070
Vertical Alluvium Alluvium on rock Rock
78 29 13
6.27 6.68 7.53
0.63 0.62 0.55
4.17 4.63 6.18
0.46 0.46 0.46
0.080 0.072 0.053
Figure 27. Normalized power spectral densities for horizontal motion. [After Elghadamsi et al. (218).]
Clough and Penzien (222) modified the KanaiTajimi power spectral density by introducing another filter to account for the numerical difficulties expected in the neighborhood of ω=0. The cause of these difficulties stems from dividing Equation 222 by ω2 and ω4, respectively, to obtain the power spectral density functions for ground velocity and displacement. The singularities close to ω=0 can be removed by passing the process through another filter that attenuates the very low frequency components. The modified power spectral density takes the form
G (ω ) =
1 + 4ξ g2 (ω / ω g )
2
[1 − (ω / ω ) ] + (2ξ ω / ω ) 2 2
g
2
g
×
(223)
g
(ω / ω1 )4 [1 − (ω / ω1 )2 ]+ 4ξ12 (ω / ω1 )2 (G0 ) Where ω1 and ξ1 are the frequency and damping parameters of the filter.
2. Earthquake Ground Motion and Response Spectra
61
Figure 28. Normalized power spectral densities for vertical motion. [After Elghadamsi et al. (218).]
Lai (221) presented empirical relationships for estimating ground frequency ωg and central frequency ωc for a given epicentral distance R in kilometers or local magnitude M L . These relationships are
ω g = 27 − 0.09 R
10 ≤ R ≤ 60
(224)
ω g = 65 − 7.5 M L 5 ≤ M L ≤ 7
(225)
ω g = 112 . ω c − 515 .
(226)
Using these relationships and the acceleration attenuation equations (see Section 2.4.1), Lai proposed a procedure for estimating a smooth power spectral density for a given strong motion duration and ground damping. Once the power spectral density of ground motion at a site is established, random vibration methods may be used to formulate probabilistic procedures for computing the response of structures. In addition, the power spectral density of ground motion may be used for other applications such as generating artificial accelerograms as discussed in Section 2.12.
2.4
FACTORS INFLUENCING GROUND MOTION
Earthquake ground motion is influenced by a number of factors. The most important factors are: 1) earthquake magnitude, 2) distance from the source of energy release (epicentral distance or distance from the causative fault), 3) local soil conditions, 4) variation in geology and propagation velocity along the travel path, and 5) earthquake source conditions and mechanism (fault type, slip rate, stress conditions, stress drop, etc.). Past earthquake records have been used to study some of these influences. While the effect of some of these parameters such as local soil conditions and distance from the source of energy release are fairly well understood and documented, the influence of source mechanism is under investigation and the variation of geology along the travel path is complex and difficult to quantify. It should be noted that several of these influences are interrelated; consequently, it is difficult to discuss them individually without incorporating the others. Some of the influences are discussed below:
62
Chapter 2
Figure 29. Peak ground acceleration plotted as a function of fault distance obtained from worldwide set of 515 strong motion records without normalization of magnitude. [After Donovan (223).]
2. Earthquake Ground Motion and Response Spectra 2.4.1
Distance
The variation of ground motion with distance to the source of energy release has been studied by many investigators. In these studies, peak ground motion, usually peak ground acceleration, is plotted as a function of distance. Smooth curves based on a regression analysis are fitted to the data and the curve or its equation is used to predict the expected ground motion as a function of distance. These relationships, referred to as motion attenuation, are sometimes plotted independently of earthquake magnitude. This was the case in the earlier studies because of the lack of sufficient number of earthquake records. With the availability of a large number of records, particularly since the 1971 San Fernando earthquake, the database for attenuation studies increased and a number of investigators reexamined their earlier studies, modified their proposed relationships for estimating peak accelerations, and included earthquake magnitude as a parameter. Donovan (223) compiled a database of more than 500 recorded accelerations from seismic events in the United States, Japan, and elsewhere and later increased it to more than 650 (224). The plot of peak ground acceleration versus fault distance for different earthquake magnitudes from his database is shown in Figure 29. Even though there is a considerable scatter in the data, the figure indicates that peak acceleration decreases as the distance from the source of energy release increases. Shown in the figure are the least square fit between acceleration and distance and the curves corresponding to mean plus and mean minus one and two standard deviations. Also presented in the figure is the envelope curve (dotted) proposed by Cloud and Perez (225). Other investigators have also proposed attenuation relationships for peak ground acceleration, which are similar to Figure 29. A summary of some of the relationships, compiled by Donovan (223) and updated by the authors, is shown in Table 24. A comparison of various relationships (224) for an earthquake magnitude
63 of 6.5 with the data from the 1971 San Fernando earthquake is shown in Figure 210. This figure is significant because it shows the comparison of various attenuation relationships with data from a single earthquake. While the figure shows the differences in various attenuation relationships, it indicates that they all follow a similar trend.
Figure 210. Comparison of attenuation relations with data from the San Fernando earthquake of February 9, 1971. [After Donovan (224).]
Housner (238), Donovan (223), and Seed and Idriss (239) have reported that at farther distances from the fault or the source of energy release (farfield), earthquake magnitude influences the attenuation, whereas at distances close to the fault (nearfield), the attenuation is affected by smaller but not larger earthquake magnitudes. This can be observed from the earthquake data in Figure 29.
64
Chapter 2
Table 24. Typical Attenuation Relationships Data Source 1. San Fernando earthquake February 9, 1971 2. California earthquake
Relationship*
Reference
log PGA = 190 / R
Donovan (223) Blume (226)
1.83 2
PGA = y 0 /(1 + ( R ' / h ) ) where log y0 = (b + 3) + 0.81M – 0.027M2 and b is a site factor
3. California and Japanese earthquakes PGA =
0.0051 10
Kanai (227)
( 0.61 M − p log R + 0.167 −1.83 / R )
TG where P = 1.66 + 3.60/R and TG is the fundamental period of the site 4. Cloud (1963) 5. Cloud (1963) 6. U.S.C. and G.S. 7. 303 Instrumental Values 8. Western U.S. records 9. U.S., Japan 10. Western U.S. records, USSR, and Iran
11. Western U.S. records and worldwide 12. Western U.S. records and worldwide 13. Western U.S. records
14. Italian records
15. Western U.S. and worldwide (soil sites)
PGA = 0.0069 e PGA = 1.254 e
1.64 M
0.8 M
/(1.1e
1.1 M
/( R + 25 )
Milne and Davenport (228) Esteva (229) Cloud and Perez (225)
2
+R )
2
log PGA = ( 6.5 − 2 log( R ' +80 )) / 981 PGA = 1.325 e
0.67 M
PGA = 0.0193 e PGA = 1.35 e
/( R + 25 )
0.8 M
0.58 M
/( R
2
Donovan (223)
1.6
+ 400 )
/( R + 25 )
1.52 0.732 M
ln PGA = −3.99 + 1.28 M − 1.75 ln[ R = 0.147 e ] M is the surface wave magnitude for M greater than or equal to 6, or it is the local magnitude for M less than 6. log PGA = −1.02 + 0.249 M − log
R
2
log PGA = 0.49 + 0.23 ( M − 6 ) − log
+ 7.3 R
2
2
− 0.00255 R
+8
2
− 0.0027 R
2
+ 7.3
2
+8
2
2
ln PGA = ln α ( M ) − β ( M ) ln( R + 20 ) M is the surface wave magnitude for M greater than or equal to 6, or it is the local magnitude for smaller M. R is the closest distance to source for M greater than 6 and hypocentral distance for M smaller than 6. α(M) and β(M) are magnitudedependent coefficients. 2
2
ln PGA = −1.562 + 0.306 M − log R + 5.8 + 0.169 S S is 1.0 for soft sites or 0.0 for rock. For M less than 6.5, 0.418 M ln PGA = −2.611 + 1.1M − 1.75 ln[ R + 0.822 e ]
Donovan (223) Donovan (223) Campbell (230)
Joyner and Boore (231) Joyner and Boore (232) Idriss (233)
Sabetta and Pugliese (234) Sadigh et al. (235)
For M greater than or equal to 6.5, ln PGA = −2.611 + 1.1M − 1.75 ln[ R + 0.316 e 16. Western U.S. and worldwide (rock sites)
0.629 M
] Sadigh et al. (235)
For M less than 6.5, ln PGA = −1.406 + 1.1M − 2.05 ln[ R + 1.353 e
0.406 M
]
For M greater than or equal to 6.5, ln PGA = −1.406 + 1.1M − 2.05 ln[ R + 0.579 e 17. Worldwide earthquakes
ln PGA = −3.512 + 0.904 M − 1.328 ln
2
0.537 M
R + [ 0.149 e
] Campbell and Bozorgnia (236)
0.647 M 2
]
+ [1.125 − 0.112 ln R − 0.0957 M ] F + [ 0.440 − 0.171 ln R ] S sr + [ 0.405 − 0.222 ln R ] S hr 18. Western North American earthquakes
F = 0 for strikeslip and normal fault earthquakes and 1 for reverse, reverseoblique, and thrust fault earthquakes. Ssr = 1 for soft rock and 0 for hard rock and alluvium Shr = 1 for hard rock and 0 for soft rock and alluvium
ln PGA = b + 0.527 ( M − 6.0 ) − 0.778 ln
R
2
+ ( 5.570 )
2
− 0.371 ln
Vs
Boore et al. (237)
1396 where b = 0.313 for strikeslip earthquakes = 0.117 for reverseslip earthquakes = 0.242 if mechanism is not specified Vs is the average shear wave velocity of the soil in (m/sec) over the upper 30 meters The equation can be used for magnitudes of 5.5 to 7.5 and for distances not greater than 80 km * Peak ground acceleration PGA in g, source distance R in km, source distance R’ in miles, local depth h in miles, and earthquake magnitude M. Refer to the relevant references for exact definitions of source distance and earthquake magnitude. 
2. Earthquake Ground Motion and Response Spectra
65
Figure 211. Strong motion stations in the Imperial Valley, California. [After Porcella and Matthiesen (240); reproduced from (239).]
Figure 212. Observed and predicted mean horizontal peak accelerations for the Imperial Valley earthquake of October 15, 1979 plotted as a function of distance from the fault. The solid curve represents the median predictions based on the observed values and the dashed curves represent the standard error bounds for the regression. [After Campbell (230).]
The majority of attenuation relationships for predicting peak ground motion are presented in terms of earthquake magnitude. Prior to the Imperial Valley earthquake of 1979, the vast majority of available accelerograms were recorded at distances of greater than approximately 10 or 15 km from the source of energy release. An array of accelerometers placed on both sides of the Imperial Fault (240) prior to this earthquake (See Figure 211) provided excellent acceleration data for small distances from the fault. The attenuation relationship from this array presented by Campbell (230) is shown in Figure 212. The figure indicates the flat slope of the acceleration attenuation curve for distances close to the source, a phenomenon which is not observed in
66
Chapter 2
Figure 213. Predicted values of peak horizontal acceleration for 50 and 84 percentile as functions of distance and moment magnitude. [After Joyner and Boore (231).]
Figure 214. Comparison of attenuation curves for the eastern and western U.S. earthquakes. (Reproduced from 239.)
the attenuation curves for farfield data. Similar observations can also be made from the attenuation curves (Figure 213) proposed by Joyner and Boore (231). The majority of attenuation studies and the relationships presented in Table 24 are primarily from the data in the western United States. Several seismologists believe that ground acceleration attenuates more slowly in the eastern United States and eastern Canada, i.e. earthquakes in eastern North America are felt at much greater distances from the epicenter than western earthquakes of similar magnitude. A comparison of the attenuation curves for the western and eastern United States earthquakes recommended by Nuttli and Herrmann (241) is shown in Figure 214. Another comparison for eastern North America prepared by Milne and Davenport (228) is presented in Figure 215. Both these figures reflect the slower attenuation of earthquake motions in the eastern United States and eastern Canada. According to Donovan (223), a similar phenomenon also exists for Japanese earthquakes. Due to the lack of
2. Earthquake Ground Motion and Response Spectra
67
ln( PGA) = −3.512 + 0.904 M W
sufficient earthquake data in the eastern United States and Canada, theoretical models which include earthquake source and wave propagation in the surrounding medium are used to study the effect of distance and other parameters on ground motion. The reader is referred to references (242 to 244) for the detailed procedure.
− 1.328 ln R s2 + [0.149 exp(0.647 M W )] 2 + [1.125 − 0.112 ln R s − 0.0957 M W ]F + [0.440 − 0.171 ln R s ]S sr + [0.405 − 0.222 ln R s ]S hr + ε (227) where PGA is the mean of the two horizontal components of peak ground acceleration (g), MW is the moment magnitude, RS is3 the closest distance to the seismogenic rupture on the fault (km), F = 0 for strikeslip and normal fault earthquakes and = 1 for reverse, reverseoblique, and thrust fault earthquakes, Ssr = 1 for soft rock and = 0 for hard rock and alluvium, Shr = 1 for hard rock and = 0 for soft rock and alluvium, and ε is the random error term with aσ zero mean and a standard deviation equal to ln(PGA) which is represented by
σ ln( PGA)
0.55 = 0.173 − 0.140 ln( PGA) 0.39
PGA < 0.068 0.068 ≤ PGA ≤ 0.21 PGA > 0.21
(228) Figure 215. Intensity versus distance for eastern and western Canada. [After Milne and Davenport (228).]
In addition to source distance and earthquake magnitude, recent attenuation relationships include the effect of source characteristics (fault mechanism) and soil conditions. As an example, Campbell and Bozorgnia (236) used 645 accelerograms from 47 worldwide earthquakes of magnitude 4.7 and greater, recorded between 1957 and 1993, to develop attenuation relationship for peak horizontal ground acceleration. The data was limited to distances of 60 km or less to minimize the influence of regional differences in crustal attenuation and to avoid the complex propagation effects at farther distances observed during the 1989 Loma Prieta and other earthquakes. The peak ground acceleration was estimated using a generalized nonlinear regression analysis and given by
with a standard error of estimate 0.021. More recently, Boore et al. (237) used approximately 270 records to estimate the peak ground acceleration in terms of 1) the closest horizontal distance Rjb (km) from the recording station to a point on the earth surface that lies directly above the rupture, 2) the moment magnitude MW, 3) the average shear wave velocity of the soil Vs (m/sec) over the upper 30 meters, and 4) the fault mechanism such that: ln( PGA) = b + 0.527 ( M W − 6.0) − 0.778 ln
3
Vs 2 2 R jb + (5.570) − 0.371 ln 1396
(229)
Seismogenic rupture zone was determined from the location of surface fault rupture, the spatial distribution of aftershocks, earthquake modelling studies, regional crustal velocity profiles, and geodetic and geologic data.
68
Chapter 2
Where b is a parameter that depends on the fault mechanism. They recommended
 0.313 for the strike  slip earthquakes b =  0.117 for the reverse  slip earthquakes  0.242
(230)
if the fault mechanism is not specified
Figure 216. Peak ground acceleration versus distance for soil sites for earthquake magnitudes of 6.5 and 7.5. [After Boore et al. (237).]
Equation 229 is used for earthquake magnitudes of 5.5 to 7.5 and distances less than 80 km. Although Equations 227 and 229 use different definitions for the source distance, the equations indicate the decaying pattern of the peak ground acceleration with distance. Figure 216 shows the variation of the peak ground acceleration with distance computed from Equation 229 for earthquakes of magnitude 6.5 and 7.5 with an unspecified fault mechanism and for soils with a shear wave velocity of 310 m/sec. Also shown in the figure is the attenuation relationship proposed by Joyner and Boore (232) (Equation 12 Table 24). The variation of peak ground velocity with distance from the source of energy release (velocity attenuation) has also been studied by several investigators such as Page et al. (211), Boore et al. (245, 246), Joyner and Boore (231), and Seed and Idriss (239). Velocity attenuation curves have similar shapes and follow similar trends as the acceleration attenuation. Typical velocity attenuation curves proposed by Joyner and Boore are shown in Figure 217. Comparisons between Figures 213 and 217 indicate that velocity attenuates somewhat faster than acceleration. The variation of peak ground displacement with fault distance or the distance from the
Figure 217. Predicted values of peak horizontal velocity for 50 and 84 percentile as functions of distance, moment magnitude, and soil condition. [After Joyner and Boore (231).]
2. Earthquake Ground Motion and Response Spectra
69
Figure 218. Duration versus epicentral distance and magnitude for soil. [After Change and Krinitzsky (247).]
source of energy release (displacement attenuation) can also be plotted. Boore et al. (245, 246) have presented displacement attenuations for different ranges of earthquake magnitude. Only a few studies have addressed displacement attenuations probably because of their limited use and the uncertainties in computing displacements accurately. Distance also influences the duration of strong motion. Correlations of the duration of strong motion with epicentral distance have been studied by Page et al. (211), Trifunac and Brady (213), Chang and Krinitzsky (247), and others. Page et al., using the bracketed duration, conclude that for a given magnitude, the duration decreases with an increase in distance from the source. Chang and Krinitzsky, also using the bracketed duration, presented the curves shown in Figures 218 and 219 for estimating durations for soil and rock as a function of distance. These figures show that
for a given magnitude, the duration of strong motion in soil is greater (approximately two times) than that in rock. Using the 90% contribution of the acceleration intensity (∫ a2 dt) as a measure of duration, Trifunac and Brady (213) concluded that the average duration in soil is approximately 1012 sec longer than that in rock. They also observed that the duration increases by approximately 1.0  l.5 sec for every 10 km increase in source distance. Although there seems to be a contradiction between their finding and those of Page et al. and ChangKrinitzsky, the contradiction stems from using two different definitions. The bracketed duration is based on an absolute acceleration level (0.05g). At longer epicentral distances, the acceleration peaks are smaller and a shorter duration is to be expected. The acceleration intensity definition of duration is based on the relative measure of the percentile
70
Chapter 2
Figure 219. Duration versus epicentral distance and magnitude for rock. [After Chang and Krinitzsky (247).]
contribution to the acceleration intensity. Conceivably, a more intense shaking within a shorter time may result in a shorter duration than a much less intense shaking over a longer time. According to Housner (238), at distances away from the fault, the duration of strong shaking may be longer but the shaking will be less intense than those closer to the fault. Recently, Novikova and Trifunac (248) used the frequency dependent definition of duration developed by Trifunac and Westermo (216) to study the effect of several parameters on the duration of strong motion. They employed a regression analysis on a database of 984 horizontal and 486 vertical accelerograms from 106 seismic events. Their study indicated an increase in duration by 2 sec for each 10 km of epicentral distance for low frequencies (near 0.2 Hz). At high frequencies (15 to 20 Hz), the increase in duration drops to 0.5 sec per each 10 km. NearSource Effects. Recent studies have indicated that nearsource ground motions
contain large displacement pulses (ground displacements which are attained rapidly with a sharp peak velocity). These motions are the result of stress waves moving in the same direction as the fault rupture, thereby producing a longduration pulse. Consequently, near source earthquakes can be destructive to structures with long periods. Hall et al. (249) have presented data of peak ground accelerations, velocities, and displacements from 30 records obtained within 5 km of the rupture surface. The ground accelerations varied from 0.31g to 2.0g while the ground velocities ranged from 0.31 to 1.77 m/sec. The peak ground displacements were as large as 2.55 m. Figures 220 and 221 offer two examples of nearsource earthquake ground motions. The first was recorded at the LADWP Rinaldi Receiving Station during the Northridge earthquake of January 17, 1994. The distance from the recording station to the surface projection of the rupture was less than 1.0 km. The figure shows a unidirectional ground
2. Earthquake Ground Motion and Response Spectra
71
Figure 220. Ground acceleration, velocity, and displacement timehistories recorded at the LADWP Rinaldi Receiving Station during the Northridge earthquake of January 17, 1994.
displacement that resembles a smooth step function and a velocity pulse that resembles a finite delta function. The second example, shown in Figure 221, was recorded at the SCE Lucerne Valley Station during the Landers earthquake of June 28, 1992. The distance from the recording station to the surface projection of the rupture was approximately 1.8 km. A positive and negative velocity pulse that resembles a single longperiod harmonic motion is reflected in the figure. Nearsource ground displacements similar to that shown in Figure 221 have also been observed with a zero permanent displacement. The two figures clearly show the nearsource ground displacements caused by sharp velocity pulses. For further details, the reader is referred to the work of Heaton and Hartzell (250) and Somerville and Graves (251).
2.4.2
Site geology
Soil conditions influence ground motion and its attenuation. Several investigators such as Boore et al. (245 and 246) and Seed and Idriss (239) have presented attenuation curves for soil and rock. According to Boore et al., peak horizontal acceleration is not appreciably affected by soil condition (peak horizontal acceleration is nearly the same for both soil and rock). Seed and Idriss compare acceleration attenuation for rock from earthquakes with magnitudes of approximately 6.6 with acceleration attenuation for alluvium from the 1979 Imperial Valley earthquake (magnitude 6.8). Their comparison shown in Figure 222 indicates that at a given distance from the source of energy release, peak accelerations on rock are somewhat greater than those on alluvium. Studies from other earthquakes indicate that this is generally the case for
Figure 221. Ground acceleration, velocity, and displacement timehistories recorded at the SCE Lucerne Valley Station during the Landers earthquake of June 28, 1992.
Figure 222. Comparison of attenuation curves for rock sites and the Imperial Valley earthquake of 1979. [After Seed and Idriss (239).]
acceleration levels greater than approximately 0.1g. At levels smaller than this value, accelerations on deep alluvium are slightly greater than those on rock. The effect of soil condition on peak acceleration is illustrated by
Seed and Idriss in Figure 223. According to this figure, the difference in acceleration on rock and on stiff soil is not that significant. Even though in specific cases, particularly soft soils, soil condition can affect peak accelerations, Seed and Idriss conclude that the influence of soil condition can generally be neglected when using acceleration attenuation curves. In a more recent study, Idriss (252), using the data from the 1985 Mexico City and the 1989 Loma Prieta earthquakes, modified the curve for soft soil sites as shown in Figure 224. In these two earthquakes, soft soils exhibited peak ground accelerations of almost 1.5 to 4 times those of rock for the acceleration range of 0.05g to 0.1g. For rock accelerations larger than approximately 0.1g, the acceleration ratio between soft soils and rock tends to decrease to about 1.0 for rock accelerations of 0.3g to 0.4g. The figure indicates that large rock accelerations are amplified through soft soils to a lesser degree and may even be slightly deamplified.
2. Earthquake Ground Motion and Response Spectra
Figure 223. Relationship between peak accelerations on rock and soil. [After Seed and Idriss (239).]
Figure 224. Variation of peak accelerations on soft soil compared to rock for the 1985 Mexico City and the 1989 Loma Prieta earthquakes. [After Idriss (252).]
Figure 225. Variation of site amplification factors (ratio of peak ground acceleration on rock to that on alluvium) with distance. [After Campbell and Bozorgnia (236).]
73 The effect of site geology on peak ground acceleration can be seen in Equation 227 proposed by Campbell and Bozorgnia (236). The ratios of peak ground acceleration on soft rock and on hard rock to that on alluvium (defined as site amplification factors) were computed from Equation 227 and are shown in Figure 225. The figure indicates that rock sites have higher accelerations at shorter distances and lower accelerations at longer distances as compared to alluvium sites, with ground accelerations on soft rock consistently higher than those on hard rock. Recent studies on the influence of site geology on ground motion use the average shear wave velocity to identify the soil category. Boore et al. (237) used the average shear wave velocity for the upper 30 meters of the soil layer to characterize the soil condition in the attenuation relationship in Equation 229. The equation indicates that for the same distance, magnitude, and fault mechanism, as the soil becomes stiffer (i.e. a higher shear wave velocity), the peak ground acceleration becomes smaller. The recent UBC code and NEHRP recommended provisions use shear wave velocities to identify the different soil profiles with a shear wave velocity of 1500 m/sec or greater defining hard rock and a shear wave velocity of 180 m/sec or smaller defining soft soil (Section 2.9). There is a general agreement among various investigators that the soil condition has a pronounced influence on velocities and displacements. According to Boore et al. (245 and 246) , Joyner and Boore (231), and Seed and Idriss (239) ; larger peak horizontal velocities are to be expected for soil than rock. A statistical study of earthquake ground motion and response spectra by Mohraz (253) indicated that the average velocity to acceleration ratio for records on alluvium is greater than the corresponding ratio for rock. Using the frequency dependent definition of duration proposed by Trifunac and Westermo (216) , Novikova and Trifunac (248) determined that for the same epicentral distance and earthquake magnitude, the strong motion duration for
74
Chapter 2
records on a sedimentary site is longer than that on a rock site by approximately 4 to 6 sec for frequencies of 0.63 Hz and by about 1 sec for frequencies of 2.5 Hz. The records on intermediate sites, furthermore, exhibited a shorter duration than those on sediments. They indicated that for frequencies of 0.63 to 21 Hz, the influence of the soil condition on the duration is noticeable. 2.4.3
Magnitude
Different earthquake magnitudes have been defined, the more common being the Richter magnitude (local magnitude) M L , the surface wave magnitude M S , and the moment magnitude M W (see Chapter 1). As expected, at a given distance from the source of energy release, large earthquake magnitudes result in large peak ground accelerations, velocities, and displacements. Because of the lack of adequate data for earthquake magnitudes greater than 7.5, the influence of the magnitude on peak ground motion and duration is generally determined through extrapolation of data from earthquake magnitudes smaller than 7.5. Attenuation relationships are also presented as a function of magnitude for a given source distance as indicated in Equations 227 and 229. Both equations show that for a given distance, soil condition, and fault mechanism, the larger the earthquake magnitude, the larger is the peak ground acceleration. Figure 216, plotted using Equation 229, confirms this observation. The influence of earthquake magnitude on the duration of strong motion has been studied by several investigators. Housner (238 and 254) presents values for maximum acceleration and duration of strong phase of shaking in the vicinity of a fault for different earthquake magnitudes (Table 25). Donovan (223) presents the linear relationship in Figure 226 for estimating duration in terms of magnitude. His estimates compare closely with those presented by Housner in Table 25. Using the bracketed duration (0.05g), Page et al. (211) give estimates of duration for various earthquake magnitudes near a fault (Table 26). Chang and Krinitzsky
(247)
give approximate upperbound for duration for soil and rock (Table 27). Their values for soil are close to those presented by Page et al., and the ones for rock are consistent with those given by Housner and by Donovan. The study by Novikova and Trifunac (248) which uses the frequency dependent definition of duration presents a quadratic expression for the duration in terms of earthquake magnitude. Their study indicates that the duration of strong motion does not depend on the earthquake magnitude at frequencies less than 0.25 Hz. For higher frequencies, the duration increases exponentially with magnitude.
Figure 226. Relationship between magnitude and duration of strong phase of shaking. [After Donovan (223).]
Table 25. Maximum Ground Accelerations and Durations of Strong Phase of Shaking [after Housner (254)] Magnitude Maximum Duration Acceleration (%g) (sec) 5.0 9 2 5.5 15 6 6.0 22 12 6.5 29 18 7.0 37 24 7.5 45 30 8.0 50 34 8.5 50 37
2. Earthquake Ground Motion and Response Spectra Table 26. Duration of Strong Motion Near Fault [after Page et al. (211)] Magnitude Duration (sec) 5.5 10 6.5 17 7.0 25 7.5 40 8.0 60 8.5 90 Table 27. Strong Motion Duration for Different Earthquake Magnitudes [after Chang and Krinitzsky (247)] Magnitude Rock Soil 5.0 4 8 5.5 6 12 6.0 8 16 6.5 11 23 7.0 16 32 7.5 22 45 8.0 31 62 8.5 43 86
2.4.4
75 earthquakes. Joyner and Boore (259) believe this ratio should be 1.25. Recent attenuation relationships include the effects of fault mechanism on ground motion as indicated in Equations 227 and 229. Equation 227 by Campbell and Bozorgnia (236) indicates that reverse, reverseoblique, and thrust fault earthquakes result in larger ground accelerations than strikeslip and normal fault earthquakes. Figure 227, computed from Equation 227, shows the variation of peak ground acceleration with distance for earthquakes with different magnitudes and fault mechanisms on alluvium. Similar observations can also be made from Equation 229 by Boore et al. (237) where reverse earthquakes result in higher accelerations than strikeslip earthquakes.
Source characteristics
Factors such as fault mechanism, depth, and repeat time have been suggested by several investigators as being important in determining ground motion amplitudes because of their relation to the stress state at the source or to stress changes associated with the earthquake. Based on the state of stress in the vicinity of the fault, many investigators believe that large ground motions are associated with reverse and thrust faults whereas smaller ground motions are related to normal and strikeslip faults. The above observations agree with the study by McGarr (255, 256) who concluded that ground acceleration from reverse faults should be greater than those from normal faults, with strikeslip faults having intermediate accelerations. McGarr also believes that ground motions increase with fault depth. Kanamori and Allen (257) presented data showing that higher ground motions are associated with faults with longer repeat times since they experience large average stress drops. Using empirical equations, Campbell (258) found that peak ground acceleration and velocity in reverseslip earthquakes are larger by about 1.4 to 1.6 times than those in strikeslip
Figure 227. Peak ground acceleration versus distance for different magnitudes and fault mechanisms. [After Campbell and Bozorgnia (236).]
2.4.5
Directivity
Directivity relates to the azimuthal variation of the angle between the direction of rupture propagation (or radiated seismic energy) and sourcetosite vector, and its effect on earthquake ground motion. Large ground accelerations and velocities can be associated with small angles since a significant portion of the seismic energy is channeled in the direction of rupture propagation. Consequently, when a large urban area is located within the small angle, it will experience severe damage.
76
Chapter 2
According to Faccioli (260), in the Northridge earthquake of January 17, 1994; the rupture propagated in the direction opposite from downtown Los Angeles and San Fernando Valley, causing moderate damage. In the HyogokenNanbu (Kobe) earthquake of January 17, 1995, the rupture was directed toward the densely populated City of Kobe resulting in significant damage. The stations that lie in the direction of the earthquake rupture propagation will record shorter strong motion durations than those located opposite to the direction of propagation (261). Boatwright and Boore (262), believe that directivity can significantly affect strong ground motion by a factor of up to 10 for ground accelerations. Joyner and Boore (259) indicate, however, that it is not clear how to incorporate directivity into methods for predicting ground motion in future earthquakes since the angle between the direction of rupture propagation and the sourcetorecordingsite vector is not known a priori. Moreover, for sites close to the source of a large magnitude earthquake, where a reliable estimate of ground motion is important, the angle changes during the rupture propagation. Most ground motion prediction studies do not explicitly include a variable representing directivity.
2.5
EVALUATION OF SEISMIC RISK AT A SITE
Evaluating seismic risk is based on information from three sources: 1) the recorded ground motion, 2) the history of seismic events in the vicinity of the site, and 3) the geological data and fault activities of the region. For most regions of the world this information, particularly from the first source, is limited and may not be sufficient to predict the size and recurrence intervals of future earthquakes. Nevertheless, the earthquake engineering community has relied on this limited information to establish some acceptable levels of risk. The seismic risk analysis usually begins by developing mathematical models, which are
used to estimate the recurrence intervals of future earthquakes with certain magnitude and/or intensity. These models together with the appropriate attenuation relationships are commonly utilized to estimate ground motion parameters such as peak acceleration and velocity corresponding to a specified probability and return period. Among the earthquake recurrence models mostly used in practice is the GutenbergRichter relationship (263, 264) known as the Richter law of magnitude which states that there exists an approximate linear relationship between the logarithm of the average number of annual earthquakes and earthquake magnitude in the form
log N (m) = A − Bm
(231)
where N(m) is the average number of earthquakes per annum with a magnitude greater than or equal to m, and A and B are constants determined from a regression analysis of data from the seismological and geological studies of the region over a period of time. The GutenbergRichter relationship is highly sensitive to magnitude intervals and the fitting procedure used in the regression analysis (265, 266, 233) . Figure 228 shows a typical plot of the GutenbergRichter relationship presented by Schwartz and Coppersmith (266) for the southcentral segment of the San Andreas Fault. The relationship was obtained from historical and instrumental data in the period 19001980 for a 40kilometer wide strip centered on the fault. The box shown in the figure represents recurrence intervals based on geological data for earthquakes of magnitudes 7.58.0 (267). It is apparent from the figure that the extrapolated portion of the GutenbergRichter equation (dashed line) underestimates the frequency of occurrence of earthquakes with large magnitudes, and therefore, the model requires modification of the Bvalue in Equation 231 for magnitudes greater than approximately 6.0 (233) .
2. Earthquake Ground Motion and Response Spectra
77 where Ai and Bi are known constants for the ith subsource. 3. Assuming that the design ground motion is specified in terms of the peak ground acceleration a and the epicentral distance from the ith subsource to the site is Ri, the magnitude ma,i of an earthquake initiated at this subsource may be estimated from
ma , i = f ( Ri , a )
(233)
where f(Ri, a) is a function which can be obtained from the attenuation relationships. Substituting Equation 233 into Equation 232, one obtains
log N i (ma ,i ) = Ai − Bi [ f ( Ri , a )]
Figure 228. Cumulative frequencymagnitude plot. The box in the figure represents range of recurrence based on geological data for earthquake magnitudes of 7.58. [After Schwartz and Coppersmith (266); reproduced from Idriss (233).]
Cornell (268) introduced a simplified method for evaluating seismic risk. The method incorporates the influence of all potential sources of earthquakes. His procedure as described by Vanmarcke (269) can be summarized as follows: 1. The potential sources of seismic activity are identified and divided into smaller subsources (point sources). 2. The average number of earthquakes per annum Ni(m) of magnitudes greater than or equal to m from the ith subsource is determined from the GutenbergRichter relationship (Equation 231) as
log N i (m) = Ai − Bi m
(232)
(234)
Assuming the seismic events are independent (no overlapping), the total number of earthquakes per annum Na which may result in a peak ground acceleration greater than or equal to a is obtained from the contribution of each subsource as
N a = ∑ N i (ma ,i )
(235)
all
4. The mean return period Ta in years is obtained as
Ta =
1 Na
(236)
In the above expression, Na can be also interpreted as the average annual probability λa that the peak ground acceleration exceeds a certain acceleration a. In a typical design situation, the engineer is interested in the probability that such a peak exceeds a during the life of structure tL. This probability can be estimated using the Poisson distribution as
P = 1 − e − λa t L
(237)
78 Another distribution based on a Bayesian procedure (270) was proposed by Donovan (223). The distribution is more conservative than the Poisson distribution, and therefore more appropriate when additional uncertainties such
Chapter 2 as those associated with the long return periods of large magnitude earthquakes are encountered. It should be noted that other ground motion parameters in lieu of acceleration such as spectral ordinates may be
Figure 229. Instrumental or estimated epicentral locations within 100 kilometers of San Francisco. [After Donovan (223).]
2. Earthquake Ground Motion and Response Spectra used for evaluating seismic risk. Other procedures for seismic risk analysis based on more sophisticated models have also been proposed (see for example Der Kiureghian and Ang, 271). The evaluation of seismic risk at a site is demonstrated by Donovan (223) who used as an example the downtown area of San Francisco. The epicentral data and earthquake magnitudes he considered in the evaluation were obtained over a period of 163 years and are depicted in Figure 229. The data is associated with three major faults, the San Andreas, Hayward, and Calaveras. Using attenuation relationships for competent soil and rock, Donovan computed the return periods for different peak accelerations (see Table 28). He then computed the probability of exceeding various peak ground accelerations during a fiftyyear life of the structure which is shown in Figure 230. Plots such as those in Figure 230 may be used to estimate the peak acceleration for various probabilities. For example, if the structure is to be designed to resist a moderate earthquake with a probability of 0.6 and a severe earthquake with a probability of between 0.1 and 0.2 of occurring at least once during the life of the structure, the peak accelerations using Figure 230(b) for rock, are 0.15g and 0.4g, respectively.
79 map, which shows contours of peak acceleration on rock having a 90% probability of not being exceeded in 50 years. The Applied Technology Council ATC (274) used this map to develop similar maps for effective peak acceleration (Figure 232) and effective peak velocityrelated acceleration (Figure 233). The effective peak acceleration Aa and the effective peak velocityrelated acceleration Av are defined by the Applied Technology Council (274) based on a study by McGuire (275). They are obtained by dividing the spectral accelerations between periods of 0.1 to 0.5 sec and the spectral
Table 28. Return Periods for Peak Ground Acceleration in the San Francisco Bay Area [after Donovan (223)] Peak Return Period (years) Acceleration Soil Rock 0.05 4 8 0.10 20 30 0.15 50 60 0.20 100 100 0.25 250 200 0.30 450 300 0.40 2000 700
2.5.1
Development of seismic maps
Using the seismic risk principles of Cornell , Algermissen and Perkins (272, 273) developed isoseismal maps for peak ground accelerations and velocities. Figure 231 is a (268)
Figure 230. Estimated probabilities for a fifty year project life. [After Donovan (223).]
Figure 231. Seismic risk map developed by Algermissen and Perkins. (Reproduced from 274.)
2. Earthquake Ground Motion and Response Spectra
Figure 232. Contour map for effective peak acceleration (ATC, 274).
81
Figure 232. (continued)
Figure 233. Contour map of effective peak velocityrelated acceleration (ATC, 274).
Figure 233. (continued)
velocity at a period of approximately 1.0 sec by a constant amplification factor (2.5 for a 5% damped spectrum). It should be noted that the effective peak acceleration will generally be smaller than the peak acceleration while the effective peak velocityrelated acceleration is generally greater than the peak velocity (275). The Aa and Av maps developed from the ATC study are in many ways similar to the AlgermissenPerkins map. The most significant difference is in the area of highest seismicity in California. Within such areas, the AlgermissenPerkins map has contours of 0.6g whereas the ATC maps have no values greater than 0.4g. This discrepancy is due to the difference between peak acceleration and effective peak acceleration and also to the decision by the participants in the ATC study to limit the design value to 0.4g based on scientific knowledge and engineering judgment. The ATC maps were also provided with the contour lines shifted to coincide with the county boundaries. The 1985, 1988, 1991 and 1994 National Earthquake Hazard Reduction Program (NEHRP) Recommended Provisions for Seismic Regulations for New Buildings (276 to 279) include the ATC Aa and Av maps which correspond to a 10% probability of the ground motion being exceeded in 50 years (a return period of 475 years). The 1991 NEHRP provisions (278) also introduced preliminary spectral response acceleration maps developed by the United States Geological Survey (USGS) for a 10% probability of being exceeded in 50 years and a 10% probability of being exceeded in 250 years (a return period of 2,375 years). These maps, which include elastic spectral response accelerations corresponding to 0.3 and 1.0 sec periods, were introduced to present new and relevant data for estimating spectral response accelerations and reflect the variability in the attenuation of spectral acceleration and in fault rupture length (278). The 1997 NEHRP recommended provisions (280) provide seismic maps for the spectral response accelerations at the short period range (approximately 0.2 sec) and at a
period of 1.0 sec. The maps correspond to the maximum considered earthquake, defined as the maximum level of earthquake ground shaking that is considered reasonable for design of structures. In most regions of the United States, the maximum considered earthquake is defined with a uniform probability of exceeding 2% in 50 years (a return period of approximately 2500 years). It should be noted that the use of the maximum considered earthquake was adopted to provide a uniform protection against collapse at the design ground motion. While the conventional approach in earlier editions of the provisions provided for a uniform probability that the design ground motion will not be exceeded, it did not provide for a uniform probability of failure for structures designed for that ground motion. The design ground motion in the 1997 NEHRP provisions is based on a lower bound estimate of the margin against collapse which was judged, based on experience, to be 1.5. Consequently, the design earthquake ground motion was selected at a ground shaking level that is 1/1.5 or 2/3 of the maximum considered earthquake ground motion given by the maps. The 1997 NEHRP Guidelines for the Seismic Rehabilitation of Buildings (281), known as FEMA273, introduce the concept of performancebased design. For this concept, the rehabilitation objectives are statements of the desired building performance level (collapse prevention, life safety, immediate occupancy, and operational) when the building is subjected to a specified level of ground motion. Therefore, multiple levels of ground shaking need to be defined by the designer. FEMA273 provides two sets of maps; each set includes the spectral response accelerations at short periods (0.2 sec) and at long periods (1.0 sec). One set corresponds to a 10% probability of exceedance in 50 years, known as Basic Safety Earthquake 1 (BSE1), and the other set corresponds to a 2% probability of exceedance in 50 years, known as Basic Safety Earthquake 2 (BSE2), which is similar to the Maximum Considered Earthquake of the 1997 NEHRP provisions (280) . FEMA273 also presents a method for
86
Chapter 2
adjusting the mapped spectral accelerations for other probabilities of exceedance in 50 years using the spectral accelerations at 2% and 10% probabilities. The Aa and Av maps, developed during the ATC study, were also used, after some modifications, in the development of a single seismic map for the 1985, 1988, 1991, 1994, and 1997 editions of the Uniform Building Code (282 to 286). The UBC map shows contours for five seismic zones designated as 1, 2A, 2B, 3, and 4. Each seismic zone is assigned a zone factor Z, which is related to the effective peak acceleration. The Z factors for the five zones are 0.075, 0.15, 0.20, 0.30, and 0.40 for zones 1, 2A, 2B, 3, and 4; respectively. The only change in the UBC seismic map occurred in the 1994 edition (285) reflecting new knowledge regarding the seismicity of the Pacific Northwest of the United States.
2.6
ESTIMATING GROUND MOTION
In the late sixties and early seventies, the severity of the ground motion was generally specified in terms of peak horizontal ground acceleration. Most attenuation relationships were developed for estimating the expected peak horizontal acceleration at the site. Although structural response and to some extent damage potential to structures can be related to peak ground acceleration, the use of the peak acceleration for design has been questioned by several investigators on the premise that structural response and damage may relate more appropriately to effective peak acceleration Aa and effective peak velocityrelated acceleration Av. Early Studies by Mohraz et al. (29), Mohraz (253), Newmark and Hall (287), and Newmark et al. (288) recommended using ground velocity and displacement, in addition to ground acceleration, in defining spectral shapes and ordinates. Prior to the 1971 San Fernando earthquake where only a limited number of records was available, Newmark and Hall (289, 290)
recommended that a maximum horizontal ground velocity of 48 in/sec and a maximum horizontal ground displacement of 36 in. be used for a unit (1.0g) maximum horizontal acceleration. Newmark also recommended that the maximum vertical ground motion be taken as 2/3 of the corresponding values for the horizontal motion. With the availability of a large number of recorded earthquake ground motion, particularly during the 1971 San Fernando earthquake, several statistical studies (29, 291, 253) were carried out to determine the average peak ground velocity and displacement for a given acceleration. These studies recommended two ratios: peak velocity to peak acceleration v/a and peak acceleration–displacement product to the square of the peak velocity ad/v2 be used in estimating ground velocities and displacements. Certain response spectrum characteristics such as the sharpness or flatness of the spectra can be related to the ad/v2 ratio as discussed later. According to Newmark and Rosenbleuth (292), for most earthquakes of practical interest, ad/v2 ranges from approximately 5 to 15. For harmonic oscillations, ad/v2 is one and for steadystate square acceleration waves, the ratio is one half. A statistical study of v/a and ad/v2 ratios was carried out by Mohraz (253) who used a total of 162 components of 54 records from 16 earthquakes. A summary of the v/a and ad/v2 ratios for records on alluvium, on rock, and on alluvium layers underlain by rock are given in Table 29. It is noted that v/a ratios for rock are substantially lower than those for alluvium with the v/a ratios for the two intermediate categories falling between alluvium and rock. Table 29 also shows that the v/a ratios for the vertical components are close to those for the horizontal components with the larger of the two peak accelerations. The 50 percentile v/a ratios for the larger of the two peak accelerations from Table 29 (24 (in/sec)/g for rock and 48 (in/sec)/g for alluvium) and those given by Seed and Idriss (239) (22 (in/sec)/g for rock and 43 (in/sec)/g for alluvium) are in close agreement. The ad/v2 ratios in Table 29
2. Earthquake Ground Motion and Response Spectra
87
Table 29. Summary of Ground Motion Relationships [after Mohraz (253)] Soil Category Group* v/a (in/sec)/g ad/v2 L 24 5.3 Rock S 27 5.2 28 6.1 V
d/a (in/g) 8 10 12
avertical/(ahorizontal)L 0.48
<30 ft of alluvium underlain by rock
L S V
30 39 33
4.5 4.2 6.8
11 17 19
0.47
30200 ft of alluvium underlain by rock
L S V
30 36 30
5.1 3.8 7.6
12 13 18
0.40
23 29 27
0.42
48 3.9 L S 57 3.5 V 48 4.6 * L: Horizontal components with the larger of the two peak accelerations S: Horizontal components with the smaller of the two peak accelerations V: Vertical components Alluvium
indicate that, in general, the ratios for alluvium are smaller than those for rock and those for alluvium layers underlain by rock. The d/a ratios are also presented in Table 29. The values indicate that for a given acceleration, the displacements for alluvium are 2 to 3 times those for rock. The table also includes the ratio of the vertical acceleration to the larger of the two peak horizontal accelerations where it is apparent that the ratios are generally close to each other indicating that soil condition does not influence the ratios. The ratio of the vertical to horizontal acceleration of 2/3 which Newmark recommended is too conservative, but its use was justified to account for the variations greater than the median and the uncertainties in the ground motion in the vertical direction (291). Statistical studies of v/a and ad/v2 ratios for the Loma Prieta earthquake of October 17, 1989 were carried out by Mohraz and Tiv (293). They used approximately the same number of horizontal components of the records on rock and alluvium that Mohraz (253) used in his earlier study. Their study indicated a mean v/a ratio of 51 and 49 (in/sec)/g and a mean ad/v2 ratio of 2.8 and 2.6 for rock and alluvium, respectively. The differences in v/a and ad/v2
ratios from the Loma Prieta and previous earthquakes indicate that each earthquake is different and that site condition, magnitude, epicentral distance, and duration influence the characteristics of the recorded ground motion.
2.7
EARTHQUAKE RESPONSE SPECTRA
Response spectrum is an important tool in the seismic analysis and design of structures and equipment. Unlike the power spectral density which presents information about input energy and frequency content of ground motion, the response spectrum presents the maximum response of a structure to a given earthquake ground motion. The response spectrum introduced by Biot (21, 22) and Housner (23) describes the maximum response of a damped singledegreeoffreedom (SDOF) oscillator at different frequencies or periods. The detailed procedure for computing and plotting the response spectrum is discussed in Chapter 3 of this handbook and in a number of publications (see for example 254, 222, 294, 295). It was customary to plot the response spectrum on a tripartite paper (fourway logarithmic paper) so that at a given frequency
88
Chapter 2
Figure 234. Comparison of pseudovelocity and maximum relative velocity for 5% damping for the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940.
or period, the maximum relative displacement SD, the pseudovelocity PSV, and the pseudoacceleration PSA can all be read from the plot simultaneously. The parameters PSV and PSA which are expressed in terms of SD and the circular natural frequency ω as PSV = ωSD and PSA = ω2SD have certain characteristics that are of practical interest (287). The pseudovelocity PSV is close to the maximum relative velocity SV at high frequencies (frequencies greater than 5 Hz), approximately equal for intermediate frequencies (frequencies between 0.5 Hz and 5 Hz) but different for low frequencies (frequencies smaller than 0.5 Hz) as shown in Figure 234. In a recent study by Sadek et al. (296), based on a statistical analysis of 40 damped SDOF structures with period range of 0.1 to 4.0 sec subjected to 72 accelerograms, it was found that the maximum relative velocity SV is equal to the pseudovelocity PSV for periods in the neighborhood of
0.5 sec (frequency of 2 Hz). For periods shorter than 0.5 sec, SV is smaller than PSV while for periods longer than 0.5 sec, SV is larger and increases as the period and damping ratio increase. A regression analysis was used to establish the following relationship between the maximum velocity and pseudovelocity responses:
SV = a v T bv PSV where
(238)
av = 1.095 + 0.647 β − 0.382 β 2 ,
bv = 0.193 + 0.838 β − 0.621β 2 , T is the natural period, and β is the damping ratio. The relationship between SV and PSV is presented in Figure 235.
2.5
2
1.5
1 β=0.02 β=0.10 β=0.20 β=0.40 β=0.60
0.5
β=0.05 β=0.15 β=0.30 β=0.50
0 0
0.5
1
1.5
2
2.5
3
3.5
4
Period (s)
Figure 235. Mean ratio of maximum relative velocity to pseudovelocity for SDOF structures with different damping ratios. [After Sadek et al. (296).]
Figure 236. Comparison of pseudoacceleration and maximum absolute acceleration for 5% damping for the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940.
90
Chapter 2 3.5
3
β=0.02
β=0.05
β=0.10
β=0.15
β=0.20
β=0.30
β=0.40
β=0.50
β=0.60 2.5
2
1.5
1
0.5 0
0.5
1
1.5
2
2.5
3
3.5
4
Period (s)
Figure 237. Mean ratio of maximum absolute acceleration to pseudoacceleration for SDOF structures with different damping ratios. [After Sadek et al. (296).]
Figure 239. Acceleration, velocity, and displacement amplifications plotted as a function of period for 5% damping for the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940.
For zero damping, the pseudoacceleration PSA is equal to the maximum absolute acceleration SA, but for dampings other than zero, the two are slightly different. For the inherent damping levels encountered in most engineering applications, however, the two can be considered approximately equal (see Figure 236). When a structure is equipped with supplemental dampers to provide large damping ratios, the difference between PSA and SA becomes significant, especially for structures with long periods. Using the results of a statistical analysis of 72 earthquake records, Sadek et al. (296) described the relationship between PSA and SA as: Figure 238. Acceleration, velocity, and displacement amplifications plotted as a function of frequency for 5% damping for the S00E component of El Centro, the imperial Valley earthquake of May 18, 1940.
SA = 1 + a a T ba PSA
(239)
2. Earthquake Ground Motion and Response Spectra and a a = 2.436 β 1.895 The relationship ba = 0.628 + 0.205 β .
where
between SA and PSA is presented in Figure 237. Arithmetic and semilogarithmic plots have also been used to represent response spectra. Building codes have presented design spectra in terms of acceleration amplification as a function of period on an arithmetic scale. Typical acceleration, velocity, and displacement amplifications for the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940 are shown in Figures 238 and 239 the former plotted as a function of frequency and the latter as a function of period. To show how ground motion is amplified in different regions of the spectrum, the peak ground displacement, velocity, and acceleration
91 for the S00E component of El Centro are plotted together on the response spectra, Figure 240. Several observations can be made from this figure. At small frequencies or long periods, the maximum relative displacement is large, whereas the pseudoacceleration is small. At large frequencies or short periods, the relative displacement is extremely small, whereas the pseudoacceleration is relatively large. At intermediate frequencies or periods, the pseudovelocity is substantially larger than those at either end of the spectrum. Consequently, three regions are usually identified in a response spectrum: the low frequency or displacement region, the intermediate frequency or velocity region, and the high frequency or acceleration region. In each region, the corresponding ground motion is amplified the most. Figure 240 also shows
Figure 240. Response spectra for 2, 5, and 10% damping for the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940, together with the peak ground motions.
92
Chapter 2
that at small frequencies (0.05 Hz or less), the spectral displacement approaches the peak ground displacement indicating that for very flexible systems, the maximum displacement is equal to that of the ground. At large frequencies (2530 Hz), the pseudoacceleration approaches the peak ground acceleration, indicating that for rigid systems, the absolute acceleration of the mass is the same as the ground. As indicated in Figure 240, the response spectra for a given earthquake record is quite irregular and has a number of peaks and valleys. The irregularities are sharp for small damping ratios, and become smoother as damping increases. As discussed previously, the ratio of ad/v2 influences the shape of the spectrum. A small ad/ν2 ratio results in a pointed or sharp spectrum while a large ad/ν2 ratio results in a flat spectrum in the velocity region. Response spectra may shift toward high or low frequency regions according to the frequency content of the ground motion. While response spectra for a specified earthquake record may be used to obtain the response of a structure to an earthquake ground motion with similar characteristics, they cannot be used for design because the response of the same structure to another earthquake record will undoubtedly be different. Nevertheless, the recorded ground motion and computed response spectra of past earthquakes exhibit certain similarities. For example, studies have shown that the response spectra from accelerograms recorded on similar soil conditions reflect similarities in shape and amplifications. For this reason, response spectra from records with common characteristics are averaged and then smoothed before they are used in design.
2.8
FACTORS INFLUENCING RESPONSE SPECTRA
Earthquake parameters such as soil condition, epicentral distance, magnitude, duration, and source characteristics influence the shape and amplitudes of response spectra. While the effects of some parameters may be studied independently, the influences of several factors are interrelated and cannot be discussed
individually. Some of these influences are discussed below: 2.8.1
Site geology
Prior to the San Fernando earthquake of 1971, accelerograms were limited in number and therefore not sufficient to determine the influence of different parameters on response spectra. Consequently, most design spectra were based on records on alluvium but they did not refer to any specific soil condition. Studies by Hayashi et al. (297) and Kuribayashi et al. (298) on the effects of soil conditions on Japanese earthquakes had shown that soil conditions significantly affect the spectral shapes. Other studies by Mohraz et al. (29) and Hall et al. (291) also referred to the influence of soil condition on spectral shapes. The 1971 San Fernando earthquake provided a large database to study the influence of many earthquake parameters including soil condition on earthquake ground motion and response spectra. In 1976, two independent studies, one by Seed, Ugas, and Lysmer (299), and the other by Mohraz (253) considered the influence of soil condition on response spectra. The study by Seed et al. used 104 horizontal components of earthquake records from 23 earthquakes. The records were divided into four categories: rock, stiff soils less than about 150 ft deep, deep cohesionless soil with depths greater than 250 ft, and soft to medium clay and sand. The response spectra for 5% damping4 were normalized to the peak ground acceleration of the records and averaged at various periods. The average and the mean plus one standard deviation (84.1 percentile) spectra for the four categories from their study is presented in Figures 241 and 242. The ordinates in these plots represent the acceleration amplifications. Also shown in Figure 242 is the Nuclear Regulatory Commission (NRC) design spectrum proposed 4
they limited their study to 5% damping, although the conclusions can easily be extended to other damping coefficients.
2. Earthquake Ground Motion and Response Spectra by Newmark et al. (288, 2100), see Section 2.9. It is seen that soil condition affects the spectra to a significant degree. The figures show that for periods greater than approximately 0.4 to 0.5 sec, the normalized spectral ordinates (amplifications) for rock are substantially lower than those for soft to medium clay and for deep cohesionless soil. This indicates that using the spectra from the latter two groups may overestimate the design amplifications for rock.
Figure 241. Average acceleration spectra for different soil conditions. [After Seed et al. (299).]
The study by Mohraz (253) considered a total of 162 components of earthquake records divided into four soil categories: alluvium, rock, less than 30 ft of alluvium underlain by rock, and 30  200 ft of alluvium underlain by rock. Figure 243 presents the average acceleration amplifications (ratio of spectral ordinates to peak ground acceleration) for 2% damping for the horizontal components with the larger of the two peak ground accelerations. Consistent with the study by Seed et al. (299), the figure shows that soil condition influences the spectral shapes to a significant degree. The acceleration amplification for alluvium extends over a larger frequency region than the amplifications for the other three soil categories. A comparison of acceleration amplifications for 5% damping from the Seed and Mohraz studies is shown in Figure 244. The figure indicates a remarkably close agreement even though the records used in the
93 two studies are somewhat different. Normalized response spectra corresponding to the mean plus one standard deviation (84.1 percentile) for the four soil categories from the Mohraz study are given in Figure 245. The plot indicates that for short periods (high frequencies) the spectral ordinates for alluvium are lower than the others, whereas, for intermediate and long periods they are higher.
Figure 242. Mean plus one standard deviation acceleration spectra for different soil conditions. [After seed et al. (299).]
Figure 243. Average horizontal acceleration amplifications for 2% damping for different soil categories. [After Mohraz (253).]
94
Figure 244. Comparison of the average horizontal acceleration amplifications for 5% damping for rock. [After Mohraz (253).]
Chapter 2 greater than 6 to formulate a relationship for pseudovelocity in terms of various earthquake parameters. The response spectra for 5% damping were computed for four site categories; rock, soft rock or stiff soil, medium stiff soil, and soft soil classified as soil class A through D, respectively. A regression analysis was performed for periods in the range of 0.1 to 4.0 sec. Their proposed equation for the pseudovelocity (PSV) in cm/sec is given as
ln( PSV ) = a + bM s + d ln[ R + c1 exp(c 2 M s )] + eF
Figure 245. Mean plus one standard deviation response spectra for 2% damping for different soil categories, normalized to 1.0g horizontal ground acceleration. [After Mohraz (253).]
Recent studies indicate that the spectral shape not only depends on the three peak ground motions, but also on other parameters such as earthquake magnitude, sourcetosite distance, soil condition, and source characteristics. Similar to ground motion attenuation relationships (Section 2.4), several investigators have used statistical analysis of the spectra at different periods to develop equations for computing the spectral ordinates in terms of those parameters. For example, Crouse and McGuire (2101) used 238 horizontal accelerograms from 16 earthquakes between 1933 and 1992 with surface wave magnitudes
(240)
where MS is the surface wave magnitude, R is the closest distance from the site to the fault rupture in km, and F is the fault type parameter which equals 1 for reverseslip and 0 for strikeslip earthquakes. The parameters a, b, c1, c2, d and e are given in tabular form for different periods and soil categories (2101). Parameters b, c1, and c2 are greater than zero whereas d is less than zero for all periods and different soil conditions. Figure 246 presents the spectral shapes for the four soil categories at a distance of 10 km from the source for a strikeslip earthquake of magnitude 7. The figure indicates higher spectral values for softer soils. A similar study was carried out by Boore et al. (237) using the average shear wave velocity Vs (m/sec) in the upper 30 m of the surface to classify the soil condition. In their study, the pseudoacceleration response PSA in g is given by
ln( PSA) = b1 + b2 ( M W − 6) + b3 ( M W − 6) 2 + b5 ln R 2jb + h 2 + bv ln
(241)
Vs VA
where MW and Rjb are the moment magnitude and distance (see section 2.4.1), respectively. The parameter b1 is related to the fault type and is listed for different periods for strikeslip and reverseslip earthquakes, and the case where the
2. Earthquake Ground Motion and Response Spectra
95
Figure 2 46. Response spectra for 5% damping for different soil conditions for a magnitude 7 strikeslip earthquake. [After Crouse and McGuire (2101).]
fault mechanism is not specified. Factors b2, b3, b5, bv, VA, and h for different periods are also presented in tabular form (237). The parameters b2, VA, and h are always positive whereas b3, b5, and bv are always negative. Consistent with the study by Crouse and McGuire (2101), Equation 241 indicates that, for the same distance, magnitude, and fault mechanism, as the soil becomes stiffer (a higher shear wave velocity), the pseudoacceleration becomes smaller since bv is always negative. 2.8.2
Magnitude
In the past, the influence of earthquake magnitude on response spectra was generally taken into consideration when specifying the peak ground acceleration at a site. Consequently, the spectral shapes and amplifications in Figures 241 and 242 were obtained independent of earthquake magnitude. Earthquake magnitude does, however, influence spectral amplifications to a certain degree. A study by Mohraz (2102) on the influence of earthquake magnitude on response
amplifications for alluvium shows larger acceleration amplifications for records with magnitudes between 6 and 7 than those with magnitudes between 5 and 6 (see Figure 247). While the study used a limited number of records and no specific recommendation was made, the figure indicates that earthquake magnitude can influence spectral shapes and may need to be considered when developing design spectra for a specific site. Equations 240 and 241 in the previous section include the influence of earthquake magnitude on the pseudovelocity and pseudoacceleration, respectively. The equations indicate that spectral ordinates increase with an increase in earthquake magnitude. Figure 248 presents the spectral ordinates computed using Equation 241 by Boore et al. (237) for soil with a Vs = 310 m/sec at a zero source distance for earthquakes with magnitudes 6.5 and 7.5 and an unspecified fault mechanism. The figure indicates that the effect of magnitude is more pronounced at longer periods and it also shows a comparison with the spectra computed from an earlier study by Joyner and Boore (232).
96
Chapter 2
Figure 247. Effect of earthquake magnitude on spectral shapes. [After Mohraz (2102).]
acceleration divided by the peak ground acceleration) for the records on rock and on alluvium for the three groups are shown in Figure 249. The plots indicate that for sites on rock, the amplifications for the nearfield are substantially smaller than those for mid or farfield for periods longer than 0.5 sec. For shorter periods, however, the amplifications for the nearfield are larger. The effect of distance is less pronounced for records on alluvium. Equation 240 proposed by Crouse and McGuire (2101) shows that the spectral ordinates decay with the logarithm of the distance (parameter d in the equation is always negative) for a given soil, earthquake magnitude, and source characteristics. A similar trend is also observed from Equation 241 by Boore et al. (237) . Figure 250 shows the pseudovelocity response computed using Equation 241 for sites on soil for a magnitude of 7.5 at various source distances for strikeslip and reverseslip fault mechanisms. The figure indicates that the spectral ordinates decrease with distance. Since the spectral shapes are nearly parallel to each other for the distance range of 10 to 80 km, it may be concluded that distance does not significantly affect the spectral shape but influences the spectral ordinates through attenuation of ground acceleration. 2.8.4
Figure 248. Pseudovelocity spectra for 5% damping on soil and earthquake magnitudes 6.5 and 7.5 at a zero distance. [After Boore et al. (237).]
2.8.3
Distance
Recent studies have considered the effect of distance on the shape and amplitudes of the earthquake spectra. Using the data from the Loma Prieta earthquake of October 17, 1989; Mohraz (2103) divided the records into three groups: nearfield (distance less than 20 km), midfield (distance between 20 to 50 km) and farfield (distance greater than 50 km). The average acceleration amplification (pseudo
Source characteristics
Fault mechanism may influence the spectral ordinates. Using Equation 240, Crouse and McGuire (2101), computed the ratios of the spectral ordinates for a reverseslip fault to ordinates for strikeslip fault for two soil categories: soft rock or stiff soil (site class B) and medium stiff soil (site class C). The ratios, plotted in Figure 251, show that the spectral ordinates for reverseslip faults are greater than the ordinates for strike slip faults for short periods but not for long periods. Crouse and McGuire concluded, however, that it is difficult to attach any significance on the influence of fault mechanism on the spectral shape. Similar trends and conclusion can also be depicted from
2. Earthquake Ground Motion and Response Spectra
97
Figure 249. Average acceleration amplification for 5% damping for different distances from the 1989 Loma Prieta earthquake for sites on (a) rock and (b) alluvium. [After Mohraz (2103).]
Figure 251. Ratio of reverseslip to strikeslip spectral ordinates for soft rock or stiff soil referenced as site class B and medium stiff soil referenced as site class C. [After Crouse and McGuire (2101).] Figure 250. Pseudovelocity spectra for 5% damping on soil and for earthquake magnitude 7.5 at different distances. [After Boore et al. (237).]
Figure 250 by Boore et al. (237) where the reverseslip faults result in a larger response for short periods and the strikeslip faults result in a larger response for long periods. The difference between the response from the two fault mechanisms, however, is not that significant.
2.8.5
Duration
While earthquake response spectra provide the best quantitative description of the intensity and frequency content of ground motion, they do not provide information on the duration of strong shaking  a parameter that many researchers and practitioners consider to be important in evaluating the damaging effects of an earthquake. The influence of the duration of
98
Chapter 2
strong motion on spectral shapes has been studied by Peng et al. (2104) who used a random vibration approach to estimate sitedependent probabilistic response spectra. Their study shows that long durations of strong motion increase the response in the low and intermediate frequency regions. This is consistent with the fact that accelerograms with long durations have a greater probability of containing longperiod wave components which can result in a large response in the long period or low frequency region of the spectrum.
2.9
EARTHQUAKE DESIGN SPECTRA
Because the detailed characteristics of future earthquakes are not known, the majority of earthquake design spectra are obtained by averaging a set of response spectra from records with similar characteristics such as soil condition, epicentral distance, magnitude, source mechanism, etc. For practical applications, design spectra are presented as smooth curves or straight lines. Smoothing is carried out to eliminate the peaks and valleys in the response spectra that are not desirable for design because of the difficulties encountered in determining the exact frequencies and mode shapes of structures during severe earthquakes when the structural behavior is most likely nonlinear. It should be noted that in some cases, determining the shape of the design spectra for a particular site is complicated and caution should be used in arriving at a representative set of records. For example, long period components of strong motion have a pronounced effect on the response of flexible structures. Recent strong motion data indicates that long period components are influenced by factors such as distance, source type, rupture propagation, travel path, and local soil conditions (250, 2105, 2106). In addition, the direction and spread of rupture propagation can affect motion in the nearfield. For these reasons, the selection of an appropriate set of records in arriving at representative design spectra is important and may require selection
of different sets of records for different regions of the spectrum. The difference between response spectra and design spectra should be kept in mind. A response spectrum is a plot of the maximum response of a damped SDOF oscillator with different frequencies or periods to a specific ground motion, whereas a smooth or a design spectrum is a specification of seismic design force or displacement of a structure having a certain frequency or period of vibration and damping (2107). Since the peak ground acceleration, velocity, and displacement for various earthquake records differ, the computed response cannot be averaged on an absolute basis. Various procedures are used to normalize response spectra before averaging is carried out. Among these procedures, two have been most commonly used: 1) normalization according to spectrum intensity (2108) where the areas under the spectra between two given frequencies or periods are set equal to each other, and 2) normalization to peak ground motion where the spectral ordinates are divided by peak ground acceleration, velocity, or displacement for the corresponding region of the spectrum. Normalization to other parameters such as effective peak acceleration and effective peak velocityrelated acceleration has also been suggested and used in development of design spectra for seismic codes. Table 210. Relative Values of Spectrum Amplification Factors (after Newmark and Hall, 290) Percent of Amplification Factor for Critical Displacement Velocity Acceleration Damping 0 2.5 4.0 6.4 0.5 2.2 3.6 5.8 1 2.0 3.2 5.2 2 1.8 2.8 4.3 5 1.4 1.9 2.6 7 1.2 1.5 1.9 10 1.1 1.3 1.5 20 1.0 1.1 1.2
The first earthquake design spectrum was developed by Housner (2109, 2110). His design spectra shown in Figure 252 are based on the
2. Earthquake Ground Motion and Response Spectra characteristics of the two horizontal components of four earthquake ground motions recorded at El Centro, California in 1934 and 1940, Olympia, Washington in 1949, and Taft, California in 1952. The plots are normalized to 20% acceleration (0.2g) at zero period (ground acceleration). For any other acceleration, the plots or the information read from them are simply scaled up or down by multiplying them by the ratio of the desired acceleration to 0.2g. In the late sixties, Newmark and Hall (289, 2
99 90)
recommended straight lines be used to represent earthquake design spectra. They suggested that three amplifications (acceleration, velocity, and displacement) which are constant in the high, intermediate, and low frequency regions of the spectrum (Table 210) together with peak ground acceleration, velocity, and displacement of 1.0g, 48 in/sec, and 36 in. be used to construct design spectra. Their recommended ground motions and the amplifications were based on
Figure 252. Design spectra scaled to 20% ground acceleration. [After Housner (2110).]
100
Chapter 2
Figure 253. Design spectra normalized to 1.0g. [After Newmark and Hall (290).]
the characteristics of several earthquake records without considering soil condition. The spectral ordinates which are obtained by multiplying the three ground motions by the corresponding amplifications are plotted on a tripartite (fourway logarithmic) paper as shown in Figure 253. The spectral displacement, spectral velocity, and spectral acceleration are plotted parallel to maximum ground displacement, ground velocity, and ground acceleration, respectively. The frequencies at the intersections of spectral displacement and velocity, and spectral velocity and acceleration define the three amplified regions of the spectrum. At a frequency of
approximately 6 Hz, the spectral acceleration is tapered down to the maximum ground acceleration. It is assumed that the spectral acceleration for 2% damping intersects the maximum ground acceleration at a frequency of 30 Hz. The tapered spectral acceleration lines for other dampings are parallel to the one for 2%. The normalized design spectra in Figure 253 can be used for design by scaling the ordinates to the desired acceleration. In the early seventies with increased activity in the design and construction of nuclear power plants in the United States, the Atomic Energy Commission AEC (later renamed the Nuclear
2. Earthquake Ground Motion and Response Spectra
101
Figure 254. NRC horizontal design spectra scaled to 1.0g ground acceleration. A, B, C, and D are control frequencies corresponding to 33, 9, 2.5, and 0.25 HZ, respectively.
Regulatory Commission) funded two studies one by John A. Blume and Associates (2111) and the other by N. M. Newmark Consulting Engineering Services (29) to develop recommendations for horizontal and vertical design spectra for nuclear power plants. These studies which used a statistical analysis of a number of recorded earthquake ground motions and computed response spectra were the basis for the Nuclear Regulatory Commission (NRC) Regulatory Guide 1.60 (288, 2100). The studies recommended that the mean plus one standard deviation (84.1 percentile) response be used for
the design of nuclear power plants and equipment. The NRC design spectra are constructed using a set of amplifications corresponding to four control frequencies (Figure 254). The spectra are normalized to 1.0g horizontal ground acceleration. While the NRC spectra were developed for design of nuclear power plants, they were also used to develop and compare design spectra for other applications. In 1978, the Applied Technology Council ATC (274) recommended a smooth version of the normalized spectral shapes proposed by
102 Seed et al. (299) be used in developing earthquake design spectra for buildings. The spectral shapes in Figures 241 and 242 were smoothed using four control periods (239). In addition, the four soil categories were reduced to three: rock and stiff soils (soil type 1), deep cohesionless or stiff clay soils (soil type 2), and soft to medium clays and sands (soil type 3). The ATC spectra which was adopted by the Seismology Committee of the Structural Engineers Association of California, SEAOC (2112) is presented in Figure 255. A comparison of the spectral shapes from the study by Mohraz (253) and those proposed by ATC is shown in Figure 256. The 1985, 1988, 1991, and 1994 editions of the Uniform Building Code (282 to 285) use the spectral shapes for the three soil conditions recommended by ATC. The design spectra for a given site is computed by multiplying the spectral shapes in Figure 255
Chapter 2 by the seismic zone factor Z (or the effective peak acceleration) obtained from the seismic maps.
Figure 256. Normalized spectral curves recommended for use in building codes. (Reproduced from 239).
Figure 255. Comparison of spectral shapes for 5% damping proposed by Mohraz with those recommended by SEAOC.
2. Earthquake Ground Motion and Response Spectra The 1985, 1988, and 1991 NEHRP recommended provisions (276 to 278) present design spectra using the effective peak acceleration Aa and the effective peak velocityrelated acceleration Av. These two factors which are obtained from seismic maps are used to define the constant acceleration and velocity segments of the design spectrum, respectively. Since Aa and Av for the vast majority of the sites in the United States are the same, the computed spectra are similar to the UBC spectra. While the 1985 NEHRP provisions included the three soil categories defined by ATC (274), the 1988 NEHRP provisions (277) and the 1988 Uniform Building Code (283) included a fourth soil category S4 based on the experience from the Mexico City earthquake of September 19, 1985 where most of the underlying soil is very soft5. Flexible structures (periods in the neighborhood of 2 sec) in that earthquake experienced large acceleration amplifications which resulted in severe and widespread damage. Consequently, it was recommended to compute the spectral shape in the velocity region from that of rock using an amplification of 2. A new procedure for constructing design spectra and computing the base shears was recommended in the 1991 NEHRP provisions (278) by obtaining the spectral acceleration ordinates at periods of 0.3 and 1.0 sec from the spectral maps (see Section 2.5). The ordinate at 0.3 sec is used for the constant acceleration zone whereas the ordinate at 1.0 sec is divided by the period T for the velocity zone. The spectral ordinates from the maps are modified according to the soil category of the site. The maps in the 1991 NEHRP provisions were provided for the soil category S2 (deep cohesionless or stiff clay soils). The provisions recommended that the spectral ordinates corresponding to the 1.0 sec period be reduced by a factor of 0.8 for soil type S1 and amplified by factors of 1.3 and 1.7 for soil types S3 and S4, respectively. 5
the shaking was most intense within a region underlain by an ancient dry lake bed composed of soft clay deposits.
103
Figure 257. Twofactor approach for constructing sitedependent design spectra recommended by the 1994 NEHRP recommended provisions.
In 1992, a workshop on site response during earthquakes was held by the National Center for Earthquake Engineering Research (NCEER), the Structural Engineers Association of California (SEAOC), and the Building Seismic Safety Council (BSSC). The workshop (2113) recommended that the spectral amplifications at different periods should depend not only on the soil condition but also on the intensity of shaking due to soil nonlinearities. Consequently, a twofactor approach was suggested for constructing the design spectra in order to account for the dependence of the spectral shape on the shaking intensity. The twofactor approach was introduced in the 1994 NEHRP provisions (279), see Figure 257. The approach uses new seismic coefficients Ca and Cv in terms of the effective peak acceleration Aa and the effective peak velocityrelated acceleration Av such that
Ca = Aa Fa and Cv = Av Fv
(242)
where Fa and Fv are site amplification coefficients that vary according to soil condition and shaking intensity (seismic zone). The provisions included tables for computing coefficients Fa and Fv as well as Ca and Cv. Six soil categories, designated as A through F, were introduced in the provisions. The first five are based primarily on the average shear wave
104 velocity6 Vs (m/sec) in the upper 30 meters of the soil profile and the sixth is based on a site specific evaluation. The categories include: (A) hard rock (Vs > 1500), (B) rock (760 < Vs ≤ 1500), (C) very dense soil and soft rock (360 < Vs ≤ 760), (D) stiff soil profile (180 < Vs ≤ 360), (E) soft soil profile (Vs ≤ 180), and (F) soils requiring sitespecific evaluations such as liquefiable and collapsible soils, sensitive clays, peats and highly organic clays, very high plasticity clays, and very thick soft/medium stiff clays. The site coefficients Fa and Fv are based primarily on the work of Borcherdt (2114) who used the strong motion data from the Loma Prieta earthquake of October 17, 1989 to compute average amplification factors normalized to firm to hard rock (NEHRP site class B) for shortperiods (0.10.5 sec), intermediateperiods (0.51.5 sec), midperiods (0.42.0 sec), and longperiods (1.55.0 sec). Data for ground accelerations of approximately 0.1g were used in an empirical procedure to find amplifications Fa and Fv. Amplification factors for ground accelerations greater than 0.1g (0.2g, 0.3g, and 0.4g) were computed by extrapolation of amplification estimates at 0.1g since few strong motion records were available for ground motions greater than 0.1g for soft soil. The extrapolations were based on results from laboratory experiments and numerical modeling. The amplifications were in good agreement with those computed by Seed et al. (2115) based on a numerical modeling of the data from the Loma Prieta records and those by Dobry et al. (2116) based on a parametric study of several hundred soil profiles. The amplifications Fa and Fv corresponding to short and mid periods with respect to firm to hard rock for different shaking intensities are shown in Figure 258. The figure indicates that site amplifications decrease with an increase in shear wave velocity and an increase in ground accelerations. Borcherdt also presented the site 6
in addition to the shear wave velocity, other parameters such as average standard penetration, undrained shear strength, and plasticity index are used in the classification.
Chapter 2 amplifications in terms of the average shear wave velocity Vs in the upper 30 meters of the soil profile as:
Fa = (V 0 / Vs )
ma
Fv = (V 0 / Vs )
mv
(243)
Where VO is the average shear wave velocity for a referenced soil profile (VO = 1050 m/sec for firm to hard rock). Parameters ma and mv represent the influence of the ground motion intensity on amplification (see Figure 258). Substitution for VO results in
Fa = (1050 / Vs ) ma Fv = (1050 / Vs )
mv
(244)
The coefficients Fa and Fv recommended by Borcherdt were the basis for those presented in the 1994 NEHRP provisions by computing the coefficients for each site category by substituting the appropriate value for Vs. Borcherdt also provided values for the coefficients Fa and Fv for constructing design spectra in association with the spectral accelerations at periods of 0.3 and 1.0 sec. Since seismic maps for spectral accelerations are for deep cohesionless or stiff clay soils, the coefficients are presented with reference to soft to firm rocks and stiff clays. For this case, Equation 243 can be used to compute the coefficients Fa and Fv using a V0= 450 m/sec. After the Northridge earthquake of January 17, 1994, Borcherdt (2117) computed coefficients Fa and Fv for accelerograms recorded on different soils in the Los Angeles area. The results indicate that the coefficients are in good agreement with those suggested in his earlier study (2114) and also those included in the 1994 NEHRP provisions (279) for small shaking intensities. For large intensities, however, the coefficients computed from the Northridge data are greater than those recommended previously.
2. Earthquake Ground Motion and Response Spectra
105
Figure 258. Variation of shortperiod Fa and longperiod Fv amplification factors normalized to firm to hard rock with mean shear wave velocity. [After Borcherdt (2114).]
The 1997 Uniform Building Code (286) used a method similar to that in the 1994 NEHRP provisions to construct the design spectrum. The design spectrum, Figure 259, is defined in terms of the seismic coefficients Ca and Cv. These coefficients are presented for the five UBC seismic zones for different soil categories, which are the same as those used in the 1994 NEHRP provisions. The only difference between the design spectra in the 1997 UBC code and the 1994 NEHRP provisions is that the former includes the nearsource factors.
These factors were introduced to amplify the spectral ordinates for sites close to a seismic source in the zone with the highest seismicity (zone 4). The nearsource factors depend on the distance to the closest active fault and the source type (maximum magnitude, rate of seismic activity, and slip rate). Design spectra presented in the 1997 NEHRP recommended provisions (280) can be constructed from the maps of spectral response accelerations at short periods SS (defined as 0.2 sec) and at 1.0 sec period S1 corresponding to
106 the maximum considered earthquake (see Section 2.5). Since the maps are provided for rock (site class B), the spectral accelerations for other soil categories are adjusted by multiplying the spectral accelerations for rock by the site coefficients Fa and Fv in the short and the mid to long period ranges, respectively. Similar to the 1994 provisions, Fa and Fv depend on the soil category and the shaking intensity and are given in tabular form based on the study by Borcherdt (2114). To construct the spectra for the design earthquake, the adjusted spectral ordinates at the maximum considered earthquake are multiplied by 2/3 (see Section 2.5).
Figure 259. Design spectrum recommended by the 1997 Uniform Building Code (286).
The 1997 NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA273 (281), uses a procedure similar to that of the 1997 NEHRP Provisions (280) to establish the 5% damped design spectra. In addition, FEMA273 uses damping modification factors in the short and longperiod ranges to reduce the spectral ordinates for damping ratios larger than 5% due to the use of supplemental damping devices in the structure.
Chapter 2
2.10
INELASTIC RESPONSE SPECTRA
Structures subjected to severe earthquake ground motion experience deformations beyond the elastic range. To a large extent, the inelastic deformations depend on the intensity of excitation and loaddeformation characteristics of the structure and often result in stiffness deterioration. Because of the cyclic characteristics of ground motion, structures experience successive loadings and unloadings and the forcedisplacement or resistancedeformation relationship follows a sequence of loops known as hysteresis loops. The loops reflect a measure of a structure’s capacity to dissipate energy. The shape and orientation of the hysteresis loops depend primarily on the structural stiffness and yield displacement. Factors such as structural material, structural system, and connection configuration influence the hysteretic behavior. Consequently, arriving at an appropriate mathematical model to describe the inelastic behavior of structures during earthquakes is a difficult task. A simple model which has extensively been used to approximate the inelastic behavior of structural systems and components is the bilinear model shown in Figure 260. In this model, unloadings and subsequent loadings are assumed to be parallel to the original loading curve. Strain hardening takes place after yielding initiates. Elasticplastic (elastoplastic) model is a special case of the bilinear model where the strain hardening slope is equal to zero (α = 0). Other hysteretic models such as stiffness and strength degrading have also been suggested. The elasticplastic model results in a more conservative response than other models. Because of its simplicity, it was widely used in the development of inelastic response spectra. Response spectra modified to account for the inelastic behavior, commonly referred to as the inelastic spectra, have been proposed by several investigators. The use of the inelastic spectra in analysis and design, however, has been limited to structures that can be modeled
2. Earthquake Ground Motion and Response Spectra
Figure 260. Bilinear forcedisplacement relationship.
107 as a singledegreeoffreedom. Procedures for utilizing inelastic spectra in the analysis and design of multidegreeoffreedom systems have not yet been developed to the extent that can be implemented in design. Similar to elastic spectra, inelastic spectra were usually plotted on tripartite paper for a given damping and ductility7 or yield deformation. When the spectra are plotted for various ductilities, computations are repeated for several yield deformations using an iterative procedure to achieve the target ductility. Depending on the parameter plotted, different names have been used to identify the spectrum (Riddell and
Figure 261. Inelastic yield spectra for the S90W component of El Centro, the Imperial Valley earthquake of May 18, 1940. Elasticplastic systems with 5% damping. [After Riddell and Newmark (2118).] 7
ratio of maximum deformation to yield deformation
108
Chapter 2
Figure 262. Total deformation spectra for the S90W component of El Centro, the Imperial Valley earthquake of May 18, 1940. Elasticplastic systems with 5% damping. [After Riddell and Newmark (2118).]
Newmark, 2118). In the inelastic yield spectrum (IYS), the yield displacement is plotted on the displacement axis; in the inelastic acceleration spectrum (IAS), the maximum force per unit mass is plotted on the acceleration axis; and in the inelastic total displacement spectrum (ITDS), the absolute maximum total displacement is plotted on the displacement axis. For elasticplastic behavior, the inelastic yield spectrum and the inelastic acceleration spectrum are identical. Examples of inelastic spectra for a 5% damped elasticplastic system for the S90W component of El Centro, the Imperial Valley earthquake of May 18, 1940 are shown in Figures 261 and 262. The figures indicate that for inelastic yield and acceleration spectra, the curves for various ductilities fall below the elastic curve (ductility of one), whereas for the inelastic total deformation spectra, they primarily fall above
the elastic, particularly in the acceleration region. It should be noted that increasing the ductility ratio smoothes the spectra and minimizes the sharp peaks and valleys that are present in the plots. A different presentation of inelastic spectra was proposed by Elghadamsi and Mohraz (2119). The spectrum, referred to as the yield displacement spectrum (YDS), is plotted similar to the inelastic total deformation spectrum except that it is plotted for a given yield displacement instead of a given ductility. The ductility is obtained as the ratio of the maximum displacement to the yield displacement for which the spectrum is plotted. Their procedure offers an efficient computational technique, particularly when statistical studies are used to obtain inelastic design spectra.
2. Earthquake Ground Motion and Response Spectra
109 inelastic systems). A" and AO are obtained by multiplying A' and AO by µ. The RiddellNewmark study (2118) also considered bilinear and stiffness degrading models and concluded that using the elasticplastic spectrum for inelastic analysis is generally on the conservative side. 2.10.1
Figure 263. Construction of inelastic acceleration and inelastic total displacement spectra from the elastic spectrum. [After Newmark and Hall (289).]
Before the RiddellNewmark study of inelastic response, the most common procedure for estimating inelastic earthquake design spectra was the one proposed by Newmark (2120, 2121) and Newmark and Hall (289). Based on results similar to those in Figures 261 and 262, and studies by Housner (2122) and Blume (2123 to 2125) , Newmark (2121) observed that: 1) at low frequencies, an elastic and an inelastic system have the same total displacement, 2) at intermediate frequencies, both systems absorb the same total energy, and 3) at high frequencies, they have the same force. These observations resulted in the recommendation by Newmark for constructing inelastic spectra from the elastic by dividing the ordinates of the elastic spectrum by two coefficients in terms of ductility µ. Figure 263 shows the construction of the inelastic spectrum from the elastic. The solid lines DVAAo represent the elastic response spectrum. The solid circles at the intersections of the lines correspond to frequencies which remain constant in obtaining the inelastic spectrum. The lines D'V'A'AO represent the inelastic acceleration spectrum whereas the lines DVA''AO'' show the total displacement spectrum. D' and V' are obtained by dividing D and V by µ. A' is obtained by dividing A by (2 µ − 1) (to insure that the same energy is absorbed by the elastic and the
Deamplification factors
When inelastic deformations are permitted in design, the elastic forces can be reduced if adequate ductility is provided. Riddell and Newmark (2118) presented a set of coefficients referred to as “deamplification factors” by which the ordinates of the elastic design spectrum are multiplied to obtain the inelastic yield spectrum. Lai and Biggs (2126), using artificial accelerograms with variable durations of strong motion, presented a set of coefficients referred to as “inelastic acceleration response ratios” by which the ordinates of the elastic spectrum are divided to give the inelastic yield spectrum. Since these two approaches are the inverse of one another, the reciprocal of the LaiBiggs coefficients represent deamplification factors. Deamplification factors can also be obtained from the NewmarkHall (289) and from the ElghadamsiMohraz (2119) procedures for estimating inelastic spectra. Comparisons of the deamplification factors from the four procedures are shown in Figure 264 for a 5% damping ratio and ductilities of 2 and 5. The figure indicates that the RiddellNewmark deamplification factors are in general the smallest (largest reduction in the elastic force) compared to the other three. Both RiddellNewmark and NewmarkHall deamplification ratios remain constant over certain frequency segments, whereas those from LaiBiggs and ElghadamsiMohraz follow parallel patterns. While the deamplification ratios are affected by ductility, they are practically not influenced by damping. Since the elastic spectral ordinates decrease significantly with an increase in damping, the decrease in inelastic spectral ordinates with
110
Chapter 2
Figure 264. Comparison of deamplification factors for 5% damping. [After Elghadamsi and Mohraz (2119).]
damping stems primarily from the elastic spectral ordinates. Elghadamsi and Mohraz (2119) also presented deamplification factors for alluvium and rock. Typical deamplification factors for alluvium and rock for 5% damping is presented in Figure 265. According to the figure, deamplifications are not significantly affected by the soil condition. The influence of the duration of strong motion on the inelastic behavior of structures has also been studied. In a nondeterministic study of nonlinear structures, Penzien and Liu (2127) concluded that structures with elasticplastic and stiffness degrading behavior are more sensitive to the duration of strong motion than elastic structures. Using a random vibration approach and the extreme value theory, Peng et al. (2128) incorporated the duration of strong motion in estimating the maximum response of structures with elasticplastic behavior. The effect of duration of strong motion on deamplification factors from Peng’s study is shown in Figure 266 which indicates that for a longer duration of strong motion, one should use a larger de
amplification (smaller reduction in elastic force). It should be noted that Lai and Biggs (2126) conclude that inelastic response spectra are not significantly affected by strong motion duration. They emphasize, however, that this conclusion is valid only when ground motion with varying strong motion durations are compatible with the same prescribed elastic response spectrum.
Figure 265. Deamplification factors for alluvium and rock for 5% damping. [After Elghadamsi and Mohraz (2119).]
2. Earthquake Ground Motion and Response Spectra
111 that use multiple lines of framing in each principal direction of the building. The ductility factor Rµ is defined as the ratio of the elastic to the inelastic displacement for a system with an elastic fundamental period T and specified ductility µ such that
R µ (T , µ ) =
Figure 266. Effect of strong motion duration on deamplification factors for systems with 2% damping. [After page et al. (1128).]
2.10.2
Response modification factors
Current seismic codes recommend force reduction factors and displacement amplification factors to be used in design to account for the energy absorption capacity of structures through inelastic action. The force reduction factors (referred to as Rfactors) are used to reduce the forces computed from the elastic design spectra. A recent study by the Applied Technology Council (2129) proposes the following expression for computing the Rfactors:
V R = e = Rs Rµ R R V
(245)
Where Ve is the base shear computed from the elastic response (elastic design spectrum), and V is the design base shear for the inelastic response. The response modification factor R is the product of the following terms: 1. the perioddependent strength factor Rs which accounts for the reserve strength of the structure in excess of the design strength, 2. the perioddependent ductility factor Rµ which accounts for the ductile capacity of the structure in the inelastic range, and 3. the redundancy factor RR which accounts for the reliability of seismic framing systems
u y (T , µ = 1) u y (T , µ )
(246)
where uy is the yield displacement. Stated differently, Rµ is the ratio of the maximum inelastic force to the yield force required to limit the maximum inelastic response to a displacement ductility µ, or the inverse of the deamplification factors presented in Section 2.10.1. The relationship between displacement ductility and ductility factor has been the subject of several studies in recent years. Earlier studies by Newmark and Hall (287, 289) provided expressions for estimating the ductility factor Rµ for elasticplastic systems irrespective of the soil condition. The expressions are
Rµ (T ≤ 0.03 sec, µ ) = 1.0 Rµ (0.12 sec ≤ T ≤ 0.5 sec, µ ) = 2 µ − 1 (247) Rµ (T ≥ 1.0 sec, µ ) = µ A linear interpolation may be used to estimate Rµ for the intermediate periods. The expressions are plotted in Figure 264 for ductility ratios of 2 and 5. Using a statistical study of 15 ground motion records from earthquakes with magnitudes 5.7 to 7.7, Krawinkler and Nassar (2130, 2131) developed relationships for estimating Rµ for rock or stiff soils for 5% damping. Their proposed relationship is
Rµ (T , µ ) = [c( µ − 1) + 1]1/ c where
(248)
112
Chapter 2
Ta
b c= + 1+ T a T
(249)
shows that the differences between these relationships are relatively small and may be ignored for engineering purposes.
and a and b are parameters that depend on the strain hardening ratio α. They recommend a = 1.00, 1.01, and 0.80 and b = 0.42, 0.37, and 0.29 for strain hardening ratios of 0% (elasto plastic system), 2%, and 10%, respectively. Miranda and Bertero (2132) using 124 accelerograms recorded on different soil conditions, developed equations for estimating Rµ for rock, alluvium, and soft soil for 5% damping. Their equation is given by
Rµ ( T , µ ) =
µ −1 +1 Φ
(250)
where Φ (T , µ ) = 1 +
2.11 1 T (10 − µ )
−
1
2 exp[ −1.5(ln T − 0.6) ]
2T for rock sites
Φ (T , µ ) = 1 +
1 T (12 − µ )
−
2
2 exp[ −2(ln T − 0.2) ]
5T for alluvium sites
Φ (T , µ ) = 1 +
Figure 267. Variation of the ductility factor with period for ductility ratios of 2, 4, and 6. [Reproduced from ATC19 (2129).]
Tg 3T
−
3T g 4T
exp[ −3(ln
T
2 − 0.25) ]
Tg for soft soil sites
(2.51) and Tg is the predominant period of the ground motion defined as the period at which the relative velocity of a linear system with 5% damping is maximum throughout the entire period range. A comparison of the NassarKrawinkler and MirandaBertero relationships for rock and alluvium for ductility ratios of 2, 4, and 6 is presented in Figure 267. The figure
ENERGY CONTENT AND SPECTRA
While the linear and nonlinear response spectra, presented in previous sections, have been used for decades to compute design displacements and accelerations as well as base shears, they do not include the influences of strong motion duration, number of response cycles and yield excursions, stiffness and strength degradation, or damage potential to structures. There is a need to reexamine the current analysis and design procedures; especially with the use of innovative protective systems such as seismic isolation and passive energy dissipation devices. In particular, the concept of energybased design is appealing where the focus is not so much on the lateral resistance of the structure but rather on the need to dissipate and/or reflect seismic energy imparted to the structure. In addition, energy approach is suitable for implementation within the framework of performancebased design since the premise behind the energy concept is that earthquake damage is related to the structure’s ability to dissipate energy. Housner (2122) was the first to recommend energy approach for earthquake resistant design. He pointed out that ground motion
2. Earthquake Ground Motion and Response Spectra transmits energy into the structure; some of this energy is dissipated through damping and nonlinear behavior and the remainder stored in the structure in the form of kinetic and elastic strain energy. Housner approximated the input energy as onehalf of the product of the mass and the square of the pseudovelocity, 1 2 m( PSV ) 2 . His study provided the impetus for later developments of energy concepts in earthquake engineering. For a nonlinear SDOF system with preyield frequency and damping ratio of ω and β , respectively; subjected to ground acceleration a(t) the equation of motion is given by:
x&& + 2 βωx& + Fs [x( t )] = −a( t )
113
E S = Recoverable elastic strain energy (257)
Fs2 = 2ω 2 E H = Dissipative plastic strain energy x
= ∫ Fs dx − 0
t Fs2 Fs2 & = F x dt − s 2ω 2 ∫0 2ω 2
(258)
The energy terms in the above equations are given in energy per unit mass. Through the remainder of this section, the term “energy” refers to the energy per unit mass.
(252)
where Fs [x( t )] is the nonlinear restoring force per unit mass. Integrating Equation (252) over the entire relative displacement history, results in the following energy balance equation:
E I = EK + ED + ES + EH
(253)
where
E I = Input energy = x
t
(254)
∫ a(t )dx = ∫ a(t ) x&dt E K = Kinetic energy = x
∫ &x&dx = 0
x& 2 2
(255)
E D = Dissipative damping energy x
t
= 2 βω ∫ x&dx = 2 βω ∫ x& 2 dt
(256)
Figure 268. Energy time histories for a low and a high frequency, elasticplastic structure subjected to El Centro ground motion. [After Zahrah and Hall (2133).]
Figure 268 presents the energy response computed by Zahrah and Hall (2133) as a
114 function of time for two elasticplastic SDOF structures; a low frequency (0.1 Hz) and a high frequency (5 Hz) structure; both with a 5% damping and a ductility of 3.0 subjected to the 1940 ElCentro ground motion. In these plots, the difference between the input energy and the dissipated energy (sum of damping and hysteretic) represents the stored energy (sum of strain and kinetic). The stored energy becomes vanishingly small at the end of motion and the energy dissipated in the structure becomes almost equal to the energy imparted to it. The larger peaks and troughs in the energy response of a lowfrequency structure as compared to a highfrequency structure indicate that for lowfrequency structures, a larger portion of the energy imparted to the structure is stored in the form of strain and kinetic energies. Zahrah and Hall (2133) introduced an energy spectrum as a plot of the numerical value of the input energy E I at the end of motion as a function of period or frequency for different damping and ductility ratios. Examples of such spectra are shown in Figure 269 for linear structures with different damping ratios using the ElCentro record and for nonlinear structures with 2% damping and ductility ratios of 2 and 5 using Taft ground motion. Zahrah and Hall indicated that for linear structures under the same ground motion, input energy spectra are generally similar in shape to response spectra and that the quantity 1 2 m( PSV ) 2 for an undamped structure is a good estimate of the amount of input energy imparted to the structure. For damped structures, however, this quantity underestimates the input energy. They also indicated that the energy spectral shapes for nonlinear systems are similar to those of linear systems and that the amount of energy input is nearly the same for a linear and a nonlinear structure (with moderate ductility) with the same frequency.
Chapter 2
(a)
(b) Figure 269. Input energy spectra for (a) linear systems with 2, 5, and 10% damping using El Centro ground motion and (b) elsticplastic systems with 2% damping and ductility ratios of 2 and 5 using Taft ground motion. [After Zahrah and Hall (2133).]
According to Uang and Bertero (2134), the energy equations in (253) through (258) should be considered as “relative energy equations” since the integrations are performed for equations of motion using the relative displacements. For this system of equations, the relative input energy is defined as the work done by the static equivalent lateral force on a fixedbase system. Uang and Bertero introduced the “absolute energy equations” by integrating the equation of motion using the absolute displacements. For
2. Earthquake Ground Motion and Response Spectra
115
Figure 270. (a) Absolute and (b) relative energy time histories for elasticplastic systems with 5% damping and ductility ratios of 5 subjected to the 1986 San Salvador earthquake. [After Uang and Bertero (2134).]
the absolute energy terms; E D , E S , and E H are the same as their relative counterparts while the absolute input energy is given as ∫ &x&t dx g
structure using the relative and absolute energy terms. In addition, Uang and Bertero (2134) converted the input energy to an equivalent velocity such that
and the absolute kinetic energy is given as
x&t2 / 2 ; where xt and x g are the absolute and ground displacement; respectively. The absolute input energy represents the work done by the total base shear on the foundation displacement. The difference between the absolute and relative, input and kinetic energies is given by:
E I ,abs − E I ,rel = E K ,abs − E K ,rel =
x& g2 2
+ x&x& g
(259)
Figure 270 shows energy timehistories for a short and a longperiod elasticplastic
VI = 2 E I
(260)
where E I can be the relative or absolute input energy per unit mass. Figure 271 presents the relative and absolute input energy equivalent velocity spectra along with the peak ground velocity for three earthquake records. As the plots indicate, the relative and absolute input energies are very close for the midrange periods (in the vicinity of predominant periods of ground motion). For longer and shorter periods, however, the difference between relative and absolute energies is significant. The figure also shows that the absolute and relative equivalent velocities converge to the
116
Chapter 2
Absolute equivalent velocity
Relative equivalent velocity
Peak ground velocity
Figure 271. Absolute and relative input energy equivalent velocity spectra for elasticplastic systems with 5% damping and ductility ratio of 5 using three earthquake records. [After Uang and Bertero (2134).]
peak ground velocity at very short and very long periods, respectively. Subsequently, Uang and Bertero concluded that the absolute input energy can be used as a damage index for shortperiod structures, while the relative input energy is more suitable for longperiod structures. Their study also showed, using energy spectra, that the input energy is insensitive to the ductility ratio. Finally, Uang and Bertero (2134) believed that for linear structures, Housner’s use of 1 2 m( PSV ) 2 to estimate input energy reflects the maximum elastic energy stored in the structure without consideration of damping energy. It should be noted that at the time of this writing, the energy concept outlined in this section does not provide the basis for seismic design, despite the body of knowledge that has been developed. Further research is required to reliably estimate both the energy demand and energy capacity of structures in order to implement energy approaches in seismic design procedures.
2.12
ARTIFICIALLY GENERATED GROUND MOTION
One major drawback in using the response spectrum method in analysis and design of structures lies in the limitation of the method to provide temporal information on structural response and behavior. Such information is sometimes necessary in arriving at a satisfactory design. For example, the response spectrum procedure can be used to estimate the maximum response in each mode of vibration, and procedures such as square root of the sum of the squares can be used to combine the modal responses. When the natural frequencies are close to each other, however, the square root of the sum of the squares can result in inaccurate estimate of the response. In such cases, the complete quadrature combination8 CQC, or a timehistory analysis may be used. If inelastic deformation is permitted in design, the inelastic spectra and the deamplification factors presented in the previous sections 8
An improved procedure for computing modal responses referred to as complete quadrature combination CQC was proposed by Der Kiureghian (see Chapter 3).
2. Earthquake Ground Motion and Response Spectra cannot be used to compute the response of structures modeled as multidegreeoffreedom, and one therefore relies on a timehistory analysis for computing the inelastic response. In many cases, structures house equipment are sensitive to floor vibrations during an earthquake. It is sometimes necessary to develop floor response spectra from the timehistory response of the floor. In addition, when designing critical or major structures such as power plants, dams, and highrise buildings, the final design is usually based on a complete timehistory analysis. The problem which often arises is what representative accelerogram should be used. Artificially generated accelerograms which represent earthquake characteristics such as a given magnitude, epicentral distance, and soil condition of the site have been used for this purpose as well as in research. For example, Penzien and Liu (2127) used artificial accelerograms to investigate the statistical characteristics of inelastic systems and Lai and Biggs (2126) used them to obtain inelastic acceleration and displacement response ratios. Random models have been used to simulate earthquake ground motion and generate artificial accelerograms. Both stationary and nonstationary random processes have been suggested (see for example 2135 to 2138). Other studies have proposed sitedependent power spectral density from recorded ground motion, which can be utilized in generating artificial accelerograms. One of the first attempts in generating artificial accelerograms was by Housner and Jennings (2135) who modeled ground motion as a stationary Gaussian random process with a power spectral density from undamped velocity spectra of recorded accelerograms. They developed a procedure for generating a random function that has the same properties of strong earthquake ground motion and used it to generate eight artificial accelerograms of 30 sec duration which exhibit the same statistical properties of real ground motion. The detailed description of the procedures for generating artificial accelerograms is
117 beyond the scope of this chapter. It may, however, be useful to briefly mention the basic elements, which are generally needed to generate an artificial accelerogram. In most cases, these elements consist of a power spectral density or a zerodamped response spectrum, a random phase angle generator, and an envelope function. The simulated motion is then obtained as a finite sum of several harmonic excitations. Usually an iterative procedure is needed to check the consistency of the artificial motion by examining its frequency content through its response spectrum or its power spectral density. A typical artificial accelerogram and integrated velocity and displacement generated from the KanaiTajimi (219, 220) power spectral density for alluvium using the peak acceleration and the duration of strong motion of the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940 is shown in Figure 272.
2.13
SUMMARY AND CONCLUSION
The stateoftheart in strong motion seismology and ground motion characterization has advanced significantly in the past three decades. One can now estimate, with reasonable accuracy, the design ground motion and spectral shapes at a given location. Earthquake magnitude, source distance, site geology, fault characteristics, duration of strong motion, etc. influence ground motion and spectral shapes. While building codes and seismic provisions account for some of these influences such as site geology, magnitude, and distance, others such as fault characteristics, travel path, and duration require further studies before they can be implemented. Response spectrum is used extensively in seismic design of structures. Recent codes recommend acceleration amplifications in terms of seismic coefficients, which account for site geology, shaking intensity, and distance for constructing design spectra and computing the design lateral forces.
118
Chapter 2
Figure 272. Acceleration  time history and integrated velocity and displacement generated from the KanaiTajimi power spectral density for alluvium using the peak ground acceleration and the duration of the S00E component of El Centro, the Imperial Valley earthquake of May 18, 1940.
In moderate and strong earthquakes, structures can experience nonlinear behavior and dissipate a portion of the seismic energy through inelastic action. To account for the energy absorption capacity of the structure, seismic codes allow the use of response modification factors, referred to as Rfactors, to reduce the elastic design forces and amplify the elastic displacements (drifts). Although the application of inelastic spectra is limited to structures which can be modeled as singledegreeoffreedom, inelastic spectra can be used to estimate the ductility demands which are needed to compute response modification or Rfactors. In special cases such as design of critical or essential structures, a timehistory analysis may be warranted. Determination of a representative
set of accelerograms which reflects the earthquake characteristics expected at the site is important. Artificially generated ground motion may be used to determine representative accelerograms. In most cases, particularly for critical and essential structures, the advice of geologists, seismologists, geotechnical engineers, and structural engineers should be obtained before ground motion and spectral shape estimates are finalized for design.
ACKNOWLEDGMENT The authors wish to thank Dr. Fawzi E. Elghadamsi who coauthored this chapter in the first edition of the handbook. His contributions, some of which are reflected in this edition, are gratefully acknowledged.
2. Earthquake Ground Motion and Response Spectra
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Benjamin, J. R., “Probabilistic Models for Seismic Force Design,” J. Structural Div., ASCE, Vol. 94, ST5, 11751196, 1968. DerKiureghian, A. and Ang A. HS., “A FaultRupture Model for Seismic Risk Analysis,” Bull. Seism. Soc. Am., Vol. 67, No. 4, 11731194, 1977. Algermissen, S. T. and Perkins, D. M., “A Technique for Seismic Risk Zoning, General Considerations and Parameters,” Proc. Microzonation Conf., 865877, Seattle, Washington, 1972. Algermissen, S. T. and Perkins, D. M., “A Probabilistic Estimate of Maximum Acceleration in Rock in Contiguous United States,” USGS Open File Report, 76416, 1976. Applied Technology Council, National Bureau of Standards, and National Science Foundation, “Tentative Provisions for the Development of Seismic Regulations for Buildings,” ATC Publication 306, NBS Publication 510, NSF Publication 788, 1978. McGuire, R. K., “Seismic Structural Response Risk Analysis, Incorporating Peak Response Progressions on Earthquake Magnitude and Distance,” Report R7451, Dept. of Civil Engineering, Mass. Inst. of Technology, Cambridge, Mass., 1975. NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1985 Edition, Building Seismic Safety Council, Washington, D.C., 1985. NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1988 Edition, Building Seismic Safety Council, Washington, D.C., 1988. NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1991 Edition, Building Seismic Safety Council, Washington, D.C., 1991. NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1994 Edition, Building Seismic Safety Council, Washington, D.C., 1994. NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1997 Edition, Building Seismic Safety Council, Washington, D.C., 1997. NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA273, Building Seismic Safety Council, Washington, D.C., 1997. Uniform Building Code, 1985 Edition, International Conference of Building Officials, Whittier, California, 1985. Uniform Building Code, 1988 Edition, International Conference of Building Officials, Whittier, California, 1988.
122 284
285
286
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288
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290
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294
295
296
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Chapter 2 Uniform Building Code, 1991 Edition, International Conference of Building Officials, Whittier, California, 1991. Uniform Building Code, 1994 Edition, International Conference of Building Officials, Whittier, California, 1994. Uniform Building Code, 1997 Edition, International Conference of Building Officials, Whittier, California, 1997. Newmark, N. M. and Hall, W. J., “Earthquake Spectra and Design,” Earthquake Engineering Research Institute, Berkeley, California, 1982. Newmark, N. M., Blume, J. A., and Kapur, K. K., “Seismic Design Criteria for Nuclear Power Plants,” J. Power Div., ASCE, Vol. 99, No. PO2, 287303, 1973. Newmark, N. M. and Hall, W. J., “Seismic Design Criteria for Nuclear Reactor Facilities,” Proc. 4th World Conf. Earthquake Eng., B4, 3750, Santiago, Chile, 1969. Newmark, N. M. and Hall, W. J., “Procedures and Criteria for Earthquake Resistant Design,” Building Practices for Disaster Mitigation, National Bureau of Standards, U.S. Department of Commerce, Building Research Series 46, 209236, 1973. Hall, W. J., Mohraz B., and Newmark, N. M., “Statistical Studies of Vertical and Horizontal Earthquake Spectra,” Nathan M. Newmark Consulting Engineering Services, Urbana, Illinois, 1975. Newmark, N. M. and Rosenblueth, E., Fundamentals of Earthquake Engineering, PrenticeHall, Englewood Cliffs, N.J., 1971. Mohraz, B. and Tiv, M., “Spectral Shapes and Amplifications for the Loma Prieta Earthquake of October 17, 1989,” Proc. 3rd U.S. Conf. Lifeline Earthquake Eng., 562571, Los Angeles, California, 1991. Trifunac, M. D., Brady, A. G., and Hudson, D. E., “Analysis of StrongMotion Earthquake Accelerograms, Vol. III, Response Spectra, Parts A through Y,” Earthquake Eng. Research Laboratory, California Institute of Technology, Pasadena, California, 19721975. Chopra, A. K., “Dynamics of Structures  A Primer,” Earthquake Engineering Research Institute, Berkeley, California, 1981. Sadek, F., Mohraz, B., and Riley, M. A., “Linear Procedures for Structures with VelocityDependent Dampers,” Journal of Structural Engineering, ASCE, Vo. 128, No. 8, 887895, 2000. Hayashi, S., Tsuchida, H., and Kurata, E., “Average Response Spectra for Various Subsoil Conditions,” Third Joint Meeting, U.S.  Japan Panel on Wind and Seismic Effects, UJNR, Tokyo, 1971.
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2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
Kuribayashi, E., Iwasaki, T., Iida, Y., and Tuji, K., “Effects of Seismic and Subsoil Conditions on Earthquake Response Spectra,” Proc. International Conf. Microzonation, Seattle, Wash., 499512, 1972. Seed, H. B., Ugas, C., and Lysmer, J., “SiteDependent Spectra for EarthquakeResistance Design,” Bull. Seism. Soc. Am., Vol. 66, No. 1, 221243, 1976. Atomic Energy Commission, “Design Response Spectra for Seismic Design of Nuclear Power Plants,” Regulatory Guide 1.60, Directorate of Regulatory Standards, Washington, D.C., 1973. Crouse, C. B. and McGuire, J. W., “Site Response Studies for Purpose of Revising NEHRP Seismic Provisions,” Earthquake Spectra, Vol. 12, No. 3, 407439, 1996. Mohraz, B., “Influences of the Magnitude of the Earthquake and the Duration of Strong Motion on Earthquake Response Spectra,” Proc. Central Am. Conf. on Earthquake Eng., San Salvador, El Salvador, 1978. Mohraz, B., “Recent Studies of Earthquake Ground Motion and Amplification,” Proc. 10th World Conf. Earthquake Eng., Madrid, Spain, 66956704, 1992. Peng, M. H., Elghadamsi, F. E., and Mohraz, B., “A Simplified Procedure for Constructing Probabilistic Response Spectra,” Earthquake Spectra, Vol. 5, No. 2, 393408, 1989. Singh, J. P., “Earthquake Ground Motions: Implications for Designing Structures and Reconciling Structural Damage,” Earthquake Spectra, Vol. 1, No. 2, 239270, 1985. “Reducing Earthquake Hazards: Lessons Learned from Earthquakes,” Earthquake Engineering Research Institute, Publication No. 8602, Berkeley, California, 1986. Housner, G. W. and Jennings, P. C., “Earthquake Design Criteria,” Earthquake Engineering Research Institute, Berkeley, California, 1982. Housner, G. W., “Spectrum Intensities of StrongMotion Earthquakes,” Proc. of the Symposium on Earthquakes and Blast Effects on Structures, Earthquake Engineering Research Institute, 1952. Housner, G. W., “Behavior of Structures During Earthquakes,” J. Eng. Mech. Div., ASCE Vol. 85, No. EM4, 109129, 1959. Housner, G. W., “Design Spectrum,” Chapter 5 in Earthquake Engineering, R.L. Wiegel, Editor, PrenticeHall, Englewood Cliffs, N.J., 1970. Blume, J. A., Sharpe, R. L., and Dalal, J. S., “Recommendations for Shape of Earthquake Response Spectra,” John A. Blume & Associates, San Francisco, California, AEC Report Wash1254, 1972.
2. Earthquake Ground Motion and Response Spectra 2112 Seismology Committee, Structural Engineers Association of California, “Recommended Lateral Force Requirements,” 1986. 2113 Martin, G. M., Editor, Proceedings of the NCEER/SEAOC/BSSC Workshop on Site Response During Earthquakes and Seismic Code Provisions, University of Southern California, Los Angeles, 1994. 2114 Borcherdt, R. D., “Estimates of SiteDependent Response Spectra for Design (Methodology and Justification),” Earthquake Spectra, Vol. 10, No. 4, 617653, 1994. 2115 Seed, R. B., Dickenson, S. E., and Mok, C. M., “Recent Lessons Regarding Seismic Response Analyses of Soft and Deep Clay Sites,” in Proc. 4th JapanU.S. Workshop on Earthquake Resistant Design of Lifeline Facilities and Countermeasures for Soil Liquefaction, National Center for Earthquake Engineering Research, State University of New York at Buffalo, Vol. I, 131145, 1992. 2116 Dobry, R., Martin, G. M., Parra, E., and Bhattacharyya, Study of Ratios of Response Spectra Soil/Rock and of Site Categories for Seismic Codes, National Center for Earthquake Engineering Research, State University of New York at Buffalo, 1994. 2117 Borcherdt, R. D., “Preliminary Amplification Estimates Inferred from StrongGroundMotion Recordings of the Northridge Earthquake of January 17, 1994,” Proc. Of the International Workshop on Site Response Subjected to Strong Earthquake Motions, Japan Port and Harbour Research Institute, Vol. 2, 2146, Yokosuka, Japan, 1996. 2118 Riddell, R., and Newmark, N. M., “Statistical Analysis of the Response of Nonlinear Systems Subjected to Earthquakes,” Civil Engineering Studies, Structural Research Series 468, Department of Civil Engineering, University of Illinois, Urbana, Illinois, 1979. 2119 Elghadamsi, F. E and Mohraz, B., “Inelastic Earthquake Spectra,” J. Earthquake Engineering and Structural Dynamics, Vol. 15, 91104, 1987. 2120 Blume, J. A., Newmark, N. M., and Corning, L. H., “Design of Multistory Reinforced Concrete Buildings for Earthquake Motions,” Portland Cement Association, 1961. 2121 Newmark, N. M., “Current Trends in the Seismic Analysis and Design of HighRise Structures,” Chapter 16 in Earthquake Engineering, R.L. Wiegel, Editor, PrenticeHall, Englewood Cliffs, N.J., 1970. 2122 Housner, G. W., “Limit Design of Structures to Resist Earthquakes,” Proc. 1st World Conf. Earthquake Engineering, 51 to 513, Berkeley, Calif., 1956.
123 2123 Blume, J. A., “A Reserve Energy Technique for the Earthquake Design and Rating of Structures in the Inelastic Range,” Proc. 2nd World Conf. Earthquake Engineering, Vol. II, 10611084, Tokyo, Japan 1960. 2124 Blume, J. A., “Structural Dynamics in EarthquakeResistant Design,” Transactions, ASCE, Vol. 125, 10881139, 1960. 2125 Blume, J. A., discussion of “Electrical Analog for Earthquake Yield Spectra,” J. Engineering Mechanics Div., ASCE, Vol. 86, No. EM3, 177184, 1960. 2126 Lai, S. P. and Biggs, J. M., “Inelastic Response Spectra for Aseismic Building Design,” J. Struct. Div., ASCE, Vol. 106, No. ST6, 12951310, 1980. 2127 Penzien, J. and Liu, S. C., “Nondeterministic Analysis of Nonlinear Structures Subjected to Earthquake Excitations,” Proc. 4th World Conf. Earthquake Engineering, A1, 114129, Santiago, Chile, 1969. 2128 Peng, M. H., Elghadamsi, F. E., and Mohraz, B., “A Stochastic Procedure for Seismic Analysis of SDOF Structures,” Civil and Mechanical Engineering Dept., School of Engineering and Applied Science, Southern Methodist University, Dallas, TX, 1987. 2129 Applied Technology Council ATC, Structural Response Modification Factors, ATC19 Report, Redwood City, California, 1995. 2130 Krawinkler, H. and Nassar, A. A., “Seismic Design Based on Ductility and Cumulative Damage Demands and Capacities,” Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings, Edited by Fajfar and Krawinkler, Elsevier Applied Science, New York, 1992. 2131 Nassar, A. A. and Krawinkler, H., Seismic Demands for SDOF and MDOF Systems, John A. Blume Earthquake Engineering Center, Report No. 95, Stanford University, Stanford, California, 1991.
124 2131 Miranda, E. and Bertero, V. V., “Evaluation of Strength Reduction Factors for EarthquakeResistant Design,” Earthquake Spectra, Vol. 10, No. 2, 357379, 1994. 2132 Zahrah, T. F. and Hall, W. J., “Earthquake Energy Absorption in SDOF Structures,” Journal of Structural Engineering, ASCE, Vol. 110, No. 8, 17571772, 1984. 2133 Uang, C. M. and Bertero, V. V., “Evaluation of Seismic Energy Structures,” Earthquake Engineering and Structural Dynamics, Vol. 19, 7790, 1990. 2134 Housner, G. W. and Jennings, P. C., “Generation of Artificial Earthquakes,” J. Engineering Mechanics Div., ASCE, Vol. 90, 113150, 1964. 2135 Shinozuka, M. and Salo, Y., “Simulation of Nonstationary Random Process,” J. Engineering Mechanics Div., ASCE, Vol. 93, 1140, 1967. 2136 Amin, M. and Ang, A. H.  S., “Nonstationary Stochastic Model of Earthquake Ground Motion,” J. Engineering Mechanics Div., ASCE, Vol. 74, No. EM2, 559583, 1968. 2137 Iyengar, R. N. and Iyengar, K. T. S., “A Nonstationary Random Process Model for Earthquake Accelerograms,” Bull. Seism. Soc. Am., Vol. 59, 11631188, 1969.
Chapter 2
Chapter 4 Dynamic Response of Structures
James C. Anderson, Ph.D. Professor of Civil Engineering, University of Southern California, Los Angeles, California
Key words:
Dynamic, Buildings, Harmonic, Impulse, SingleDegreeof –Freedom, Earthquake, Generalized Coordinate, Response Spectrum, Numerical Integration, Time History, MultipleDegreeofFreedom, Nonlinear, Pushover, Instrumentation
Abstract:
Basic principles of structural dynamics are presented with emphasis on applications to the earthquake resistant design of building structures. Dynamic characteristics of single degree of freedom systems are discussed along with their application to single story buildings. The response of these systems to harmonic and impulse loading is described and illustrated by application to simple structures. Consideration of the earthquake response of these systems leads to the concept of the elastic response spectrum and the development of design spectra. The use of procedures based on a single degree of freedom is extended to multiple degree of freedom systems through the use of the generalized coordinate approach. The determination of generalized dynamic properties is discussed and illustrated. A simple numerical integration procedure for determining the nonlinear dynamic response is presented. The application of matrix methods for the analysis of multiple degree of freedom systems is discussed and illustrated along with earthquake response analysis. A response spectrum procedure suitable for hand calculation is presented for elastic response analyses. The nonlinear static analysis for proportional loading and the nonlinear dynamic analysis for earthquake loading are discussed and illustrated with application to building structures. Finally, the use of the recorded response from buildings containing strong motion instrumentation for verification of analytical models is discussed.
183
184
Chapter 4
4. Dynamic Response of Structures
4.1
185
Introduction
The main cause of damage to structures during an earthquake is their response to ground motions which are input at the base. In order to evaluate the behavior of the structure under this type of loading condition, the principles of structural dynamics must be applied to determine the stresses and deflections, which are developed in the structure. Structural engineers are familiar with the analysis of structures for static loads in which a load is applied to the structure and a single solution is obtained for the resulting displacements and member forces. When considering the analysis of structures for dynamic motions, the term dynamic simply means “timevarying”. Hence the loading and all aspects of the response vary with time. This results in possible solutions at each instant during the time interval under consideration. From an engineering standpoint, the maximum values of the structural response are usually the ones of particular interest, specially in the case of structural design. The purpose of this chapter is to introduce the principles of structural dynamics with emphasis on earthquake response analysis. Attention will initially be focused on the response of simple structural systems, which can be represented in terms of a single degree of freedom. The concepts developed for these systems will then be extended to include generalized singledegreeoffreedom (SDOF) systems using the generalizedcoordinate approach. This development in turn leads to the consideration of the response of structures having multiple degrees of freedom. Finally, concepts and techniques used in nonlinear dynamicresponse analysis will be introduced.
4.2
Dynamic Equilibrium
The basic equation of static equilibrium used in the displacement method of analysis has the form,
p = kv
(41)
where p is the applied force, k is the stiffness resistance, and v is the resulting displacement. If the statically applied force is now replaced by a dynamic or timevarying force p(t), the equation of static equilibrium becomes one of dynamic equilibrium and has the form
p(t ) = mv&&(t ) + cv&(t ) + kv (t )
(42)
where a dot represents differentiation with respect to time. A direct comparison of these two equations indicates that two significant changes, which distinguish the static problem from the dynamic problem, were made to Equation 41 in order to obtain Equation 42. First, the applied load and the resulting response are now functions of time, and hence Equation 42 must be satisfied at each instant of time during the time interval under consideration. For this reason it is usually referred to as an equation of motion. Secondly, the time dependence of the displacements gives rise to two additional forces which resist the applied force and have been added to the righthand side. The equation of motion represents an expression of Newton’s second law of motion, which states that a particle acted on by a force (torque) moves so that the time rate of change of its linear (angular) momentum is equal to the force (torque):
p( t ) =
d dv (m ) dt dt
(43)
where the rate of change of the displacement with respect to time, dv/dt, is the velocity, and the momentum is given by the product of the mass and the velocity. Recall that the mass is equal to the weight divided by the acceleration of gravity. If the mass is constant, Equation 43 becomes
p( t ) = m
d dv ( ) = mv&&(t ) dt dt
(44)
186
Chapter 4
which states that the force is equal to the product of mass and acceleration. According to d’Alembert’s principle, mass develops an inertia force, which is proportional to its acceleration and opposing it. Hence the first term on the righthand side of Equation 42 is called the inertia force; it resists the acceleration of the mass. Dissipative or damping forces are inferred from the observed fact that oscillations in a structure tend to diminish with time once the timedependent applied force is removed. These forces are represented by viscous damping forces, that are proportional to the velocity with the constant proportionality referred to as the damping coefficient. The second term on the righthand side of Equation 42 is called the damping force. Inertia forces are the more significant of the two and are a primary distinction between static and dynamic analyses. It must also be recognized that all structures are subjected to gravity loads such as selfweight (dead load) and occupancy load (live load) in addition to dynamic base motions. In an elastic system, the principle of superposition can be applied, so that the responses to static and dynamic loadings can be considered separately and then combined to obtain the total structural response. However, if the structural behavior becomes nonlinear, the response becomes loadpathdependent and the gravity loads must be considered concurrently with the dynamic base motions. Under strong earthquake motions, the structure will most likely display nonlinear behavior, which can be caused by material nonlinearity and/or geometric nonlinearity. Material nonlinearity occurs when stresses at certain critical regions in the structure exceed the elastic limit of the material. The equation of dynamic equilibrium for this case has the general form
p(t ) = mv&&(t ) + cv&(t ) + k (t )v (t )
(45)
in which the stiffness or resistance k is a function of the yield condition in the structure,
which in turn is a function of time. Geometric nonlinearity is caused by the gravity loads acting on the deformed position of the structure. If the lateral displacements are small, this effect, which is often referred to as Pdelta, can be neglected. However, if the lateral displacements become large, this effect must be considered. In order to define the inertia forces completely, it would be necessary to consider the accelerations of every mass particle in the structure and the corresponding displacements. Such a solution would be prohibitively timeconsuming. The analysis procedure can be greatly simplified if the mass of the structure can be concentrated (lumped) at a finite number of discrete points and the dynamic response of the structure can be represented in terms of this limited number of displacement components. The number of displacement components required to specify the position of the mass points is called the number of dynamic degrees of freedom. The number of degrees of freedom required to obtain an adequate solution will depend upon the complexity of the structural system. For some structures a single degree of freedom may be sufficient, whereas for others several hundred degrees of freedom may be required.
4.3
SINGLEDEGREEOFFREEDOM SYSTEMS
4.3.1
TimeDependent Force
The simplest structure that can be considered for dynamic analysis is an idealized, onestory structure in which the single degree of freedom is the lateral translation at the roof level as shown in Figure 41. In this idealization, three important assumptions are made. First, the mass is assumed to be concentrated (lumped) at the roof level. Second, the roof system is assumed to be rigid, and third, the axial deformation in the columns is neglected. From these assumptions it follows that all lateral resistance is in the resisting elements such as columns, walls, and diagonal
4. Dynamic Response of Structures
187
braces located between the roof and the base. Application of these assumptions results in a discretized structure that can be represented as shown in either Figure 4lb or 41c with a timedependent force applied at the roof level. The total stiffness k is simply the sum of the stiffnesses of the resisting elements in the story level. The forces acting on the mass of the structure are shown in Figure 41d. Summing the forces acting on the free body results in the following equation of equilibrium, which must be satisfied at each instant of time:
f i + f d + f s = p (t ) where fi = inertia force = mu&&
(46)
fd = damping (dissipative) force= cv& fs = elastic restoring force = kv p(t ) = timedependent applied force
u&& is the total acceleration of the mass, and & v are the velocity and displacement of the v,
mass relative to the base. Writing Equation 46 in terms of the physical response parameters results in
mu&& + cv& + kv = p(t )
(47)
It should be noted that the forces in the damping element and in the resisting elements depend upon the relative velocity and relative displacement, respectively, across the ends of these elements, whereas the inertia force depends upon the total acceleration of the mass. The total acceleration of the mass can be
Figure 41. singledegreeoffreedom system subjected to timedependent force.
188
Chapter 4
expressed as
4.3.2
u&&(t ) = g&&(t ) + v&&(t )
(48)
where
v&&(t ) = acceleration of the mass relative to the base
g&&(t ) = acceleration of the base In this case, the base is assumed to be fixed &&(t ) = 0 and with no motion, and hence g u&&(t ) = v&&(t ) . Making this substitution for the acceleration, Equation 47 for a timedependent force becomes
mv&& + cv& + kv = p(t )
(49)
Earthquake Ground Motion
When a singlestory structure, shown in Figure 42a, is subjected to earthquake ground motions, no external dynamic force is applied at the roof level. Instead, the system experiences an acceleration of the base. The effect of this on the idealized structure is shown in Figure 42b and 42c. Summing the forces shown in Figure 42d results in the following equation of dynamic equilibrium:
fi + fd + f s = 0
(410)
Substituting the physical parameters for fi, fd and fs in Equation 410 results in an equilibrium equation of the form
mu&& + cv& + kv = 0
Figure 42. Singledegreeoffreedom system subjected to base motion.
(411)
4. Dynamic Response of Structures
189
This equation can be written in the form of Equation 49 by substituting Equation 48 into Equation 411 and rearranging terms to obtain
mv&& + cv& + kv = pe (t )
(412)
where
pe (t ) = effective timedependent force = − mg&&(t )
Hence the equation of motion for a structure subjected to a base motion is similar to that for a structure subjected to a timedependent force if the base motion is represented as an effective timedependent force which is equal to the product of the mass and the ground acceleration. 4.3.3
Figure 43. Rotating particle of mass.
Mass and Stiffness Properties
Most SDOF models consider structures, which experience a transactional displacement of the roof relative to the base. In this case the translational mass is simply the concentrated weight divided by the acceleration of gravity (32.2 ft/sec2 or 386.4 in./sec2). However, cases do arise in which the rotational motion of the system is significant. An example of this might be the rotational motion of a roof slab which has unsymmetrical lateral supports. Newton’s second law of motion states that the time rate of change of the angular momentum (moment of momentum) equals the torque. Considering a particle of mass rotating about an axis o, as shown in Figure 43, the moment of momentum can be expressed as
L = rmv&(t ) = mr 2
dθ dt
(413)
The torque N is then obtained by taking the time derivative:
N=
dL = I&θ& dt
(414)
where
I = mr 2 = mass moment of inertia For a rigid body, the mass moment of inertia can be obtained by summing over all the mass particles making up the rigid body. This can be expressed in integral form as
I = ∫ ρ 2 dm
(415)
where ρ is the distance from the axis of rotation to the incremental mass dm. For dynamic analysis it is convenient to treat the rigidbody inertia forces as though the translational mass and the mass moment of inertia were concentrated at the center of mass. The mass and mass moment of inertia of several common rigid bodies are summarized in Figure 44. Example 41 (Determination of Mass Properties) Compute the mass and mass moment of inertia for the rectangular plate shown in Figure 45. • Translational mass:
190
Chapter 4
Figure 44. Rigidbody mass and mass moment of inertia.
dm = µdV = µtdxdy
m = µν = µabt where µ = mass density = mass per unit volume V = total volume • Rotational mass moment of inertia:
I = ∫ ρ 2 dm ,
Where ρ 2 = x 2 + y 2
I = ∫ ρ 2 dm = 4µt ∫
a/2
∫
b/2
( x 2 + y 2 )dxdy
b 3 a + a 3b b2 + a 2 = µabt 48 12 2 2 a +b I =m 12 I = 4µt
4. Dynamic Response of Structures
191 force—displacement (stiffness) relationships of several of the more common lateral force members used in building structures. As indicated previously, the assumptions used in developing the SDOF model restrict lateral resistance to structural members between the roof and base. These might include such members as columns, diagonal braces, and walls. Stiffness properties for these elements are summarized in Figure 46. 4.3.4
Free Vibration
Figure 45. Rectangular plate of example 41.
In order to develop dynamic models of SDOF systems, it is necessary to review the
Free vibration occurs when a structure oscillates under the action of forces that are inherent in the structure without any externally
Figure 46. Stiffness properties of lateral force resisting elements.
192
Chapter 4
applied timedependent loads or ground motions. These inherent forces arise from the initial velocity and displacement the structure has at the beginning of the freevibration phase. Undamped Structures The equation of motion for an undamped SDOF system in free vibration has the form
mv&&(t ) + kv (t ) = 0
(416)
T=
2π m 1 = 2π = ω k f
(420a)
The amplitude of motion is given as: 2
. v(0) 2 p= + [v(0 )] w
(420b)
which can be written as
v&&(t ) + ω2 v (t ) = 0
(417)
where ω2 = k / m . This equation has the general solution
v (t ) = A sin ω t + B cos ω t
(418)
in which the constants of integration A and B depend upon the initial velocity v&(0) and initial displacement v(0). Applying the initial conditions, the solution has the form
v (t ) =
v&(0) sin ωt + v (0) cos ωt ω
(419)
This solution in time is represented graphically in Figure 47. Several important concepts of oscillatory motion can be illustrated with this result. The amplitude of vibration is constant, so that the vibration would, theoretically, continue indefinitely with time. This cannot physically be true, because free oscillations tend to diminish with time, leading to the concept of damping. The time it takes a point on the curve to make one complete cycle and return to its original position is called the period of vibration, T. The quantity ω is the circular frequency of vibration and is measured in radians per second. The cyclic frequency f is defined as the reciprocal of the period and is measured in cycles per second, or hertz. These three vibration properties depend only on the mass and stiffness of the structure and are related as follows:
Figure 47. Freevibration response of an undamped SDOF system.
It can be seen from these expressions that if two structures have the same stiffness, the one having the larger mass will have the longer period of vibration and the lower frequency. On the other hand, if two structures have the same mass, the one having the higher stiffness will have the shorter period of vibration and the higher frequency. Example 42 (Period of undamped free vibration) Construct an idealized SDOF model for the industrial building shown in Figure 48, and estimate the period of vibration in the two principal directions. Note that vertical cross
4. Dynamic Response of Structures bracings are made of 1inchdiameter rods, horizontal cross bracing is at the bottom chord of trusses, and all columns are W8 × 24. •Weight determination: Roof level: Composition roof 9.0 psf Lights, ceiling, mechanical 6.0 psf Trusses 2.6 psf Roof purlins, struts 2.0 psf Bottom chord bracing 2.1 psf Columns (10 ft, 9 in.) 0.5 psf Total 22.2 psf Walls: Framing, girts, windows 4.0 psf Metal lath and plaster 6.0 psf Total 10.0 psf Total weight and mass:
193 North—south:
k 231.6 = = 21.8 rad/sec m 0.485 2π 2π T= = = 0.287 sec. ω 21.8 1 f = = 3.48 Hz T
ω=
W = (22.2)(100)(75) + (10)(6)(200 + 150) W = 187,500 lb = 187.5 kips
m=
W 187.5 = = 0.485 kipssec2/in. g 386.4
•Stiffness determination: North—south (moment frames):
12 EI (12)( 29000)(82.8) = L3 (144) 3 k i = 9.6 kips/in. ki =
24
k = ∑ k i = 24(9.6) = 231.6 kips/in. i =1
Figure 48. Building of Example 42.
East—west (braced frames):
AE cos 2 θ L A = πd 2 / 4 = 0.785
ki =
L = 12 2 + 20 2 = 23.3 ft = 280 in. θ = tan −1 (12 / 20) = 31o , Cos(31o ) = 0.585 ki =
(0.785)( 29000)(0.858) 2 = 59.7 kips/in. 280
East—west:
k 358.7 = = 27.2 rad/sec m 0.485 2π 2π T= = = 0.23 sec. ω 27.2 1 f = = 4.3 HZ T ω=
6
k = ∑ k i = 6(59.7) = 358.7 kips/in. i =1
• Period determination:
Damped Structures In an actual structure which is in free vibration under the action of internal forces, the amplitude of the vibration
194
Chapter 4
tends to diminish with time and eventually the motion will cease. This decrease with time is due to the action of viscous damping forces which are proportional to the velocity. The equation of motion for this condition has the form
mv&&(t ) + cv&(t ) + kv (t ) = 0
(421)
This equation has the general solution sin ωd t + v (0) cos ωd t v (t ) = e −λωt [v&(0) + v (0)λω] ωd (422a)
defined as 2mω and is the least amount of damping that will allow a displaced oscillator to return to its original position without oscillation. For most structures, the amount of viscous damping in the system will vary between 3% and 10% of critical. Substituting an upper value of 20% into the above expression for the damped circular frequency gives the result that ω d = 0.98ω . Since the two values are approximately the same for values of damping found in structural systems, the undamped circular frequency is used in place of the damped circular frequency. In this case the amplitude of motion is given as: 2
where
λ=
C C = = percentage of critical Ccr 2mω
damping
ωd = ω 1 − λ2
=
damped
circular
frequency
. v(0) + v(0)λw 2 p= + [v(0)] (422b) wD One of the more useful results of the freevibration response is the estimation of the damping characteristics of a structure. If a structure is set in motion by some external force, which is then removed, the amplitude will decay exponentially with time as shown in Figure 49. It can further be shown that the ratio between any two successive amplitude peaks can be approximated by the expression
v (i ) = e 2 πλ v (i + 1)
(423)
Taking the natural logarithm of both sides results in
δ = ln
Figure 49. Free vibration response of a damped SDOF system.
The solution to this equation with time is shown in Figure 49. The damping in the oscillator is expressed in terms of a percentage of critical damping, where critical damping is
v (i ) = 2 πλ v (i + 1)
(424)
where the parameter δ is called logarithinic decrement. Solving for percentage of critical damping, λ, gives
λ≈
δ 2π
the the
(425)
4. Dynamic Response of Structures
195
The above equation provides one of the more useful means of experimentally estimating the damping characteristics of a structure.
4.4
Response to Basic Dynamic Loading
4.4.1
can
be
as g&&0 sin pt ,
the
equivalent force amplitude as poe = mg&&o and the frequency ratio β = p/ω. The solution for the time dependent displacement has the form
v (t ) =
Introduction
represented
mg&&o 1 × (sin pt − β sin ωt ) k (1 − β 2 ) (326b)
Time histories of earthquake accelerations are in general random functions of time. However, considerable insight into the response of structures can be gained by considering the response characteristics of structures to two basic dynamic loadings; harmonic loading and impulse loading. Harmonic loading idealizes the earthquake acceleration time history as a train of sinusoidal waves having a given amplitude. These might be representative of the accelerations generated by a large, distant earthquake in which the random waves generated at the source have been filtered by the soil conditions along the travel path. Impulse loading idealizes the earthquake accelerations as a short duration impulse usually having a sinusoidal or symmetrical (isosceles) triangular shape. The idealization may be a single pulse or it may be a pulse train containing a limited number of pulses. This loading is representative of that which occurs in the near fault region. This section will present a brief overview of the effects of harmonic loading and impulse loading on the response of building structures. 4.4.2
Harmonic Loading
For an undamped system subjected to simple harmonic loading, the equation of motion has the form
mv&& + kv = p0 sin pt
(326a)
where P0 is the amplitude and p is the circular frequency of the harmonic load. For a ground acceleration, the acceleration
where
mg&&o / k = poe / k = the static displacement 1 = dynamic amplification factor 1 − β2 sin pt = steady state response βsin ω t = transient response induced by the initial conditions From equation (426b) it can be seen that for lightly damped systems, the peak steady state response occurs at a frequency ratio near unity when the exciting frequency of the applied load equals the natural frequency of the system. This is the condition that is called resonance. The result given in Equation (426b) implies that the response of the undamped system goes to infinity at resonance, however, a closer examination in the region of β equal to unity, Clough and Penzien (44) , shows that it only tends toward infinity and that several cycles are required for the response to build up. A similar analysis for a damped system shows that at resonance, the dynamic amplification approaches a limit that is inversely proportional to the damping ratio
DA =
1 2λ
(426c)
For both the undamped and the damped cases, the response builds up with the number of cycles as shown in Figure 410a.
196
Chapter 4 4.4.3
Figure 410a. Resonance response.
The required number of cycles for the damped case can be estimated as 1/λ. The condition of resonance can occur in buildings which are subjected to base accelerations having a frequency that is close to that of the building and having a long duration. The duration of the ground shaking is an important factor in this type of response for the reasons just discussed. The Mexico City earthquakes (1957, 1979, 1985) have produced good examples of harmonic type ground motions which have a strong resonance effect on buildings. Ground motions having a period of approximately 2 seconds were recorded during the 1985 earthquake and caused several buildings to collapse in the upper floors. It must be recognized that as the response tends to build up, the effective damping will increase and as cracking and local yielding occur the period of the structure will shift. Both of these actions in the building will tend to reduce the maximum response. Since the dynamic amplification and number of cycles to reach the maximum response are both inversely proportional to the damping, the use of supplemental damping in the building to counter this type of ground motion is attractive.
Impulse Motion
Much of the initial work on impulse loads was done during the period of 19501965 and is discussed by Norris et al.(415). The force on structures generated by a blast or explosion can be idealized as a single pulse of relatively short duration. More recently it has become recognized that some earthquake motions, particularly those in the near fault region, can be idealized as either a single pulse or as a simple pulse train consisting of one to three pulses. The accelerations recorded in Bucharest, Romania during the Vrancea, Romania earthquake (1977), shown in Figure 410b, are a good example of this type of motion. It is of interest to note that this site is more than 100 miles from the epicenter, indicating that this type of motion is not limited to the near fault region.
Figure 410b. Bucharest (1977) ground acceleration.
The maximum response to an impulse load will generally be attained on the first cycle. For this reason, the damping forces do not have time to absorb much energy from the structure. Therefore, damping has a limited effect in controlling the maximum response and is usually neglected when considering the maximum response to impulse type loads. The rectangular pulse is a basic pulse shape. This pulse has a zero (instantaneous) rise time
4. Dynamic Response of Structures
197
and a constant amplitude, po, which is applied to the structure for a finite duration td. During the time period when the load is on the structure (t < td) the equation of motion has the form
mv&& + kv = po
(426d)
displacement ductility ratio which is defined as the ratio of the maximum displacement to the displacement at yield.
µ=
v max v yield
(426g)
which has the general solution
v (t ) =
p0 (1 − cos ω t ) k
(426e)
When the impulse load is no longer acting on the structure, the system is responding in free vibration and the equation of motion becomes
v (t ) =
v&(t d ) sin ω t + v (t d ) cos ω t (426f) ω
Figure 410c. Maximum elastic response, rectangular and triangular load pulses.[416]
where
t = t − td The displacement, v(td) and the velocity v&(t d ) at the end of the loading phase become the initial conditions for the free vibration phase. It can be shown that the dynamic amplification, DA, which is defined as the ratio of the maximum dynamic displacement to the static displacement, will equal 2 if td ≥ T/2 and will equal 2sin(π td /T) if t d ≤ T / 2 . For elastic response, the dynamic amplification is a function of the shape of the impulse load and the duration of the load relative to the natural period of the structure as shown in Figure 410c. For nonlinear behavior, the equation of motion becomes more complex, requiring the use of numerical methods for solution. Results of initial studies for basic pulse shapes were presented in the form of response charts(415) such as the one shown in Figure 410d which can be thought of as a constant strength response spectra. For nonlinear response, the dynamic amplification factor is replaced by the
Figure 410d. Maximum elastoplastic response, rectangular load pulse.[416]
It can also be seen that the single curve representing the elastic response becomes a family of curves for the inelastic response.
198
Chapter 4
These curves depend upon the ratio of the maximum system resistance, Rm, to the maximum amplitude of the impulse load. Note that the bottom curve in Figure 410d which has a resistance ratio of 2 represents the elastic response curve with the ductility equal to or less than unity for all values of td /T. It can also be seen that as the resistance ratio decreases, the ductility demand increases. 4.4.4
Example 43 (Analysis for Impulse Base Acceleration)
The three bay frame shown in Figure 410e is assumed to be pinned at the base. It is subjected to a ground acceleration pulse which has an amplitude of 0.5g and a duration of 0.4 seconds. It should be noted that this acceleration pulse is similar to one recorded at the Newhall Fire Station during the Northridge earthquake (1994). The lateral resistance at ultimate load is assumed to be elastoplastic. The columns are W10 × 54 with a clear height of 15 feet and the steel is A36 having a nominal yield stress of 36 ksi. Estimate the following:
For a W10 × 54 column, I = 303 in4 and Z = 66.6 in3 The lateral stiffness of an individual column is calculated as
ki =
kip 3EI 3( 29000) × 303 = = 4.5 3 3 L in (15 × 12)
and the total stiffness becomes
K = ∑ k i = 4 × 4.5 = 18.0
kip in
The mass is the weight divided by the acceleration of gravity,
m=
W 100 kips kips − sec 2 = = 0 . 26 g 386.4 in2 in sec
The period of vibration of the structure can now be calculated as
T = 2π
m 0.26 = 2π = 0.75 sec . k 18.0
and the duration ratio becomes
td 0.4 = = 0.53 T 0.75 The effective applied force, Pe is given as
Pe = mg&&o = m × 0.5g = 0.5W = 50 kips The ultimate lateral resistance of the structure occurs when plastic hinges form at the tops of the columns and a sway mechanism is formed. The nominal plastic moment capacity of a single column is
M P = Fy Z = 36. × 66.6 = 2400 in  kips and the shear resistance is
Vi =
M P 2400 = = 13.33 kips. h 180
The total lateral resistance is Figure 410e. Building elevation, resistance and loading, Example 43.
(a) the displacement ductility demand, (b) the maximum displacement and (c) the residual displacement.
R = 4Vi = 53.33 Kips The resistance to load ratio, is then given as
R 53.3 = = 1.1 Pev 50
4. Dynamic Response of Structures
199
Figure 410f. Computed displacement time history
Using this ratio and the duration ratio, td /T and entering the response spectrum given in Figure 410d, the displacement ductility demand is found to be 2.7. The displacement at yield can be obtained as
vy =
R 53.3 = = 3.0 in. K 18
and the maximum displacement is
v max = µ × lc y = 2.7 × 3.0 = 8.1in.
that structure reaches the maximum displacement on the first cycle and that from this time onward, it oscillates about a deformed position of 5.6 inches which is the plastic displacement. This can also be seen in a plot of the force versus displacement, shown in Figure 410g which indicates a single yield excursion followed by elastic oscillations about the residual displacement of 5.6 inches.
The residual or plastic deformation is the difference between the maximum displacement and the displacement at yield.
v( residual ) = v p = 8.1 − 3.0 = 5.1inches More recently, these calculations have been programmed for interactive computation on personal computers. The program NONLIN (414) can be used to do this type of calculation and to gain additional insight through the graphics that are available. Using the program, the maximum displacement ductility is calculated to be 2.85, the maximum displacement is 8.4 inches, and the plastic displacement is 5.6 in. A plot of the calculated time history of the displacement, shown in Figure 410f, indicates
Figure 410g. Computed force versus displacement.
200
Chapter 4
4.4.5
Approximate resopnse to impulse loading
In order to develop a method for evaluating the response of a structural system to a general dynamic loading, it is convenient to first consider the response of a structure to a shortduration impulse load as shown in Figure 410h, If the duration of the applied impulse load, t, is short relative to the fundamental period of vibration of the structure, T, then the effect of the impulse can be considered as an incremental change in velocity. Using the impulsemomentum relationship, which states that the impulse is equal to the change in momentum, the following equation is obtained:
v&(t ) =
1 t p(t )dt m ∫0
1 t1 p (t )dt , m ∫0
v (t − t1 ) =
1 t1 p (t ) dt sin ω (t − t1 ) mω ∫0
(427)
For a damped structural system, the freevibration response is given by Equation 422 Applying the above initial conditions to Equation 422 results in the following equation for the damped response:
v (t − t1 ) =
1 m ωd
∫
t1
p (t ) dt e −λ ω( t −t1 )
(428)
× sin ωd (t − t1 ) 4.4.6
Response to General Dynamic Loading
(426)
Following the application of the shortduration impulse load, the system is in free vibration and the response is given by Equation 419. Applying the initial conditions at the beginning of the free vibration phase,
v&(t1 ) =
Equation 419 becomes
v (t1 ) negligible
The above discussion of the dynamic response to a shortduration impulse load can readily be expanded to produce an analysis procedure for systems subjected to an arbitrary loading time history. Any arbitrary time history can be represented by a series of shortduration impulses as shown in Figure 411. Consider one of these impulses which begins at time ℑ after the beginning of the time history and has a duration dτ. The magnitude of this differential impulse is p(τ) dτ, and it produces a differential response which is given as
dv ( τ) =
p ( τ) sin ω t ′dτ mω
(429)
The time variable t ′ represents the freevibration phase following the differential impulse loading and can be expressed as
t′ = t − τ
(430)
Substituting this expression into Equation 429 results in
dv ( τ) =
Figure 410h. Short duration rectangular impulse.
p ( τ) sin ω(t − τ) dτ mω
(431)
The total response can now be obtained by superimposing the incremental responses of all the differential impulses making up the time
4. Dynamic Response of Structures
201
history. Integrating Equation 431, the total displacement response becomes
v (t ) =
1 t p( τ) sin ω(t − τ) dτ mω ∫0
(432)
which is known as the Duhamel integral. When considering a damped structural system, the differential response is given by Equation 428 and the Duhamel integral solution becomes t
v(t ) = ∫ o
p(τ ) e − λω (t −τ ) sin ω d (t − τ ) dτ mω d
the integral will require the use of numerical methods. For these two reasons, the use of a direct numerical integration procedure may be preferable for solving for the response of a dynamic system subjected to general dynamic load. This will be addressed in a later section on nonlinear response analysis. However, the Duhamelintegral result can be applied in a convenient and systematic manner to obtain a solution for the linear elastic structural response for earthquake load.
(433) 4.4.7
Earthquake Response of Elastic Structures
TimeHistory Response The response to earthquake loading can be obtained directly from the Duhamel integral if the timedependent force p(t) is replaced with the effective timedependent force Pe(t), which is the product of the mass and the ground acceleration. Making this substitution in Equation 433 results in the following expression for the displacement:
v (t ) =
V (t ) ω
(434)
where the response parameter V(t) represents the velocity and is defined as t
V (t ) = ∫ g&&( τ) e −λ ω( t − τ ) sin ωd (t − τ) dτ (435) 0
Figure 411. Differential impulse response.
Since the principle of superposition was used in the derivation of Equations 432 and 433, the results are only applicable to linear structural systems. Furthermore, evaluation of
The displacement of the structure at any instant of time during the entire time history of the earthquake under consideration can now be obtained using Equation 434. It is convenient to express the forces developed in the structure during the earthquake in terms of the effective inertia forces. The inertia force is the product of the mass and the total acceleration. Using Equation 411, the total acceleration can be expressed as
u&&(t ) = −
c k v&(t ) − v (t ) m m
(436)
202
Chapter 4
If the damping term can be neglected as contributing little to the equilibrium equation, the total acceleration can be approximated as
u&&(t ) = −ω 2 v(t )
(437)
The effective earthquake force is then given as
Q (t ) = mω2 v (t )
(438)
The above expression gives the value of the base shear in a singlestory structure at every instant of time during the earthquake time history under consideration. The overturning moment acting on the base of the structure can be determined by multiplying the inertia force by the story height:
M (t ) = hmω2 v (t )
(439)
Response Spectra Consideration of the displacements and forces at every instant of time during an earthquake time history can require considerable computational effort, even for simple structural systems. As mentioned previously, for many practical problems and especially for structural design, only the maximum response quantities are required. The maximum value of the displacement, as determined by Equation 434, will be defined as the spectral displacement
S d = v(t ) max
(440)
Substituting this result into Equations 438 and 439 results in the following expressions for the maximum base shear and maximum overturning moment in a SDOF system:
Qmax = mω2 S d
(441)
M max = hmω2 S d
(442)
An examination of Equation 434 indicates that the maximum velocity response can be approximated by multiplying the spectral displacement by the circular frequency. This response parameter is defined as the spectral pseudovelocity and is expressed as
S pv = ω S d
(443)
In a similar manner, Equation 437 indicates that the maximum total acceleration can be approximated as the spectral displacement multiplied by the square of the circular frequency. This product is defined as the spectral pseudoacceleration and is expressed as
S pa = ω2 S d
(444)
A plot of the spectral response parameter against frequency or period constitutes the response spectrum for that parameter. A schematic representation of the computation of the displacement spectrum for the northsouth component of the motion recorded at El Centro on May 18, 1940 has been presented by Chopra(41) and is shown in Figure 412. Because the three response quantities are related to the circular frequency, it is convenient to plot them on a single graph with log scales on each axis. This special type of plot is called a tripartite log plot. The three response parameters for the El Centro motion are shown plotted in this manner in Figure 413. For a SDOF system having a given frequency (period) and given damping, the three spectral response parameters for this earthquake can be read directly from the graph. Two types of tripartite log paper are used for plotting response spectra. Note that on the horizontal axis at the bottom of the graph in Figure 413, the period is increasing from left to right. For this reason, this type of tripartite log paper is often referred to as period paper. A similar plot of the response spectra for the El Centro NS ground motion is shown in Figure 414. Here it can be seen that frequency, plotted on the horizontal axis, is increasing from left to right. This type of tripartite paper is referred to as frequency paper.
4. Dynamic Response of Structures
Figure 412. Computation of deformation (or displacement) response spectrum. [After Chopra (41)].
203
204
Chapter 4
Figure 413. Typical tripartite responsespectra curves.
4. Dynamic Response of Structures
Figure 414. Response spectra, El Centro earthquake, May 18,1940, northsouth direction.
Figure 415. Sitespecific response spectra.
205
206 4.4.8
Chapter 4 Design Response Spectra
Use of the elastic response spectra for a single component of a single earthquake record
(Figure 413), while suitable for purposes of analysis, is not suitable for purposes of design. The design response spectra for a particular site should not be developed from a single
Figure 416. Smoothed sitespecific design spectra.
4. Dynamic Response of Structures acceleration time history, but rather should be obtained from the ensemble of possible earthquake motions that could be experienced at the site. This should include the effect of both near and distant earthquakes. Furthermore, a single earthquake record has a particular frequency content which gives rise to the jagged, sawtooth appearance of peaks and valleys shown in Figure 413. This feature is also not suitable for design, since for a given period, the structure may fall in a valley of the response spectrum and hence be underdesigned for an earthquake with slightly different response characteristics. Conversely, for a small change in period, the structure might fall on a peak and be overdesigned. To alleviate this problem the concept of the smoothed response spectrum has been introduced for design. Statistics are used to create a smoothed spectrum at some suitable design level. The mean value or median spectrum can generally be used for earthquakeresistant design of normal building structures. Use of this spectrum implies there is a 50% probability that the design level will be exceeded. Structures that are particularly sensitive to earthquakes or that have a high risk may be designed to a higher level such as the mean plus one standard deviation, which implies that the probability of exceedance is only 15.9%. Structures having a very high risk are often designed for an enveloping spectrum which envelopes the spectra of the entire ensemble of possible site motions. Response spectra which are representative of a magnitude6.5 earthquake at a distance of 15 miles, developed by the Applied Technology Council (42), are shown in Figure 415. The corresponding smoothed design spectra are shown in Figure 416. Newmark and Hall (43) have proposed a method for constructing an elastic design response spectrum in which the primary input datum is the anticipated maximum ground acceleration. The corresponding values for the maximum ground velocity and the maximum ground displacement are proportioned relative to the maximum ground acceleration, which is
207 normalized to 1.0g. The maximum ground velocity is taken as 48 in./sec, and the maximum ground displacement is taken as 36 in. It should be noted that these values represent motions which are more intense than those normally considered for earthquakeresistant design; however, they are approximately in the correct proportion for earthquakes occurring on competent soils and can be scaled for earthquakes having lower ground acceleration. Table 41. Relative values of spectrum amplification factors (43). Percentage Amplification factor for of critical Damping Displacement Velocity Acceleration 0 2.5 4.0 6.4 0.5 2.2 3.6 5.8 1 2.0 3.2 5.2 2 1.8 2.8 4.3 5 1.4 1.9 2.6 10 1.1 1.3 1.5 20 1.0 1.1 1.2
Three principal regions of the response spectrum are identified, in which the structural response can be approximated as a constant, amplified value. Amplification factors are applied to the ground motions in these three regions to obtain the design spectrum for a SDOF elastic system. Based on a large data base of recorded earthquake motions, amplification factors which give a probability of exceedance of about 10% or less are given in Table 41 for various values of the structural damping. The basic shape of the Newmark— Hall design spectrum using the normalized ground motions and the amplification factors given in Table 41 for 5% damping is shown in Figure 417. The displacement region is the lowfrequency region with frequencies less than 0.33 Hz (periods greater than 3.0 sec). The maximum displacement of the SDOF system is obtained by multiplying the maximum ground displacement by the displacement amplification factor given in Table 41. The velocity region is in the midfrequency region between 0.33 Hz (3.0 sec) and 2.0 Hz (0.5 sec). Maximum velocities in this region are obtained by
208
Chapter 4
multiplying the maximum ground velocity by the amplification factor for the velocity (Table 41). An amplified acceleration region lies between 2.0 Hz (0.5 sec) and 6.0 Hz (0.17 sec). The amplified response is obtained in the same manner as in the previous two cases. Structures having a frequency greater than 30 Hz (period less than 0.033 sec) are considered to be rigid and have an acceleration which is equal to the ground acceleration. In the frequency range between 6 Hz (0.17 sec) and 30 Hz (0.033 sec) there is a transition region between the ground
acceleration and the amplified acceleration region. Similar design spectra corresponding to the postulated ground motion presented in Figures 415 and 416 are shown in Figure 418. In order to further define which response spectrum should be used for design, it is necessary to estimate the percentage of critical damping in the structure. A summary of recommended damping values for different types of structures and different stress conditions is given in Table 42 as a guideline.
Figure 417. Basic New markHall design spectrum normalized to 1.0g for 5% damping (43).
4. Dynamic Response of Structures
209
Figure 418. A New markHall design spectra.
Example 44 (Construction of a NewmarkHall Design Spectrum) Construct a NewmarkHall design spectrum for a maximum ground acceleration of 0.2g, and use it to estimate the maximum base shear for the industrial building of Example 41. Assume the damping is 5 percent of critical. •Determine ground motion parameters: ground acceleration = (1.0)(0.2) = 0.2g ground velocity = (48.0)(0.2)=9.6in./sec. ground displacement=(36.0)(0.2)=7.2 in. •Amplified response parameters: acceleration = (0.2)(2.6) = 0.52g
velocity = (9.6)(1.9) = 18.2 in./sec displacement = (7.2)(1.4) = 10.0 in. The constructed design spectrum is shown in Figure 419. From Example 41: NS: T = 0.287 sec. ω = 21.8 rad/sec, f = 3.48 HZ From the design spectrum for f = 3.48 Hz: Sd=v(t)max =0.42 in.
210
Chapter 4
Table 42 Recommended Damping Values (43) Stress level Type and condition of structure Working stress,<1/2 yield point
Vital piping Welded steel, prestressed concrete, wellreinforced concrete(only slight cracking) Reinforced concrete with considerable cracking Bolted and / or riveted steel, wood structures with nailed or bolted joints.
Percentage of critical damping 12 23
Stress level
Type and condition of structure
At or just below yield point
Vital piping Welded steel, prestressed concrete(without complete loss in prestress) Prestressed concrete with no prestress left Bolted and / or riveted steel, wood structures with nailed or bolted joints. Wood structures with nailed joints
35 57
Percentage of critical damping 23 57
710 1015
1520
From Equation 442: Qmax = (0.485)(21.8)2(0.42) = 96.8 kips EW: T = 0.23 sec, ω = 27.2 rad/sec, From the design spectrum for f = 4.3 Hz: Sd = 0.28 in. From Equation 442: Qmax=(0.485)(21.8)2(0.28) = 64.5 kips
4.4
GENERALIZEDCOORDINATE APPROACH
Up to this point, the only structures which have been considered are singlestory buildings which can be idealized as SDOF systems. The analysis of most structural systems requires a more complicated idealization even if the response can be represented in terms of a single degree of freedom. The generalizedcoordinate approach provides a means of representing the response of more complex structural systems in terms of a single, timedependent coordinate, known as the generalized coordinate. Displacements in the structure are related to the generalized coordinate as
v ( x, t ) = φ( x )Y (t )
(445)
Where Y(t) is the timedependent generalized coordinate and φ(x ) is a spatial shape function which relates the structural degrees of freedom, v(x, t), to the generalized coordinate. For a generalized SDOF system, it is necessary to represent the restoring forces in the damping elements and the stiffness elements in terms of the relative velocity and relative displacement between the ends of the element: Figure 419. Response spectrum of Example 43.
∆v&( x, t ) = ∆φ( x )Y& (t )
(446)
4. Dynamic Response of Structures
∆v( x, t ) = ∆φ( x )Y (t )
211 (447)
Most structures can be idealized as a vertical cantilever, which limits the number of displacement functions that can be used to represent the horizontal displacement. Once the displacement function is selected, the structure is constrained to deform in that prescribed manner. This implies that the displacement functions must be selected carefully if a good approximation of the dynamic properties and response of the system are to be obtained. This section will develop the equations for determining the generalized response parameters in terms of the spatial displacement function and the physical response parameters. Methods for determining the shape function will be discussed, and techniques for determining the more correct displacement function for a particular structure will be presented. 4.4.1
Displacement Functions and Generalized Properties
Formulation of the equation of motion in terms of a generalized coordinate will be restricted to systems which consist of an assemblage of lumped masses and discrete elements. Lateral resistance is provided by discrete elements whose restoring force is proportional to the relative displacement between the ends of the element. Damping forces are proportional to the relative velocity between the ends of the discrete damping element. Formulation of the equation of motion for systems having distributed elasticity is described by Clough and Penzien. (44) The general equation of dynamic equilibrium is given in Equation 46, which represents a system of forces which are in equilibrium at any instant of time. The principle of virtual work in the form of virtual displacements states that If a system of forces which are in equilibrium is given a virtual displacement which is consistent with the boundary conditions, the work done is zero.
Applying this principle to Equation 46 results in an equation of virtual work in the form
f i δv + f d δ∆v + f s δ∆v − p (t ) δv = 0
(448)
where it is understood that v = v ( x, t ) and that the virtual displacements applied to the damping force and the elastic restoring force are virtual relative displacements. The virtual displacement can be expressed as
δv ( x, t ) = φ( x )δY (t )
(449)
and the virtual relative displacement can be written as
δ∆v ( x, t ) = ∆φ( x )δY (t )
(450)
where
∆v ( x, t ) = φ( xi ) Y (t ) − φ( x j ) Y (t ) = ∆φ( x )Y (t ) The inertia, damping and elastic restoring forces can be expressed as
f i = mv&& = mφY&& f d = c∆v& = c∆φY&&
(451)
f s = k∆v = k∆φY Substituting Equations 449, 450, and 451 into Equation 448 results in the following equation of motion in terms of the generalized coordinate:
m *Y&& + c *Y& + k *Y = p * (t )
(452)
where m*, c*, k*, and p* are referred to as the generalized parameters and are defined as
212
Chapter 4
Figure 420. Generalized singledegreeoffreedom system.
Where ω represents the circular frequency of the generalized system and is given as
m = ∑ mi φ = generalized mass *
2 i
i
c = ∑ ci ∆φi2 = generalized damping *
i
k * = ∑ k i ∆φi2 = generalized stiffness
(453)
i
p = ∑ pi φi = generalized force *
i
For a timedependent base acceleration the generalized force becomes
p * = g&&L
(454)
where
L = ∑ mi φi i
(455)
= earthquake participation factor It is also convenient to express the generalized damping in terms of the percent of critical damping in the following manner:
c * = ∑ ci ∆φ(i ) 2 = 2λm * ω i
(456)
ω=
k* m*
(457)
The effect of the generalizedcoordinate approach is to transform a multipledegreeoffreedom dynamic system into an equivalent singledegreeoffreedom system in terms of the generalized coordinate. This transformation is shown schematically in Figure 420. The degree to which the response of the transformed system represents the actual system will depend upon how well the assumed displacement shape represents the dynamic displacement of the actual structure. The displacement shape depends on the aspect ratio of the structure, which is defined as the ratio of the height to the base dimension. Possible shape functions for highrise, midrise, and lowrise structures are summarized in Figure 421. It should be noted that most building codes use the straightline shape function which is shown for the midrise system. Once the dynamic response is obtained in terms of the generalized coordinate, Equation 445 must be used to determine the displacements in the structure, and these in turn
4. Dynamic Response of Structures
213
Figure 421. Possible shape functions based on aspect ratio.
can be used to determine the forces in the individual structural elements. In principle, any function which represents the general deflection characteristics of the structure and satisfies the support conditions could be used. However, any shape other than the true vibration shape requires the addition of external constraints to maintain equilibrium. These extra constraints tend to stiffen the system and thereby increase the computed frequency. The true vibration shape will have no external constraints and therefore will have the lowest frequency of vibration. When choosing between several approximate deflected shapes, the one producing the lowest frequency is always the best approximation. A good approximation to the true vibration shape can be obtained by applying forces representing the inertia forces and letting the static deformation of the structure determine the spatial shape function. Example 45 (Determination of generalized parameters) Considering the fourstory, reinforcedconcrete moment frame building shown in Figure 422, determine the generalized mass, generalized stiffness, and fundamental period of vibration in the transverse direction using the following shape functions:
φ( x ) = sin( πx / 2 L) and (b) φ( x ) = x / L .All beams are 12in. × 20 in., and all columns are 14 in × 14 in. f c′ =4000
(a)
psi, and the modulus of elasticity of concrete is 3.6 × 10 6 psi. Reinforcing steel is made of grade60 bars. Floor weights (total dead load) are assumed to be 390 kips at the roof, 445 kips at the fourth and third levels, and 448 kips at the first level. Live loads are 30 psf at the roof and 80 psf per typical floor level.
Figure 422. Building of Example 45.
Assuming beams are rigid relative to columns (Figure 423),
214
Chapter 4 (a) Assuming φ( x ) = sin( πx / 2 L) : Level K 4
M
φi
0.252
1.000
∆φi
209 3
0.288
0.929
0.288
0.726
0.290
0.420
1.054 0.249
0.203
209 1
8.613 0.152
0.306
140
K ∆φi2
0.252 0.071
209 2
M φi2
19.570 0.051
0.420 M* = 0.704
24.696 K* = 53.933
k* 53.93 = = 8.75 rad/ sec * 0.704 m and Ta = 0.72 sec
ω =
(b) Assuming φ( x ) = x / L
Figure 423. Assumed shape of column deformation.
Level
12 EI∆ V= L3
K
4
140
i =1
kips (3)(12)(3.6 × 10 3 )(3201) = 209 3 in. (126)
(3)(12)(3.6 × 10 3 )(3201) kips = 140 3 in. (144)
Calculating Figure 424):
generalized
12.139 0.022
0.276
10.665 M* = 0.517 K* = 47.183
Since Ta > Tb , φ( x ) = sin( πx / 2 L) is a better approximation to the deflected shape than φ( x ) = x / L
3
K story = ∑ K i = 3K i (one frame)
K1 =
0.241
k* 47.183 ω= = * m 0.517 = 9.55 rad/sec and Tb = 0.66 sec .
12( 20) 3 = = 8000 in. 4 12
K 4 , 3, 2 =
12.240 0.077
0.290 0.276
3
12.139 0.166
0.242
209
K ∆φi2
0.252
0.288 0.517
1
M φi2
0.241 0.288 0.759
2
14(14) = 3201in. 4 12
∆φi
0.252 1.000
209
V 12 EI Ki = = 3 ∆ L
I beam
φi
209 3
I col =
M
properties
(see
4.4.2
Rayleigh’s Method
Rayleigh’s method is a procedure developed by Lord Rayleigh (45) for analyzing vibrating systems using the law of conservation of energy. Its principal use is for determining an accurate approximation of the natural frequency of a structure. The success of
4. Dynamic Response of Structures
215
Figure 424. Development of a generalized SDOF model for building of Example 44.
the technique in accomplishing this has been recognized by most building codes, which have adopted the procedure as an alternative for estimating the fundamental period of vibration. In addition to providing an estimate of the fundamental period, the procedure can also be used to estimate the shape function φ (x). In an undamped elastic system, the maximum potential energy can be expressed in terms of the external work done by the applied forces. In terms of a generalized coordinate this expression can be written as
( PE ) max
Y = 2
p *Y ∑ pi φi = 2
(458)
Similarly, the maximum kinetic energy can be expressed in terms of the generalized coordinate as
Equation 458 to Equation 459 results in the following expression for the circular frequency:
ω=
p* m *Y
(460)
Substituting this result into Equation 420 for the period results in
T = 2π
m *Y p*
(461)
Multiplying the numerator and denominator of the radical by Y and using Equation 445 results in the expression for the fundamental period:
T = 2π
∑w v g∑ p v i
2 i i
(462)
i i
( KE ) max =
ω 2Y 2 2
∑ mi φi2 = i
ω 2Y 2 m * (459) 2
According to the principle of conservation of energy for an undamped elastic system, these two quantities must be equal to each other and to the total energy of the system. Equating
which is the expression found in most building codes. The forces which must be applied laterally to obtain either the shape function φ (x) or the displacement v(x) represent the inertia forces, which are the product of the mass and the acceleration. If the acceleration is assumed to vary linearly over the height of a building with
216
Chapter 4
uniform weight distribution, a distribution of inertia force in the form of an inverted triangle will be obtained, being maximum at the top and zero at the bottom. This is similar to the distribution of base shear used in most building codes and can be a reasonable one to use when applying the Rayleigh method. The resulting deflections can be used directly in Equation 462 to estimate the period of vibration or they can be normalized in terms of the generalized coordinate (maximum displacement) to obtain the spatial shape function to be used in the generalizedcoordinate method. Example 46 (Application of Rayleigh’s Method) Use Rayleigh’s method to determine the spatial shape function and estimate the fundamental period of vibration in the transverse direction for the reinforcedconcrete building given in Example 44. We want to apply static lateral loads that are representative of the inertial loads on the building. Since the story weights are approximately equal, it is assumed that the accelerations and hence the inertial loads vary linearly from the base to the roof (see Figure 425). Note that the magnitude of loads is irrelevant and is chosen for ease of computation. The following computations (on the bottom of this page) are a tabular solution of Equation 461.
T = 2π
m *Y , p*
T = 2π
(0.666)(0.3343) = 0.712 sec 16.912
or
K
4
m
P
0.252
8.0
209 3
0.288 0.288 0.288 140
Earthquake Response of Elastic Structures
TimeHistory Analysis Substituting the generalized parameters of Equations 453 and 454 into the Duhamelintegral solution, Equation 433, results in the following solution for the displacement:
φ( x ) L V (t ) m*ω
(463)
Using Equation 437, the inertia force at any position x above the base can be obtained from
V
∆=V/k
8
0.0383
14
0.0670
18
0.0861
20
0.1429
4.0
209 1
4.4.3
6.0
209 2
Figure 425.Frame of Example 45.
v ( x, t ) =
Note that since T = 0.721 is greater than either of the periods calculated in Example 45, Level
the deflected shape given by applying the static loads is a better approximation than either of the two previous deflected shapes.
2.0
v
φ
0.3343
1.000
0.252
8.000
0.2960
0.886
0.226
5.316
0.2290
0.685
0.135
2.740
0.1429 0.000
0.428 0.000
0.053 0.666
0.856 16.912
mi φi2
Pi φi
4. Dynamic Response of Structures
217
q( x, t ) = m( x )v&&( x, t ) = m( x )ω2 v ( x, t ) (464) which, using Equation 463, becomes
q( x, t ) =
m( x )φ( x ) L ωV (t ) m*
(465)
The base shear is obtained by summing the distributed inertia forces over the height H of the structure:
Q (t ) = ∫ q( x, t )dx =
L2 ωV (t ) m*
(466)
The above relationships can be used to determine the displacements and forces in a generalized SDOF system at any time during the time history under consideration. ResponseSpectrum Analysis The maximum value of the velocity given by Equation 435 is defined as the spectral pseudovelocity (Spv), which is related to the spectral displacement (Sd) by Equation 443. Substituting this value into Equation 463 results in an expression for the maximum displacement in terms of the spectral displacement:
v ( x ) max =
φ( x ) L S d m*
(467)
The forces in the system can readily be determined from the inertia forces, which can be expressed as
q( x ) max = m( x )v&&( x ) max = m( x )ω2 v ( x ) max (468) Rewriting this result in terms of the spectral pseudoacceleration (Spa) results in the following:
q( x ) max =
φ( x )m( x ) L S pa m*
(469)
Of considerable interest to structural engineers is the determination of the base shear. This is a key parameter in determining seismic design forces in most building codes. The base shear Q can be obtained from the above expression by simply summing the inertia forces and using Equation 455:
Qmax =
L 2 S pa m*
(470)
It is also of interest to express the base shear in terms of the effective weight, which is defined as
W* =
( ∑iwi φi ) 2
∑w φ i
i
2 i
(471)
The expression for the maximum base shear becomes
Qmax = W * S pa / g
(472)
This form is similar to the basic baseshear equation used in the building codes. In the code equation, the effective weight is taken to be equal to the total dead weight W, plus a percentage of the live load for special occupancies. The seismic coefficient C is determined by a formula but is equivalent to the spectral pseudoacceleration in terms of g. The basic code equation for base shear has the form
Qmax = CW
(473)
The effective earthquake force can also be determined by distributing the base shear over the story height. This distribution depends upon the displacement shape function and has the form
qi = Qmax
mi φi L
(474)
If the shape function is taken as a straight line, the code force distribution is obtained. The overturning moment at the base of the structure
218
Chapter 4
can be determined by multiplying the inertia force by the corresponding story height above the base and summing over all story levels:
M O = ∑ hi qi
(475)
From Equation 466,
Qmax =
(0.827) 2 (0.185)(386.4) = 88.84 kips 0.666
i
Example 47 (Spectrum Analysis of Generalized SDOF System) Using the design spectrum given in Figure 426, the shape function determined in Example 46, and the reinforcedconcrete moment frame of Example 45, determine the base shear in the transverse direction, the corresponding distribution of inertia forces over the height of the structure, and the resulting overturning moment about the base of the structure.
The overturning moment is: (see Fig, 427)
T = 0.721 sec., f = 1 / T = 1.39 Hz, ω = 8.715 rad/sec. From the design spectrum Spa = 0.185g. Level
mi
φi
mi φi2
miφi
miφi/L
qmax
4
0.252
1.000
0.252
0.252
0.305
27.10
3
0.288
0.866
0.226
0.255
0.308
27.36
Vmax
27.10 54.46 2
0.288
0.685
0.135
0.197
0.238
21.14
1
0.288
0.428
0.053
0.123
0.149
13.24
0.666
0.827
75.60 88.84
Figure 427. Story shears and overturning moment (Example 46)
M o = 27.10(43.5) + 27.36(33) + 21.14( 22.5) + 13.24(12) = 2716 ft − kips The displacement is
v max = φ (ϕ / m*) S d = φ α S d where Figure 426. Design spectrum for Example 46.
4. Dynamic Response of Structures S d = S pa / ω2
and
α = ϕ / m*
(0.185)( 386.4) = 0.941 (8.715) 2 0.827 α= = 1.242 0.666 vi = (1.242)(0.941)φi = 1.168φi Sd =
4.5
v 4 = 1.168 in.
v3 = 1.035 in.
v 2 = 0.80 in.
v1 = 0.50 in.
RESPONSE OF NONLINEAR SDOF SYSTEMS
In an earlier section it was shown that the response of a linear structural system could be evaluated using the Duhamel integral. The approach was limited to linear systems because the Duhamelintegral approach makes use of the principle of superposition in developing the method. In addition, evaluation of the Duhamel integral for earthquake input motions will require the use of numerical methods in evaluating the integral. For these reasons it may be more expedient to use numerical integration procedures directly for evaluating the response of linear systems to general dynamic loading. These methods have the additional advantage that with only a slight modification they can be used to evaluate the dynamic response of nonlinear systems. Many structural systems will experience nonlinear response sometime during their life. Any moderate to strong earthquake will drive a structure designed by conventional methods into the inelastic range, particularly in certain critical regions. A very useful numerical integration technique for problems of structural dynamics is the so called stepbystep integration procedure. In this procedure the time history under consideration is divided into a number of small time increments ∆ t. During a small time step, the behavior of the structure is assumed to be linear. As nonlinear behavior occurs, the incremental stiffness is modified. In this manner, the response of the nonlinear system is approximated by a series of linear systems having a changing stiffness. The
219 velocity and displacement computed at the end of one time interval become the initial conditions for the next time interval, and hence the process may be continued step by step. 4.5.1
Numerical Formulation of Equation of Motion
This section considers SDOF systems with properties m, c, k(t) and p(t), of which the applied force and the stiffness are functions of time. The stiffness is actually a function of the yield condition of the restoring force, and this in turn is a function of time. The damping coefficient may also be considered to be a function of time; however, general practice is to determine the damping characteristics for the elastic system and to keep these constant throughout the complete time history. In the inelastic range the principle mechanism for energy dissipation is through inelastic deformation, and this is taken into account through the hysteretic behavior of the restoring force. The numerical equation required to evaluate the nonlinear response can be developed by first considering the equation of dynamic equilibrium given previously by Equation 46. It has been stated previously that this equation must be satisfied at every increment of time. Considering the time at the end of a short time step, Equation 46 can be written as
f i (t + ∆t ) + f d (t + ∆t ) + f s (t + ∆t ) = p(t + ∆t ) (476) where the forces are defined as
f i = mv&&(t + ∆t ) f d = cv&(t + ∆t ) n
f s = ∑ k i (t )∆vi (t ) = rt + k (t ) ∆v (t ) (477) i =1
∆v ( t ) = v ( t + ∆t ) − v ( t ) n −1
rt = ∑ k i (t )∆vi (t ) i =1
and in the case of ground accelerations
220
Chapter 4
p(t + ∆t ) = pe (t + ∆t ) = − mg&&(t + ∆t ) (478) Substituting Equations 477 and 478 into Equation 476 results in an equation of motion of the form mv&&(t + ∆t ) + cv&(t + ∆t ) +
∑ k ∆v i
i
= −mg&&(t + ∆t ) (479)
It should be noted that the incremental stiffness is generally defined by the tangent stiffness at the beginning of the time interval
ki =
df s dv
(480)
In addition, the dynamic properties given in Equations 477 and 478 can readily be exchanged for the generalized properties when considering a generalized SDOF system. 4.5.2
Numerical Integration
Many numerical integration schemes are available in the literature. The technique considered here is a stepbystep procedure in which the acceleration during a small time increment is assumed to be constant. A slight variation of this procedure, in which the acceleration is assumed to vary linearly during a small time increment, is described in detail by Clough and Penzien.(44). Both procedures have been widely used and have been found to yield good results with minimal computational effort. If the acceleration is assumed to be constant during the time interval, the equations for the constant variation of the acceleration, the linear variation of the velocity and the quadratic variation of the displacement are indicated in Figure 428. Evaluating the expression for velocity and displacement at the end of the time interval leads to the following two expressions for velocity and displacement:
Figure 428. Increment motion (constant acceleration).
v&(t + ∆t ) = v&(t ) + v&&(t + ∆t )
∆t ∆t + v&&(t ) (481) 2 2
v (t + ∆t ) = v (t ) + v&(t )∆t + v&&(t + ∆t )
∆t 2 ∆t 2 (482) + v&&(t ) 4 4
Solving Equation 482 for the acceleration v&&(t + ∆t ) gives
v&&(t + ∆t ) =
4 4 ∆v − v&(t ) − v&&(t ) 2 ∆t ∆t
which can be written as
(483)
4. Dynamic Response of Structures
v&&(t + ∆t ) =
4 ∆v + A(t ) ∆t 2
221 (484)
where
Moving terms containing the response conditions at the beginning of the time interval to the righthand side of the equation results in the following socalled pseudostatic form of the equation of motion:
∆v = v ( t + ∆t ) − v ( t ) 4 A(t ) = − v&(t ) − v&&(t ) ∆t Note that this equation expresses the acceleration at the end of the time interval as a function of the incremental displacement and the acceleration and velocity at the beginning of the time interval. Substituting Equation 483 into Equation 481 gives the following expression for the velocity at the end of the time increment:
v&(t + ∆t ) =
2 ∆v − v&(t ) ∆t
(485)
2 ∆v + B(t ) ∆t
(486)
where
B(t ) = − v&(t ) It is convenient to express the damping as a linear function of the mass:
c = αm = λCcr = 2mωλ
(487)
Use of this equation allows proportionality factor α to be expressed as
α = 2λω
k t ( ∆v ) = p ( t + ∆t )
(490)
where
4m 2αm + + kt ∆t ∆t 2 p(t + ∆t ) = −mg&&(t + ∆t ) − R(t ) − m[ A(t ) − αB(t )]
kt =
The solution procedure for a typical time step is as follows:
which can be written as
v&(t + ∆t ) =
4 2 m ∆v + A(t ) + αm ∆v + B (t ) + R(t ) + k∆v 2 ∆ t ∆t = mg&&(t + ∆t ) (489)
the
(488)
Substituting Equations 485, 486, and 488 into Equation 479 results in the following form of the equation for dynamic equilibrium:
1. Given the initial conditions at the beginning of the time interval, calculate the coefficients A(t) and B(t). 2. Calculate the effective stiffness. 3. Determine the effective force. 4. Solve for the incremental displacement
v = p /kt
(491)
5. Determine the displacement, velocity and acceleration at the end of the time interval:
v (t + ∆t ) = v (t ) + ∆v 2 v&(t + ∆t ) = + B (t ) ∆t 4 v&&(t + ∆t ) = 2 + A(t ) ∆t
(492)
6. The values given in Equation 492 become the initial conditions for the next time increment, and the procedure is repeated.
222
Chapter 4
The above algorithm can be easily programmed on any microcomputer. If it is combined with a data base of recorded earthquake data such as EQINFOS,(46) it can be used to gain considerable insight into the linear and nonlinear response of structures that can be modeled as either a SDOF system or as a generalized SDOF system. It also forms the background material for later developments for multipledegreeoffreedom systems. An important response parameter that is unique to nonlinear systems is the ductility ratio. For a SDOF system, this parameter can be defined in terms of the displacement as
µ=
v (max) v ( plastic) = 1.0 + v ( yield) v ( yield)
m2 m3 . . .
v1 v 2 v3 . . . mn vn (494)
MULTIPLEDEGREEOFFREEDOM SYSTEMS
In many structural systems it is impossible to model the dynamic response accurately in terms of a single displacement coordinate. These systems require a number of independent displacement coordinates to describe the displacement of the mass of the structure at any instant of time. 4.6.1
m1 { fi } =
(493)
As can be seen from the above equation, the ductility ratio is an indication of the amount of inelastic deformation that has occurred in the system. In the case of a SDOF system or generalized SDOF system the ductility obtained from Equation 493 usually represents the average ductility in the system. The ductility demand at certain critical regions, such as plastic hinges in critical members, may be considerably higher.
4.6
diagonal matrix of mass properties in which either the translational mass or the mass moment of inertia is located on the main diagonal.
Mass and Stiffness Properties
In order to simplify the solution it is usually assumed for building structures that the mass of the structure is lumped at the center of mass of the individual story levels. This results in a
It is also convenient for building structures to develop the structural stiffness matrix in terms of the stiffness matrices of the individual story levels. The simplest idealization for a multistory building is based on the following three assumptions: (i) the floor diaphragm is rigid in its own plane; (ii) the girders are rigid relative to the columns and (iii) the columns are flexible in the horizontal directions but rigid in the vertical. If these assumptions are used, the building structure is idealized as having three dynamic degrees of freedom at each story level: a translational degree of freedom in each of two orthogonal directions, and a rotation about a vertical axis through the center of mass. If the above system is reduced to a plane frame, it will have one horizontal translational degree of freedom at each story level. The stiffness matrix for this type of structure has the tridiagonal form shown below: For the simplest idealization, in which each story level has one translational degree of freedom, the stiffness terms ki in the above equations represent the translational story stiffness of the ith story level. As the assumptions given above are relaxed to include axial deformations in the columns and flexural deformations in the girders, the stiffness term ki in Equation 495 becomes a submatrix of stiffness terms, and the story displacement vi
4. Dynamic Response of Structures
k1 k 2 { fs} =
223
− k2 k1 + k 2
− k3
− k3
k2 + k3
− k4
.
.
.
.
.
.
.
.
.
.
. − kn
becomes a subvector containing the various displacement components in the particular story level. The calculation of the stiffness coefficients for more complex structures is a standard problem of static structural analysis. For the purposes of this chapter it will be assumed that the structural stiffness matrix is known. 4.6.2
Mode Shapes and Frequencies
The equations of motion for undamped free vibration of a multipledegreeoffreedom (MDOF) system can be written in matrix form as
[ M ]{v&&} + [ K ]{v} = {0}
(496)
Since the motions of a system in free vibration are simple harmonic, the displacement vector can be represented as
{v} = {v} sin ω t
(497)
Differentiating twice with respect to time results in
{v&&} = − ω2 {v}
(498)
Substituting Equation 498 into Equation 496 results in a form of the eigenvalue equation,
v1 v2 v 3 . . . − k n v n −1 k n −1 + k n v n
([ K ] − ω [ M ]){v} = {0} 2
(3 − 95)
(499)
The classical solution to the above equation derives from the fact that in order for a set of homogeneous equilibrium equations to have a nontrivial solution, the determinant of the coefficient matrix must be zero:
det([ K ] − ω2 [ M ]) = {0}
(4100)
Expanding the determinant by minors results in a polynomial of degree N, which is called the frequency equation. The N roots of the polynomial represent the frequencies of the N modes of vibration. The mode having the lowest frequency (longest period) is called the first or fundamental mode. Once the frequencies are known, they can be substituted one at a time into the equilibrium Equation 499, which can then be solved for the relative amplitudes of motion for each of the displacement components in the particular mode of vibration. It should be noted that since the absolute amplitude of motion is indeterminate, N1 of the displacement components are determined in terms of one arbitrary component. This method can be used satisfactorily for systems having a limited number of degrees of freedom. Programmable calculators have programs for solving the polynomial equation and for doing the matrix operations required to determine the mode shapes. However, for
224
Chapter 4
problems of any size, digital computer programs which use numerical techniques to solve large eigenvalue systems(47) must be used. Example 48 (Mode Shapes and Frequencies) It is assumed that the response in the transverse direction for the reinforcedconcrete moment frame of Example 44 can be represented in terms of four displacement degrees of freedom which represent the horizontal displacements of the four story levels. Determine the stiffness matrix and the mass matrix, assuming that the mass is lumped at the story levels. Use these properties to calculate the frequencies and mode shapes of the fourdegreeoffreedom system. •Stiffness and mass matrices: The stiffness coefficient kij is defined as the force at coordinate i due to a unit displacement at coordinate j, all other displacements being zero (see Figure 429): where B = ω2/800 •Characteristic equation:
ω2 B1 = 0.089 = 1 , ω1 = 8.438 , T1 = 0.744 sec 800 ω22 B2 = 0.830 = , ω2 = 25.768 , T1 = 0.244 sec 800 ω2 B3 = 2.039 = 3 , ω3 = 40.388 , T3 = 0.155 sec 800 ω24 B4 = 3.225 = , ω4 = 50.800 , T4 = 0.124 sec 800 •Mode shapes (see Figure 429) are obtained by substituting the values of Bi, one at a time, into the equations
([ K ] − ω2 [ M ]){v} = {0} and determining N1 components of the displacement vector in terms of the first component, which is set equal to unity. This results in the modal matrix
1.00 1.00 1.00 1.00 0.91 0.20 − 1.07 − 1.78 [Φ ] = 0.74 − 0.78 − 0.75 1.75 0.47 − 1.05 1.24 − 0.92
[ K ] − ω2 [ M ] = 0 B 4 − 6.183B 3 + 11.476 B 2 − 6.430 B + 0.486 = 0 Solution:
Solution of the above problem using the computer program ETABS (412) gives the following results: 209 − 209 − 209 418 [K ] = − 209 0 0 0 1.01 0 1 0 1.15 [M ] = 0 4 0 0 0
0 0 418 − 209 − 209 349 0 0 0 0 1.15 0 0 1.16 0 − 209
0 0 − 1.05 1.05 − 1.01B − 1.05 2 . 09 1 . 15 B 1 . 05 0 − − [ K ] − ω2 [ M ] = 200 0 2.09 − 1.15B − 1.05 − 1.05 0 0 1.74 − 1.16 B − 1.05
4. Dynamic Response of Structures
0.838 0.268 {T } = 0.152 0.107 1.00 1.00 1.00 1.00 0.91 0.20 − 1.07 − 1.78 [Φ ] = 1.75 0.74 − 0.78 0.75 0.47 − 1.05 1.24 − 0.92 This program assumes the floor diaphragm is rigid in its own plane but allows axial deformation in the columns and flexural deformations in the beams. Hence, with these added degrees of freedom (fewer constraints) the fundamental period increases. However, comparing the results of this example with those of Example 45, it can be seen that for this structure a good approximation for the firstmode response was obtained using the generalized SDOF model and the static deflected shape.
225 4.6.3
Equations of Motion in Normal Coordinates
Betti’s reciprocal work theorem can be used to develop two orthogonality properties of vibration mode shapes which make it possible to greatly simplify the equations of motion. The first of these states that the mode shapes are orthogonal to the mass matrix and is expressed in matrix form as
{φn }T [ M ]{φm } = {0}
( m ≠ n ) (4101)
Using Equations 499 and 4101, the second property can be expressed in terms of the stiffness matrix as
{φn }T [ K ]{φm } = {0}
(m ≠ n ) (4102)
which states that the mode shapes are orthogonal to the stiffness matrix. It is further assumed that the mode shapes are also orthogonal to the damping matrix:
{φn }T [C ]{φm } = {0}
(m ≠ n)
(4103)
Sufficient conditions for this assumption have been discussed elsewhere.(48) Since any MDOF system having N degrees of freedom also has N independent vibration mode shapes, it is possible to express the displaced shape of the structure in terms of the amplitudes of these shapes by treating them as generalized coordinates (sometimes called normal coordinates). Hence the displacement at a particular location, vi, can be obtained by summing the contributions from each mode as N
vi = ∑ φinYn
(4104)
n =1
Figure 429. Stiffness determination and mode shape(Example 48).
In a similar manner, the complete displacement vector can be expressed as
226
Chapter 4 N
{v} = ∑ {φn }Yn = [Φ ]{Y }
(4105)
n =1
It is convenient to write the equations of motion for a MDOF system in matrix form as
[ M ]{v&&} + [C ]{v&} + [ K ]{v} = {P(t )} (4106) which is similar to the equation for a SDOF system, Equation 49. The differences arise because the mass, damping, and stiffness are now represented by matrices of coefficients representing the added degrees of freedom, and the acceleration, velocity, displacement, and applied load are represented by vectors containing the additional degrees of freedom. The equations of motion can be expressed in terms of the normal coordinates by substituting Equation 4105 and its appropriate derivatives into Equation 4106 to give [ M ][Φ ]{Y&&} + [C ][Φ ]{Y& } + [ K ][Φ ]{Y } = {P (t )} (4107) Multiplying the above equation by the transpose of any modal vector {φn} results in the following: {φn }T [ M ][Φ ]{Y&&} + {φn }T [C ][Φ ]{Y& } (4108) + {φn }T [ K ][Φ ]{Y } = {φn }T {P (t )} Using the orthogonality conditions of Equations 4101, 4102, and 4103 reduces this set of equations to the equation of motion for a generalized SDOF system in terms of the generalized properties for the n th mode shape and the normal coordinate Yn:
M n*Y&&n + C n*Y&n + K n*Y = Pn* (t )
(4109)
where the generalized properties for the nth mode are given as
M n* = generalized mass = {φn }T [ M ]{φn } C n* = generalized damping = {φn }T [C ]{φn } = 2λ n ωn M n* K n* = generalized stiffness = {φn }T [ K ]{φn } = ω2n M n* Pn* (t ) = generalized loading = {φn }T {P(t )} (4110) The above relations can be used to further simplify the equation of motion for the nth mode to the form
P * (t ) Y&&n + 2λ n ωnYn + ωn2Yn = n * Mn
(4111)
The importance of the above transformations to normal coordinates has been summarized by Clough and Penzien,(44) who state that The use of normal coordinates serves to transform the equations of motion from a set of N simultaneous differential equations which are coupled by off diagonal terms in the mass and stiffness matrices to a set of N independent normal coordinate equations. It should further be noted that the expressions for the generalized properties of any mode are equivalent to those defined previously for a generalized SDOF system. Hence the use of the normal modes transforms the MDOF system having N degrees of freedom into a system of N independent generalized SDOF systems. The complete solution for the system is then obtained by superimposing the independent modal solutions. For this reason this method is often referred to as the modalsuperposition method. Use of this method also leads to a significant saving in computational effort, since in most cases it will not be necessary to use all N modal responses to accurately represent the response of the structure. For most structural systems the lower modes make the primary contribution to the total response. Therefore, the response can
4. Dynamic Response of Structures
227
usually be represented to sufficient accuracy in terms of a limited number of modal responses in the lower modes. 4.6.4
EarthquakeResponse Analysis
TimeHistory Analysis As in the case of SDOF systems, for earthquake analysis the timedependent force must be replaced with the effective loads, which are given by the product of the mass at any level, M, and the ground acceleration g(t). The vector of effective loads is obtained as the product of the mass matrix and the ground acceleration:
Pe (t ) = [ M ]{Γ}g&&(t )
(4112)
time t can be obtained by the Duhamel integral expression
Yn ( t ) =
ϕ nVn (t ) M n* ωn
(4115)
where Vn(t) represents the integral t
Vn (t ) = ∫ g&&( τ)e −λ n ωn ( t − τ ) sin ωn (t − τ)dτ (4116) 0
The complete displacement of the structure at any time is then obtained by superimposing the contributions of the individual modes using Equation 4105 N
where {Γ} is a vector of influence coefficients of which component i represents the acceleration at displacement coordinate i due to a unit ground acceleration at the base. For the simple structural model in which the degrees of freedom are represented by the horizontal displacements of the story levels, the vector {Γ} becomes a unity vector, {1}, since for a unit ground acceleration in the horizontal direction all degrees of freedom have a unit horizontal acceleration. Using Equation 4108, the generalized effective load for the nth mode is given as
Pen* (t ) = L n g (t )
(4113)
Where L n = {φn } [ M ]{Γ} T
Substituting Equation 4113 into Equation 4111 results in the following expression for the earthquake response of the nth mode of a MDOF system:
Y&&n + 2λ n ωnY&n + ωn2Yn = ϕ n g&&(t ) / M n* (4114) In a manner similar to that used for the SDOF system, the response of this mode at any
{v (t )} = ∑ {φn }Yn (t ) = [Φ ]{Y (t )} (4117) n =1
The resulting earthquake forces can be determined in terms of the effective accelerations, which for each mode are given by the product of the circular frequency and the displacement amplitude of the generalized coordinate:
ϕ ω V (t ) Y&&ne (t ) = ωn2Yn (t ) = n n *n Mn
(4118)
The corresponding acceleration in the structure due to the n th mode is given as
{v&&ne (t )} = {φn }Y&&ne (t )
(4119)
and the corresponding effective earthquake force is given as
{qn (t )} = [ M ]{v&&n (t )} = [ M ]{φn }ωn ϕ nVn (t ) / M n*
(4120)
The total earthquake force is obtained by superimposing the individual modal forces to obtain
228
Chapter 4 N
q(t ) = ∑ qn (t ) = [ M ][Φ ]ω2Y (t ) (4121)
{qn (t )} =
n =1
The base shear can be obtained by summing the effective earthquake forces over the height of the structure: H
Qn (t ) = ∑ qin (t ) = {1}T {qn (t )} i =1
(4122)
= M en ωnVn (t ) where M en = L n2 / M n* is the effective mass for the nth mode. The sum of the effective masses for all of the modes is equal to the total mass of the structure. This results in a means of determining the number of modal responses necessary to accurately represent the overall structural response. If the total response is to be represented in terms of a finite number of modes and if the sum of the corresponding modal masses is greater than a predefined percentage of the total mass, the number of modes considered in the analysis is adequate. If this is not the case, additional modes need to be considered. The base shear for the nth mode, Equation 4122, can also be expressed in terms of the effective weight,Wen, as
Qn (t ) =
Wen ωnVn (t ) g
H
Wen
Wφ i =1 i in
∑
H
i =1
Wi φ
)
2 in
(4125)
ResponseSpectrum Analysis
The above equations for the response of any mode of vibration are exactly equivalent to the expressions developed for the generalized SDOF system. Therefore, the maximum response of any mode can be obtained in a manner similar to that used for the generalized SDOF system. By analogy to Equations 434 and 443 the maximum modal displacement can be written as
Yn (t ) max =
Vn (t ) max = S dn ωn
(4126)
Making this substitution in Equation 4115 results in
Yn max = ϕ n S dn / M n*
(4127)
The distribution of the modal displacements in the structure can be obtained by multiplying this expression by the modal vector
{vn }max = {φn }Yn max = (4123)
{φn }L n S dn (4128) M n*
The maximum effective earthquake forces can be obtained from the modal accelerations as given by Equation 4120:
where
(∑ =
4.6.5
[ M ]{φn }Qn (t ) Ln
2
(4124)
The base shear can be distributed over the height of the building in a manner similar to Equation 474, with the modal earthquake forces expressed as
{qn }max =
[ M ]{φn }ϕ n S pan M n*
(4129)
Summing these forces over the height of the structure gives the following expression for the maximum base shear due to the nth mode:
Qn max = ϕ 2n S pan / M n*
(4130)
4. Dynamic Response of Structures
229
which can also be expressed in terms of the effective weight as
Qn max = Wen S pan / g
(4131)
where Wen is defined by Equation 4124. Finally, the overturning moment at the base of the building for the nth mode can be determined as
M o = h [M ]{φn }L n S pan / M n*
(4132)
Since this combination assumes that the maxima occur at the same time and that they also have the same sign, it produces an upperbound estimate for the response, which is too conservative for design application. A more reasonable estimate, which is based on probability theory, can be obtained by using the squarerootofthesumofthesquares (SRSS) method, which is expressed as
r≈
N
∑r
2 n
(4134)
n =1
where h is a row vector of the story heights above the base.
4.6.6
Modal Combinations
Using the responsespectrum method for MDOF systems, the maximum modal response is obtained for each mode of a set of modes, which are used to represent the response. The question then arises as to how these modal maxima should be combined in order to get the best estimate of the maximum total response. The modalresponse equations such as Equations 4117 and 4121 provide accurate results only as long as they are evaluated concurrently in time. In going to the responsespectrum approach, time is taken out of these equations and replaced with the modal maxima. These maximum response values for the individual modes cannot possibly occur at the same time; therefore, a means must be found to combine the modal maxima in such a way as to approximate the maximum total response. One such combination that has been used is to take the sum of the absolute values (SAV) of the modal responses. This combination can be expressed as N
r ≤ ∑ rn n =1
(4133)
This method of combination has been shown to give a good approximation of the response for twodimensional structural systems. For threedimensional systems, it has been shown that the completequadraticcombination (CQC) method (49) may offer a significant improvement in estimating the response of certain structural systems. The complete quadratic combination is expressed as
r≈
N
N
∑∑ r p r i
ij j
(4135)
i =1 j =1
where for constant modal damping
pij =
8λ2 (1 + ζ )ζ 3 / 2 (1 − ζ 2 ) 2 + 4λ2 ζ(1 + ζ) 2
(4136)
and
ζ = ω j / ωi λ = c / ccr Using the SRSS method for twodimensional systems and the CQC method for either two or threedimensional systems will give a good approximation to the maximum earthquake response of an elastic system without requiring a complete timehistory analysis. This is particularly important for purposes of design.
230
Chapter 4
Table 33. Computation of response for model of Example 48 Modal Modal Response Param n=1 2 3 eter ω= 8.44 25.77 40.39 1.212 0.289 0.075 αn = 1.190 0.155 0.062 Sd = 1.00 1.00 1.00 0.91 0.20 1.07 φ= Response 0.74 0.78 0.75 0.47 1.05 1.24 Quantity Displacement n=4 1.44 0.045 0.019 vn=φnαnSdn 3 1.31 0.009 0.020 (Eq.3.128) 2 1.07 0.035 0.014 1 0.68 0.047 0.023 Acceleration
4 50.80 0.010 0.039 1.00 1.78 1.75 0.92 0.002 0.003 0.003 0.001
SAV 1.506 1.342 1.122 0.751
Combined Response SRSS 1.441 1.310 1.071 0.682
CQC 1.441 1.310 1.071 0.682
n= 4 3
102.6 93.3
29.9 6.0
31.0 32.6
5.1 7.7
168.6 139.6
111.4 99.3
110.7 98.9
2 1
76.2 48.4
23.2 31.2
22.8 37.5
7.7 2.6
129.9 119.7
83.2 68.8
83.3 70.0
n=4 3 2 1
25.91 26.82 21.91 14.03
7.54 1.72 6.68 9.05
7.83 9.38 6.56 11.35
1.30 2.23 2.23 0.75
42.6 40.2 37.4 35.2
28.1 28.6 23.9 20.2
27.9 28.4 23.9 20.6
Shear Qn=Σqn
n= 4 3 2 1
25.91 52.73 74.64 88.67
7.54 9.26 2.58 6.47
7.83 1.55 8.11 3.24
1.30 0.93 1.30 0.55
42.6 64.5 86.6 98.9
28.1 53.6 75.1 89.0
28.0 53.5 75.1 89.0
Overturning Moment (ftkips)
n= 4 3 2 1
272.1 825.7 1609.4 2673.4
79.2 176.4 203.5 125.9
82.2 65.9 19.2 19.7
13.7 3.9 17.5 24.1
447.2 1071.9 1849.6 2843.1
295.4 846.9 1622.4 2676.5
293.6 845.3 1621.3 2675.7
v&&n = ω 2 n vn
Inertia force qn = Mv&&n
Example 49 (Response Spectrum Analysis) Use the design response spectrum given in Example 47 and the results of Example 48 to perform a responsespectrum analysis of the reinforced concrete frame. Determine the modal responses of the four modes of vibration, and estimate the total response using the SAV, SRSS, and CQC methods of modal combination. Present the data in a tabular form suitable for hand calculation. Finally, compare the results with those obtained in Example 46 for a generalized SDOF model. From Example 47,
0 0 1.01 0 0 1.15 0 0 1 [M ] = 0 1.15 0 4 0 0 0 1.16 0 8.44 25.77 r {ω } = 40.39 sec 50.80
4. Dynamic Response of Structures
1.00 1.00 1.00 1.00 0.91 0.20 − 1.07 − 1.78 [Φ ] = 0.74 − 0.78 − 0.75 1.75 0.47 − 1.05 1.24 − 0.92 1.34 ω 4.10 {f}= = Hz 2π 6.43 8.09 10.0 4.0 Sv = in./sec 2.5 2.0 S dn = S vn / ωn From Equation 4128,
{vn }max = {φn }(ϕ n / M n* ) S dn = {φ} αS dn {qn } = [ M ]{v&&n } = [ M ] ω2 {v n } N
Qn = ∑ qni i =1
For CQC combination, λ = 0.05 = constant for all modes 1.0000 0.0062 .0025 .0017 0.0062 1.0000 0.0452 0.0193 pij = 0.0025 0.0452 1.0000 0.1582 0.0017 0.0193 0.1582 1.0000 The computation of the modal and the combined response is tabulated in Table 43. The results are compared with those obtained for the SDOF model in Table 44. Table 44. Comparison of results obtained from MDOF and SDOF models. Response MDOF SDOF parameter (Example 39) (Example 37) Period (sec) 0.744 0.721 Displacements(in) Roof 1.44 1.17 3rd 1.31 1.04 2nd 1.07 0.80
231 Response parameter 1st Inertia force (kips) Roof 3rd 2nd 1st Base shear (kips) Overturning moment (ftkips)
4.7
MDOF (Example 39) 0.68
SDOF (Example 37) 0.50
28.1 28.6 23.9 20.2 89.0 2678
27.1 27.4 21.1 13.2 88.8 2716
NONLINEAR RESPONSE OF MDOF SYSTEMS
The nonlinear analysis of buildings modeled as multiple degree of freedom systems (MDOF) closely parallels the development for single degree of freedom systems presented earlier. However, the nonlinear dynamic time history analysis of MDOF systems is currently considered to be too complex for general use. Therefore, recent developments in the seismic evaluation of buildings have suggested a performancebased procedure which requires the determination of the demand and capacity. Demand is represented by the earthquake ground motion and its effect on a particular structural system. Capacity is the structure's ability to resist the seismic demand. In order to estimate the structure's capacity beyond the elastic limit, a static nonlinear (pushover) analysis is recommended (417). For more demanding investigations of building response, nonlinear dynamic analyses can be conducted. For dynamic analysis the loading time history is divided into a number of small time increments, whereas, in the static analysis, the lateral force is divided into a number of small force increments. During a small time or force increment, the behavior of the structure is assumed to be linear elastic. As nonlinear behavior occurs, the incremental stiffness is modified for the next time (load) increment. Hence, the response of the nonlinear system is approximated by the response of a sequential series of linear systems having varying stiffnesses.
232
Chapter 4
Static Nonlinear Analysis Nonlinear static analyses are a subset of nonlinear dynamic analyses and can use the same solution procedure without the time related inertia forces and damping forces. The equations of equilibrium are similar to Equation 41 with the exception that they are written in matrix form for a small load increment during which the behavior is assumed to be linear elastic.
[ K ]{∆v} = {∆P}
approximate the first mode of vibration. These forces are increased in a proportional manner by a specified load factor. The lateral loading is increased until either the structure becomes unstable or a specified limit condition is attained. The results from this type of analysis are usually presented in the form of a graph plotting base shear versus roof displacement. The pushover curve for a sixstory steel building (418) is shown in Figure 429a and the sequence of plastic hinging is shown in Figure 429b.
(4136a)
For computational purposes it is convenient to rewrite this equation in the following form
[ K t ]{∆v} + {Rt } = {P}
(4136b)
where Kt is the tangent stiffness matrix for the current load increment and Rt is the restoring force at the beginning of the load increment which is defined as n −1
{Rt } = ∑ [ K ti ]{∆vi } i =1
Figure 429b Sequence of Plastic Hinge Formation, Six Story Steel Building.
The equations of equilibrium for a multiple degree of freedom system subjected to base excitation can be written in matrix form as
[ M ]{v&&} + [C ]{v&} + [ K ]{v} (Eq.4137) = −[ M ]{Γ}g&&(t )
Figure 429a. Pushover Curve, Six Story Steel Building.
The lateral force distribution is generally based on the static equivalent lateral forces specified in building codes which tend to
This equation is of the same form as that of Eq. 476 for the single degree of freedom system. The acceleration, velocity and displacement have been replaced by vectors containing the additional degrees of freedom. The mass has been replaced by the mass matrix which for a lumped mass system is a diagonal matrix with the translational mass and rotational mass terms on the main diagonal. The incremental stiffness has been replaced by the incremental stiffness matrix and the damping has been replaced by the damping matrix. This
4. Dynamic Response of Structures
233
latter term requires some additional discussion. In the mode superposition method, the damping ratio was defined for each mode of vibration. However, this is not possible for a nonlinear system because it has no true vibration modes. A useful way to define the damping matrix for a nonlinear system is to assume that it can be represented as a linear combination of the mass and stiffness matrices of the initial elastic system
[C ] = α[ M ] + β[ K ]
(Eq 4138)
viscous damping. Therefore, an exact expresentation of damping is not as important in a nonlinear system as it is in a linear system. One should be aware of the characteristics of the damping function to insure that important components of the response are not lost. For instance, if the coefficients are selected to give a desired percentage of critical damping in the lower modes and the response of the higher modes is important, the higher mode response may be over damped and its contribution to the total response diminished.
Where α and β are scaler multipliers which may be selected so as to provide a given percentage of critical damping in any two modes of vibration of the initial elastic system. These two multipliers can be evaluated from the expression ωj α = 2− 1 β ω j
− ωi ωω 1 2i j2 ω − ωi ωi j
λi (Eq.4139) λ j
where ωi and ωj are the percent of critical damping in the two specified modes. Once the coefficients α and β are determined, the damping in the other elastic modes is obtained from the expression
λk =
βω k α + 2 ωk 2
(Eq. 4140)
A typical damping function which was used for the nonlinear analysis of a reinforced concrete frame (410) is shown in Figure 430. Although the representation for the damping is only approximate it is justified for these types of analyses on the basis that it gives a good approximation of the damping for a range of modes of vibration and these modes can be selected to be the ones that make the major contribution to the response. Also in nonlinear dynamic analyses the dissipation of energy through inelastic deformation tends of overshadow the dissipation of energy through
Figure 430. Damping functions for a framed tube.
Substituting Eq. 4138 into Eq. 4137 results in
[ M ]{v&&} + α [ M ]{v&} + β [ K i ]{v&} + [ K ]{v} = −[ M ]{ Γ} g&&(t ) (Eq. 4141) where Ki refers to the initial stiffness.
234
Chapter 4
Representing the incremental stiffness in terms of the tangent stiffness, Kt, and rearranging some terms, results in [ K ]{v} = [ K t ]{∆v} = {Rt } + [ K t ]{∆v} (Eq. 4142)
~ [ K ] = [C 0 [ M ] + C1 [ K i ] + [ K t ]] ~ {P} = {P(t )} − {Rt }
− [ M ] {{ At } + α{Bt }} − β [ K i ]{Bt }
4 2α + 2 ∆t ∆t 2β C1 = ∆t
where
C0 = n −1
{Rt } = ∑ [ K ti ]{∆vi } i =1
Using the stepbystep integration procedure in which the acceleration is assumed to be constant during a time increment, equations similar to Eqs. 4.84 and 486 can be developed for the multiple degree of freedom system which express the acceleration and velocity vectors at the end of the time increment in terms of the incremental displacement vector and the vectors of initial conditions at the beginning of the time increment:
{v&&(t )} = (
4 ){∆v} + { At } ∆t 2
(Eq. 4143)
{v&(t )} = (
2 ){∆v} + {Bt } ∆t
(Eq. 4144)
{v (t )} = {v (t − ∆t )} + {∆v}
(Eq. 4144a)
where
{ At } = −
where
4 {v&(t − ∆t )} − {v&&(t − ∆t )} (Eq. 4145) ∆t
{Bt } = −{v&(t − ∆t )}
(Eq. 4146)
Substituting Eqs. 4142 through 4146 into Eq. 4141 and rearranging some terms leads to the pseudostatic form
~ ~ [ K ]{∆v} = {P}
(Eq. 4147)
The incremental displacement vector can be obtained by solving Eq. 4147 for {∆v} This result can then be used in Eqs. 4143, 4144 and 4144a to obtain the acceleration vector, the velocity vector and the displacement vector at the end of the time interval. These vectors then become the initial conditions for the next time interval and the process is repeated. Output from a nonlinear response analysis of a MDOF system generally includes response parameters such as the following: an envelope of the maximum story displacements, an envelope of the maximum relative story displacement divided by the story height (sometimes referred to as the interstory drift index (IDI), an envelope of maximum ductility demand on structural members such as beams, columns, walls and bracing, an envelope of maximum rotation demand at the ends of members, an envelope of the maximum story shear, time history of base shear, moment versus rotation hysteresis plots for critical plastic hinges, time history plots of story displacements and time history plots of energy demands (input energy, hysteretic energy, kinetic energy and dissapative energy). For multiple degree of freedom systems, the definition of ductility is not as straightforward as it was for the single degree of freedom systems. Ductility may be expressed in terms of such parameters as displacement, relative displacement, rotation, curvature or strain. Example 410.Seismic Response Analyses The following is a representative response analysis for a six story building in which the lateral resistance is provided by moment resistant steel frames on the perimeter. The
4. Dynamic Response of Structures structure has a rectangular plan with typical dimensions of 228′ × 84′ as shown in Figure 431. The building was designed for the requirements of the 1979 Edition of the Uniform Building Code (UBC) with the seismic load based on the use of static equivalent lateral forces. Elastic Analyses As a first step in performing the analyses, the members of the perimeter frame will be stress checked for the design loading conditions and the dynamic properties of the building will be determined. This will help to insure that the analytical model of the building is correct and that the gravity loading which will be used for the nonlinear response analysis is also reasonable. This will be done using a three dimensional model of the lateral force system and the ETABS (411) computer program. This program is widely used on the west coast for seismic analysis and design of building systems. An isometric view of the perimeter frame including the gravity load is shown in Figure 432. The location of the concentrated and distributed loads depends upon the framing system shown in Figure 431. Using the postprocessor program STEELER (412), the lateral force system is stress checked using the AISCASD criteria. The stress ratio is calculated as the ratio of the actual stress in the member to the allowable stress. Applying the gravity loads in combination with the static equivalent lateral forces in the transverse direction produces the stress ratios shown in Figure 433. This result includes the effect of an accidental eccentricity which is 5% of the plan dimension. The maximum stress ratio in the columns is 0.71 and the maximum in the beams is 0.92. These values are reasonable based on standard practice at the time the building was designed. Ideally, the stress ratio should be just less than one, however, this is not always possible due to the finite number of steel sections that are available. Modal analyses indicate that the first three lateral modes of vibration in each direction
235 represent more than 90% of the participating mass. In the transverse direction, these modes have periods of vibration of 1.6, 0.6 and 0.35 seconds. In the longitudinal direction, the periods are slightly shorter. Dynamic analyses are conducted using the same analytical model and considering an ensemble of five earthquake ground motions recorded during the Northridge earthquake. A representative time history of one of these motions is shown in Figure 434. The corresponding stress ratios in the perimeter frame are shown in Figure 435 for earthquake motion applied in the transverse direction. Stress ratios in the beams of the transverse frames range from 2.67 to 4.11 indicating substantial inelastic behavior. Stress ratios in excess of 1.12 are obtained in all of the columns of the transverse frames, however, it should be recalled that there is a factor of safety of approximately 1.4 on allowable stress and plastic hinging. Nonlinear Analyses In order to estimate the lateral resistance of the building at ultimate load, a static, nonlinear analysis (pushover) is conducted for proportional loading. The reference lateral load distribution is that specified in the 1979 UBC. This load distribution is then multiplied by a load factor to obtain the ultimate load. The nonlinear model is a two dimensional model in which the plasticity is assumed to be concentrated in plastic hinges at the ends of the members. The results of the pushover analysis are usually represented in terms of a plot of the roof displacement versus the base shear as shown in Figure 436. This figure indicates that first yielding occurs at a base shear of approximately 670 kips and a roof displacement of approximately 7.25 inches. The UBC 1979 static equivalent lateral forces for this frame results in a base shear of 439 kips which implies a load factor of 1.52 on first yield. At a roof displacement of 17.5 inches, a sway mechanism forms with all girders hinged and
Figure 431 Typical floor framing plan ~ Fourth & fifth floors
Figure 432. Gravity Loading Pattern, ETABS
Figure 433. Calculated Stress Ratios, Design Loads, ETABS
Figure 434. Recorded Base Acceleration, Sta. 322, NS
240
Chapter 4
Figure 435. Calculated Stress Ratios, Sta. 322 Ground Motion
Figure 436. Static Pushover Curve
hinges at the base of the columns. At this displacement, the pushover curve is becoming almost horizontal indicating a loss of most of the lateral stiffness. This behavior is characterized by a large increase in displacement for a small increase in lateral load since lateral resistance is only due to strain hardening in the plastic hinges. The ultimate load is taken as 840 kips which divided by the code base shear for the frame (439 kips) results in a load factor of 1.91 on ultimate. Note that the elastic dynamic analysis for the acceleration shown in Figure 434 results in a displacement at the roof of 16.7 inches. Comparing this to the pushover curve (Figure 436) indicates that the structure should be well into the inelastic range based on the displacement response.
Figure 437. Calculated Nonlinear Dynamic Response.
242
Chapter 4
Figure438. Nonlinear Displacement, Roof Level
The nonlinear dynamic response of a structure is often presented in terms of the following response parameters: (1) envelope of maximum total displacement, (2) envelope of maximum story to story displacement divided by the story height (interstory drift index), (3) maximum ductility demand for the beams and columns, (4) envelopes of maximum plastic hinge rotation, (5) moment versus rotation hysteresis curves for critical members and (6) envelopes of maximum story shear. Representative plots of four of these parameters are shown in Figure 437. The lateral displacement envelope (Figure 437a) indicates that the maximum displacement at the roof level is 12.3 inches which is less than the 16.7 inches obtained from the elastic dynamic analysis. The interstory drift and total beam rotation curves are shown in Figure 4 37b which indicates that the interstory drift ranges from 0.01 (1%) to 0.024 (2.4%). The beam rotation can be seen to range between 0.016 and 0.025. The curvature ductility demands of the beams and columns is shown in Figure 437c.
The maximum ductility demand for the columns is 1.8 and for the beams it is 3.3. The hysteretic behavior of a plastic hinge in a critical beam is shown in a plot of moment versus rotation in Figure 4 37d. A final plot, Figure 438, shows the nonlinear displacement time history of the roof. This figure illustrates the displacement of a pulse type of input. After some lessor cycles during the first 7 seconds of the time history, the structure sustains a strong displacement at approximately 8 seconds which drives the roof to a displacement of 12 inches relative to the base. Note the acceleration pulse at this time in the acceleration time history (Figure 434). Following this action, the structure begins to oscillate about a new, deformed position at four inches displacement. This is a residual displacement, which the structure will have following the earthquake and is characteristic of inelastic behavior. Additional details of this analysis example can be found in the literature (413) .
4. Dynamic Response of Structures
243
Figure 439. Location of Strong Motion Instrumentation
4.8
VERIFICATION OF CALCULATED RESPONSE
The dynamic response procedures discussed in the previous sections must have the ability to reliably predict the dynamic behavior of structures when they are subjected to critical seismic excitations. Hence, it is necessary to compare the results of analytical calculations with the results of largescale experiments. The best largescale experiment is when an earthquake occurs and properly placed instruments record the response of the building to ground motions recorded at the base. The instrumentation (accelerometers) placed in a sixstory reinforced concrete building by the California Strong Motion Instrumentation Program (CSMIP) is indicated in Figure 439. The lateral force framing system for the
building, shown in Figure 440, indicates that there are three moment frames in the transverse (EW) direction and two moment frames in the longitudinal (NS) direction. Note that the transverse frames at the ends of the building are not continuous with the longitudinal frames. It is assumed that the floor diaphragms are rigid in their own plane. During the Loma Prieta earthquake the instrumentation recorded thirteen excellent records of building response having a duration of more than sixty seconds (419) . Since the response was only weakly nonlinear, the calculations can be made using the ETABS program, however, similar analyses can also be conducted with a nonlinear response program (420).
244
Chapter 4
Figure 440. ETABS Building Model
To improve the evaluation of the recorded response, spectral analyses are conducted in both the time domain (response spectra) and frequency domain (Fourier spectra). A further refinement of the Fourier analysis can be attained by calculating a Fourier amplitude spectra for a segment (window) of the recorded time history. The fixed duration window is then shifted along the time axis and the process is
repeated until the end of the time history record. This results in a “moving window Fourier amplitude spectra” (MWFAS) which indicates the changes in period of the building response during the time history as shown in Figure 441. In this example a tensecond window was used with a fivesecond shift for the first sixty seconds of the recorded response. In general, the length of the “window” should be at least 2.5 times the fundamental period of the structure. If the connections (offsets) are assumed to be rigid, the initial stiffness of the building prior to any cracking of the concrete can be estimated using the analytical model with member properties of the gross sections. This results in a period of 0.71 seconds in the EW direction and 0.58 seconds in the NS direction. This condition can also be evaluated by the results obtained from the initial window of the MWFAS. An examination of Figure 441 indicates an initial period of 0.71 in the EW direction and 0.58 seconds in the NS direction. Identical results were also obtained from ambient vibration tests conducted by Marshall, et al. (421).
Figure 41. Variation of Building Period with Time
4. Dynamic Response of Structures
245
Figure 442. Time History Comparisons of Acceleration, Displacement
During the strong motion portion of the response, cracking in the concrete and limited yielding of the tension steel will cause the period of vibration to lengthen. In order to represent this increased flexibility in the elastic analytical model, the flexibility of the individual members can be reduced to an effective value or the rigid offsets at the connections (413) can be reduced in length. For this example, the rigid offsets were reduced by fifty percent. This results in a period of 1.03 seconds in the EW direction and 0.89 seconds in the NS direction which are in the range of values obtained from the MWFAS. Considering the entire duration of the recorded response, the Fourier amplitude spectra indicates a period of 1.05 seconds in the EW direction and 0.85 in the NS direction. Corresponding values obtained from a response spectrum analysis
indicate 1.0 EW and 0.90 NS. It can be concluded that for this building, all of these values are in good agreement. The MWFAS also indicate an increase in period of approximately fifty percent in both principal directions during the earthquake. This amount of change is not unusual for a reinforced concrete building (422), however, it does indicate cracking and possible limited yielding of the reinforcement. The time histories of the acceleration and displacement at the roof level are shown in Figure 442. This also shows a good correlation between the measured and the calculated response.
246
Chapter 4
REFERENCES 41
42
43
44
45 46
47
48
49
410
411
412
413
414
415
Chopra, A, K., Dynamics of Structures, A Primer, Earthquake Engineering Research institute, Berkeley, CA, 1981. Applied Technology Council, An Evaluation of a Response Spectrum Approach to Seismic Design of Buildings, ATC2, Applied Technology Council, Palo Alto, CA, 1974. Newmark, N. M., and Hall, W. J., "Procedures and Criteria for Earthquake Resistant Design", Building Practices for Disaster Mitigation, U. S. Department of Commerce, Building Science Series 46, 1973. Clough, R. W., and Penzien, J., Structural Dynamics, McGrawHill, Inc., New York, NY, 1975. Rayleigh, Lord, Theory of Sound, Dover Publications, New York, NY, 1945. Lee, V. W., and Trifunac, M. D., "Strong Earthquake Ground Motion Data in EQINFOS, Part I," Report No.8701, Department of Civil Engineering, USC, Los Angeles, CA, 1987. Bathe, K. J., and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1976. Caughy,T. K., "Classical Normal Modes in Damped Linear Dynamic systems," Journal of Applied Mechanics, ASME, Paper No. 59A62, June, 1960. Wilson, E. L., Der Kiureghian, A.Rand Bayo, E. P., "A Replacement for the SRSS Method in Seismic Analysis", Earthquake Engineering and Structural Dynamics, Vol. 9, 1981. Anderson, J. C., and Gurfinkel, G., "Seismic Behavior of Framed Tubes," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 4, No. 2, OctoberDecember, 1975. Habibullah, A., "ETABS, Three Dimensional Analysis of Building Systems," User's Manual, Version 6.0, Computers & Structures, Inc. Berkeley, California, 1994. Habibullah, A., "ETABS Design Postprocessors," Version 6.0, Computers & Structures, Inc. Berkeley, California, 1994. Anderson, J. C., "Moment Frame Building", Buildings Case Study Project, SSC 9406, Seismic Safety Commission, State of California, Sacramento, California, 1996. Charney, F.A., NONLIN, Nonlinear Dynamic Time History Analysis of Single Degree of Freedom Systems, Advanced Structural Concepts, Golden, Colorado, 1996. Norris, C.H., Hansen, R.J., Holley, M.J., Biggs, J.M., Namyet, S., and Minami, J.K., Structural Design for Dynamic Loads, McGrawHill Book Company, New York, New York, 1959.
416 U.S. Army Corps of Engineers, Design of Structures to Resist the Effects of Atomic Weapons, EM 1110345415, 1957. 417 Applied Technology Council, “Seismic Evaluation and Retrofit of Concrete Buildings”, ATC40, Applied Technology Council, Redwood City, California, 1991. 418 Anderson, J.C. and Bertero, V.V., “Seismic Performance of an Instrumented Six Story Steel Building”, Report No. UCB/EERC91/111, Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, California, 1991. 419 California Division of Mines and Geology, “CSMIP StrongMotion Records from the Santa Cruz Mountains (Loma Prieta) California Earthquake of October 17, 1989”, Report OSMS 8906. 420 Anderson, J.C. and Bertero, V.V., “Seismic Performance of an Instrumented Six Story Steel Building”, Report No. UCB/EERC93/01, Earthquake Engineering Research Center, University of California at Berkeley. 421 Marshall, R.D., Phan, L.T. and Celebi, M., “Full Scale Measurement of Building Response to Ambient Vibration and the Loma Prieta Earthquake”, Proceedings, Fifth National Conference on Earthquake Engineering, Vol. II, Earthquake Engineering Research Institute, Oakland, California, 1994. 422 Anderson, J.C., Miranda, E. and Bertero, V.V., “Evaluation of the Seismic Performance of a ThirtyStory RC Building”, Report No. UCBEERC93/01, Earthquake Engineering Research Center, University of California, Berkeley.
Chapter 5 Linear Static Seismic Lateral Force Procedures
Roger M. Di Julio Jr., Ph.D., P.E. Professor of Engineering, California State University, Northridge
Key words:
Code Philosophy, Design Base Shear, Design Story Forces, Design Drift Limitations, Equivalent Static Force Procedure, Near Fault Factors, Seismic Zone Factors, UBC97, IBC2000, Regular and Irregular Structures, Torsion and Pdelta Effects, Site Soil Factors, Importance Factors
Abstract:
The purpose of this chapter is to review and compare the sections of current seismic design provisions, which deal with the specification of seismic design forces. Emphasis will be on the equivalent static force procedures as contaned in the 2000 edition of the International Building Code and the 1997 Edition of the Uniform Building Code. There are two commonly used procedures for specifying seismic design forces: The "Equivalent Static Force Procedure" and "Dynamic Analysis". In the equivalent static force procedure, the inertial forces are specified as static forces using empirical formulas. The empirical formulas do not explicitly account for the "dynamic characteristics" of the particular structure being designed or analyzed. The formulas were, however, developed to adequately represent the dynamic behavior of what are called "regular" structures, which have a reasonably uniform distribution of mass and stiffness. For such structures, the equivalent static force procedure is most often adequate. Structures that do not fit into this category are termed "irregular". Common irregularities include large floortofloor variation in mass or center of mass and soft stories. Such structures violate the assumptions on which the empirical formulas, used in the equivalent static force procedure, are based. Therefore, its use may lead to erroneous results. In these cases, a dynamic analysis should be used to specify and distribute the seismic design forces. Principles and procedures for dynamic analysis of structures were presented in Chapter 4.
247
248
Chapter 5
5. Linear Static Seismic Lateral Force Procedures
5.1
INTRODUCTION
In order to design a structure to withstand an earthquake the forces on the structure must be specified. The exact forces that will occur during the life of the structure cannot be known. A realistic estimate is important, however, since the cost of construction, and therefore the economic viability of the project depends on a safe and cost efficient final product. The seismic forces in a structure depend on a number of factors including the size and other characteristics of the earthquake, distance from the fault, site geology, and the type of lateral load resisting system. The use and the consequences of failure of the structure may also be of concern in the design. These factors should be included in the specification of the seismic design forces. There are two commonly used procedures for specifying seismic design forces: The "Equivalent Static Force Procedure" and "Dynamic Analysis". In the equivalent static force procedure, the inertial forces are specified as static forces using empirical formulas. The empirical formulas do not explicitly account for the "dynamic characteristics" of the particular structure being designed or analyzed. The formulas were, however, developed to adequately represent the dynamic behavior of what are called "regular" structures, which have a reasonably uniform distribution of mass and stiffness. For such structures, the equivalent static force procedure is most often adequate. Structures that do not fit into this category are termed "irregular". Common irregularities include large floortofloor variation in mass or center of mass and soft stories. Such structures violate the assumptions on which the empirical formulas, used in the equivalent static force procedure, are based. Therefore, its use may lead to erroneous results. In these cases, a dynamic analysis should be used to specify and distribute the seismic design forces. A dynamic analysis can take a number of forms, but should account for the irregularities of the structure by modeling its "dynamic
249 characteristics" including natural frequencies, mode shapes and damping. The purpose of this chapter is to review and compare the sections of current seismic design provisions, which deal with the specification of seismic design forces. Emphasis will be on, as in the documents discussed, the equivalent static force procedure. The following seismic design provisions are included in the discussion, which follows: 1. The Uniform Building Code, Volume 2, “Structural Engineering Design Provisions” issued by the International Conference of Building Officials, 1997 edition, referred to as UBC97. 2. “International Building Code”, IBC2000 Edition, Published by the International Code Council, INC., referred to as IBC2000 IBC2000 is based on "NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, Part I Provisions, prepared by the Building Seismic Safety Council for the Federal Emergency Management Agency (FEMA), 1997 edition, referred to as FEMA302. The commentary on this document is contained in Part 2 Commentary, designated FEMA303.
5.2
CODE PHILOSOPHY
The philosophy of a particular document indicates the general level of protection that it can be expected to provide. Most code documents clearly state that their standards are minimum requirements that are meant to provide for life safety but not to insure against damage. The codespecified forces are generally lower than the actual forces that would occur in a large or moderate size earthquake. This is because the structure is designed to carry the specified loads within allowable stresses and deflections, which are considerably less than the ultimate or yield capacity (when using working stress design) of the materials and system. It is assumed that the larger loads that actually occur will be accounted for by the factors of safety and by the redundancy and
250
Chapter 5
ductility of the system. Life safety is thereby insured but structural damage may be sustained.
5.3
V = 0.11C a IW
UBC97 PROVISIONS
UBC97, basically provides for the use of the equivalent static force procedure or a dynamic analysis for regular structures under 240 feet tall and irregular structures 65 feet or less in height. A dynamic analysis is required for regular structures over 240 feet tall, irregular structures over 65 feet tall, and buildings that are located on poor soils (type SF) and have a period greater than 0.7 seconds. Although UBC97 allows for both working stress design and alternately strength or load and resistance factor design, the earthquake loads are specified for use with the latter. This is a departure from previous editions where the earthquake loads were specified at the working stress level. 5.3.1
Design Base Shear V
The design base shear is specified by the formula:
V=
Cv I W RT
(51)
Where, T is the fundamental period of the structure in the direction under consideration, I is the seismic importance factor, Cv is a numerical coefficient dependent on the soil conditions at the site and the seismicity of the region, W is the seismic dead load, and R is a factor which accounts for the ductility and overstrength of the structural system. Additionally the base shear is dependent on the seismic zone factor, Z. The base shear as specified by Equation 51 is subject to three limits: The design base shear need not exceed:
V=
2.5C a I W R
And cannot be less than:
(52)
(53)
Where Ca is another seismic coefficient dependent on the soil conditions at the site and regional seismicity. Additionally in the zone of highest seismicity (zone 4) the design base shear must be greater than:
V=
0.8ZN v I W R
(54)
Where Nv is a nearsource factor that depends on the proximity to and activity of known faults near the structure. Faults are identified by seismic source type, which reflect the slip rate and potential magnitude of earthquake generated by the fault. The near source factor Nv is also used in determining the seismic coefficient Cv for buildings located in seismic zone 4. 5.3.2
Seismic Zone Factor Z
Five seismic zones, numbered 1 2A, 2B, 3 and 4 are defined. The zone for a particular site is determined from a seismic zone map (See Figure 51). The numerical values of Z are: Zone 1 Z 0.075
2A 0.15
2B 0.2
3 0.3
4 0.4
The value of the coefficient thus normalized can be viewed as the peak ground acceleration, in percent of gravity, in each zone. 5.3.3
Seismic Importance Factor I
The importance factor I is used to increase the margin of safety for essential and hazardous facilities. For such structures I=1.25. Essential structures are those that must remain operative immediately following an earthquake such as emergency treatment areas and fire stations. Hazardous facilities include those housing toxic or explosive substances (See Table 51).
5. Linear Static Seismic Lateral Force Procedures 5.3.4
5.3.6
Building Period T
The building period may be determined by analysis or using empirical formulas. A single empirical formula may be used for all framing systems: 3
T = Ct hn 4
(55)
where
0.035 for steel moment frames 0.030 for concrete moment frames Ct = 0.030 for eccentric braced frames 0.020 for all other buidlings hn = the height of the building in feet. If the period is determined using Rayleigh's formula or another method of analysis, the value of T is limited. In Seismic Zone 4, the period cannot be over 30% greater than that determined by Equation 55 and in Zones 1, 2 and 3 it cannot be more than 40% greater. This provision is included to eliminate the possibility of using an excessively long period to justify an unreasonably low base shear. This limitation does not apply when checking drifts. 5.3.5
251
Structural System Coefficient R
The structural system coefficient, R is a measure of the ductility and overstrength of the structural system, based primarily on performance of similar systems in past earthquakes. The values of R for various structural systems are found in Table 52. A higher number has the effect of reducing the design base shear. For example, for a steel special moment resisting frame the factor has value of 8.5, while and ordinary moment resisting frame the value is 4.5. This reflects the fact that a special moment resisting frame is expected to perform better during an earthquake.
Seismic Dead Load W
The dead load W, used to calculate the base shear, includes not only the total dead load of the structures but also partitions, 25% of the floor live load in storage and warehouse occupancies and the weight of snow when the design snow load is greater than 30 pounds per square foot. The snow load may be reduced by up to 75% if its duration is short. The rationale for including a portion of the snow load in heavy snow areas is the fact that in these areas a significant amount of ice can build up and remain on roofs. 5.3.7
Seismic Coefficients Cv and Ca
The seismic coefficients Cv & Ca are measures of the expected ground acceleration at the site. They may be found in Tables 53 and 54. The coefficient, and hence the expected ground accelerations are dependent on the seismic zone and soil profile type. They therefore reflect regional seismicity and soil conditions at the site. Additionally in seismic zone 4 they also depend on the seismic source type and near source factors Na and Nv. These factors reflect local seismicity in the region of highest seismic activity. 5.3.8
Soil Profile Type S
The soil profile type reflects the effect of soil conditions at the site on ground motion. They are found in Table 55 and are labeled SA, through SF.
252
Chapter 5
Figure 51. Seismic Zone Map of the United States Table 51 Seismic Importance Factor Occupancy Category 1. Essential facilities
2. Hazardous facilities
3. Special occupancy structures
4. Standard occupancy 5. Miscellaneous
Occupancy or Functions of Structure Group I, Division 1 Occupancies having surgery and emergency treatment areas. Fire and police stations. Garages and shelters for emergency vehicles and emergency aircraft. Structures and shelters in emergencypreparedness centers. Aviation control towers. Structures and equipment in government communication centers and other facilities required for emergency response. Standby powergenerating equipment for Category 1 facilities. Tanks or other structures containing housing or supporting water or other firesuppression material or equipment required for the protection of Category 1, 2 or 3 structures. Group H, Divisions 1, 2, 6 and 7 Occupancies and structures therein housing or supporting toxic or explosive chemicals or substances. Nonbuilding structures housing, supporting or containing quantities of toxic or explosive substances that, if contained within a building, would cause that building to be classified as a Group H, Division 1, 2 or 7 Occupancy. Group A, Divisions 1, 2 and 2.1 Occupancies. Buildings housing Group E, Divisions 1 and 3 Occupancies with a capacity greater than 300 students. Buildings housing Group B Occupancies used for college or adult education with a capacity greater than 500 students. Group I, Divisions 1 and 2 Occupancies with 50 or more resident incapacitated patients, but not included in Category 1. Group I, Division 3 Occupancies. All structures with an occupancy greater than 5,000 persons. Structures and equipment in powergenerating stations, and other public utility facilities not included in Category 1 or Category 2 above, and required for continued operation. All structures housing occupancies or having functions not listed in Category 1, 2 or 3 and Group U Occupancy towers. Group U Occupancies except for towers.
Seismic Importantce Factor, I 1.25
1.25
1.00
1.00 1.00
Table 52. Structural Systems Basic Structural System 1. Bearing wall system
LateralForceResisting System Description 1. Lightframed walls with shear panels a. Wood Structural panel walls for structures three stories or less b. All other lightframed walls 2. Shear walls a. Concrete b. Masonry 3. Light steelframed bearing walls with tension only bracing 4. Braced frames where bracing carries gravity load a. Steel b. Concrete c. Heavy timber 2. Building frame system 1. Steel eccentrically braced frame (EBF) 2. Lightframed walls with shear panels a. Wood structural panel walls for structures three stories or less b. All other lightframed walls 3. Shear walls a. Concrete b. Masonry 4. Ordinary braced frames a. Steel b. Concrete c. Heavy timber 5. Special concentrically braced frames a. Steel 3. Momentresisting 1. Special momentresisting frame (SMRF) frame system a. Steel b. Concrete 2. Masonry momentresisting wall frame (MMRWF) 3. Concrete intermediate momentresisting frame (IMRF) 4. Ordinary momentresisting frame (OMRF) a. Steel b. Concrete 5. Special truss moment frames of steel (STMF) 4. Dual systems 1. Shear walls a. Concrete with SMRF b. Concrete with steel OMRF c. Concrete with concrete IMRF d. Masonry with SMRF e. Masonry with steel OMRF f. Masonry with concrete IMRF g. Masonry with masonry MMRWF 2. Steel EBF a. With steel SMRF b. With steel OMRF 3. Ordinary braced frames a. Steel with steel SMRF b. Steel with steel OMRF c. Concrete with concrete SMRF d. Concrete with concrete IMRF 4. Special concentrically braced frames a. Steel with steel SMRF b. Steel with steel OMRF 5. Cantilevered column 1. Cantilevered column elements building systems 6. Shear wallframe 1. Concrete interaction systems
R 5.5 4.5 4.5 4.5 2.8 4.4 2.8 2.8 7.0 6.5 5.0 5.5 5.5 5.6 5.6 5.6 6.4 8.5 8.5 6.5 5.5 4.5 3.5 6.5 8.5 4.2 6.5 5.5 4.2 4.2 6.0 8.5 4.2 6.5 4.2 6.5 4.2 7.5 4.2 2.2 5.5
254
Chapter 5
The soil profile types are broadly defined in generic terms, for example “Hard Rock” for type SA. They are also defined by the physical properties of the soil determined by standard tests including; shear wave velocity, standard penetration test, and undrained shear strength. 5.3.9
are used in conjunction with the soil profile type to determine the seismic coefficients Cv and Ca (See Tables 53 and 54). For example, for seismic source type A at a distance to the fault of less than 2km, Na = 1.5 (See Table 57). This is then used with Table 54 to determine the seismic coefficient, Ca.
Seismic Source Type A, B and C 5.3.11
The seismic source type is used to specify the capability and activity of faults in the immediate vicinity of the structure. It is used only in seismic zone 4. The seismic source types, labeled A, B or C, are found in Table 56. They are defined in terms of the slip rate of the fault and the maximum magnitude earthquake it is capable of generating. For example, the highest seismic risk is posed by seismic source type A, which is defined by a maximum moment magnitude of 7.0 or greater and a slip rate of 5mm/year or greater. 5.3.10
Near Source Factors Na and Nv
The near source factors Na and Nv are found in Tables 57 and 58. In seismic zone 4, they
Distribution of Lateral Force Fx
The base shear V, as determined from Equations 51 through 54 are distributed over the height of the structure as a force at each level Fi, plus an extra force Ft at the top: n
V = Ft + ∑ Fi i =1
The extra force at the top is:
Ft = 0.07TV ≤ 0.25V if T > 0.7 sec . (57a) Ft = 0.0 if T ≤ 0.7 sec . (57b) Ft accounts for the greater participation of higher modes in the response of longer period structures.
Table 53. Seismic Coefficient CV Soil Profile Type SA SB SC SD SE SF
Seismic Zone Factor, Z Z = 0.075 Z = 0.15 Z = 0.2 Z = 0.3 Z = 0.4 0.06 0.12 0.16 0.24 0.32NV 0.08 0.15 0.20 0.30 0.40NV 0.13 0.25 0.32 0.45 0.56NV 0.18 0.32 0.40 0.54 0.64NV 0.26 0.50 0.64 0.84 0.96NV Sitespecific geotechnical investigation and dynamic site response analysis shall be performed.
Table 54. Seismic Coefficient Ca Soil Profile Type SA SB SC SD SE SF
(56)
Seismic Zone Factor, Z Z = 0.075 Z = 0.15 Z = 0.2 Z = 0.3 Z = 0.4 0.06 0.12 0.16 0.24 0.32Na 0.08 0.15 0.20 0.30 0.40Na 0.09 0.18 0.24 0.33 0.40Na 0.12 0.22 0.28 0.36 0.44Na 0.19 0.30 0.34 0.36 0.36Na Sitespecific geotechnical investigation and dynamic site response analysis shall be performed.
5. Linear Static Seismic Lateral Force Procedures
The remaining portion of the total base shear (V  Ft) is distributed over the height, including the top, by the formula:
Fx =
(V − Ft )(wx hx )
(58)
n
∑ wi hi
i =1
255 Where, w is the weight at a particular level and h is the height of a particular level above the shear base. At each floor, the force is located at the center of mass. For equal story heights and weights, Equation 58 distributes the force linearly, increasing towards the top. Any significant variation from this triangular distribution indicates an irregular structure.
Table 55. Soil Profile Types Soil Profile Type SA
Soil Profile Name/Generic Description Hard Rock
SB
Rock
SC
Very Dense Soil and Soft Rock
SD
Stiff Soil Profile
SE
Soft Soil Profile
SF
Average Soil Properties for Top 100 Feet (30 480 mm) of Soil Profile Shear Wave Velocity, Standard Penetration Undrained Shear feet/second (m/s) Test, (blows/foot) Strength, psf (kPa) > 5,000 (1,500) __ __ 2,500 to 5,000 (760 to 1,500) 1,200 to 2,500 > 50 >2,000 (360 to 760) (100) 600 to 1,200 15 to 50 1,000 to 2,000 (180 to 360) (50 to 100) < 600 < 15 < 1,000 (180) (50) Soil Requiring Sitespecific Evaluation.
Table 56. Seismic Source Type Seismic Source Type A
Seismic Source Description Faults that are capable of producing large magnitude events and that have a high rate of seismic activity.
Seismic Source Definition Maximum Moment Slip Rate, SR Magnitude, M (mm/year) M ≥ 7.0
SR ≥ 5
B
All faults other than Types A and C.
M ≥ 7.0 M < 7.0 M ≥ 6.5
SR < 5 SR > 2 SR < 2
C
Faults that are not capable of producing large magnitude earthquakes and that have a relatively low rate of seismic activity.
M < 6.5
SR ≤ 2
Table 57. NearSource Factor Na Seismic Source Type
≤ 2 km 1.5 1.3 1.0
A B C
Closest Distance to Known Seismic Source 5 km 1.2 1.0 1.0
≥ 10 km 1.0 1.0 1.0
Table 58. NearSource Factor NV Seismic Source Type A B C
≤ 2 km 2.0 1.6 1.0
Closest Distance to Known Seismic Source 5 km 10 km 1.6 1.2 1.2 1.0 1.0 1.0
≥ 15 km 1.0 1.0 1.0
256
Chapter 5
5.3.12
Story Shear and Overturning Moment Vx and Mx
PDelta effects must be included in determining member forces and story displacements where significant.
The story shear at level x is the sum of all the story forces at and above that level: n
Vx = Ft + ∑ Fi
(59)
i=x
The overturning moment at a particular level Mx is the sum of the moments of the story forces above, about that level. Hence:
M x = Ft (hn − hx ) + ∑ Fi (hi − hx )
Torsion and PDelta Effect
Accidental torsion, due to uncertainties in the mass and stiffness distribution, must be added to the calculated eccentricity. This is done by adding a torsional moment at each floor equal to the story shear multiplied by 5% of the floor dimension, perpendicular to the direction of the force. This procedure is equivalent to moving the center of mass by 5% of the plan dimension, in a direction perpendicular to the force. If the deflection at either end of the building is more than 20% greater than the average deflection, it is classified as torsionally irregular and the accidental eccentricity must be amplified using the formula: 2
δ Ax = MAX ≤ 3.0 1.2δ AVG
The seismic design forces and hence the base shear as determined from Equations 51 through 54, must be multiplied be a reliability/ redundancy factor for the lateral load resisting system:
(510)
Design must be based on the overturning moment as well as the shear at each level. 5.3.13
Reliability / Redundancy Factor ρ
1≤ ρ = 2 −
n
i=x
5.3.14
(511)
where δavg = the average displacement at level x δmax = the maximum displacement at level x
20 rmax AB
≤ 1.5
(512)
Where, AB is the ground floor area of the structure in square feet and rmax is the maximum elementstory shear ratio. The element story shear ratio (ri) at a particular level is the ratio of the shear in the most heavily loaded member to the total story shear. The maximum ratio, rmax is defined as the largest value of ri in the lower twothirds of the building. Special provisions for calculating r, for different lateral load resisting systems, are demonstrated in the examples that follow. For special momentresisting frames, if ρ exceeds 1.25, additional bays must be added. For the purposes of determining drift (displacement), and in seismic zones 0, 1 and 2, ρ =1.0. 5.3.15
Drift Limitations
The deflections due to the design seismic forces are called the design level response displacements, ∆s. The seismic forces used to determine ∆s may be calculated using a reliability/redundancy factor equal to one, ignoring the limitation represented by Equation 53, and using an analytically determined period greater than the limits outlined in section 5.3.4. The maximum inelastic response is defined as:
5. Linear Static Seismic Lateral Force Procedures
∆ M = .7 R∆ s
(513)
Where, R is the structural system coefficient defined in Table 52. Deflection control is specified in terms of the story drift, which is defined as the lateral displacement of one level relative to the level below. The story drift is determined from the maximum inelastic response as defined by Equation 513. The displacement must include both translation and torsion. Hence, the drift must be checked in the plane of the lateral load resisting elements, generally at the ends of the building. PDelta displacements must be included where significant.
257 For structures with a period less than 0.7 seconds, the maximum story drift is limited to:
∆ a ≤ .025h
(514)
Where, h is the story height. For structures with a period greater than 0.7 seconds:
∆ a ≤ .020h
5.3.16
(515)
Irregular Structures
UBC97 quantifies the notion of irregularity, which it breaks into two broad categories:
Table 59. Vertical Structural Irregularities Irregularity Type and Definition Stiffness irregularitysoft story 1. A soft story is one in which the lateral stiffness is less that 70 percent of that in the story above or less than 80 percent of the average stiffness of the three stories above. Weight (mass) irregularity 2. Mass irregularity shall be considered to exist where the effective mass of any story is more than 150 percent of the effective mass of an adjacent story. A roof that is lighter than the floor below need not be considered. Vertical geometric irregularity 3. Vertical geometric irregularity shall be considered to exist where the horizontal dimension of the lateralforceresisting system in any story is more than 130 percent of that in an adjacent story. Onestory penthouses need not be considered. Inplane discontinuity in vertical lateralforceresisting element 4. An inplane offset of the lateralloadresisting elements greater than the length of those elements. Discontinuity in capacityweak story 5. A weak story is one in which the story strength is less than 80 percent of that in the story above. The story strength is the total strength of all seismicresisting elements sharing the story shear for the direction under consideration.
Table 510. Plan Structural Irregularities Irregularity Type and Definition Torsional irregularityto be considered when diaphragms are not flexible 1. Torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at one end of the structure transverse to an axis is more than 1.2 times the average of the story drifts of the two ends of the structure. Reentrant corners 2. Plan configurations of a structure and its lateralforceresisting system contain reentrant corners, where both projections of the structure beyond a reentrant corner are greater than 15 percent of the plan dimension of the structure in the given direction. Diaphragm discontinuity 3. Diaphragms with abrupt discontinuities or variations in stiffness, including those having cutout or open areas greater than 50 percent of the gross enclosed area of the diaphragm, or changes in effective diaphragm stiffness of more than 50 percent from one story to the next. Outofplane offsets 4. Discontinuities in a lateral force path, such as outofplane offsets of the vertical elements. Nonparallel systems 5. The vertical lateralloadresisting elements are not parallel to or symmetric about the major orthogonal axes of the lateralforceresisting system.
258
Chapter 5
vertical structural and plan structural irregularity. Vertical irregularities include soft or weak stories, large changes in mass from floor to floor and large discontinuities in the dimensions or inplane locations of lateral load resisting elements. Plan irregular buildings include those which undergo substantial torsion when subjected to seismic loads, have reentrant corners, discontinuities in floor diaphragms, discontinuity in the lateral force path, or lateral load resisting elements which are not parallel to each other or to the axes of the building. The precise definitions of these irregularities are found in Tables 59 and 510. For a more detailed discussion of irregularity, see Chapter Six.
5.3.17
Dynamic Lateral Force Procedure
UBC97 requires that, if the base shear determined by a dynamic analysis using a sitespecific spectra is less than that specified by the static lateral force procedure, it must be scaled to equal that determined by the equivalent static force procedure. Similarly, if the base shear obtained from a dynamic analysis is greater than that specified by the static lateral force procedure, it may be scaled down. In this manner, the dynamic characteristics of the structure are modeled, and thus the forces are distributed properly, while the code level forces are maintained. If a sitespecific spectrum is not available, the spectra provided in UBC97 (see Figure 52) can be used.
Figure 52. Design Response Spectra
5. Linear Static Seismic Lateral Force Procedures 5.3.18
259 V ≥ .11 Ca IW
Examples
Equation 5 –3
Example 51:
V ≥ .11 (.3) (1) (5900) = 194.7 < 804.5
Determine the UBC97 design seismic forces for a threestory concrete shear wall office building. It is located in Southeastern California on rock with a shear wave velocity of 3000 ft/ sec. The story heights are 13 feet for the first floor and 11 feet for the second and third floors. The story dead loads are 2200, 2000 and 1700 kips from the bottom up. The plan dimensions are 180 feet by 120 feet. The walls in the direction under consideration are 120 feet long and are without openings. The shear walls do not carry vertical loads. Sample calculations are presented and a complete tabulation is found in Table 511.
V = 804.5 k • Vertical Distribution: T< 0.7 sec Ft = 0.0
Equation 57b
Fx = (V − Ft ) w x hx ∑ wi hi n
F3 = 804.5 (59.5) / 136.1 = 351.7 k F2 = 804.5 (48) / 136.1 = 283.7 k F1 = 804.5 (28.6) / 136.1 = 169.1 k • Story Shear:
• Base Shear:
n
Vx = Ft + ∑ Fi
C IW V= v RT
Equation 51
I=1.0 R=5.5 (Shear Walls) Seismic zone 3 Z = .3 Soil Profile Type SB Cv = .3
Table 51 Table 52 Figure 51 Section 5.3.2 Table 55 Table 53
Equation 55 T = .02(35) 4 = .29 Seconds W = 1700 + 2000 + 2200 = 5900 k
Equation 59
i=x
V3 = 351.7 k V2 =351.7 +283.7 = 635.4 k V3=351.7 +283.7 = 169.1 = 804.5 k • Overturning Moment:
M x = Ft (hn − hx ) + ∑ Fi (hi − hx ) n
.3(1.0) (5900) = 1109.2 k 5.5(.29 )
M3 = 351.7 (11) = 3869 ftk M2 = 351.7 (22) +283.7 (11) = 10,858 ftk M1 = 351.7 (35) +283.7(24)+ 169.1(13)=21,317 Table 511: Example 51 Level
V ≤ 2.5 Ca = .3
Ca I W R
Eq. 510
i=x
3
V=
Equation 58
i =1
Equation 52 Table 54
2.5(.3)(1) (5900) = 804.5 k < 1109.2 V≤ 5.5
hx
wx
(ft)
(k)
wXhX x103
Fi+Ft
VX
Mx
(k)
(k)
(ftk)
3
35
1700.
59.5
351.7
351.7
3869.
2
24
2000.
48.0
283.7
635.4
10858
1
13
2200.
28.6
169.1
804.5
21317.
5900.
136.1
804.5
Σ
• Allowable Inelastic Story Displacement: T ≤ .7 seconds
260
Chapter 5
∆a ≤ .025 h
Equation 514
2nd & 3rd Floors: ∆a ≤ .025 (11x12) = 3.3 inches 1st Floor: ∆a ≤ .025 (11 x13) = 3.56 inches • Equivalent Elastic Story Displacement:
∆≤
.025h .025 = h = .0065h .7 R .7(5.5)
Eq. 513
frame office building located in Los Angeles, California on very dense soil and soft rock. The building is located 5km from a fault capable of large magnitude earthquakes and that has a moderate slip rate (M>7, SR>2mm/yr). The story heights are all thirteen feet. The plan area is 100 feet by 170 feet. The total dead load is 100 pounds per square foot at all levels. The moment frames consist of two four bay frames in the transverse direction and two seven bay frames in the longitudinal direction. Sample calculations are presented and a complete tabulation is found in Table 512.
2nd & 3rd Floor: ∆ ≤ .0065 (11 x 12) = .858 inches
• Base Shear:
1st Floor: ∆ ≤ .0065 (13 x 12) = 1.01 inches
V=
• Reliability / Redundancy Factor:
I = 1.0 R = 8.5 (SMRF) Seismic Zone 4 Z = .4 Soil Profile Type Sc Seismic Source Type B Nv = 1.2 Cv = .56 Nv = .56 (1.2) = .67
For shear walls, ri is the maximum value of the product of the wall shear and 10/lw, divided by the total shear, where lw is the length of wall in feet (120 ft). An approximation of rmax can be obtained by assuming that half the story shear is carried by each wall.
rmax
T =.035 (117)
(V 2)(10 120) = .04 = S
20 rmax A B
ρ= 2 – 20 /. 04
Equation 51
3
4
= 1.25 seconds
Table 51 Table 52 Figure 51 Section 5.3.2 Table 55 Table 56 Table 58 Table 53 Equation 55
W = .1 (170) 100 = 1700 k / floor W = 9 (1700) = 15,300 k
VS
AB = 120 x 180 = 21,600 ft2
ρ = 2
C v IW RT
V=
(.67 )(1.0)W = .063W = .063(15,300) = 964.8K 8.5(1.25)
Equation 512
21,600 = –1.4 < ρmin = 1.0
ρ = 1.0 Example 52: Determine the UBC97 design seismic forces for a nine story ductile moment resisting steel
V ≤ 2.5
Ca I W R
Na = 1.0 Ca = .4 Na = .4 (1.0) = .4 V ≤ 2.5
Equation 52 Table 57 Table 54
(.4)(1.0) (15,300) = 1800 k > 964.8 k 8.5
V ≥ .11 Ca IW
Equation 53
5. Linear Static Seismic Lateral Force Procedures V ≥ .11(.4)(1.0)(15,300)= 673.2k < 964.8k
261 Table 412: Example 42 Level
hx
wx
wxhx
Fi+Ft
Vx
Mx
(ft)
(k)
x103
(k)
(k)
(ftk)
9
117
1700.
198.9
260.1
260.1
3381.
8
104
1700.
176.8
156.2
416.6
8793.
7
91
1700.
155.7
137.6
553.9 15994.
6
78
1700.
132.6
117.2
671.1 24718.
Since the building is in zone 4:
.8ZN v I V≥ W R V≥
Equation 54
.8(.4)(1.2 )(1.0) (15,300) = 691.2 k < 964.8 k 8.5
V = 964.8 k • Vertical Distribution: T > .07 sec Ft=.07TV=.07(1.25)(964.8)= 84.4k .25V = .25 (964.8) = 241.2 > 84.4 Ft = 84.4 k (VFt) = 964.8 – 84.4 = 880.4
Fx
5
65
1700.
110.5
97.6
768.7 34711.
4
52
1700.
88.4
78.1
846.8 45720.
3
39
1700.
66.3
58.6
905.4 57490.
2
26
1700.
45.2
39.9
945.3 69779.
1
13
964.8 82321.
Σ
Eq.57a
∑ wi hi
i =1
F9+Ft= 880.4(1700)(117)/996,500+84.4=260.1k F8 = 880.4 (1700) (104) / 996,500 = 156.2 k (See Table 512) • Story Shear: n
Vx = Ft + ∑ Fi
Equation 59
i=x
19.5 964.8
.02h .02h = = .00336h .7 R .7(8.5)
• Reliability / Redundancy Factor For moment frames, ri is normally 70% of the shear in two adjacent interior columns. An approximation for ri can be obtained by assuming all interior columns carry equal shear and external columns carry half as much.
ρ=2−
20 rmax AB
AB = 100 x 170 = 17,000 ft2
Overturning Moment
Transverse Direction: Two 4 Bay Frames n
i=x
Eq. 510
M9 = 260.1 (13) = 3381 ft.k, M8 = 260.1 (26) + 156.2 (13) = 8,793 ftk • Allowable Inelastic Story Displacement: T > 0.7 seconds
Eq. 513
∆ ≤ .00336 (13 x 12) = .52 inches
V9 = 260.4 k V8 = 260.4 + 156.2 = 416.6 k
M x = Ft (hn − hx ) + ∑ Fi (hi − hx )
Eq. 515
• Equivalent Elastic Story Displacement:
Equation 58
n
22.1 996.5
∆a<.02h= .02(13x12)= 3.12 inches
∆a ≤
(V − Ft )(wx hx ) =
1700. 15300.
Equation 512
rmax = .7(V 8 + V 8)/ V = .175 20 ρ = 2− = 1.12 ≤ 1.25 .175 17,000 ρmax = 1.25
for special moment frame ok.
ρ = 1.12 Longitudinal Direction: Two 7 Bay Frames
262
Chapter 5
rmax = .7(V 14 + V 14 )/ V = .1 ρ = 2−
20 = .47 < ρ min = 1.0 .1 17,000
ρ = 1.0
5.4
IBC2000 PROVISIONS
IBC2000 is broadly similar to UBC97, but does contain significant differences. These include ground accelerations specified on a local basis by a set seismic risk maps. The concept of a seismic use group, which is related to the importance factor in UBC97, is introduced. In addition to defining the importance factor it is used to designate the seismic design category and to establish the allowable story drift. The seismic design category determines the analysis procedures to be used and height and system limitations. 5.4.1
Seismic Use Group I, II, III
Each structure is assigned to a seismic use group based on the occupancy of the building and the consequences of severe earthquake damage. Three seismic hazard groups are defined: GROUP III...."having essential facilities that are required for postearthquake recovery and those containing substantial quantities of hazardous substances ". These facilities include fire and police stations, hospitals, medical facilities having emergency treatment facilities, emergency preparedness centers, operation centers, communication centers, utilities required for emergency backup, and structures containing significant toxic or explosive substances. GROUP II...."have a substantial public hazard due to occupancy or use...". These include high occupancy buildings and utilities not required for emergency backup. GROUP I  All other buildings.
5.4.2
Occupancy Importance Factor I
An occupancy importance factor is assigned based on the seismic use group. This factor is used to increase the design base shear for structures in seismic use groups II and III. The values of the importance occupancy factor are: Seismic Use Group I II III 5.4.3
I 1.0 1.25 1.50
Maximum Considered Earthquake Ground Motion
Regional seismicity is specified by a series of maps. The maps provide the spectral response accelerations at short periods, Ss and at a period of one second, S1 (see Figures 53 and 54). In areas of low seismic activity (Ss ≤ .15g, S1 ≤ .04g) the acceleration need not be determined. 5.4.4
Site Class
The soil conditions at the site determine the structures “site class”. These are virtually identical to the soil profile types in UBC97 (see Table 55). 5.4.5
Site Coefficients Fa and Fv
The regional seismicity, as expressed by the maximum considered earthquake ground motion, Ss and S1, must be modified for the soil conditions at the site. These are defined by the site class. The maximum considered earthquake spectral response accelerations adjusted for site class effects, are: SMS = Fa Ss
(516 a)
SM1 = Fv S1
(516 b)
Figure 53. IBC 2000 Spectral Map for Short Period Range (T=0.3 Sec)
264
Chapter 5
Figure 54 IBC 2000 Spectral Map for Intermediate Period Range (T=1.0 Sec).
Table 513. Values of Fa as a Function of Site Class and Mapped Short Period Maximum Considered Earthquake Spectral Acceleration Mapped Maximum Considered Earthquake Spectral Response Acceleration at Short Periods SS ≤ 0.25 SS=0.75 SS=1.00 SS ≥ 1.25 SS=0.5 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.2 1.2 1.1 1.0 1.0 1.6 1.4 1.2 1.1 1.0 2.5 1.7 1.2 0.9 a a a a a a
Site Class A B C D E F a
Sitespecific geotechnical investigation and dynamic site response analyses shall be performed.
Table 514. Values of Fv as a Function of Site Class and Mapped 1 Second Period Maximum Considered Earthquake Spectral Acceleration Site Class A B C D E F
Mapped Maximum Considered Earthquake Spectral Response Acceleration at 1 SecondPeriod S1=0.3 S1=0.4 S1 ≥ 0.5 S1 ≤ 0.1 S1=0.2 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.7 1.6 1.5 1.4 1.3 2.4 2.0 1.8 1.6 1.5 3.5 3.2 2.8 2.4 a a a a a a
Where, Fa and Fv are the site coefficients defined in Tables 513 and 514. For site class F, and site class E in regions of high seismicity (Ss>1.25g or S1>5g), a sitespecific geotechnical investigation must be performed. 5.4.6
Design Spectral Response Accelerations SDS and SD1
The spectral accelerations for the design earthquake are: SDS =2/3SMS
(517a)
SD1 =2/3SM1
(517b)
height and irregularity, those components of the structure that must be designed for seismic loads, and the types of analysis required. The seismic design categories, designated A through F, are presented in Tables 515 and 516. They depend on the seismic use group and the design spectral acceleration coefficients, SDS and SD1. The structure is assigned the more severe of the two values taken for these tables. 5.4.8
Design Base Shear V
IBC2000 specifies the design base shear by the formula: V = C sW
These are the accelerations determine the design base shear.
used
to
The base shear is a percentage, Cs of the total dead load W. 5.4.9
5.4.7
(518)
Total Dead Load W
Seismic Design Category
The structure must be assigned a seismic design category, which determines the permissible structural systems, limitations on
The seismic dead load consists of the total weight of the structure, plus partitions and permanent equipment. It also includes 25% of floor live load in areas used for storage, and the
266
Chapter 5
snow load if it is greater than 30 lb/ft2. The snow load may be reduced by up to 80% if its duration is short. 5.4.10
Seismic Response Coefficient CS
The seismic response determined from the formula:
coefficient
is
(521)
Additionally for structures in seismic design categories E and F, and for structures with a 1 second spectral response greater than or equal to .6g, it cannot be less than:
Cs =
S Cs = DS R I
.5S1 R I
(522)
(519) where
where SDS = the design spectral acceleration in the short period range R = the response modification factor from Table 517 and defined below I = the occupancy importance factor The coefficient Cs, as specified by Equation 519, is subject to three limits. It need not exceed:
Cs =
C s = .044 S DS I
S D1 TR I
(520)
SD1 = the design spectral response at a 1.0 second period T = the fundamental period of the structure S1 = the maximum considered earthquake spectral response acceleration at a 1 second period 5.4.11
Building Period T
The building period can be estimated using the empirical formula: Ta = Ct hn3/4
(523)
where It must be greater than:
Table 515. Seismic Design Category Based on Short Period Response Accelerations Value of SDS SDS < 0.167g 0.167g ≤ SDS < 0.33g 0.33g ≤ SDS < 0.50g 0.50g ≤ SDS
I A B C Da
Seismic Use Group II A B C Da
III A C D Da
a Seismic Use Group I and II structures located on sites with mapped maximum considered earthquake spectral response acceleration at 1 second period, S1, equal to or greater than 0.75g shall be assigned to Seismic Design Category E and Seismic Use Group III structure located on such sites shall be assigned to Seismic Design Category F.
Table 416. Seismic Design Category Based on 1Second Period Response Accelerations Value of SDS SD1 < 0.067g 0.067g ≤ SD1 < 0.133g 0.133g ≤ SD1 < 0.20g 0.20g ≤ SD1
I A B C Da
Seismic Use Group II A B C Da
III A C D Da
5. Linear Static Seismic Lateral Force Procedures
0.035 for steel moment frames 0.030 for concrete moment frames Ct = 0.030 for eccentric braced frames 0.020 for all other buidlings
267 allowable period used to specify the base shear is: SD1
≥ 0.4 0.3 0.2 0.15 ≤ 0.1
hn = the height of the building in feet. An alternate formula is provided for steel and concrete moment frame buildings twelve stories or less in height and with story heights ten feet or greater: Ta =0.1 N
Tmax/Ta 1.2 1.3 1.4 1.5 1.7
This provision insures that an excessively long analytically determined period is not used to justify an unrealistically low design base shear. When determining drifts these limits do not apply.
(524) 5.4.12
Response Modification Factor R
where, N is the number of stories. The period may also be determined by an analysis. The period used to determine the base shear is subject to an upper limit, which is based on the design spectral response acceleration at a period of one second, SD1. The relationship between SD1 and the maximum
The response modification factor, R serves the same function as the structural system coefficient in UBC–97. It reduces the design loads to account for the damping and ductility of the structural system. An abbreviated set for values for R is found in Table 517.
Table 517 Design Coefficients and Factors for Basic SeismicForceResisting Systems Basic SeismicForceResisting System Bearing Wall Systems
Response Modifications Coefficient, R
Deflection Amplication Factor, Cd
Special reinforced concrete shear walls
5.5
5
Ordinary reinforced concrete shear walls
4.5
4
Special steel concretrically braced frames
6
5
Special reinforced concrete shear walls
6
5
Special steel moment frames
8
5.5
Ordinary steel moment frames
4
3.5
Building Frame Systems
Moment Resisting Frame Systems
Dual Systems with Intermediate Moment Frames Capable of Resisting at Least 25% of Prescribed Seismic Forces Special reinforced concrete shear walls
6
5
5.5
4.5
Special steel moment frames
2.5
2.5
Ordinary steel moment frames
1.25
2.5
Ordinary reinforced concrete shear walls Inverted Pendulum Systems and Cantilevered Column Systems
268
Chapter 5
5.4.13
Vertical Distribution of Force FX
The seismic force at any level is a portion of the total base shear: Fx = Cvx V
(525)
where
C vx =
w x hx
k
n
∑w h
k
(526)
i i
where τ = 1.0 for the top 10 stories τ = 0.8 for the 20th story from the top and below and is interpolated between 0.8 and 1.0 for stories in between. Part of the reasoning behind this reduction is that the design story forces are an envelope of the maximums at each floor, and it is unlikely that they will all reach a maximum simultaneously.
i =1
5.4.15 where wi,wx= the portion of the dead load at or assigned to level i or x hi,hx= height above the base to level i or x k = an exponent related to the building period as follows: For buildings with a period of 0.5 seconds or less, k=1.0. If the period is 2.5 seconds or more, k=2.0. For buildings with a period between 0.5 and 2.5 seconds, it may be taken as 2.0 or determined by linear interpolation between 1.0 and 2.0. For k=1.0 the distribution is a straight line. This is reasonable for short buildings with a regular distribution of mass and stiffness. Hence, k=1.0 for buildings with a period of 0.5 seconds or less. For k=2.0 the distribution is a parabola with the vertex at the base. This is reasonable for tall regular buildings where the participation of higher modes is significant. Hence, k=2.0 for buildings with a period of 2.5 seconds or more. This effect is accounted for by the force Ft, placed at the roof in UBC97. 5.4.14
Overturning Moment Mx
n
i=x
For buildings, other than masonry, over four stories the allowable drifts are: ∆ ≤ ∆a=0.010 hsx ∆ ≤ ∆a=0.015 hsx ∆ ≤ ∆a=0.020 hsx
Use Group III Use Group II Use Group I
(528) (529) (530)
For buildings four stories or less and height, other than masonry, the allowable drifts are: ∆ ≤ ∆a=0.015 hsx Use Group III (528a) Use Group II (529a) ∆ ≤ ∆a=0.020 hsx ∆ ≤ ∆a=0.025 hsx Use Group I (530a) where ∆ = the design interstory displacement ∆a = the allowable story displacement hsx= the height of the story below level x The design interstory displacement ∆, is the difference in the deflections δx, at the top and bottom of the story under consideration. It is based on the calculated deflections and is evaluated by the formula:
δx =
C d δ xe I
(531)
where
IBC2000 allows for a reduction in the design overturning moment:
M x = τ ∑ Fi (hi − hx )
Drift Limitations
(527)
Cd = the deflection amplification factor δxe = the deflections determined by an elastic analysis. I =the occupancy importance factor
5. Linear Static Seismic Lateral Force Procedures The deflection amplification factor Cd is assigned values from 1.25 to 5.5 and accounts for the ductility of the system and the properties of the materials from which it is constructed (see Table 517). In determining these deflections the period determined by an analysis may be used to calculate the base shear without considering the limitation on the period discussed in Section 5.4.11. This has the implication that lower story forces may be used to determine deflections than are used to determine member forces. A similar provision is contained in UBC97. Where significant, PDelta and torsional deflections must be considered in satisfying the drift limitation. This is discussed further in the next section. 5.4.16
Torsion and PDelta Effect
Torsion is accounted for in same manner as in UBC97. The torsional moment resulting from the location of the center of mass plus that resulting from an assumed movement of five percent of the plan dimension must be accounted for.
269 For buildings with torsional irregularity, in seismic design categories C through F, the five percent accidental torsion must be amplified using Equation 511. For this purpose a building is irregular if the diaphragm is rigid and the maximum interstory displacement is more than 1.2 times the average. The PDelta effect must be included in the computation of story shears, story drifts and member forces when the value of the "stability coefficient" has a value, for any story, such that: θ = Px∆/VxhsxCd > 0.10
(532)
where ∆= the design story drift Vx= the seismic force acting between level x and x1 hsx = the story height below level x Px = total gravity load at and above level x Cd = the deflection amplification factor The stability coefficient can be visualized as the ratio of the PDelta moment (Px∆) to the lateral force story moment (Vxhsx). Hence if the
Table518. Plan Structural Irregularities 1a
1b
2
3
4 5
Irregularity Type and Description Torsional Irregularity—to be considered when diaphragms are not flexible Torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at one end of the structure transverse to an axis is more than 1.2 times the average of the story drifts at the two ends of the structure. Extreme Torsional Irregularity – to be considered when diaphragms are not flexible Extreme torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at one end of the structure transverse to an axis is more than 1.4 times the average of the story drifts at the two ends of the structure. Reentrant Corners Plan configurations of a structure and its lateral forceresisting system contain reentrant corners, where both projections of the structure beyond a reentrant corner are greater than 15 percent of the plan dimension of the structure in the given direction. Diaphragm Discontinuity Diaphragms with abrupt discontinuities or variations in stiffness, including those having cutout or open areas greater than 50 percent of the gross enclosed diaphragm area, or changes in effective diaphragm stiffness of more than 50 percent from one story to the next. OutofPlane Offsets Discontinuities in a lateral force resistance path, such as outofplane offsets of the vertical elements. Nonparallel Systems The vertical lateral forceresisting elements are not parallel to or symmetric about the major orthogonal axes of the lateral forceresisting system.
270
Chapter 5
Table 519. Vertical Structural Irregularities 1a
1b
2
3
4
5
Irregularity Type and Description Stiffness Irregularity—Soft Story A soft story is one in which the lateral stiffness is less than 70 percent of that in the story above or less than 80 percent of the average stiffness of the three stories above. Stiffness Irregularity—Extreme Soft Story An extreme soft story in one in which the lateral stiffness is less than 60 percent of that in the story above or less than 70 percent of the average stiffness of the three stories above. Weight (Mass) Irregularity Mass Irregularity shall be considered to exist where the effective mass of any story is more than 150 percent of the effective mass of an adjacent story. A roof that is lighter than the floor below need not be considered. Vertical Geometric Irregularity Vertical geometric irregularity shall be considered to exist where the horizontal dimension of the lateral forceresisting system in any story is more than 130 percent of that in an adjacent story. InPlane Discontinuity in Vertical Lateral Force Resisting Elements An inplane offset of the lateral forceresisting elements greater than the length of those elements or a reduction in stiffness of the resisting element in the story below. Discontinuity in Capacity—Weak Story A weak story is one in which the story lateral strength is less than 80 percent of that in the story above. The story strength is the total strength of all seismicresisting elements sharing the story shear for the direction under consideration.
PDelta moment is equal to 10 percent of the story moment at any floor the PDelta effect should be considered. The code also specifies an upper limit on the stability coefficient. 5.4.17
Irregularity
IBC2000 defines irregularity in a manner similar to UBC97, but goes further by assigning a building to a seismic design category based on its irregularity. It distinguishes between the two broad categories of plan and vertical irregularity. Plan irregularities include: a nonsymmetrical geometric configuration, reentrant corners, significant torsion due to eccentricity between mass and stiffness, nonparallel lateral force resisting elements, out of plane offsets and discontinuous diaphragms. Vertical irregularities include: soft and weak stories, large changes in massstiffness ratios between adjacent floors, large changes in plan dimension from floor to floor and significant horizontal offsets in the lateral load system. The definitions of plan and vertical structural irregularities and their assigned seismic design categories are found in Tables 518 and 519.
5.4.18
Reliability Factor ρ
The reliability factor ρ is identical to and serves the same function as in UBC97 (See section 5.3.14, Equation 512). It is assigned a value of 1.0 for seismic design categories A, B and C. For special moment resisting frames in Seismic Design Category D, ρ cannot exceed 1.25. For special moment resisting frames in Seismic Design Categories E and F, ρ cannot exceed 1.1. 5.4.19
Analysis Procedures
The minimum level of structural analysis is dependent on the seismic design category. For buildings in category A, the design lateral force at all floors is 1 % of gravity. Buildings in categories B and C, whether regular or irregular, may be analyzed using the equivalent lateral force procedure. The analysis procedure for buildings in categories D, E & F is specified as follows. Regular buildings up to 240 feet in height may be analyzed using the equivalent lateral force procedure. Buildings that are either over 240 feet tall, irregular, located on poor soils, or
5. Linear Static Seismic Lateral Force Procedures close to known faults in areas of high seismicity require various types of dynamic analysis. 5.4.20
Dynamic Analysis
Provisions are included for a simplified two dimensional version of modal analysis which is applicable to regular structures with independent orthogonal seismic force resisting systems. For such structures the motion is predominantly planar and a two dimensional model may be appropriate. For irregular structures or with interacting seismic force resisting systems a three dimensional model is required. The required base shear is equal to that determined by Equation 518, where the period used may be 20 percent longer than the maximum period allowed in the equivalent lateral force procedure (see Section 5.4.11). The justification for this is that a modal analysis is more accurate than a static analysis. Although the total force on the building does not change appreciably its distribution over the height is more accurately modeled. 5.4.21
271 SD1= 2 3 SM1 = 2 3 (.2) = .133g Eq. 517b seismic design cat. C Tables 515,16 R = 6 (special shear walls) Table 517 Cs=
SDS .333 = =.0555 R I 61
Equation 519
Ta =.02hn3/4=.02(35)3/4=.29 Sec
CS ≤
SD1 .133 = = .076 T(R I ) .29(6 1)
Example 53:
CS ≥ .044SD1 I = .044(.333)(1) = .0147 Eq.521 CS = .0555 V = .0555 (5900) = 327.5k • Vertical Distribution: Fx = CvxV
Cvx =
Equation 525
w x hx
k
n
∑w h
Equation 526
k
i i
i =1
Rework Example 51 using IBC2000 and special reinforced concrete shear walls.
• Overturning Moment:
• Base Shear:
M x = τ ∑ Fi (hi − hx ) n
Equation 527
i=x
seismic use group I I = 1.0 SS = .5g S1 = .2g site class B Fa = 1.0 Fv = 1.0 SMS = Fa SS = .5g SM1 = FvS1 = .2g SDS= 2 3 SMS = 2 3 (.5) = .333g
Eq.520
T < 0.5 sec k = 1.0 Since k=1, the procedure is identical to Example 51. See Table 520.
Examples
V = C SW
Eq.523
Equation 518 Section 5.4.1 Section 5.4.2 Figure 53 Figure 54 Table 55 Table 513 Table 514 Eq. 516a Eq. 516b Eq. 517a
τ = 1.0 for top ten stories Since τ=1, the procedure is the same as for Example 51. See Table 520. Table 520: Example 53 Level 3 2 1 Σ
hx (ft) 35 24 13
wx (k) 1700 2000 2200 5900
wxhx x103 59.5 48.0 28.6 136.1
Cvx (k) 0.44 0.35 0.21 1.00
Fx (k) 144.1 114.6 68.8 327.5
Vx (k) 144.1 258.7 327.5
Mx (ftk) 1585 4431 8688
• Allowable Inelastic Story Displacements:
272
Chapter 5
S1 ≥ .75 g
Tables 515,16; footnote a. seismic design category E R = 8 (special moment frame) Table 517
seismic use group I less than four stories
∆ a = 0.025h sx
Equation 530a Cs =
st
1 Floor: ∆ = 0.025(13)(12) = 3.9 inches 2nd and 3rd Floors: ∆ = 0.025(11)(12)= 3.3 inches
T = .035(117 ) 4 = 1.25 sec 3
• Equivalent Elastic Story Displacement:
δ=
Cd δ xe
Equation 531
I
Cd = 5
S DS 1.37 = = .171 R I 81
Cs ≤
Equation 523 Equation 520
C s ≥ .044 S DS I = .044(1.37 )(1) = .0603 Eq. 521
Table 517
Cs ≥
δ = 5 δ xe
S D1 .7 = = .07 TR I 1.25(8 1)
Equation 519
.5S1 .5(.81) = = .051 R I 81
Equation 522
Cs = .07g V = Cs W = .07 (15,300) = 1071 k
1st Floor: ∆ = ∆ a 5 = 3.9/5 = 0.78 inches 2nd and 3rd Floors: ∆ = ∆ a 5 = 3.3/5= 0.66 inches
• Vertical Distribution:
Cvx =
• Reliability Factor:
w x hx
k
n
∑w h
k
Equation 526
i i
seismic design category C ρ = 1.0
i =1
Section 5.4.18
Interpolate to find k:
Example 54: Rework Example 52 using IBC2000. • Base Shear:
V = C SW
Equation 518
seismic use group I Section 5.4.1 I = 1.0 Section 5.4.2 Ss = 2.05g Figure 53 Figure 54 S1 = 0.81g site class C Table 55 Fa = 1.0 Table 513 Table 514 Fv = 1.3 SMS = Fa Ss = 1.0 (2.05) = 2.05 Eq. 516a SM1 = FV S1 = 1.3 (.81) = 1.05 Eq. 516b Eq. 517a SDS= 2/3SMS= 2/3(2.05)= 1.37g SD1= 2/3 SM1= 2/3 (1.05) = 0.7g Eq. 517b
k=1.0+(1.25.5)/(2.5.5)=1.375 h91.375=1171.375=697.8 Cv9=1700(697.8)/1700(3000.9)=.233 F9=.233(1071)=250 k See Table 521. The story shear is determined by the same procedure as UBC97. • Overturning Moment:
M x = τ ∑ Fi (hi − hx ) n
i=x
Equation 527
τ = 1.0 for top ten stories Since τ=1.0 the procedure is the same as for UBC97. See Table 521.
5. Linear Static Seismic Lateral Force Procedures Table 521: Example 54 hx1.375
Cvx
Fx (k)
Vx (k)
Mx (ftk)
1700
697.8
.233
250
250
3250
104
1700
593.5
.198
212
462
9256
91
1700
493.9
.165
177
639
17563
6
78
1700
399.6
.133
142
781
27716
5
65
1700
311.0
.104
111
892
39312
4
52
1700
228.8
.076
81
973
51967
3
39
1700
155.1
.051
55
1028
65325
2
26
1700
88.2
.029
31
1059
79092
1
13
1700
35.0
.011
12
1071
90870
15300.
3000.9
1.0
1071
hx (ft)
wx
9
117
8 7
Level
Σ
(k)
• Allowable Inelastic Story Displacements: seismic use group I ∆a=.02hsx=.02(13)(12)=3.12 inches
Eq. 530
• Equivalent Elastic Story Displacements: Cd = 5.5 δ = Cdδxe/ I= 5.5 δxe ∆ ≤ 2.34/5.5 = 0.567 inches
Table 517 Equation 531
• Reliability Factor: The calculations are the same as for UBC97 (See example 52): ρ = 1.0 ρ = 1.12
Longitudinal Transverse
But in seismic design category E: ρmax = 1.1
Section 5.4.18
Therefore, we need more transverse bays. Note that ρ will be even higher using actual shears.
5.5
CONCLUSION
Basic linear static lateral force procedures of the 1997 UBC, the 1997 NEHRP, and the 2000 IBC codes were discussed. Numerical examples were provided to highlight practical applications of these procedures.
273
274
Chapter 5
Chapter 6 Architectural Considerations
Christopher Arnold FAIA, RIBA Building systems Development Inc.
Key words:
Configuration, Regular Configurations, Irregular Configurations, Proportion, Setbacks, Plan Density, Perimeter Resistance, Redundancy, Symmetry, Asymmetry, SoftStories, Weak Stories, Code Provisions, Plan Irregularities, Elevation Irregularities, Architectural Implications.
Abstract:
While the provision of earthquake resistance is accomplished through structural means, the architectural design, and the decisions that create it, play a major role in determining the building's seismic performance. The building architecture must permit as effective a seismic design as possible: at the same time the structure must permit the functional and aesthetic aims of the building to be realized. The three categories are: (1) the building configuration, (2) structurally restrictive detailed architectural design, and (3) Hazardous nonstructural components. This chapter discusses one other issue that bears on the architectural decisions that affect seismic performance: that of the methods by which mutual architectural and engineering seismic design decisions are made during the building design and construction process. This, in turn, leads to some consideration of the architect/engineer relationship as it affects the seismic design problem.
275
276
Chapter 6
6. Architectural Considerations
6.1
INTRODUCTION
While the provision of earthquake resistance is accomplished through structural means, the architectural design, and the decisions that create it, play a major role in determining the building's seismic performance. The building architecture must permit as effective a seismic design as possible: at the same time the structure must permit the functional and aesthetic aims of the building to be realized. The architectural design decisions that influence the building's seismic performance can be grouped into three categories. These categories are not exclusive, and each category of decision may influence the others, but it is useful to structure the decisions in this way because it clarifies the influences and their mutual interactions. The three categories are:
• The building configuration: This is defined as the size, shape and proportions of the threedimensional form of the building. The terms building concept, or conceptual design, are often also loosely used by architects to identify the configuration, although these terms also refer to architectural characteristics such as internal planning and building organization. Strictly speaking, configuration refers only to the geometrical properties of the building form.
• Structurally restrictive detailed architectural design: This refers to the architectural design of building details, such as columns or walls, that may affect the structural detailing in ways that are detrimental to good seismic design practice.
• Hazardous nonstructural components: The design of many nonstructural components is the architect's responsibility, and if inadequately designed against seismic forces, they may present a hazard to life. In addition, they may represent a major cause of property loss, and in the case of essential facilities or
277 other services, their damage may cause loss of building function. Engineering issues in the design of these components are dealt with in Chapter 14. This chapter discusses one other issue that bears on the architectural decisions that affect seismic performance: that of the methods by which mutual architectural and engineering seismic design decisions are made during the building design and construction process. This, in turn, leads to some consideration of the architect/engineer relationship as it affects the seismic design problem.
6.2
CONFIGURATION CHARACTERISTICS AND THEIR EFFECTS
6.2.1
Configuration Defined
For our purposes building configuration can be defined as building size and shape: the latter includes the characteristic of proportion. In addition, our definition includes the nature, size and location of the structural elements, because these are often determined by the architectural design of the building, and are a subject of mutual agreement between architect and engineer. This extended definition of configuration is necessary because of the interaction of these elements in determining the seismic performance of the building. In addition, architectural decisions may influence the nature, size and location of nonstructural components that may affect structural performance, either by altering the stiffness of structural members or changing the mass distribution in the building.. These elements are generally part of the initial concept of the building but they may be added later, when the building is in operation. This particularly applies to infill walls, which may have a dramatic effect on the effective height, stiffness, and load distribution of columns. In this chapter they are discussed later as separate
278
Chapter 6
issues, apart from their relationship to configuration. These include such elements as walls, columns, service cores, and staircases, and also the quantity and type of the exterior wall elements. 6.2.2
Origins and Determinants of Configuration
The building configuration, or concept, is influenced by three main factors:
• urban design, business and real estate issues. • planning and functional concerns. • image and style The selected configuration is the result of a decision process that balances these varying requirements and influences and, within a budget, resolves conflicts into an architectural concept. In very general terms three basic categories of architecture can be distinguished based on their main objective:
• Economical containers the "decorated shed": warehouses, industrial plants, some department stores and commercial buildings
• Problem/solving,
functional
facility

hospitals, residential,
educational,
laboratories,
• Prestigious and/or highstyle image corporate headquarters, some public buildings and university buildings, museums, entertainment, and some retail stores. These categories also bear some relationship to the architects, or firms, that design them, for there is much covert specialization in architecture. This can cause client confusion: when the client who wants an economical container goes to a prestige architect, or when the client with a difficult planning problem goes to the container architect. Building function and planning produce a demand for certain settings and kinds of space division, connected by a circulation pattern for the movement of people, supplies, and equipment. These demands ultimately lead to certain building arrangements, dimensions and determinants of configuration. Urban design and planning requirements may affect the exterior form of the building. A height limit may set a certain maximum height; the street pattern may, particularly in a dense urban situation, determine the plan shape of the building, at least for its lower floors. City
Figure 61. Setback regulations, New York
6. Architectural Considerations planning requirements sometimes dictate the need for open first floors, for vertical setbacks, or other characteristics of architectural form. Urban design includes issues such as zoning and planning regulations, which by defining setbacks, height limits and sunangle requirements often define the building envelope. For example, recent studies have argued convincingly that early skyscraper form was predominantly determined by local landuse patterns, municipal codes and zoning (Figure 61). For example, the striking differences in form between the skyscrapers of Chicago and New York were due to the imposition of a 130 feet height limit on the former, and no limits on the latter. Zoning laws in New York, in 1916, spawned the buildings with "weddingcake" setbacks, while a 1923 law in Chicago permitted a tower to rise above the old height limit, but restricted its total volume(61). Engineers can accept the problems of zoning and building function in determining configuration, because they fit into the engineer's rationalist concept of the world. It is the third influence, the need for the building to present an attractive, interesting, unique, or even sensational image to the outside observer, and often the occupants, that engineers feel the trouble begins. Here is where the irrational artist takes over, and the laws of physics and economy may be violated. It is important to understand the need for the architect sometimes to provide a distinctive image for the building. If this need did not exist the owner might go to an engineer or contractor to obtain a simple economical building, and indeed, many owners do so.. Up until the early years of the 20th. century for a Western architect the common acceptance required a historical style typically mediaeval or renaissance  even when totally new building types such as railroad stations or skyscrapers were conceived. In engineering and materials terms these traditional forms were all derived from masonry structure: the need to keep the blocks of masonry in compression, and the creation of devices such as arches and vaults, to
279 enable the masonry to achieve larger spans than were possible by using slabs of masonry as beams or lintels. These masonry determined forms survived well into the 20th. century, even when buildings were supported by concealed steel frames, and arches had become a structural anachronism. Moreover, the prevailing historical architectural styles preferred symmetricalness, and decreed that buildings should be massive at the base, with smaller openings, and their mass should decrease with the upper floors.
Figure 62. The International Style
The revolution in architectural aesthetics that began in the 1920's, and is often called the "International Style" was based on exploiting the forms that could be created by use of frame structures, combined with a desire to strip architecture of its decoration and adherence to historic styles The International Style in architecture was not alone in extoling the virtues of unadorned structure and absence of decoration in its glorification of the beauties of Euclidean geometry. The same thing was going on in the world of painting and sculpture, and these arts were being stripped of their traditional content in favor of simplicity, geometry, and new materials. As architects began to exploit the aesthetics of an architecture based on engineered frames, the seeds of seismic configuration problem were sown. Loadbearing masonry buildings were very limited in the extent to which configuration irregularities were possible: with short spans redundancy was always present: the
280 extensive use of walls, both in exteriors and interiors, meant that, even though the masonry was unreinforced, unit stresses were very low. Large cantilevers and setbacks were not possible. But with the steel or concrete frame all these limitations were unnecessary: the building structure could be unbelievably slender (because now the columns and beams were analyzed and sized by engineers), first floor walls could be omitted, so that the building seemed to float in space. Lightness and grace were sought, rather than ornamented mass. (Figure 62) Buildings could even cantilever out safely so that they could become larger as they rose: the inverted pyramid could be built. These possibilities were eagerly explored by a new generation of architects: with them came other ideas: the rejection of symmetricalness of plan in favor of a more exciting and more rational disposition of elements (rational because the building elements were allowed to occur where planning function was most efficient, instead of being forced into [sometimes] inefficient symmetry). Examples of the International Style were limited to a few avantgarde buildings in all countries before World War 2, and then bloomed in the rich economic years that began in the 50's. The United States , Western Europe, Latin America, the Soviet Union and Japan exploded in a fury of development, almost all constructed in their regional versions of the International Style. These years of intensive development saw the world's cities grow into huge metropolises: they were also years in which seismic design as it related to the new, spare, framed buildings was inadequately understood, and it took earthquakes in Latin America, Mexico and the United States (in Alaska, 1964, and San Fernando, 1971) to make engineers realize that such buildings were unforgiving and intolerant of the very irregularities that architects had embraced with such enthusiasm. This architecture of the 50's to the 70's has left us with a legacy of poor seismic configurations that present a serious problem in
Chapter 6 reducing the earthquake threat to our cities. The problem is exacerbated when it is allied to the engineering design problem of the use of the nonductile reinforced concrete frame structure, which was the norm up to about 1975. This historical discourse is relevant to seismic design, because it shows that:
• the minimalist structural frame provided the basis for an architectural aesthetic which was in tune with the spirit of the age, aesthetically, economically and politically. • what we now call discontinuities and irregularities were critical elements of the new architectural aesthetic. • these elements were made possible by the use of the engineered structural frame, and by a new level of architect/engineer collaboration. It is, however, worth mentioning, that the new style originated , was promoted and developed in Western Europe, predominantly France and Germany, which, of course, are essentially non seismic zones. A more complete discussion of the origins and influence of the International Style will be found in Reference 62. 6.2.3
Configuration Influences in General
Configuration largely determines the ways in which seismic forces are distributed throughout the building, and also influences the relative magnitude of those forces. For a given ground motion, the major determinant of the total inertial force in the building is , following Newton's Second Law of Motion, the building mass (approximated on the earth's surface by its weight). While the size and shape of the building (together with the choice of materials), establish its weight the building square footage and volume are determined by the building program (and the budget) : the listing of required spaces and the activities and equipment that they contain. But for any given program an almost infinite variety of
6. Architectural Considerations configurations can provide a solution, and it is the variables in these configurations that affect the distribution of inertial forces due to ground shaking . Thus the discussion of configuration influence on seismic performance becomes the identification of configuration variables that affect the distribution of forces. These variables represent irregularities, or deviations from a "regular" configuration that is an optimum, or ideal, with respect to dealing with lateral forces. 6.2.4
281 64). For convenience, the building is arbitrarily shown as three stories: a one story building might be better seismically, all other things being equal, but with a multistory building we can show some necessary attributes of such a building.
The Optimum Seismic Configuration.
It is easiest to define a regular building by providing an example: the design discussed below represents an essentially perfectly regular building, which in turn represents an "optimum" seismic design. Its characteristics are such that deviations from the design progressively detract from its intrinsic seismic capabilities: these deviations result in "irregularities" and a familiar list of configuration irregularities can be identified. The discussion of these irregularities from an engineering and architectural viewpoint form the main body of this section.. Architecture implies occupancy: thus a solid block of concrete, which might be an optimum seismic design, is sculpture, not architecture. The great pyramid of Gizeh is architecture, and certainly approaches an optimum seismic design, but architecturally it is very uneconomic in its use of space and volume in housing only two small rooms within an enormous volume of unreinforced masonry (Figure 63). Our optimal seismic design is compromised by the need also to be reasonably optimal architecturally that is, in its ability to be a functional and economically viable architectural concept. Our design shows the three basic ways of achieving seismic resistance, and these are also part of the optimization, so the building is seismically optimized architecturally, in its configuration, and also demonstrates the best arrangement of its seismic resisting elements, in complete harmony with the architecture (Figure
Figure 63. The great pyramid of Gizeh
Considered purely as architecture this little building is quite acceptable, and would be simple and economical to construct. It is also a prototypical International Style building. Depending on its exterior treatment  its materials, and the care and refinement with which they are disposed it could range from a very economical functional building to an elegant architectural jewel; it is not complete, architecturally, of course, because stairs, elevators etc. must be added, and the building is not spatially interesting , although its interior could be configured with nonstructural components to provide almost any quality of room that was desired with the exception of interesting and/or unusual spatial volumes more than one story in height. What are the characteristics of this design that make it regular, and also make it so good considering only architectural configuration and the disposition of the seismic resisting elements? Any engineer will recognize them, but it is worth while listing them, because they are specific attributes whose existence or absence thereof can be quickly ascertained in any actual design. These attributes, and their effects, are:
282
Chapter 6
Figure 64. The optimal seismic design
• Low heightto base ratio • • • • • • • •
¾ Minimizes tendency to overturn Equal floor heights ¾ Equalizes column/wall stiffness Symmetrical plan shape ¾ Reduces torsion Identical resistance on both axes ¾ Balanced resistance in all directions Uniform section and elevations ¾ Eliminates stress concentrations Maximum torsional resistance ¾ Seismic resisting elements at perimeter Short spans ¾ Low unit stress in members Redundancy ¾ Toleration of failure of some members Direct load paths, no cantilevers ¾ No stress concentrations
6.3
METHODS OF ANALYSIS
6.3.1
Methods of Analysis and the Regular Building
An important aspect of a building's response to ground motion is the method of analysis used to establish the seismic forces. The estimate of total forces and their distribution is both a function of and a determinant of the lateral forceresisting system employed in the building. The great majority of designs estimate lateral forces through use of the static equivalent lateral force method (ELF) established in typical seismic codes , which involve estimating a base shear and then distributing the resulting forces through the structural elements of the building. It is
6. Architectural Considerations important to recognize that the forces derived from an equivalent force method used according to a typical seismic code and many other code provisions, assume a regular building, comparable to our ideal form described above. This assumption is noted in the Commentary to the 1997 NEHRP Recommended Provisions for Seismic Regulations for New Buildings(63): "The Provisions were basically derived for buildings having regular configurations. Past earthquakes have repeatedly shown that buildings having irregular configurations suffer greater damage than buildings having regular configurations. This situation prevails even with good design and construction" The Commentary to the 1990 Recommended Lateral Force Requirements of the Structural Engineers Association of California (Ref.64), discusses the design basis for regular buildings in some detail. Two important concepts apply for regular structures. First, the linearly varying lateral force distribution given by the ELF formulas are a reasonable and conservative representation of the actual response force distribution due to earthquake ground motions. Second, when the design of the elements in the lateral force resisting system is governed by the specified seismic load combinations, the cyclic inelastic deformation demands will be reasonably uniform in all elements, without large concentrations in any part of the system. The acceptable level of inelastic deformation demand for the system is therefore reasonably represented by the Rw value for the system. However, "when a structure has irregularities, then these concepts, assumptions and approximations may not be reasonable or valid, and corrective design factors and procedures are necessary to meet the design objectives". It is safe to say, based on studies of building inventories, that over half the buildings that have been designed in the last few decades do not conform to the simple uniform building configuration upon which the code is based. For new designs, the simple equivalent lateral force
283 analysis of the code must often be augmented by engineering judgment based on experience. Progressive evolution of seismic codes has resulted in increasing force levels and the consideration of additional parameters in estimating force levels, but the impact of configuration irregularity, which was first introduced into the Uniform Building Code in 1973, long remained a matter of judgment. However, starting in 1988 the UBC quantified some configuration parameters, to establish the condition of regularity or irregularity, and laid down some specific analytical requirements for irregular structures. 6.3.2
Irregular Configurations: Code Definitions and Methods of Analysis
In the Commentary to the 1980 SEAOC Recommended Lateral Force Requirements and Commentary(65), over 20 types of "irregular structures or framing systems" were noted as examples of designs that should involve extra analysis and dynamic consideration rather than use of the normal equivalent lateral force method. These types are illustrated in Figure 65, which is a graphical interpretation of the SEAOC list. Scrutiny of these conditions shows that the majority of irregularities are configurational issues within the terms of our definition. This list of irregularities defined the conditions, but provided no quantitative basis for establishing the relative significance of a given irregularity. These irregularities vary in the importance of their effects, and their influence also varies in accord with the particular geometry or dimensional basis of the condition. Thus, while in an extreme form the reentrant corner is a serious type of plan irregularity, in a lesser form it may have little significance (Figure 66). The determination of the point at which a given irregularity becomes serious is a matter of judgment.
Figure 65. Graphic interpretation of "Irregular Structures or Framing Systems" from the commentary to the "SEAOC Recommended Lateral Force Requirements and Commentary" (a) Buildings with Irregular Configuration (b) Buildings with abrupt changes in lateral resistance (c) Buildings with abrupt changes in lateral stiffness (d) Unusual or novel structural features.
284
6. Architectural Considerations The SEAOC Commentary explained the difficulty of going beyond this basic listing as follows: Due to the infinite variation of irregularities (in configuration) that can exist, the impracticality of establishing definite parameters and rational rules for the application of this Section are readily apparent. However, in the most recent version of the SEAOC Requirements and Commentary, and starting in the 1988 revisions to the Uniform Building Code, (which is based on the SEAOC document), an attempt has been made to quantify some critical irregularities, and to define geometrically or by use of dimensional ratios the points at which the specific irregularity becomes an issue of such concern that remedial measures must be taken.
Figure 66. The reentrant corner plan : a range of significance
The code approach to reducing the detrimental effect of irregularity is to require more advanced methods of analysis where such conditions occur  more specifically, where the ELF analysis method must be augmented or cannot be used. While this may provide a more accurate diagnosis, and in some instances strengthening of certain members, it does not correct the condition: this must still be done by design means based on understanding of the effects of the condition on building response. The code requirements relating to the definition of regularity and irregularity, and the determination of the analysis methods required have now become complex, and for design purposes the relevant sections of the applicable code should be referred to. The outline that follows focuses on identifying the irregular conditions for which the ELF method can be
285 used, must be augmented or where a more complex method is necessary. The irregularity type references are to the 1997 NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings as illustrated in Figure 67. This figure is a graphic interpretation of Table 5.2.3.1 and Table 5.2.3.2 in the Provisions. The terminology and configuration requirements in the UBC and the NEHRP Provisions are essential similar. The ELF method can be used for the following irregular structural types, with the noted augmentations: 1. All structures in Seismic Design Category A (in the NEHRP Provisions the Seismic Design Category is a classification assigned to a structure based on its seismic use group, or occupancy, and the severity of the design earthquake ground motion at the site). 2. Structures with reentrant corners ( plan irregularity type 2), diaphragm discontinuity (type 3) outofplane offsets (type 4) , in Seismic Design Categories D, E and F, must provide for an increase in design forces of 25% for connection of diaphragms to vertical elements and to collectors, and connection of collectors to vertical elements. 3. Structures with nonparallel systems (plan irregularity type 5) in Seismic Design Category C,D,E and F, must be analyzed for seismic forces applied in the critical direction, or satisfy the following combination of loads: 100% of forces in one direction plus 30% of the forces in the perpendicular direction. 4. Structures with outofplane offsets (plan irregularity type 4) and inplane discontinuity in vertical lateral force resisting elements (vertical irregularity type 4) must have the design strength to resist the maximum axial forces that can develop in accordance with specially defined load combinations.
Figure 67. Irregularities defined in the 1997 NEHRP Provisions
286
6. Architectural Considerations
Other buildings with plan or vertical irregularities as defined in the Tables, that are not required to use modal analysis as identified below, may use the ELF procedure with "dynamic characteristics given special consideration" : the engineer must use judgment in computing forces. Buildings with certain types of vertical irregularity may be analyzed as regular buildings in accordance with normal ELF procedures. These buildings are generally referred to as setback buildings. The following procedure may be used: 1. The base and lower portions of a building having a setback vertical configuration may be analyzed as indicated in (2) below if all of the following conditions are met: a.The base portion and the tower portion , considered as separate buildings, can be classified as regular and. b.The stiffness of the top story of the base is at least five times that of the first story of the tower. Where these conditions are not met, the building shall be analyzed using modal analysis. 2. The base and tower portions of the building may be analyzed as separate buildings in accordance with the following: a.The tower may be analyzed in accordance with the usual ELF procedure with the base taken at the top of the base portion. b.The base portion then must be analyzed in accordance with the ELF procedure using the height of the base portion of hn and with the gravity load and base shear of seismic forces the tower portion acting at the top level of the base portion.
287 Modal Analysis is required in the following instances: 1. Buildings which are in Seismic Design Category D, E or F, are over 65 feet in height, and have: soft stories (vertical irregularity type 1a) extreme soft stories (vertical irregularity type 1b) mass irregularities (vertical irregularity type 2) vertical geometrical irregularity (vertical irregularity type 3) Exceptions: vertical structural irregularities of types 1a, 1b or 2 do not apply where no story drift ratio under design lateral load is greater than 130 percent of the story drift ratio of the next story above 2 Buildings , with torsional irregularity (plan irregularity type 1a) in Seismic Design Category D, E or F and extreme torsional irregularity ((plan irregularity type 1b) in Seismic Design Category D. In addition an increase in design forces of 25% is required for connection of diaphragms to vertical elements and to collectors, and connection of collectors to vertical elements, and a torsion amplification factor. 3. All structures over 240 feet in height. The following irregular structures are not permitted: Weak story structures (vertical irregularity type 5) over 2 floors or 30 feet in height with a weak story less than 65% of the strength of the story above, in Seismic Design Categories, B, C, D, E and F. Extreme soft story structures (vertical irregularity type 1b) and extreme torsional irregularity structures (plan irregularity
288 type 1b) in Seismic Design categories E and F. The Commentary to the NEHRP Provisions also provides a procedure which may reduce the need to perform modal analysis. "The procedures defined in the Provisions include a simplified modal analysis which takes account of irregularity in mass and stiffness distribution over the height of the building. It would be adequate, in general, to use the ELF procedure for buildings whose seismic resisting system has the same configuration in all stories and all floors, and whose floor masses and cross sectional areas and moments of inertia of structural members do not differ by more than 30% in adjacent floors and in adjacent stories. For other buildings, the following criteria should be applied to decide whether modal analysis procedures should be used: 1. The story shears should be computed using the ELF procedure. 2. On this basis, approximately dimension the structural members, and then compute the lateral displacement of the floors. 3.
Replace the hxk term in the vertical distribution of seismic forces equation with these displacements and recompute the lateral forces to obtain new story shears.
4. If at any story the recomputed story shear differs from the corresponding value as obtained from the normal ELF procedure by more than 30%, the building should be analyzed using the modal analysis procedure. If the difference is less than this value, the building may be designed for the story shear obtained in the application of the present criterion and the modal analysis procedures are not required."
Chapter 6 This procedure greatly reduces the likelihood that the considerably more complex modal analysis procedure will be required for the building analysis: this is of major importance because building irregularity is quite likely to be present in buildings of modest size and tight budget, and costly analysis procedures are not welcome to the owner. In addition, the 1997 NEHRP Provisions make further predominantly nonquantitative comments about the use of the Equivalent Lateral Force procedure for irregular buildings: "The ELF procedure is likely to be inadequate in the following cases: 1. Buildings with irregular mass and stiffness properties in which case the simple formulas for vertical distribution of lateral forces may lead to erroneous results: 2. Buildings (regular or irregular) in which the lateral motions in two orthogonal directions and the torsional motions are strongly coupled, and 3. Buildings with irregular distribution of story strengths leading to possible concentration of ductility demand in a few stories of the building. In such cases, a more rigorous procedure which considers the dynamic behavior of the structure should be employed. The Provisions Commentary points out that the ELF procedure, and both versions of the modal analysis procedure (a simple version and a general version with several degrees of freedom per floor which are described in the Provisions) are all likely to err systematically on the unsafe side if story strengths are distributed irregularly over height. This points to the importance of eliminating such irregularities if possible, but often they will be present because of detailed architectural requirements: if they cannot be eliminated, the engineer must use his judgment to assess their effects on the analysis
6. Architectural Considerations Even if the modal analysis procedure is used there are limitations to the information that the analysis provides. The procedure adequately addresses vertical irregularities of stiffness, mass or geometry. Other irregularities must be carefully considered on a judgmental basis, and so the engineer must rely on his experiential and conceptual knowledge of the building's response in order to effectively accommodate all irregularities.
6.4
GENERAL BUILDING CHARACTERISTICS
6.4.1
Introduction
These are issues relating to the building configuration as a whole and apply to all configurations. Irregularity as defined in current seismic codes , and as discussed above, covers the majority of configuration variables that have a significant effect on the seismic performance of the building. Although definitions vary, there is general agreement on those configuration irregularities that are important. However, the code listing is not complete: issues of building proportion and size are not included, nor are issues such as the building plan density or its redundancy the subject of code provisions, although the latter is briefly mentioned.. These are discussed below. The problem of pounding, which combines the issue of drift with that of building adjacency, and as such may present an architectural problem, is discussed in Section 6.9 below. 6.4.2
Size, Proportion and Symmetry
• Building size: It is possible to introduce configuration irregularities into a wood frame house that would be serious problems in a large building, and yet produce a safe structure with the inclusion of relatively inexpensive and unobtrusive provisions. This is because a small
289 wood frame structure is light in weight and inertial forces will be low. In addition, spans are short and relative to the floor area, there will probably be a large number of walls to share the loads. For a larger building, the violation of basic layout and proportion principles exacts an increasingly severe cost, and as the forces become greater, good performance cannot be relied upon as in an equivalent building of better configuration. As the absolute size of a structure increases, the number of alternatives for the arrangement of its structure decreases. A bridge span of 300ft. may be built as a beam, arch, truss, or suspension system, but a span of 3000 ft. can only be designed as a suspension structure. And as the size increases the structural discipline becomes more rigorous: architectural flourishes that are perfectly acceptable at the size of a house become physically impossible at the size of a suspension bridge.(Figure 68).
Figure 68. The designer's suspension bridge
In looking at the influence of building size on seismic performance, the influence of both the dynamic environment and the characteristics of ground motion result in more complexity than does the influence of size on vertical forces. Increasing the height of a building may seem equivalent to increasing the span of a cantilever beam, and so it is (all other things being equal). The problem with the analogy is that as a building grows taller its period will tend to increase, and a change in period means a change in the building response. The effect of the building period must be considered in relation to the period of ground motion, and if amplification occurs, the effect of an increase in height may be quite disproportionate to the increase itself. Thus
290 doubling the building height from 6 to 10 stories may, if amplification occurs, result in a four or fivefold increase in seismic forces. The earthquake in Mexico City in 1985 resulted in major response and amplification in buildings in the 6 to 20 story range, with generally reduced response in wellbuilt buildings below and above these heights. Although a 100ft. height limit throughout Japan was enforced until 1964, a 150ft 13 story limit was the maximum in Los Angeles until 1957, and the limit was 80 ft and later 100 ft on San Francisco, height is rarely singled out as a variable to be used to reduce the building response. Two recent exceptions to this may be noted. After the Armenian earthquake of 1988, planners of the reconstruction of the city of Leninakan limited the height of new buildings to three stories, because of the ground conditions and the bad experience with taller buildings. This decision is especially interesting because it required a major shift in planning and architectural thinking: prior to this, almost all Sovietstyle housing consisted of medium to highrise blocks. After the Mexico City earthquake of 1985 a number of damaged buildings were "topped" as part of the repair strategy: a number of floors were removed, thus changing the building period to something less in tune with the long period ground motions that the city experiences. The present approach is generally not to legislate seismic height limits (except insofar as seismic codes impose height limits relating to types of construction), but to enforce more specific seismic design and performance criteria. Generally, urban design, realestate or programmatic factors will be more significant, and earthquake performance must be engineered with the height predetermined by these factors. It is easy to visualize the overturning forces associated with height as a seismic problem (although the issue is more that of the aspect ratio of shear walls rather than the building as a whole), but large plan areas can be detrimental also. When the plan becomes extremely large, even if it is symmetrical and of simple shape,
Chapter 6 the building can have trouble responding as one unit to the ground motion. Unless there are numerous interior lateralforce resisting elements, largeplan buildings impose unusually severe requirements on their diaphragms, which have large lateral spans, and can build up large forces to be resisted by shear walls or frames. The solution is to add walls or frames to reduce the span of the diaphragm, although it is recognized that this may introduce problems in the use of the building. In a very large building, seismic separations may be necessary to subdivide the building and keep the diaphragm forces within bounds, in which case the seismic separations may also act as thermal expansion joints. An interesting example of a correct "intuitive" response to this problem is that of the design of the Imperial Hotel, Tokyo, by the architect Frank Lloyd Wright in the early 1920s. He subdivided this large complex building, with long wings and many reentrant corners, into small regular boxes, each about 35 ft. by 60 ft in plan. In doing this, he appears to have been concerned about the possibility of differential settlement caused by a travelling wave on the site. In the use of this concept, to which he attributed in large measure the success of the building in surviving the 1923 Kanto earthquake, Wright was well ahead of his time. The shortpile foundation scheme, which Wright claimed as a major invention, probably had much less to do with the building's good performance(66).
• Building Proportion In seismic design, the proportions of a building may be more important than its absolute size. For tall buildings, the slenderness ratio (height/least depth) of a building, calculated in the same way as for an individual member, is a more important consideration than just height alone. Dowrick(67) suggests attempting to limit the height/depth ratio to 3 or 4, explaining:
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"The more slender a building the worse the overturning effects of an earthquake and the greater the earthquake stresses in the outer columns, particularly the overturning compressive forces which can be very difficult to deal with." As urban land becomes more expensive, there is a trend towards designing very slender "sliver" buildings which, although not necessarily very high, may have a large height/depth ratio. Nowhere is this trend more apparent than in Japanese cities, where multistory buildings may be built on sites that are of the order of 15 to 20 ft wide (Figure 69). However, the same economic forces often dictate that these buildings will be built very close together, so that they will tend to respond as a unit rather than as individual freestanding buildings, although more recent Japanese
buildings have incorporated relatively large separations to reduce the risk of pounding.
• Building Symmetry The term symmetry denotes a geometrical property of building plan configuration. Structural symmetry means that the center of mass and center of resistance are located at, or close to, the same point (unless live loads affect the actual center of mass). The single admonition that appears in all codes and in textbooks that discuss configuration is that symmetrical forms are preferred to asymmetrical ones. The two basic reasons are that eccentricity between the centers of mass and resistance will produce torsion and stress concentrations. However, a building with reentrant corners is not necessarily asymmetrical (a cruciform
Figure 69. Slender buildings, Tokyo, Japan
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Figure 610. False symmetry: offset structural core
building may be symmetrical) but it is irregular, as defined, for example, in current seismic codes. Thus symmetry is not sufficient on its own, and only when it is combined with simplicity is it beneficial. Nevertheless, it is true that as the building becomes more symmetrical, its tendency to suffer torsion and stress concentration will reduce, and performance under seismic forces will tend to be less difficult to analyze. This suggests that when good seismic performance must be achieved with maximum economy of design and construction, the symmetrical, simple shapes are much to be preferred. But these tendencies must not be mistaken for an axiom that a symmetrical building will not suffer torsion. The effects of symmetry refer not only to the overall building shape, but to its details of design and construction. Study of building performance in past earthquakes indicates that performance is sensitive to quite small variations in symmetry within the overall form.. This is particularly true in relation to shearwall design and where service cores are designed to act as major lateral resistant elements. It is possible to have a building which is geometrically symmetrical in exterior form, but highly asymmetrical in the arrangement of its structural systems. The most common form of this condition (sometimes termed "false symmetry") is the building with interior structural cores that, for planning reasons, are unsymmetrically arranged. This can be a major
source of undesirable torsional response. (Figure 610) Experience in the Mexico City earthquake of 1985 showed that many buildings that were symmetrical and simple in overall plan suffered severely because of asymmetrical location of service cores and escape staircases. Moreover, as soon as a structure begins to suffer damage (cracking in shear walls or columns, for example), its distribution of resistance elements changes, so that even the most symmetrical of structures becomes dynamically asymmetrical and subject to torsional forces. Finally, it must be recognized that architectural requirements will often make the symmetrical design impossible. In these circumstances, it may be necessary, depending on the size of the building and the type of asymmetry, to subdivide the building into simple elements. There is a tendency, as noted above, for the very tall building to tend towards symmetry and simplicity. The seismic problems are most apparent in the low to mediumheight building, where considerable choice exists as to plan form and the disposition of the major masses of the building. 6.4.3
Plan Density, Perimeter Resistance, and Redundancy
The size and density of structural elements in the buildings of former centuries is strikingly greater than in today's buildings. Structural
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Figure 611. Structural plan density
technology has allowed us to push this trend continually further. Earthquake forces are generally greater at the base of the building. The bottom story is required to carry its own lateral load in addition to the shear forces of all the stories above, which is analogous to the downward buildup of vertical gravity loads. At this same lowest level, programmatic and aesthetic criteria are often imposed on the building that demand the removal of as much solid material as possible. This requirement is the opposite of the most efficient seismic configuration, which would provide the greatest intensity of vertical resistant elements at the base, where they are most needed. An interesting statistical measure in this regard is the ground level vertical plan density, defined as the total area of all vertical structural elements divided by the gross floor area. The most striking characteristic of the modern framed building is the tremendous reduction of structural plan density compared to historic buildings. For instance, a typical 10 to 20 story, moment resistant steel frame building will
touch the ground with its columns over 1% or less of its plan area, and combined frame shearwall designs will typically reach structural plan densities of only 2%. The densely filledin "footprints" of buildings of previous eras present a striking contrast: the structural plan density can go as high as 50%, in the case of the Tag Mail: the ratio for St. Peter's in Rome is about 25%, and for Chartres Cathedral 15%. The 16story Monadnock Building in Chicago, which used exterior bearing walls of brick 6 ft. thick at the ground level, has a ratio of 15% (Figure 611). Analogous to structural plan density is the measure of the extent of walls in a structure. Surveys of damaged buildings in Japan and Turkey have indicated a clear relationship between the length of walls in a boxtype system building and the extent of damage. This relationship has been incorporated in the seismic codes of these and other countries to provide prescriptive guidance for the design of simple structures. In Figure 612, although both configurations are symmetrical and contain the same amount of shear wall, the location of walls is
294 significantly different. The walls on the right form greater lever arms for resisting overturning and torsional moments. In resisting torsion, with the center of twist of a symmetrical building located at or near the geometrical center, the further the resisting material is placed from the center, the greater the lever arm through which it acts, and hence the greater the resisting moment that can be generated. Placing resisting members on the perimeter whenever possible is always desirable, whether the members are walls, frames, or braced frames, and whether they have to resist direct lateral forces, torsion, or both.
Figure 612. Location of lateral resistance elements
The design characteristic of redundancy plays an important role in seismic performance, and is significant in several aspects, most especially because the redundant design will almost certainly offer direct load paths and in this it tends to result in higher plan density as discussed above. In addition, historic buildings tended to be highly redundant, because short spans required many points of support, and thus each supporting member incurs much lower stresses, often even within the capability of unreinforced masonry. Thus, the very limitations of traditional materials forced the designers into good design practices such as redundancy, direct load paths and high plan density. The detailing of connections is often cited as a key factor in seismic performance, since the more integrated and interconnected a structure is, the more load distribution possibilities there are.
Chapter 6
6.5
SEISMIC SIGNIFICANCE OF TYPICAL CONFIGURATION IRREGULARITIES
6.5.1
Introduction
The discussion of configuration issues that follows incorporates all the codedefined issues but, in going back to our original definition of configuration, categorizes configuration problems in ways that relates the seismic implications to those of their architectural origins as decisions made at the conceptual stages of the design. For each configuration issue, five issues are outlined: definition of the condition, its seismic effects, its architectural implications, historical performance in past earthquakes, and solutions. The notes on architectural effects discuss the origin and purpose of the condition in architectural terms: the discussion of solutions deals with conceptual design approaches, and is most relevant for the consideration of existing buildings.
6.6
PLAN CONFIGURATION PROBLEMS
6.6.1
Reentrant Corners
• Definition The reentrant , or "inside" corner is the common characteristic of overall building configurations that, in plan, assume the shape of an L, T, H, +, or combination of these shapes.
• Seismic Effects There are two related problems created by these shapes. The first is that they tend to produce variations of rigidity, and hence differential motions, between different parts of the building, resulting in a local stress concentration at the "notch" of the reentrant
6. Architectural Considerations corner. In Figure 613, if the ground motion occurs with a northsouth emphasis at the Lshaped building shown, the wing oriented northsouth will, for geometrical reasons, tend to be stiffer than the wing oriented eastwest. The northsouth wing, if it were a separate building, would tend to deflect less than the eastwest wing, but the two wings are tied together and attempt to move differentially at their notch, pulling and pushing each other.(Figure 614). For ground motions along the other axis, the wings reverse roles, but the differential problem remains.
Figure 613. Separated buildings
295 The result is rotation, which tends to distort the form in ways that will vary in nature and magnitude depending on the nature and direction of the ground motion, and result in forces that are very difficult to analyze and predict. The stress concentration at the notch and the torsional effects are interrelated. The magnitude of the forces and the seriousness of the problem will be dependent on:
• • • •
the mass of the building the structural systems the length of the wings and their aspect ratios the height of the wings and their height/depth ratios
In addition, it is not uncommon for wings of a reentrant corner building to be of different height, so that the vertical discontinuity of a setback in elevation is combined with the horizontal discontinuity of the reentrant corner, resulting in an even more serious problem. The reentrant corner is perhaps the major irregularity that will be found in older buildings, including unreinforced masonry. In addition, in such buildings it is rare to find seismic separations at the intersections of the wings, so the prospects for torsion and stress concentration are high, when the wings are long and tall.
• Architectural Implications
Figure 614. The Lshaped building
The second problem is torsion. This is because the center of mass and center of rigidity in this form cannot geometrically coincide for all possible earthquake directions.
Reentrant corners create a useful set of building shapes, enabling large plan areas to be accommodated in compact form, while still providing a high percentage of perimeter rooms with access to light and air. Thus such configurations are common for highdensity housing and hotel projects, in which habitable rooms must be provided with windows. Concerns for daylighting and natural ventilation that were prevalent during the energy crisis of the 1970's resulted in something of a revival of interest in the increased use of narrow buildings and the traditional set of reentrant corner
296 configurations. The courtyard form, most appropriate for hotels and apartment houses in tight urban sites, has always remained useful. In its contemporary form the courtyard often becomes a glasscovered atrium, but the structural form is the same.
• Historical Performance Examples of damage to reentrant corner buildings are common, and this problem was one of the first to be identified by observers. It had been identified before the turn of the century, and by the 1920s was generally acknowledged by the experts of the day. Naito (68) attributed significant damage in the 1923 Kanto earthquake to this factor. The same damage phenomena were reported for the 1925 Santa Barbara and 1964 Alaska earthquakes (Figure 615), and for the 1985 Mexico City earthquake Large wood frame apartment houses with many reentrant corners are common in Los Angeles and suffered badly in the Northridge earthquake of 1994.
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• Solutions There are two basic alternative solutions to this problem: to separate the building structurally into simple shapes, or to tie the building together strongly at lines of stress concentration and locate resistance elements to reduce torsion. If a decision is made to use separation joints, they must be designed and constructed correctly to achieve the intent. Structurally separated entities of a building must be fully capable of resisting vertical and lateral forces on their own. To design a separation joint, the maximum drift (or some reasonable criterion) of the two units must be calculated by the structural engineer. The worst case is when the two units would lean towards one another simultaneously, and hence the dimension of the separation space must allow for the sum of the deflections. In a tall building the relative motion between portions of the building will become very large, and create major problems of architectural detailing.
Figure 615. Damage concentrated at the intersection of two wings of an Lshaped school, Anchorage, Alaska, 1964
6. Architectural Considerations One of these is to preserve integrity against fire and smoke spread. The MGM Grand Hotel in Las Vegas is a Tshaped building in plan, with seismic joints approximately 12 in. in dimension. In the fire of 1983 these joints allowed smoke to propagate to the upper floors, resulting in many deaths. Several considerations arise if it is decided to dispense with separation joints and tie the building together. Collectors at the intersection can transfer forces across the intersection areas, but only if the design allows for these beam like members to extend straight across without interruption. Walls in this same location are even more efficient than collectors. (Figure 616).
297 Since the free end of the wing tends to distort most under tension, it is desirable to place resisting members at this location. The use of splayed rather than rightangle reentrant corners lessens the stress concentration at the notch, which is analogous to the way a rounded hole in a steel beam creates less stress concentration problems than a rectangular hole, or the way a tapered cantilever beam is more desirable than one that is abruptly notched (Figure 617). 6.6.2
Variations in Perimeter Strength and Stiffness
• Definition This section discusses the detrimental effects of wide variations in strength and stiffness in building elements that provide seismic resistance and are located on the building perimeter
• Seismic Effects If arranged to provide balanced resistance perimeter resistance elements are particularly effective in reducing torsional effects because of their long lever arm relative to the center of resistance. If the resistance is not balanced, the detrimental effects can be extreme. Figure 617. Solutions to the Lshaped building
Figure 616. Splay in plan relieves reentrant corner problem: analogies to beam
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This problem may occur in buildings whose configuration is geometrically symmetrical and simple, but nonetheless irregular for seismic design purposes. If there is wide variation in strength and stiffness around the perimeter, the centers of mass and resistance will not coincide, and torsional forces will tend to cause the building to rotate around the center of resistance. This effect is illustrated in Figure 618.
of the openfront building. The weaknesses of openfront designs have been discussed by Degenkolb(69): Figure 619 shows the plans of three similar buildings , each with three shear walls so arranged that there is an open end and therefore major torsions in the building. If the buildings are similar, with uniform shear elements (uniform distribution of stiffness) and considering only shear deformations, it can rather simply be proved that the torsional deflection of the open end varies as the square of the length of the building.
• Architectural Implications
Figure 618. Torsional response
A common instance of this problem is that
A common example of this condition occurs in store front design, particularly on corner lots, and in freestanding commercial and industrial buildings with varied openings around the perimeter. A special case is that of fire stations that require large doors for the movement of equipment. In these buildings it is particularly important to avoid major distortion of the front opening, for example if the doors jam and cannot be opened, the fire station is out of action at a time when its equipment is most needed. Tiltup concrete industrial and warehouse buildings, in which lateral resistance is provided by the perimeter walls, often also require a variety of openings for entrances, loading docks, and office windows, with a
Figure 619. Open front design: torsional deflection varies as the square f the length
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299
Figure 620. J.C.Penney department store, Anchorage, Alaska, 1964Note: unbalanced location of perimeter walls, particularly on third, forth and fifth floors, leading to severe torsional forces and near collapse.
consequent variation in seismic resistance around the perimeter.
• Historical Performance A classical instance of this problem occurred in the J.C.Penney Department Store in Anchorage, Alaska, in the 1964 earthquake. The building was so badly damaged that it had to be demolished. The store was a fivestory building of reinforcedconcrete construction. The exterior walls were a combination of pouredinplace concrete, concrete block, and precast concrete nonstructural panels which were heavy, but unable to take large stresses. The first story had shear walls on all four elevations. The upper stories, however, had a structurally open north wall, resulting in Ushaped shear wall bracing system (similar to a typical openfront store) which, when subjected to eastwest lateral forces, would result in large torsional forces (Figure 620).
A special case is also that of apartment house and hotels that are oriented to a view, such as a beach. which implies the need for large openings on the view elevation. The El Faro building was a small apartment house located facing the beach in the Chilean resort town of Vina del Mar. In order to exploit the view, two elevations are open: the stairs and elevator shaft are concentrated to the rear of the building and their walls provide the seismic resistance. The result is a wide eccentricity between the centers of mass and resistance. In the Chilean earthquake of 1985, this building rotated and very nearly collapsed: it was subsequently demolished. (Figure 621)
• Solutions The objective of any solution to this problem is to reduce the possibility of torsion, and to balance the resistance around the
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perimeter. Four alternative strategies can be employed, and are shown in Figure 622.
Figure 621. El Faro apartments, Vina del Mar, Chile, 1985
The first approach is to design a frame structure with approximately equal strength and stiffness for the entire perimeter. The opaque portions of the perimeter can be constructed of nonstructural cladding material that will not affect the seismic performance of the frame. This can be done either by using lightweight cladding, or by ensuring that heavy materials (such as concrete or masonry) are isolated from the frame. A second approach is to increase the stiffness of the open facades by adding shear walls at or near the open face. This solution is, of course, dependent on a design which permits this solution. A third solution is to use a very strong momentresisting or braced frame at the open front, which approaches the solid walls in stiffness. The ability to do this will be dependent on the size of the facades: along steel frame can never approach a long concrete wall in stiffness. This is, however, a good solution for wood frame structures, such as apartment houses with a ground floor garage space, because even a rather long steel frame can be made to approach plywood walls in stiffness. Finally, the possibility of torsion may be accepted and the structure designed to resist it. This solution will only apply to small structures with stiff diaphragms, which can be designed to act as a unit. 6.6.3
Nonparallel Systems
• Definition The vertical load resisting elements are not parallel or symmetric about the major orthogonal axes of the lateralforce resisting system.
• Seismic Effects
Figure 622. Solutions to open front buildings
This condition results in a high probability of torsional forces under a ground motion , because the centers of mass and resistance cannot coincide for all directions of ground motion. Moreover, the narrower portions of the
6. Architectural Considerations building will tend to be more flexible than the wider ones, which will increase the tendency to torsion. The problem is often exacerbated by perimeters with variations of strength and stiffness (Figure 623). A characteristic form of this condition is the triangular or wedgeshaped building that results from street intersections at an acute angle. These forms often employ a solid, stiff party wall in combination with more open flexible facing the street. The result is a form that is very prone to torsion.
301 Commentary , but it is identified as irregular in the 1988 UBC, the 1990 SEAOC Commentary, and subsequent codes and provisions.
Figure 623. Wedge shaped plan: invitation to torsion
• Architectural Implications Nonrectiliner forms have become increasingly fashionable in the last few years as a reaction against the rectangular "box". Forms that are triangular, polygonal, or curved have become commonplace, even in very large buildings. However, in some instances the desired forms can be achieved by nonstructural elements attached to a structure which may be essentially regular and rectilinear. (Figure 624) Extreme forms of non rectilinearity are a feature of "deconstructionist" architecture, which is discussed in Section 6.11. The traditional , trapezoidal or "flatiron" form resulting from the streetlayout constraints is still common in highdensity urban locations.
• Historical Performance This form has been fairly recently identified as a problem configuration. The form was not identified as irregular in the 1890 SEAOC
Figure 624. Form achieved by nonstructural attachments to main
Many buildings of this type were constructed in Mexico City, resulting from the high density and street layout of the city, and instances of poor performance were observed in the 1985 earthquake. Many buildings suffered severe distortion, particularly wedgeshaped buildings with stiff party walls opposite the apex of the triangular form (Figure 625). In many cases the condition was exacerbated by other irregularities such as a soft story.
• Solutions Since 1988 the UBC and the NEHRP Provisions place some special requirements on the design of these types of configuration. Particular care must be exercised to reduce the effects of torsion. In general, opaque walls should be designed as frames clad in
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Figure 625. Distortion in wedgeshaped building, Mexico City, 1985
lightweight materials, to reduce the stiffness discrepancy between these walls and the rest of the structure. Alternatively, special design solutions may be introduced to increase the torsional resistance of the narrow parts of the building, although this may be difficult to achieve while still retaining open facades or internal areas. 6.6.4
Diaphragm Configuration
• Definition The diaphragm configuration is the shape and arrangement of horizontal resistance elements that transfer forces between vertical resistance elements.
• Seismic Effects
Diaphragms perform a crucial role in distributing forces to the vertical seismicresisting elements. The diaphragm acts as a horizontal beam, and its edges act as flanges. Diaphragm penetration and geometrical irregularities are analogous to such irregularities in other building elements, leading to torsion and stress concentration. The size and location of these penetrations is critical to the effectiveness of the diaphragm. The reason for this is not hard to see when the diaphragm is visualized as a beam: it is obvious that openings cut in the tension flange of a beam will seriously weaken its loadcarrying capacity. In a vertical load system. a penetration in a beam flange would occur in either a tension or a compression area: in a lateral load system, the hole will be in a region of both tension and compression, since the loading alternates in direction.
6. Architectural Considerations When diaphragms form part of a resistant system, they may act in either a flexible or stiff manner. This depends partly on the size of the diaphragm (its area between enclosing resistance members or stiffening beams), and also on its material. The flexibility of a diaphragm, relative to the shear walls whose forces it transmits, also has a major influence on the nature and magnitude of those forces.
303
• Ensure that multiple penetrations are spaced sufficiently far from one another to allow reinforcing elements to develop their required capacity • Ensure that collectors and drag struts are uninterrupted by openings
6.7
Vertical Configuration Problems
6.7.1
Soft and Weak Stories
• Architectural Implications Diaphragms are generally floors or roofs, and so have major architectural functions aside from their seismic role. The shape of the diaphragm is dependent on the overall plan form of the building, and how it can be subdivided by walls or collectors. In addition, however, architectural requirements such as staircases, elevators and duct shafts, skylights, and atria result in variety of diaphragm penetrations. In some cases, as in the need for elevators in an Lshaped building, the logical planning location for elevators (at the hinge of the L) is also the area of greatest seismic stress.
• Historical Performance Failures specifically due to diaphragm design are difficult to identify, but there is general agreement that poor diaphragm layout is a potential contributor to failure.
• Solutions Diaphragm penetrations are a form of irregularity specifically called out in the 1990 SEAOC Commentary that requires engineering judgment. In addition, current codes and provisions specifically define such penetrations, and impose some additional requirements on the diaphragm design in such cases. The general approach to the design of penetrations in diaphragms is to:
• Ensure that penetrations do not interfere with diaphragm attachment to walls or frames.
• Definition A soft story is one that shows a significant decrease in lateral stiffness from that immediately above. A weak story is one in which there is a significant reduction in strength compared to that above.
• Seismic Effects The condition may occur at any floor, but is most critical when it occurs at the first story, because the forces are generally greatest at this level. The essential characteristics of a weak or soft first story consist of a discontinuity of strength or stiffness, which occurs at the secondstory connections. This discontinuity is caused because lesser strength, or increased flexibility, in the first story structure results in extreme deflections in the first story, which, in turn, result in a concentration of forces at the second story connections. If all the stories are approximately equal in strength and stiffness, the entire building deflection under earthquake forces is distributed approximately equally to each story. If the first story is significantly less strong or more flexible, a large portion of the total building deflection tends to concentrate there, with consequent concentration of forces at the secondstory connections .(Figure 626)
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• The use of a discontinuous shear wall, in which shear forces are resisted by walls that do not continue to the foundations, but stop at second floor level, thus creating a similar condition to that of the second item above.
• Discontinuous load paths, created by a change of vertical and horizontal structure at the second story.
Figure 626. The softstory effect
In more detail, the softstory problem may result from four basic conditions. These are diagrammed in Figure 627 and are:
• A firststory structure significantly taller than upper floors, resulting in less stiffness and more deflection in the first story.
• An abrupt change of stiffness at the second story, though the story heights remain approximately equal. This is caused primarily by material choice: the use, for instance, of heavy precast concrete elements above an open first story.
The above characteristics, individually or in combination are readily identifiable in existing buildings provided that the building structure can be studied in its entirety, either in the field or by reference to accurate asbuilt construction documents.
• Architectural Implications A taller first story often has strong programmatic justification, when large spaces, such as meeting rooms or a banking hall, must be provided at ground level. Similarly, an open ground floor often meets urban design needs by providing both real and symbolic access to a plaza or street, or by providing space at the base of a building. The changes in proportion provided by a high story, or the "floating box" concept (now somewhat outdated), are very real aesthetic tools for the architect, although engineers may find such concepts hard to rationalize in their terms.
Figure 627. Types of soft story
6. Architectural Considerations Engineers must accept that some form of variation in the first story will remain a desirable architectural characteristic for the foreseeable future: whether it is "soft" or "weak" in seismic terms is a matter for the architect and engineer to resolve.
• Historical Performance The general type of soft first story configuration was early identified as a problem. Failures in masonry buildings in the 1925 Santa Barbara earthquake were identified by Dewell and Willis(610) as softfirststory failures. In more recent times, with extensive use of frame structures, damage to reinforcedconcrete buildings in Caracas (1967) clearly identified the risk to tall buildings with this condition. In the Mexico City earthquake of 1985, researchers determined that soft first stories were a major contributor to 8% of serious failures, and the actual percentage is probably greater because many of the total collapses were precipitated by this condition. The particular case of the discontinuous shear wall has led to clearly diagnosed failures in United States buildings. Olive View hospital, a new structure that was badly damaged in the 1971 San Fernando earthquake, represents a classic case of the problem. The vertical configuration of the main building was a twostory layer of rigid frames on which was supported a fourstory shear wallframe structure (Figure 628). The second floor extended out to form a large plaza.
Figure 628. Olive View hospital, San Fernando, 1971 (a) elevation of stair towers (b) section through main building
305 The severe damage occurred in the softstory portion: the upper floors moved so much as a unit that the columns at ground level could not accommodate such a huge displacement between their bases and tops and failed. The largest amount by which a column was left permanently out of plumb was 2 1/2 feet. Though not widely identified, the stair towers at Olive View also show a clear and separate example of a discontinuous shearwall failure. These sevenstory towers were independent structures, and proved incapable of standing up on their own: three stair towers overturned completely, while the fourth leaned outwards 10 degrees. The six upper stories were rigid reinforced concrete walls, but the bottom story was composed of six freestanding reinforcedconcrete frames, which failed. The exception was the north tower, whose walls came down to the foundation directly without any discontinuity; this was the only tower to remain standing. Olive View hospital was demolished after the earthquake, and a new hospital built on the same site. The performance of the Imperial County Services Building, El Centro, in the Imperial Valley Earthquake of 1979, provides another example of the effects of architectural characteristics on seismic resistance. The building was a reinforcedconcrete structure built in 1969. In this mild earthquake the building suffered a major structural failure, resulting in column fracture and shortening (by compression) at one endthe eastof the building. (Figure 629). The origin of this failure lies in the discontinuous shear wall at that end of the building. The fact that this failure originated in the configuration is made clear by the architectural difference between the east and west ends: this is an example of the large effect on seismic performance of a relatively small design variation between the two ends of the building.. The difference in location of the small groundfloor shear walls was sufficient to create a major difference in response to the rotational forces on the large end shear walls (Figure 630).
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• Solutions If a high first story is desired, either:
• Introduce bracing that stiffens the columns up to a level comparable to the superstructure. • Add columns at the first story to increase stiffness, or • Change the design of the firststory columns to increase stiffness. Figure 629. Imperial County Services Building, El Centro, California. failure of end bay at discontinuous shear wall, (Imperial valley earthquake of 1979)
If a large opaque wall is required in a location that could create a soft first story:
• Insure that such a wall is not part of the lateral load resisting system • Reduce the mass of the wall by use of light material and hollow construction • If a heavy wall is necessary, then insure that the wall is detached in such a way that the superstructure is free to deflect in a comparable way to the first floor
Figure 630. Imperial County Service Building, plan and elevations
A more recent instance is that of a medical office building in the Northridge earthquake of 1994, constructed at about the same time as the previous two buildings discussed. The simple rectangular building had discontinuous shear walls at each end. These proved inadequate to deal with the forces, with consequent severe torsional damage at each end of the building, (Figure 631) This building also had a structural discontinuity at the second floor that caused the "pancaking" of the second floor.
If the architect insists on such material and design constraints that a major discontinuous shear wall is the only solution, the engineer should refuse to do it. The liabilities involved in using such a proven failure mechanism are too great. If the lateral resistance system is based on the use of an interior core (for a highrise office building, for example), the perimeter columns may be tall, but there is no soft first story, provide the core is brought down to the ground. In such a building it is not difficult, if the coreplan dimensions are sufficient, to insure that the stiffness of a tall first story is adequate to prevent structural discontinuity at the second floor. One condominium building a good example of architect engineer collaboration. That building achieved an elegant exterior appearance which appeared to be a soft first floor. However, the seismic resistance was provided by a strong interior box shear wall structure that enabled the taller first floor to be accommodated with ease. The building suffered
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Figure 631. Discontinuous shear wall failure, office building, Northridge
virtually no damage in the strong Chilean earthquake of 1985. It should be noted that in the 1997 NEHRP Provisions structures with a weakstory discontinuity in capacity that is less than 65% of the story above are not permitted over 2 stories or 30 feet in height in Seismic Design Categories B,C,D,E and F. 6.7.2
Columns: Variations in Stiffness, Short Columns, and Weak Column/Strong Beam.
• Definition This section considers the use of columns of varying stiffness, by reason of either differences in length or deliberate or inadvertent bracing: the use of columns that are significantly weaker than connecting beams: and the use of columns
in one floor that are significantly shorter than those on other floors.
• Seismic Effects Seismic forces are distributed in proportion to the stiffness of the resisting members. Hence, if the stiffness of the supporting columns (or walls) varies, those that are stiffer (usually shorter) will "attract" the most forces. The effect of this phenomenon is explained in Figure 632. The important point is that stiffness (and hence forces) varies approximately as the cube of the column length. Similarly, a uniform arrangement of short columns supporting a floor will attract greater forces to that floor, with a corresponding possibility of failure. Typically such an arrangement may also involve deep and stiff spandrel beams, making the columns significantly weaker than the beams.
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Figure 632. Effect of variations of column stiffness
Such a design is in conflict with a basic principle of seismic design, which is to design a structure in such a way that under severe seismic forces, beams will deform plastically before columns. This is based on the reasoning that as beams progress from elastic to inelastic behavior they start to deform permanently. This action will dissipate and absorb some of the seismic energy. Conversely, if the column fails first and begins to deform and buckle, major vertical compressive loads may quickly lead to total collapse. Mixing of columns of varying stiffness on different facades may also lead to torsional effects, since the building assumes the attributes of varying perimeter resistance discussed above.
• Architectural Implications The origin of variations in column stiffness generally lies in architectural considerations. Hillside sites, infilling of portions of frames with nonstructural but stiff material to create high strip windows, desire to raise a portion of the building of the ground on tall "pilotis", while leaving other areas on shorter columns, or stiffening some columns with a mezzanine or a
Chapter 6 loft, while leaving others at their full, unbraced height. These issues are important because their effects may be counterintuitive. For example, infilling may be done as a remodel activity later in the building life for which the engineer is not consulted, because intuition may suggest to the designer that he is strengthening it in the act of shortening it rather than introducing a serious stress concentration for which the structure was not designed. For vertical forces a reduction in the effective length of a column is beneficial because it reduces the likelihood of buckling, but the effect under lateral forces is quite different. Variations in openings in different facades are often required from a daylighting or energyconservation requirement. Where openings are created by variations in structural arrangement, rather than by variations in cladding, some of these conditions may well arise.
• Historical Performance Significant column failures, sometimes leading to collapse, have been attributed to these conditions in a number of recent earthquakes, particularly in Japan, Latin America, and Algeria. Many Japanese schools, employing short columns on one side of an elevation, or using a weak column, strongbeam configuration, suffered severe damage in the Tokaichioki earthquake in 1968 and the 1978 Miyagikenoki earthquake. (Figure 633) In Latin America, the problem has frequently been caused by inadvertent stiffening of columns through nonstructural infill which, when combined with high glazing, creates short columns. In the El Asnam (Algeria) earthquake of 1980, many apartment structure failures were caused by short columns used at ground level to provide a ventilated open space (called a "vide sanitaire") in a semibasement location . The significant failure of a large condominium and hotel structure in the Guam earthquake of 1993 has been ascribed in part to the creation of a
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Figure 633. Short column failure, school, Japan: Miyagiken oki, 1978
short column condition by the introduction of nonstructural stiffening elements(611) (Figure 634)
• Solutions The general solution is to match the detailed seismic design carefully to the architectural requirements. The weakcolumn, strongbeam condition can be avoided by insuring that deep spandrels are isolated from the columns; in the same way the lengths of columns around a facade can be kept approximately equal. Horizontal bracing can be inserted to equalize the stiffness of a set of columns of varying height (Figure 635). Heavy nonstructural walls must be isolated from columns to insure that a shortcolumn condition is not created. (Figure 636).
6.7.3
Vertical Setbacks
• Definition A vertical setback is a horizontal, or near horizontal, offset in the plane of an exterior facade of a structure.
• Seismic Effects The problem with this shape lies in the general problem of discontinuity: the abrupt change of strength and stiffness. In the case of this complex configuration, it is most likely to occur at the line of the setback, or "notch". The seriousness of the setback effect depends on the relative proportions and absolute size of the separate parts of the building. In addition, the symmetry or
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Figure 634. Short column failure, Guam, 1993
Figure 635. Horizontal bracing to stiffen a high open end entrance
6. Architectural Considerations B u ild in g S tru ctu re G ap Steel A n g les C o n crete B lo ck
Figure 636. Heavy nonstructural wall isolated from structure at top and side
asymmetry in plan of the tower and base affect the nature of the forces. If the tower or base or both are dynamically asymmetrical, then torsional forces will be introduced into the structure, resulting in great complexity of analysis and behavior. The setback configuration can also be visualized as a vertical reentrant corner. Stresses must go around a corner , because a notch has been cut out, preventing a more direct route. Hence, the smaller the steps or notches in a setback, the smaller the problem. A smooth taper avoids the notch problem altogether. A tapering beam will not experience stress concentrations, whereas a notched beam will. Setbacks with shear walls in the tower portion that are not continued to the ground are highly undesirable. Besides the change of stiffness where the shear wall enters the base structure, the shear wall will transmit large forces to the top diaphragm of the base. Although, typically, setbacks occur in a single building, the condition can also be created by adjoining buildings of different heights which have inadequate or nonexistent seismic separations.
• Architectural Implications Setbacks may be introduced for several reasons. The three most common are zoning requirements that require upper floors to be set back to admit light and air to adjoining sites, program requirements that require smaller
311 floors at the upper levels, or stylistic requirements relating to building form. Setbacks relating to zoning were common a few decades ago when daylighting was a major concern, and resulted in characteristic shapes of older highrise buildings in New York and other large cities. Stylistic fashions replaced these forms with those of simple rectangular solids, made possible by advances in artificial lighting and airconditioning. Now, there is a renewed interest in setback shapes for stylistic reasons, while at the same time energy conservation requirements have reinstated a functional interest in setbacks for daylighting reasons. An interesting example of this stylistic trend is that of the new planning code for San Francisco, which specifically mandates setbacks for large buildings in the downtown area. These represent relatively minor variations in the vertical plane of the facade, rather than the abrupt rising tower on a base, which is of more serious seismic consequence. The trend is, however, away from vertical structural continuity at the perimeter and thus introduces complexity and cost into the structural solution. A type of setback configuration only made possible by modern framed construction is that of the building that grows larger with height. This type is termed inverted setback or inverted pyramid depending on its form. Its geometrical definition is the same as that of the setback, but, because of the problems of overturning, its extremes of shape are less. Nevertheless. some surprising demonstrations of this shape have appeared, and it appears to be one whose image has a powerful design appeal (Figure 637).
• Historical Performance Although commonly identified as a configuration problem, severe failures of modern buildings attributed to this condition are few. While traditional towers, primarily churches, have suffered their share of failures, the number of those that have survived severe
Figure 637. Dallas City Hall : an inverted pyramid
damage is remarkable. An example from the Kobe earthquake of 1995 shows a failure in a setback building at the plane of weakness created by a combination of the setbacks and adjoining openings in the wall (Figure 638) While there have been recorded failures of invertedsetback buildings, notably in the Agadir (Morocco) earthquake of 1960, some of the more striking examples have performed well. This is probably because the appearance of instability inherent in this form results in special attention being paid to its structural design. Typically, such buildings devote a much larger percentage of their construction cost to structure than more conventional buildings.
• Solutions Setbacks have long been recognized as a problem, and so the Uniform Building Code has attempted to mandate special provisions for them currently, the earthquake regulations of the Code refer to setback configurations as follows:
Buildings having setbacks wherein the plan dimensions of the tower in each direction is at least 75% of the corresponding plan dimension of the lower part may be considered as uniform buildings without setbacks, provided other irregularities as defined in this section do not exist. An appendix to the 1990 SEAOC Commentary to this section includes a lengthy discussion of the setback problem and an approach to its analysis .: In general, conceptual solutions to the setback problem are analogous to those for its horizontal counterpart, the reentrant corner plan. The first type of solution consists of a complete seismic separation in plan, so that portions of the building are free to react independently. For this solution, the guidelines for seismic separation, discussed elsewhere, should be followed.
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Figure 638. Failure of setback building along a plane of weakness
When the building is not separated, the analysis proposed in the appendix to the 1990 SEAOC Commentary provides the best guidelines, with some necessary interpretations to fit the particular case. Particular attention should be paid to avoiding vertical column discontinuity, so that setbacks should be arranged to coincide with normal bay sizes (which may result in a series of small bays). Any large building with major setback conditions should be subjected to special analysis, or at least to careful investigation of probable dynamic behavior. Finally, the inverted setback configuration of any extreme form and size should be avoided in seismic areas, unless the owner is willing to assume the considerable additional structural costs that will be incurred. The 1997 NEHRP Provisions, as noted earlier, permit vertical setback configurations to be analyzed using the simple ELF method if the stiffness on the top story of the base is at least
five times that of the first story of the tower. The UBC permits use of the standard ELF method for a twostage analysis of tower and base if the average story stiffness of the base is at least 10 times greater than the average story stiffness of the tower.
6.8
STRUCTURALLY RESTRICTIVE ARCHITECTURAL DETAILING
6.8.1
Components and Connections
• Definition By structurally restrictive detailing we mean detailed architectural design of a component that prevents good seismic design practice in the structural design. 313
314
• Seismic Effects This problem represents a micro version of typical overall building configuration problems. Architectural detailing may place dimensional or location constraints on structural design resulting in weakness or eccentricity of force actions that can lead to stress concentration or local torsion. The problem is most critical at beamcolumn connections, which are highly stressed, but often represent a critical element in the aesthetic scheme of the building. Structural detailing ideally provides for direct load transfer and minimum local eccentricity, with forces resolved at a point. Architectural detailing may result in inadequate size and eccentric or discontinuous load paths (Figure 639). The problem is particularly critical for reinforcedconcrete structures, where constraints may provide inadequate room for proper placing of reinforcing.
Chapter 6 expression of structural forces, and be easy to accommodate, or they may directly contradict structural action and lead to weakness. Recesses are often designed by architects to accentuate the line at which materials meet one another, particularly when the materials are different or meet at right angles, as in a columnslab junction.
Figure 640. Facades: differences in architectural emphasis
• Historical Performance:
Figure 639. Eccentric load paths created by architectural detailing of structural connection
• Architectural Implications Detailed design is an important element in architectural expression. As an example, the design of the perimeter beamconlumn connection can provide the building with a predominantly horizontal, vertical, or neutral emphasis. (Figure 640). But the structural implications of these variations may not be understood by the architect. Another example is the use of taper or the insertion of recesses in columns. Tapered columns may be a correct
Specific performance attributable to this condition is difficult to document but the problem is generally recognized by engineers. Two well documented cases do exist where architectural detailing contributed to failure. The first is that of the column design of the Olive View Hospital, damaged in the 1971 San Fernando earthquake (discussed previously as an example of softfirststory failure.). A significant difference in performance was observed between corner and internal columns in this building. The twelve Lshaped corner columns were completely shattered and their loadcarrying capacity reduced almost to zero. The interior columns, of square section, had spiral ties, and although they lost most of their concrete cover, they retained loadcarrying capacity and probably saved the building from collapse. Because of their architectural form, it was not possible for the corner columns to use spiral ties (Figure 641). Higher stress and torsion in the corner columns may also have contributed to their poor performance.
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• Solutions: Close coordination between architect and engineer is necessary to insure that architectural detailing does not result in undesirable structural design constraints.
Figure 641. Exterior column sections at Olive View Hospital, San Fernando, California. Due to their shape, corner columns could not be spirally reinforced
The Imperial County Service Building at El Centro, California, suffered severe damage in the 1979 Imperial Valley earthquake, and four columns at one end of the building were badly shattered. Detailed study of these columns showed that an architectural recess had been placed at the line where the columns met the ground. (Figure 642). This recess caused a reduction in sectional area of the column and a reduction in axial loadcarrying capacity. Analytical and experimental studies have shown that this change in column section accentuated the undesirable performance of these columns(67).
Figure 642. Column detail, Imperial County Services Building, El Centro, California. Note architectural recess affecting reinforcement continuity
6.9
PROBLEMS OF ADJACENCY
6.9.1
Pounding
• Definition: Pounding is damage caused by two buildings, or different parts of a building, hitting one another.
• Seismic Effects: Pounding as characterized in Codes and Guidelines and in most analytical research studies takes the form of in plane displacements of two adjacent buildings, as in the investigation of a row of adjacent buildings by Athanassiadou et al(612). Empirical observation shows that building separations are complex in their basic conditions and in their effects, and lack of separation is not necessarily detrimental. Observation has shown that the end buildings of a row of adjacent buildings tend to suffer more damage than interior buildings. Analytical pounding studies consider regular buildings in elevation. In fact, the sway characteristics of buildings are much influenced by irregularities, particular that of soft first stories, that can lead to extreme displacements or even collapse. Some of these characteristics are shown in Figure 643. Similarly, analytical studies have always assumed regular buildings in plan. Since adjacent buildings with little or no separation will generally be found in the older sections of down town, building plans are often very irregular, leading to torsional effects under ground motion. These characteristics are shown in Figure 644.
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Figure 643. Irregularities may create extreme displacements or collapse
Study of pounding damage in Kobe in 1995 showed that very large deflections were often caused by design flaws (such as a soft first story) or near source extreme shaking velocities, or a combination of the two. In addition, many instances of large building deflections (or "leaning") related to ground/foundation failures. These effects are not accounted for in code type separation requirements, which assume a uniform deflection for the height of the building, related only to ground motion. Observation has also shown that, in some cases, the close proximity of buildings may act as a support, particularly for buildings in midblock, and increasing the space between buildings might serve, in some cases, to increase deflections and damage rather than reducing them. A probable instance of this was observed by the author in Mexico City, in 1985. In this instance, a tall slender building with an apparent serious soft first story problem, appeared to be restrained by low, stiff buildings on either side. (Figure 3 ). Several instances of this phenomenon were observed in Kobe.
This point is very difficult to assess. The response to shaking of a number of adjacent buildings with essentially no separation between them must be equivalent to the response of a large building with a variety of strengths, stiffness and other structural characteristics which would be very difficult to analyze. The possibility of pounding is a function of the vertical deflection or drift of adjoining buildings (or parts of a building). Drift is calculated by applying the code design forces to the building and then observing the deflections that result. Since these estimated forces will be less than what we know can occur, calculated deflections must be corrected to obtain a more realistic estimate of how much the building may actually move. Alternatively, an accurate estimate of drift may be made that accounts for all foreseeable factors. Potential pounding presents some particular problems of a socioeconomic nature where existing buildings are concerned. The socioeconomic problems consist of how to involve the adjoining building owner in possibly costly studies, design and construction work that the owner may not wish to participate in or may
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Figure 644. Irregularities in plan may create additional torsional effects that impact adjoining buildings
even actively oppose them. The problems are particularly critical in the case of common structure, because rehabilitation is very difficult, if not impossible, without the neighbor's involvement and probably some degree of rehabilitation to his property. In the case of falling hazards, it would be desirable for the neighbor to mitigate them, but the extent to which the federal owner can require this are not clear. The problem of pounding is traditionally dealt with by requiring a large gap between buildings. This can, in theory, be achieved without impacting the adjoining owner. While in engineering terms it may seem obvious that it is in the adjoining owners best interest to cooperate in evaluation and mitigation, in socioeconomic terms there may be many reasons, valid or otherwise, for reluctance. The owner may have real economic constraints in incurring any costs of evaluation or mitigation, and be quite ready to accept the possible risks of inaction. In addition, the owner may have short term intentions of redeveloping or selling his property, and so not wish to incur
expenses that will be of no conceivable benefit to him. Thus, the possibility of cooperative rehabilitation will be much conditioned by how the adjoining owner sees his economic future and views unsolicited action by a neighbor that might impact it.
• Architectural Implications Pounding is included in this discussion of configuration issues because it is a matter of where buildings are located relative to other structures, which is an early architectural decision. The problem has considerable architectural implications for the construction of buildings on constricted urban sites, because to make provision for the worst case condition could result in large building separations and significant loss of usable space. While building codes place modest limits on drift (for example, 0.005 time story height) based on static analysis, actual experience with drift and calculations of realistic figures provide some startling numbers. Freeman(613) calculated
318 the actual drift on flexible buildings up to 20 stories under 0.4 g acceleration as being 0.020 0.055 times the story height. For a 12 story building this translates into 40110 in. for a 14ft story height. A separation that could accommodate two such buildings vibrating out of phase would have to be 18 ft. 4 in. wide. Clearly compromise is necessary, but nonetheless, loss of usable space measured in lineal feet becomes serious. In addition, the idea of urban buildings with spaces of 2  3 feet between them suggests a very difficult maintenance problem.
• Historical Performance: Problems of adjacency have been routinely noted by earthquake investigators over the past several decades. In the 1972 Managua earthquake, the fivestory Grant Hotel suffered a complete collapse of its third floor when battered by the roof level of the adjacent twostory building. In the 1964 Alaska earthquake, the 14story Anchorage Westward Hotel pounded against its low rise ball room and an adjoining sixstory wing, although separated by a 4in, gap. The pounding was severe enough in the high rise to dislocate some of the metal floor decking from its steel supports. In recent earthquakes, pounding has continued to be a serious issue. The earthquake that struck Mexico City in 1985 has revealed the fact that pounding was present in over 40% of 330 collapsed or severely damaged buildings surveyed, and in 15% of all cases it led to collapse. Many instances of pounding were observed in the Kobe earthquake of 1995.
• Solutions: Perhaps due to the high incidence of pounding damage observed in the 1985 Mexico City earthquake a number of researchers have studied pounding problems in recent years. Two recent studies, by Jeng et al,(614) and the study by Athanassiadou et al,(612) are representative, and both contain a full set of references to other
Chapter 6 studies of the problem. Jeng et al. present a new method for estimating the likely minimum building separation necessary to preclude seismic pounding: two 10 story concrete frame buildings are analyzed by way of example. Athanassiadou et al. studied the seismic response of adjacent buildings in series, with similar or different dynamic characteristics, using SDOF systems subjected to base motions. These, and other studies, confirm the results of empirical surveys, and to provide quantitative information that is necessary for code and design practice development, although as yet the quantitative data is not readily transferable to code values. To assume that code limits on drift provide an accurate estimate of possible drift is unrealistic, but accurate estimates may provide very large worst case figures. Blume, Corning and Newmark suggest an alternative method(615) : Compute the required separation as the sum of the deflections computed for each building separately on the basis of an increment in deflection for each story equal to the yieldpoint deflection for that story, arbitrarily increasing the yield deflections of the two lowest stories by multiplying them by a factor of 2. An earlier edition of the Uniform Building Code contained a rule of thumb intended for the relatively stiff structures of that day(616): separations should be " one inch plus one half inch for each ten feet of height above twenty feet". It should be noted that, notwithstanding the high cost of land in Japanese cities, new structures in Kobe seem to be providing a generous allowance for differential drift. A possible alternative approach is to place an energyabsorbing material between the buildings; this obvious simple approach seems to have been little studied. Many buildings in Mexico City were, in fact, protected from collapse because they were erected hard up against adjoining buildings on
6. Architectural Considerations both sides, so that whole blocks of buildings acted as a unit, and the group was stronger than the individual structures. As evidence of this, Mexican studies showed that 42% of severely damaged buildings were corner buildings, lacking the protection of adjoining structures. This finding suggests the need for serious research on the subject of allowable drift, pounding, and the design and construction of closely spaced buildings. 6.9.2
Other Adjacency Problems
Two other problems of adjacency give cause for concern: one is that of damage caused to a building by falling portions of an adjoining building: in the 1989 Loma Prieta earthquake a death was caused in downtown Santa Cruz when a portion of unreinforced masonry wall fell through the roof of a lower adjoining building, and six deaths were caused in San Francisco when part of a masonry wall fell on some parked cars. The other adjacency problem is that created by structural elements  generally walls or columns  that are common to adjoining buildings: while instances of damage caused by this condition are not specifically identified, there is a clear problem when an owner wishes to rehabilitate a building which has structural elements common to an adjoining building that is not undergoing related rehabilitation.
6.10
6.10.1
THE ARCHITECT/ENGINEER RELATIONSHIP ArchitectEngineer Interaction
In the United States the architect/engineer relationship is delicate because typically the engineer is employed by the architect, and if he complains too much about the architect's design he may be replaced. An architect who finds his design criticized by his engineer can generally find an alternative engineer who will
319 accommodate him. It is extremely hard to ascertain whether this second engineer reaches this accommodation because he is more ignorant than his colleague, more of a gambler, or more inventive and clever. There are, of course, many instances where architects and engineers have built up close relationships and communicate fruitfully, with the engineer participating at an early stage of design. However, even in these instances the pressure of business often means that, for financial reasons, the engineer is not employed until the building schematic design is complete. This applies particularly to private work, where the developer must have a design perhaps only a three dimensional sketch in order to procure financing, and he does not want to incur additional consultant costs until the financing is secured. The following description is of the preliminary design process of a large U.S. architectural for a client in the Pacific Rim: ".. we developed a method whereby we would send a team of three people for a week, working in the client's office, or from a hotel room, but having client input into daily charettes, lots of alternatives in sketch form, not spending many hours of presentation, but spending the hours on design. At the end of the week we would generally have a viable concept that the client had signed off."(617) Thus the schematic design for a multimillion dollar project is completed in a week: presumably the design is then brought to the engineer for him to insert a structure. Obviously, in this instance, much depends on the knowledge and experience of this three person team to ensure that the design is structurally reasonable. More risky is when analogous processes are conducted by a single architect with a desire to produce a design that will amaze his client. In seeking improved architect/engineer interaction a number of conditions must apply:
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• The engineer must communicate directly with the architectural design person or team
• The architect must take seriously his shared responsibility with the engineer for the seismic performance of the building. Recent experiences, such as Northridge and Kobe, should encourage this attitude. • Mutual respect and cooperation: an adversarial relationship will not be productive. • Common language and understanding: The architect must have some understanding of seismic engineering terms such as acceleration, amplification, base shear, brittle failure, damping (and so on through the engineering glossary). At the same time the architect should have a general understanding of the characteristics of typical seismic structural concepts: shear walls, bracing, moment frames, diaphragms, baseisolation etc. The new concepts of performance based seismic design should also be understood. In turn the engineer must understand the architect's functional needs and aspirations.
• Collaboration must occur at the onset of a project: before architectural concepts are developed or very early on in their conception • Business conditions that restrict early architect/engineering interaction must be alleviated (by the use of a general consulting retainer fee, for example, recovered from those projects that are achieved). • If the architect does not want to interact with his engineer, or if for some reason is prevented from doing so, then he should work with simple regular forms, close to the optimal seismic design While it is reasonable for engineers to ask that architects become better informed about seismic design and the consequences of their configuration decisions, the engineer must understand that while for them seismic design is
of paramount importance, for architects and their clients it takes very low priority as far as their own interests. For the architect, seismic design and safety is taken care of by the engineer: it is no more a subject of concern than provision for vertical forces, which never comes up for discussion between owner and architect, and seldom between architect and engineer. The architect is preoccupied with issues of codes and regulations relating to planning and design far removed from seismic problems, but of great importance and interest to his client. Similarly, the architect is continuously evaluating planning options, materials issues and both functional and aesthetic concerns upon which his client is constantly questioning him. Above all, the work must be done on time and on budget, and the architect would also like the job to be profitable. Architects vary greatly in their interests: the stereotype of the architect as an unworldly aesthete is seldom true. Some architects are brilliant salespeople and business managers: some are very close to engineers, and interested in how the building is engineered and constructed: some are excellent project managers and will ensure that budgets and schedules are kept: some are inspiring managers of people and will run an exciting and enjoyable office: some are brilliant at the design of details, the behavior of materials and the development of construction documents: and some are thoughtful and inventive designers. The large, wellrun office will have a mix of the above in its staff. The small office must try and find a few people that combine the above roles. As the profession of architecture becomes more complex, specialization is becoming more common: even large firms cannot play all roles, and the small office must specialize in a limited type of design. The advent of CAD and other information systems has extended the range for the small practitioner, but these systems need large capital investments that produce their own forms of limitation.
6. Architectural Considerations
6.11
Future Images
6.11.1
Beyond the International Style
The tenets of the International Style began to be seriously questioned in the mid1970's, both in print by architectural critics and historians and in practice by architects beginning to bring new design approaches to the drawing board and to construction. This questioning finally bore fruit in an architectural style known broadly as "Postmodern". Although this term was criticized by critics and the architects who were seen to be designing in this style, the term became a useful mark of identity. In general, postmodernism meant:
321 that approximates our optimal structure: the sensation is all in the nonstructural surface treatments. Designed as an economical design/build project the building has recently undergone seismic retrofit unrelated to its configurational characteristics.
• the revival of surface decoration on buildings • a return to symmetry in overall form • the use of classical forms, such as arches, decorative columns, pitched roofs, in nonstructural ways, and generally in simplified variations of the original elements. • a revival of exterior color as an element, with a palette of characteristic colors (e.g. dark green, pink, Chinese red, bright yellow, buff etc) Developments of postmodernism also involved both the revival of full, scholarly, classical revival as a style., and also very personal images by a few prominent architects in terms of scale and forms, which were derived from a variety of sources, such as Victorian engineering, ancient Egyptian architecture and nonEuclidean geometry. In seismic terms, this change in stylistic acceptance was, if anything, beneficial. The return to classical forms and symmetry was helpful to the structure as a whole, and almost all of the decorative elements were nonstructural. Inspection of an early icon of postmodernism, the Portland office building designed by Michael Graves, (Figure 645). shows an extremely simple and ordinary structure. Indeed, the Portland building, which created a sensation when completed, has a form
Figure 645. Office building, Portland, Oregon. Michael Graves, architect, 1979
It should be noted, however, that an interest in seismic design had no influence on the development of postmodernism  it is, and was a strictly aesthetic and cultural movement. At the same time that postmodernism was making historical architectural style legitimate again, another style evolved in parallel:. This style, originally christened "hitech" (the term has not stuck) returned to the celebration of engineering and new industrial techniques and materials as the stuff of architecture. This style developed primarily in Europe, notably in England and France, and was exemplified in a few seminal works, such as the Pompidou Center in Paris, the Lloyds building in London, and the Hong Kong and Shanghai bank in Hong Kong. These buildings proclaimed a new version of the functionalism of the thirties, updated to provide flexibility, adaptability and advanced servicing for an uncertain future, using exposed structure with beautiful castings as connections. In truth, these buildings are as
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aesthetically and stylistically conceived as any postmodern or classical revival building. The rise of postmodernism released architects from the straitjacketed moralities of the International Style. As a result, at present a kind of aesthetic bedlam reigns, and several competing private styles coexist, competing for clients  and finding them. The leading exponents of the new styles form an architectural jetset, cruising the world dropping off their stylistic gems to clients and countries that can afford them. The importance of wellpublicized designs by fashionable architects is that they create new accepted styles. Architects are very responsive to form and design and once a form gains credence practicing architects the world over begin to reproduce it. Today's New York corporate headquarters highrise becomes tomorrow's suburban Savings and Loan Office, as became clear in the adoption of the metal and glass curtain for building exteriors. The first two highly publicized curtain walls were that of the United Nations building and the Lever Brothers building, both in New York city in the early 50's: by the mid 60's every town in America had its stock of bluegreen glazed commercial buildings. So, to predict the design vernacular of the future it is necessary to look at what is being done in highstyle architecture, and in particular, to try and guess which forms seem to catching the imagination of architects and starting to be reproduced at a more modest level . Amid the bedlam of design voices, three influential trends can be discerned.. 6.11.2
of the late President Mitterand's "grand projects". This is a single office building, some 34 stories tall, designed as a cubical arch, framing the end of the Defense development on the perimeter of Paris. The arch is in line with the main axis through Paris to the Louvre, on which lies the Arch De Triomphe. The horizontal bridge structure provides exhibition and meeting spaces.
Figure 646. Grande Arche of the Defense, Paris, Johan Otto von Spreckelsen, architect
Influential Trends
• The bridge building: The bridge building form is that of twin highrise buildings connected at the roof with horizontal occupied space that acts as a bridge. The concept is that of a single building. The prototypical form of this, that has seized architect's imaginations, is that of the Grand Arch of the Defense (Figure 646), in Paris, one
A similar form is that of the Umeda Sky Building (Figure 647) in Osaka, Japan. This building incorporates a midair garden , midair escalators and a midair bridge to connect the two parts of the building. The architect, Hiroshi Hara, sees this form as the beginning of an approach to a threedimensional network to our congested cities. This building is in a fairly severe seismic zone and is carefully designed for earthquake resistance.
6. Architectural Considerations
323 in Los Angeles, (Figure 649) shows his warped and non vertical forms applied to a skyscraper. Despite its flourishes, the building is essentially rectilinear with the warped elements achieved by nonstructural addons to the main structure.
• The Deconstructed Building Deconstruction is a term applied to the work of a number of architects presently working around the world: the term is derived from the language and literary movement of the same name that originated in literary criticism. The principles of deconstruction were first formulated by the French philosopher and critic Jacques Derrida, in the early 1970's and have since revolutionized literary criticism and the study of language and meaning. Because deconstructed buildings essentially ignore the limitations of constructability, few have yet been built. One of the architects most commonly associated with deconstruction is the Iraqi, Zaha Hadid, who works in London. Figure 650 shows her design for a normally prosaic building  a fire station completed in 1993 in Germany. Figure 647. Umeda Sky building, Osaka, Japan, Hiroshi Hara, architect, 1988
The bridge or twin tower forms have immense drama and appeal, and so we can expect to see five story versions of them appearing in our shopping malls and suburban centers.
• The warped building: A strong design trend is that of buildings that use warped forms, often combined with non vertical walls and irregular warped exterior surfaces. The most prominent exponent of these forms is the American architect Frank Gehry, who is now building these forms all over the world. His Guggenhein Museum in Bilbao, Spain, completed in 1997 is typical of his style, and has been hailed as a masterpiece by architectural critics worldwide (Figure 648). His tower for the Rapid Transport Headquarters
6.12
CONCLUSION
These examples of new trends in architecture have been selected because experience has shown the force of images created by architectural innovators, however strange they may at first appear. The architects illustrated are those among many who are having great influence in the schools of architecture and among younger professionals. Engineers may expect to be confronted by these kinds of configurations in the coming years. Engineering rationality, and even buildability, appears to have little influence on these forms. There is controversy in the profession about this, and many critics view the new architecture as akin to theater set design, in which image is everything and its method of construction and longevity is irrelevant. Be that as it may, the zeitgeist is changing, and architects will perforce have to obey it.
324
Chapter 6
Figure 648. Guggenheim Museum, Bilbao, Spain, Frank Gehry, architect, 1996
Successful engineers will understand these imperatives, enjoy the experimentation that this work represents, and assist the architects in realizing their ambitions. New methods of analysis will help, but engineers must also continue to develop their own innate feeling for how buildings perform, and be able to visualize the interaction of configuration elements that are quite unfamiliar. Meanwhile, the residue of configuration problems left by the architecture/engineering of the 50's to 70's must be dealt with. Some will disappear as aging buildings are replaced: this should be encouraged, as it is the only guaranteed way of removing the earthquake threat. For other buildings, engineers must use their ingenuity and imagination to find affordable methods of retrofit. And there need be no recriminations: these problems are the joint product of architect/engineer interaction that, in its time, was fruitful: nature always has the last word in reminding us of our collective ignorance.
Figure 649. Rapid Transport District Headquarters, Los Angeles, Frank Gehry, architect, 1995
6. Architectural Considerations
325
Figure 650. Fire Station, Vitra factory complex, WeilamRhein, Germany Zaha Hadid, architect, 1995
Simple, economical buildings will continue to be built, and our optimal seismic design will continue to be viable. It may form the basis of performance based design which, if it is to be successful, will have to be free of the kinds of irregularities that make performance prediction difficult or impossible. We may expect design to develop in ways analogous to the poetry and prose of written communication. Most discourse is carried out in prose: the serviceable language of business and news reporting. At the level of literature, prose approaches an art form, in which the subtleties of language and human behavior are explored. Out in advance, often almost unintelligible, are the poets using words and language in new and unexpected ways: but over time they reveal insights in language so compelling that our speech and even our behavior is changed. Thus the language of Shakespeare shows up in the newspaper and even the office Email.
REFERENCES 61 Willis, C., Form Follows Finance, Princeton Architectural Press, New York, 1995 62 Arnold, C., "Architectural Aspects of Seismic Resistant Design" : Proceedings, Eleventh World Conference on Earthquake Engineering, Acapulco, 1996 63 Building Seismic Safety Council, NEHRP Recommended Provisions for Seismic Regulations for New Buildings, Building Seismic Safety Council, Washington, DC (1997) 64 Seismology Committee, Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, Structural Engineers Association of California, 1990 65 Seismology Committee, Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, Structural Engineers Association of California, 1980 66 Reitherman, R.K., "Frank Lloyd Wright's Imperial Hotel: a Seismic Re evaluation" ,Proceedings, Seventh World Conference on Earthquake Engineering. Istanbul, 1980 67 Dowrick, D.J., Earthquake Resistant Design, John Wiley and Sons, London, 1989 68 Naito, T., "Earthquakeproof Construction", Bulletin of Seismic Society of America, Vol. 17, No. 2, June 1977. 69 Degenkolb, H.J., "Seismic Design: Structural Concepts" Summer Seismic Institute for Architectural faculty, AIA Research Corporation, Washington, DC, 1977
326 610 Dewell, H and Willis, B, "Earthquake Damage to Buildings," Bulletin of. Seismic Society of America, Volume 15, No. 4 , Dec. 1925 611 Comartin, C.D., ed., "Guam Earthquake of August 8, 1993: Reconnaissance Report", Earthquake Spectra, 11, Supplement B, Earthquake Engineering Research Institute, Oakland, CA, 1993 612 Athanassiadou, C.J., Penelis, G.C and Kappos, A.J.: Seismic Response of Adjacent Buildings with Similar or Different Dynamic Characteristics, Earthquake Spectra, Volume 10, Number 2, Earthquake Engineering Research Institute, Oakland, CA, 1994 613 Freeman, S.A., "Drift Limits: Are They Realistic." Structural Moments, Structural Engineers Association of Northern California, Berkeley, CA 1980 614 Jeng, V, Kasai, and Maison, B.T.: A Spectral Method to Estimate Building Separations to Avoid Pounding, Earthquake Spectra, Volume 8, Number 2, Earthquake Engineering Research Institute, Oakland, CA, 1992 615 Blume, J.A, Newmark, N.M, and Corning,L.H, Design of Multistory Concrete Buildings for Earthquake Motion ,Portland Cement Association, Skokie, IL, 1961 616 International Conference of Building Officials (ICBO), 1958 Uniform Building Code (UBC), Whittier , CA 1958 617 Fuller, L.P. "Going Global". World Architecture, 42 , London, 1996
Chapter 6
Chapter 7 Design for Drift and Lateral Stability
Farzad Naeim, Ph. D., S.E. Vice President and Director of Research and Development, John A. Martin & Associates, Los Angeles, California.
Key words:
Drift, Pdelta, Stability, Exact methods, Approximate methods, Code provisions, UBC97, ICBO2000, Bent action, Chord action, Shear deformations, Moment resisting frames, Braced Frames, Shear walls, framewall interactions, Firstorder displacements, Secondorder displacements.
Abstract:
This chapter deals with the problems of drift and lateral stability of building structures. Design for drift and lateral stability is an issue that should be addressed in the early stages of design development. In many cases, especially in tall buildings or in cases where torsion is a major contributor to structural response, the drift criteria can become a governing factor in selection of the proper structural system. The lateral displacement or drift of a structural system under wind or earthquake forces, is important from three different perspectives: 1) structural stability; 2) architectural integrity and potential damage to various nonstructural components; and 3) human comfort during, and after, the building experiences these motions. In design of building structures, different engineers attribute various meanings to the term "stability". Here, we consider only those problems related to the effects of deformation on equilibrium of the structure, as stability problems. Furthermore, we will limit the discussion to the stability of the structure as a whole. Local stability problems, such as stability of individual columns or walls, are discussed in Chapters 9, 10, and 11 of the handbook. Several practical methods for inclusion of stability effects in structural analysis as well as simplified drift design procedures are presented. These approximate methods can be valuable in evaluation of the potential drift in the early stages of design. Numerical examples are provided to aid in understanding the concepts, and to provide the reader with the "handson" experience needed for successful utilization of the material in everyday design practice.
327
328
Chapter 7
7. Design for Drift and Lateral Stability
7.1
INTRODUCTION
This chapter deals with the problems of drift and lateral stability of building structures. Design for drift and lateral stability is an issue which should be addressed in the early stages of design development. In many cases, especially in tall buildings or in cases where torsion is a major contributor to structural response, the drift criteria can become a governing factor in selection of the proper structural system. In design of building structures, different engineers attribute various meanings to the term "stability"(71). Here, we consider only those problems related to the effects of deformation on equilibrium of the structure, as stability problems. Furthermore, we will limit the discussion to the stability of the structure as a whole. Local stability problems, such as stability of individual columns or walls, are discussed in Chapters 9, 10, and 11 of the handbook. The concerns that have resulted in code requirements for limiting lateral deformation of structures are explained in Section 7.2. The concept of lateral stability, its relationship to drift and the PDelta effect, and factors affecting lateral stability of structures are discussed in Section 7.3. Several practical methods for inclusion of stability effects in structural analysis are presented in Section 7.4. Simplified drift design procedures are presented in Section 7.5. These approximate methods can be valuable in evaluation of the potential drift in the early stages of design. Section 7.6 covers the drift and PDelta analysis requirements of major United States seismic design codes. Several numerical examples are provided to aid in understanding the concepts, and to provide the reader with the "handson" experience needed for successful utilization of the material in everyday design practice. The relative lateral displacement of buildings is sometimes measured by an overall drift ratio or index, which is the ratio of maximum lateral displacement to the height of
329 the building. More commonly, however, an interstory drift ratio, angle, or index is used, which is defined as the ratio of the relative displacement of a particular floor to the story height at that level (see Figure 71). In this chapter, unless otherwise noted, the term drift means the relative lateral displacement between two adjacent floors, and the term drift index, is defined as the drift divided by the story height.
Figure 71. Definition of drift.
7.2
THE NEED FOR DRIFT DESIGN
The lateral displacement or drift of a structural system under wind or earthquake forces, is important from three different perspectives: 1) structural stability; 2) architectural integrity and potential damage to various nonstructural components; and 3)
330
Chapter 7
human comfort during, and after, the building experiences these motions. 7.2.1
Structural Stability
Excessive and uncontrolled lateral displacements can create severe structural problems. Empirical observations and theoretical dynamic response studies have indicated a strong correlation between the magnitude of interstory drift and building damage potential(72). Scholl(73) emphasizing the fact that the potential for drift related damage is highly variable, and is dependent on the structural and nonstructural detailing provided by the designer, has proposed the following generalization of damage potential in relationship to the interstory drift index δ: 1. at δ = 0.001 ; nonstructural damage is probable 2. at δ = 0.002 ; nonstructural damage is likely 3. at δ = 0.007 ; nonstructural damage is relatively certain and structural damage is likely 4. at δ = 0.015 ; nonstructural damage is certain and structural damage is likely Drift control requirements are included in the design provisions of most building codes. However, in most cases, the codes are not specific about the analytical assumptions to be used in the computation of the drifts. Furthermore, most of the codes are not clear about how the magnifying effects of stability related displacements ,such as Pdelta deformations, are to be incorporated in evaluation of final displacements and corresponding member forces. 7.2.2
for building occupants and those who are in the vicinity of the building. Past earthquakes have proven that nonstructural components can also greatly influence the seismic response of the building. Chapter 13 of the handbook is devoted to this important aspect of seismic design. 7.2.3
Human Comfort
Human comfort and motion perceptibility, which are of importance in the design of structures for wind induced motions, are relatively insignificant in seismic design, where the primary objective is to limit damage and prevent loss of life. For very essential structures, where continued operation of facilities is desired during and immediately after an earthquake, a more conservative design or application of special techniques, such as seismic isolation (see Chapter 14), may be considered. However, here again, the primary goal is to keep the system operational, and to prevent damage, rather than to provide for comfort of the occupants during strong ground motion. Some investigators have studied the behavior of building occupants during strong ground motions (74, 75, 76). Such studies can provide owners, architects, and hazard mitigation authorities, with valuable guidelines for considering these human factors in planning, design, and operation of building structures.
7.3
DRIFT, PDELTA, AND LATERAL STABILITY
7.3.1
The Concept of Lateral Stability
Architectural Integrity
Architectural systems and components, and a variety of other nonstructural items in a building, constitute a large portion of the total investment in the project. In many cases the monetary value of these items exceeds the cost of the structural system by a large margin. In addition, these nonstructural items can be potential sources of injury, and even loss of life,
To illustrate the concept of stability, consider an ideal column without geometrical or material imperfections. Furthermore, assume that there are no lateral loads, and that the column remains elastic regardless of the force magnitude. If the axial force is slowly increased, the column will undergo axial
7. Design for Drift and Lateral Stability deformation, and no lateral displacements will occur. However, when the applied forces reach a certain magnitude called the critical load (Pcr), significant lateral displacements may be observed.
Figure 72. Structural stability of an idealized column and a real frame.
Figure 72a shows the loaddeflection behavior of this ideal column. It is important to notice that when the magnitude of axial force exceeds Pcr, there are two possible paths of equilibrium: one along the original path, with no lateral displacements, and one with lateral displacements. However, equilibrium along the original path is not stable, and any slight disturbance can cause a change in the equilibrium position and significant lateral displacements. The force Pcr is called the bifurcation load or first critical load of the system. For this ideal column reaching the bifurcation point does not imply failure simply because it was assumed that it will remain
331 elastic regardless of the deflection magnitude. However, in a real column, such large deformations can cause yielding, stiffness reduction, and failure. In a structural system, buckling of critical members and the corresponding large lateral displacements, can cause a major redistribution of forces and overall collapse of the system. It is important to note that the bifurcation point, exists only for perfectly symmetric members under pure axial forces. If the same ideal column is simultaneously subjected to lateral loads, or if asymmetry of material or geometric imperfections are present, as they are in any real system, lateral displacements would be observed from very early stages of loading. When a frame under constant gravity load is subjected to slowly increasing lateral loads, the lateral displacement of the system slowly increases, until it reaches a stage that in order to maintain static equilibrium a reduction in the gravity or lateral loads is necessary (Figure 72b). This corresponds to the region with negative slope on the forcedisplacement diagram. If the loads are not reduced, the system will fail. When the same frame is subjected to earthquake ground motion, reaching the negative slope region of the loaddisplacement diagram, does not necessarily imply failure of the system (see Figure 73). In fact, it has been shown that in the case of repeated loads with direction reversals, such as those caused by earthquake ground motion, the load capacity of the system will be significantly larger than the stability load for the same system subjected to unidirectional monotonic loads(71, 78). Perhaps this is one reason for scarcity of stabilitycaused building failures during earthquakes. Exact computation of critical loads, for real buildings, is a formidable task. This is true even in a static environment, let alone the added complexities of dynamic loading and inelastic response. Exact buckling analysis is beyond the capacity and resources of a typical design office, and beyond the usual budget and timeframe allocated for structural analysis of buildings.
332
Chapter 7
In everyday structural analysis, the stability effects are accounted for either by addressing the problem at the element level (via effective length factors), or by application of one of the various PDelta analysis methods.
displacement or drift of the structure. In fact several studies(79, 710, 711, 712) have shown that by increasing lateral stiffness, the critical load of the building will increase and the chances of stability problems are reduced. Drift limitations are imposed by seismic design codes primarily to serve this purpose. 7.3.2
Figure 73. A typical loaddisplacement curve for a frame under constant gravity load and reversing lateral load.
The simplest way to minimize lateral stability problems is to limit the expected lateral
PDelta Analysis
For most practical purposes, an accurate estimate of the stability effects may be obtained by what is commonly referred to as Pdelta analysis. Overall stability failures of structures have not been common during past earthquakes. However, with the continuing trend towards lighter structural systems, and recent discoveries about the nature of nearfield ground motion(713, 714, 715), the secondorder effects are beginning to receive more attention. It is believed that, in most cases, observance of proper drift limitations will provide the necessary safeguard against the overall lateral
Figure 74. Applied loads in the undeformed and deformed states.
7. Design for Drift and Lateral Stability stability failure of the structure.
Figure 75. The Pdelta effect. (a) Equilibrium in the under formed state. (b) Immediate Pdelta effect, (c) Accumulation of the Pdelta effect.
In conventional firstorder structural analysis, the equilibrium equations are formulated for the undeformed shape of the structure. However, when a structure undergoes deformation, it carries the applied loads into a deformed state along with it (Fig. 74). The
333 changes in position of the applied forces are cumulative in nature and cause additional secondorder forces, moments, and displacements which are not accounted for in a firstorder analysis. Studies(716) have shown that the single most important secondorder effect is the Pdelta effect. Figure 75 illustrates the Pdelta effect on a simple cantilever column. In some cases, stability or secondorder effects are small and can be neglected. However, in many other cases such as tall buildings, systems under significant gravity loads, softstory buildings, or systems with significant torsional response, the secondorder effects may be quite significant and hence, should be considered in the structural analysis. Although it is true that ignoring secondorder effects is not likely to result in overall stability failure of typical buildings subjected to earthquake ground motion, these effects can frequently give rise to a series of premature material failures at the level of forces, that would seem safe by a firstorder analysis. Strong evidence relating excessive drift to seismic damage during earthquakes, supports this point.
Figure 76. Plan of the 24 story structure (717).
334 7.3.3
Chapter 7 Factors Affecting Lateral Stability
In general, the magnitude of the gravity loads and factors that increase lateral displacement, affect lateral stability of the structure. Chief among these factors are rotation at the base of the structure(712), any significant rotation at any level above the base (as that caused by formation of plastic hinges in the columns or walls), and significant asymmetry or torsion in the structure.
carrying capacity of the structure, two asymmetric models were constructed by moving the shearwall couple from grid lines three and four, to grid lines four and five in one model, and to grid lines five and six in another model. Loaddisplacement diagrams for the three configurations are shown in Figure 78, where λ is the ratio of the ultimate lateral loads to the working stress lateral loads. Gravity loads were not changed. Reduction in the ultimate lateral load carrying capacity due to induced asymmetry proved to be drastic (51% in one case and 66% in the other case).
Figure 78. Loaddisplacement relationships for various configurations of the 24 story structure (717).
Figure 77. Elevation of the 24 story structure (717).
Wynhoven and Adams(717) studied the effects of asymmetry and torsion on the ultimate load carrying capacity of a 24 story frameshear wall building with typical plan and elevation layouts as shown in Figures 76 and 77. The behavior of individual members was idealized as elasticperfectly plastic. To consider the influence of torsion on the load
7.4
PRACTICAL SECONDORDER ANALYSIS TECHNIQUES
7.4.1
The Effective Length Factor Method
This method is an attempt to reduce the complex problem of overall frame stability to a relatively simple problem of elastic stability of individual columns with various end conditions. The role of the effective length factor K, is to replace an actual column of length L and complex end conditions to an equivalent column of length KL with both ends pinned, so
7. Design for Drift and Lateral Stability
335
that the classic Euler buckling equation can be used to examine column stability. It is further assumed that if the buckling stability of each individual column has been verified by this method, then a system instability will not occur.
Figure 79. Beamcolumn models used in the development of the effective length factor equations.
The general equations for effective length factors are derived from the elastic stability analysis of simple beamcolumn models such as those shown in Figure 79. These equations are (718) : for the sidesway prevented case:
G AG B 4
π 2 G A + GB 2 + 2 K 2 π =1 tan + 2K π/k
π/k 1 − tan π / k
(71) for the sidesway permitted case:
( π / K ) 2 G AG B π G + GB − 1 tan − A K 36 6
π =0 K (72)
where GA and GB are the relative rotational stiffness of the beams to the columns, measured at ends A and B of the column under consideration:
Ic
G=
∑L
c
Ig
∑L
(73)
g
Graphical solutions to these equations are given by the well known SSRC alignment charts(719) shown in Figure 710. The SSRC Guide(719) recommends that for pinned column bases, G be taken as 10, and for column bases rigidly attached to the foundation, the value of G be taken as unity. Furthermore, when certain conditions are known to exist at the far end of a beam, the corresponding beam stiffness term in Equation 73 should be multiplied by a factor. For the sideswayprevented case, this factor is 1.5 for the far end hinged and 2.0 for the far end fixed. For the sideswaypermitted case this factor is 0.5 for the far end hinged and 0.67 for the far end fixed. Effective length factors have been incorporated in the column design interaction equations of several building design codes. The effectivelengthfactor method has been subjected to serious criticism by various researchers. The main criticism is that the effective length factor method, which is based on elastic stability analysis of highly idealized cases, can not be trusted to provide reasonable estimates of the stability behavior of real structural systems. Furthermore, several studies have shown that the lateral stability of a frame, or individual story, is controlled by the collective behavior of all the columns in the story, rather than the behavior of a single column. Hence, if a stability failure is to occur, the entire story must fail as a unit(712). Examples and evidence of the shortcomings of the effective length factor method have been documented, among others, by MacGregor and Hage(716) and ChoengSiatMoy(720, 721). In spite of this evidence, the effective length factor method has continued to survive as a part of the requirements of many building codes. Recently, new editions of some building codes are moving away from this tradition.
336
Chapter 7
Figure 710. Alignment charts for determination of effective length factors(719).
7.4.2
Approximate Buckling Analysis
a
In approximate buckling analysis, the buckling load of a single story, or that of the structure as a whole, is estimated. A magnification factor µ, which is a function of the ratio of the actual gravity load to the buckling load, is defined, and for the design of structural members, all lateral load effects are multiplied by this magnification factor. Then, member design is performed by assuming an effective length factor of one.
a
Parts of section 7.4.2 have been extracted from Reference 622 with permission from Van Nostrand Reinhold Company.
Several approximate methods have been developed for estimation of critical loads of building structures(710, 711, 712, 722). Among these, a simple method developed by Nair(722) is explained here. This method takes advantage of the fact that most multistory buildings have lateral loaddisplacement characteristics that are similar to those of either a flexural cantilever or a shear cantilever. Buildings with braced frames or shear walls, and tall buildings with unbraced frames or tubular frames, usually have lateral load deformation characteristics that approach those of a flexural cantilever. On the other hand, buildings of low or moderate height with unbraced frames (in which column axial deformations are not significant) usually have
7. Design for Drift and Lateral Stability
337
lateral loaddisplacement characteristics similar to those of a shear cantilever. The above observations can be extended to the torsional behavior of structures. If in a multistory building, torsional stiffness is provided by braced frames, shear walls, or tall unbraced frames not exhibiting tube action, the torsionrotation characteristics of the building will be similar to the lateral loaddisplacement characteristics of a flexural cantilever. If a building's torsional stiffness is provided by low to midrise unbraced frames, or by tubular frames, the building will have torsionrotation characteristics that are similar to the lateral loaddisplacement characteristics of a shear cantilever. Buildings Cantilevers
Modeled
as
∆ = 0.125 f H 4/EI
(77)
for EI = (a /H) EI0:
∆ = 0.167 fH 4/EI0
(78)
for EI = (a/H)2EI0:
∆ = 0.250 fH 4/EI0
(79)
Flexural
For a flexural cantilever of height H and constant stiffness EI, the uniformly distributed vertical load, per unit height (Figure 711), pcr , that will cause lateral buckling is given by the equation pcr = 7.84 EI / H 3
(74)
If the stiffness varies with the equation EI = (a /a/H)EI0, where EI0 is the stiffness at the base and a is the distance from the top, the critical load is given by: pcr = 5.78 EI0 /H 3
(75)
If the stiffness varies with the equation EI = (a/H)2EI0, the critical load is: pcr = 3.67 EI0 / H 3
(Eq. 76)
These solutions for critical load can be found in basic texts on elastic stability. If a uniformly distributed lateral load of f per unit height is applied to a flexural cantilever, the lateral displacement ∆ at the top is: for a constant EI:
Figure 711. Lateral loading and buckling of a flexural cantilever(722).
If the lateral load is not uniform, an approximate answer may be obtained by defining f as the equivalent uniform lateral load that would produce the same base moment as the lateral load used in the analysis. By combining Equations 74, 75, and 76 with
338
Chapter 7
Equations 77, 78 and 79, EI can be eliminated and pcr can be expressed in terms of f/∆ , as follows: for a constant EI : pcr = 0.98 fH/D
(710)
for EI = (a/H) EI0: pcr = 0.96 fH/D
(711)
for EI = (a/H )2EI0: pcr = 0.92 fH/D
r2 = (712)
From the above equations it is obvious that the relation between pcr and f/∆ is not very sensitive to stiffness variation over the height of the structure. Hence, regardless of the distribution of stiffness, the following equation is sufficiently accurate for design purposes: pcr = 0.95 fH /∆
(713)
The magnification factor µ, as previously defined, is given by:
µ=
1 1 − γp / φ pcr
(714)
where p is the actual average gravity load per unit height on the building, γ is the design load factor, and φ is the strength reduction factor. Note that p must include the load on all vertical members, including those that are not part of the lateralloadresisting system. Thus, if the lateral displacement is known from a firstorder analysis, the critical load and the corresponding magnification factor can be estimated using Equations 713 and 714. For buildings whose torsional behavior approaches that of a flexural cantilever, the following formula may be used to estimate the torsional buckling load of the structure: r2 pcr = 0.95 tH / θ
where t is an applied torsional load, per unit height of the building, θ is the rotation at the top of the building in radians, pcr is the critical vertical load for torsional buckling per unit height of the building, and r is the polar radius of gyration of the vertical loading about the vertical axis at the center of twist of the building. For a doubly symmetric structure, uniformly distributed gravity loading, and a rectangular floor plan with dimensions a and b:
(715)
a 2 + b2 12
(716)
Buildings Modeled as Shear Cantilevers If a portion of a vertical shear cantilever undergoes lateral deformation δ, over a height h, when subjected to a shear force V, the critical load for lateral buckling of that portion of the cantilever is given by Pcr = Vh/δ
(717)
When the above equation is applied to a single story of a building, h is the story height, δ is the story drift caused by the story shear force V, and Pcr is the total vertical force that would cause lateral buckling of the story (see Figure 712).
Figure 712. Lateral loading and buckling of a story in a shear cantilever type building(722).
7. Design for Drift and Lateral Stability
339
The magnification factor µ , is given by
µ=
1 1 − γP / φPcr
(718)
where P is the total gravity force in the story, γ is the load factor, and φ is the strength reduction factor. The accuracy of Equation 717, when applied to a single story of a framed structure, depends on the relative stiffness of the beams and columns, and on the manner in which the gravity loads are distributed among the columns. The error is greatest for stiff beams and slender columns and may be as high as 20%. For buildings whose torsional behavior approaches that of a shear cantilever, the following equation may be used to estimate the torsional buckling load of a particular story of the building: r2 Pcr = T h/θ
floor. The corresponding magnification factor assuming γ = φ = 1.0, is
µ=
1 = 1.106 1 − 130 / 1360
and the magnified lateral displacement at the roof is given by:
γ∆ = 1.106(0.729) = 0.806 ft An elastic stability analysis of this building indicates a critical load of 1,369 psf for NorthSouth buckling. A largedeformation analysis for combined gravity load and NorthSouth lateral loading indicates a roof displacement of 0.805 ft. (723)
(719)
where T is an applied torsional load on the story, θ is the torsional deformation of the story (in radians) due to the torque T, h is the story height, Pcr is the critical load for torsional buckling of the story, and r is the polar radius of gyration of the vertical load. Application Examples Consider the twenty story buildings shown in Figure 713. The buildings are analyzed using a linear elastic analysis program for a constant lateral load of 25 psf applied in the NorthSouth direction. The EastWest plan widths are 138 ft. The gravity load is assumed to be 130 psf on each floor. For building I, the firstorder displacement at the top is 0.729 ft. Using Equation 713:
Figure 713. Buildings analyzed in references (722) and (723).
H = 240 ft f = 0.025(138) = 3.45 kips/ft ∆ = 0.729 ft pcr =0.95(3.45)(240)/0.729=1079 kips/ft The estimated critical load of 1079 k/ft corresponds to 12,948 kips or 1,360 psf on each
For building II, the computed story drifts for the 15th, 10th, and 5th levels are 0.0522 ft, 0.0609 ft, and 0.0582 ft, respectively. The corresponding story shears at these levels are 228 kips, 435 kips, and 642 kips. Using Equation 717:
340
Chapter 7
15th story: Pcr = 228(12)/0.0522 = 52,414 kips 10th story:Pcr = 435(12)/0.0609 = 85,714 kips 5th story: Pcr = 642(12)/0.0582 = 132,371 kips The corresponding magnification factors assuming γ = φ = 1.0 are: for the 15th story:
µ=
1 = 1.165 1 − 7427 / 52,414
for the 10th story:
1 µ= = 1.189 1 − 13,616 / 85,714 for the 5th story:
µ=
1 = 1.176 1 − 19,806 / 132,371
and the magnified story drifts are: for the 15th story:
µ ∆ = 1.165(0.0522) = 0.0608 ft for the 10th story:
µ ∆ = 1.189(0.0609) = 0.0724 ft for the 5th story:
µ ∆ = 1.176(0.0582) = 0.0684 ft A largedeformation analysis of this building(723) indicates story drifts of 0.0607 ft, 0.0723 ft, and 0.0686 ft for the 15th, 10th, and 5th stories, respectively. 7.4.3
Approximate PDelta Analysis
Three methods for approximate Pdelta analysis of building structures are presented in this section: the iterative Pdelta method; the
direct Pdelta method; and the negative bracing member method. All three methods are shown to be capable of providing accurate estimates of Pdelta effects. Iterative PDelta Method The iterative Pdelta method(716, 724, 725, 726) is based on the simple idea of correcting firstorder displacements, by adding the Pdelta shears to the applied story shears. Since Pdelta effects are cumulative in nature, this correction and subsequent reanalysis should be performed iteratively until convergence is achieved. At each cycle of iteration a modified set of story shears are defined as:
∑V = ∑V + (∑ P )∆ i
1
i −1
/h
(720)
where ΣVi is the modified story shear at the end of ith cycle of iteration, ΣV1 is the firstorder story shear, ΣP is the sum of all gravity forces acting on and above the floor level under consideration, ∆i1 is the story drift as obtained from firstorder analysis in the previous cycle of iteration, and h is the story height for the floor level under consideration. Iteration may be terminated when Vi ≈ Vi −1 or
∆ i ≈ ∆ i −1 .
∑
∑
Generally for elastic structures of reasonable stiffness, convergence will be achieved within one or two cycles of iteration(716). One should note that since the lateral forces are being modified to approximate the Pdelta effect, the column shears obtained will be slightly in error (716) . This is true for all approximate methods which use sway forces to approximate the Pdelta effect.
EXAMPLE 71 For the 10 story moment resistant steel frame shown in Figure 714, modify the firstorder lateral displacements to include the Pdelta effects by using the Iterative Pdelta Method. The computed firstorder lateral displacements and story drifts for the frame are
7. Design for Drift and Lateral Stability
341
Table 71. Applied forces and computed FirstOrder Displacements for the 10story frame. Lateral disp. Level Story height Gravity force Lateral load Story shear D1, in. h, in. ΣP, kips V, kips ΣV1 ,kips 10 144 180 30.22 30.22 7.996 9 144 396 21.94 52.17 7.479 8 144 612 19.57 71.74 6.743 7 144 828 17.20 88.93 5.958 6 144 1044 14.83 103.76 5.051 5 144 1260 12.45 116.21 4.152 4 144 1476 10.08 126.30 3.238 3 144 1692 7.71 134.01 2.400 2 144 1908 5.34 139.34 1.533 1 180 2124 2.97 142.31 0.765
Story drift ∆1, in. 0.517 0.736 0.785 0.907 0.899 0.914 0.833 0.867 0.768 0.765
bent at the 8th level of the frame. The story height (h) is 12 feet (144 in.), the total gravity force at this level (ΣP) is 612 kips, the story shear (ΣV) is 71.74 kips, and the firstorder story drift is 0.785 inches (see Table 71). The PDelta Contribution to the story shear is:
(ΣP )∆1 h
=
(612)(0.785) = 3.34 kips 144
and the modified story shear is:
Figure 714. Elevation of the story moment frame used in Example 71.
shown in Table 71. The tributary width of the frame is 30 ft. The gravity load is 100 psf on the roof and 120 psf on typical floors. Use centertocenter dimensions. The calculations for this example using the iterative Pdelta method are presented in Tables 72 and 73. The convergence was achieved in two cycles of iteration. Table 73 also shows results obtained by an "exact" Pdelta analysis. To further explain the steps involved in the application of this method, let us consider the
ΣV2 = ΣV1 + (ΣP) ∆1 / h = 71.74 + 3.34 = 75.08 kips Repeating this operation for all stories results in a modified set of story shears, from which a modified set of applied lateral forces is obtained (Table 72). A new firstorder analysis of the frame subjected to these modified lateral forces results in a modified set of lateral displacements (D2) and story drifts (∆2) as shown in Table 72. The maximum displacement obtained from the second analysis was 8.478 in., which is 9% larger than the original firstorder displacement. Hence, a second iteration is necessary. Again performing the calculations for the bent at the 8th floor:
(ΣP ) ∆ 2 (612)(0.823) = = 3.50 kips h 144 ΣV3 = ΣV2 + (ΣP) ∆2 / h =71.74 + 3.50 = 75.24 kips Another firstorder analysis for the new set of lateral forces indicates a maximum displacement of 8.508 inches, which is less than
342
Chapter 7
Table 72. Iterative Pdelta method (First cycle of iteration) Level (ΣP) ∆1 / h, Modified lateral ΣV1+(ΣP) ∆1 / h, kips kips Force V2, kips 10 0.65 30.87 30.87 9 2.02 54.19 23.32 8 3.34 75.08 20.89 7 5.22 94.15 19.07 6 6.52 110.28 16.13 5 8.00 124.21 13.93 4 8.59 134.89 10.68 3 10.19 144.20 9.31 2 10.18 149.52 5.32 1 9.03 151.34 1.82
Modified lateral Disp. D2, in. 8.478 7.945 7.178 6.355 5.396 4.441 3.465 2.568 1.638 0.815
Table 73. Iterative Pdelta method (Second cycle of iteration) Level (ΣP) ∆2 / h, Modified lateral ΣV2+(ΣP) ∆2 / h, kips kips Force V3, kips 10 0.67 30.89 30.89 9 2.11 54.28 23.39 8 3.50 75.24 20.96 7 5.51 94.44 19.20 6 6.92 110.68 16.24 5 8.54 124.75 14.07 4 9.19 135.49 10.74 3 10.93 144.94 9.45 2 10.90 150.24 5.30 1 9.62 151.93 1.69
Modified story Drift ∆2, in. 0.533 0.767 0.823 0.959 0.955 0.976 0.897 0.930 0.823 0.815
Modified lateral Disp. D3, in. 8.508 (8.510) 7.975 (7.976) 7.207 (7.209) 6.382 (6.384) 5.419 (5.421) 4.461 (4.462) 3.480 (3.481) 2.580(2.581) 1.645 (1.646) 0.818 (0.819)
Modified story Drift ∆3, in. 0.534 (0.534) 0.768 (0.768) 0.825 (0.825) 0.962 (0.963) 0.959 (0.959) 0.980 (0.980) 0.900 (0.901) 0.935 (0.935) 0.827 (0.827) 0.818 (0.819)
* Values in parentheses represent results of an “exact” Pdelta analysis.
1% larger than the displacements obtained in the previous iteration. Hence, the iteration was terminated at this point. The firstorder and secondorder lateral displacements and story drifts are shown in Figures 715 and 716. As indicated by these figures, the results are virtually identical to the exact results. Direct PDelta Method The direct Pdelta method(716) is a simplification of the iterative method. Using this method, an estimate of final deflections is obtained directly from the first order deflections. The simplification is based on the assumption that story drift at the ith level is proportional only to the applied story shear at that level (ΣVi). This assumption allows the treatment of each level independent of the others. If F is the drift caused by a unit lateral load at the ith level, then the first order drift ∆1 is:
∆1=F ΣV1
(721)
After the first cycle of iteration,
F ∆ 2 = F ΣV2 = F (ΣV1 )1 + (ΣP ) h
(722)
and after the i th cycle of iteration: 2 F F ∆ i +1 = FΣV1 1 + (ΣP ) + (ΣP ) h h i F + L + ( ΣP ) h
(723)
7. Design for Drift and Lateral Stability
343 Equation 723 is a geometric series that converges if (ΣP) F/h) < 1.0, to
∆ Final =
F ΣV1 1 − F1 (ΣP ) / h
(724)
But FΣV1 = ∆1. Hence, the final secondorder deflection is:
∆ Final =
Figure 715. Lateral displacement of the 10story frame as obtained by various Pdelta methods.
Figure 716. Story drift ratios of the 10 story frame as obtained by various Pdelta methods.
∆1 1 − (ΣP)∆1 /(ΣV1 )h
(725)
Equation 725 can be expressed as ∆Final = µ∆1, where µ = 1/[1(ΣP)∆1/(ΣV1)h] is a magnification factor by which the firstorder effects should be multiplied to include the secondorder effects. All internal forces and moments related to the lateral loads should also be magnified by µ. Member design may be carried out using an effective length factor of one. An estimate of the critical load for an individual story, or the entire frame, can be obtained directly from Equation 725. Note that if (ΣP)∆1/(ΣV1)h = 1, the secondorder displacement would go to infinity. Hence, ΣP = (ΣV1)h/∆1 may be considered to be the critical load of the system. Similarly, Σ(Pr2) = ΣT1 h/θ1 can be viewed as the torsional critical load of the system. It is interesting to note that the critical loads and the magnification factor obtained here are in essence the same as those obtained in Section 7.4.2. by an approximate buckling analysis. The term (ΣP)∆1/(ΣV1)h is commonly referred to as the stability index. Similarly, a torsional stability index may be defined as Σ(Pr2)θ1/(ΣT1h). It has been suggested(716) that if the stability index is less than 0.0475 for all three axes of the building, the secondorder effects can be ignored. For values of the stability index between 0.0475 and 0.20, the direct Pdelta method can provide accurate estimates of the secondorder effects. Designs for which values of the stability index exceed 0.20 should be avoided.
344
Chapter 7
Table 74. Pdelta analysis by direct Pdelta method (Example 72) Level h, ΣP, ΣV1, ∆1, kips in. in kips 10 144 30.22 180 0.517 9 144 52.17 396 0.736 8 144 71.74 612 0.785 7 144 88.93 828 0.907 6 144 103.76 1044 0.899 5 144 116.21 1260 0.914 4 144 126.30 1476 0.838 3 144 134.01 1692 0.867 2 144 139.34 1908 0.768 1 180 142.31 2124 0.765
EXAMPLE 72 For the 10 story frame of Example 71 compute the secondorder displacements and story drifts by the direct Pdelta method. The calculations using the direct Pdelta method are shown in Table 74. For example, for the first floor which has a story height of 15 feet (180 inches), the story shear is 142.31 kips, the total gravity force is 2124 kips, and the firstorder drift is 0.765 inches. The magnification factor and the secondorder displacements are:
1 = 1.068 1 − ( 2124)(0.765) /(142.31)(180) ∆ 2 = µ∆1 = (1.068)(0.76) = 0.817 in.
µ=
A comparison with the exact results (Figures 715 and 716) reveals the remarkable accuracy of this simple technique. Negative Bracing Member Method The negative bracing member method(716, 726, 727), which was first introduced by Nixon, Beaulieu and Adams(727), provides a direct estimate of the PDelta effect via any standard firstorder analysis program. Fictitious bracing members with negative areas are inserted (Figure 717) to model the stiffness reduction due to the Pdelta effect. The cross sectional area of the negative braces for each floor level can be obtained by a
µ 1.022 1.040 1.049 1.062 1.067 1.074 1.073 1.082 1.079 1.068
∆2=µ∆1, in. 0.528 0.766 0.823 0.964 0.959 0.982 0.899 0.938 0.829 0.817
2ndOrder Disp.,in. 8.505 7.977 7.211 6.388 5.424 4.465 3.483 2.584 1.646 0.817
simple analogy to the Hooke's law (F = K∆). The additional shear due to Pdelta effect is (ΣP)∆/h, where ΣP is the total gravity force and h is the story height. The term ΣP/h is a stiffness term but it is contributing to lateral displacement instead of resisting it. Hence, it can be considered as a negative stiffness. A brace with a cross sectional area A, a length Lbr, modulus of elasticity E, making an angle α with the floor, provides a stiffness equal to (AECos2α)/Lbr against lateral displacement. By equating the brace stiffness to ΣP/h, the required area of the equivalent negative brace is obtained:
A=−
ΣP Lbr h E cos 2 α
(726)
It is important to note that, due to the horizontal and vertical forces in the braces, the axial forces and shears in the columns will be slightly in error. These errors can be reduced by making the braces as long as possible (see Figure 717).
EXAMPLE 73 For the 10 story frame of Example 71, compute the secondorder displacements and story drifts by the Negative Bracing Member Method. The modulus of elasticity of the braces is:
7. Design for Drift and Lateral Stability
345
Lbr = (60) 2 + (15) 2 = 61.847 ft. cos 2 α = (60 / 61.847) 2 = 0.9412 The negative brace area for each floor level may now be calculated using Equation 726. For example, for the fourth floor where the total gravity force is 1476 kips, the negative brace area is:
A4 = −
(1476)(734.26) (144)( 29000)(0.9615)
= −0.2699 in 2 The brace areas, and the displacements obtained using the negative braces, are shown in Table 75. The very good agreement with the "exact" results (Table 73) is evident.
Figure 717. Frame modeled with negative braces.
E = 29,000. Ksi For a typical floor,
Lbr = (60) 2 + (15) 2 = 61.188 ft. cos 2 α = (60 / 61.188) 2 = 0.9615 For the first floor,
Modified Versions of Approximate Pdelta Methods The PDelta methods presented in this chapter ignore the "CS" effect (Figure 74d). For most practical problems, the CS effects are much smaller than the Pdelta effects, and can be ignored. However, if needed, the Pdelta methods described in previous sections, can be simply modified to include this effect. The modification is achieved by multiplying the member axial forces by a flexibility factor, γ. For a single column, γ is given by(726):
Table 75. Pdelta analysis by negativebracingmember method. Level h, ΣP, Lbr, in. in kips 10 144 180 734.26 9 144 396 734.26 8 144 612 734.26 7 144 828 734.26 6 144 1044 734.26 5 144 1260 734.26 4 144 1476 734.26 3 144 1692 734.26 2 144 1908 734.26 1 144 2124 742.16
E cos2α 27,884 27,884 27,884 27,884 27,884 27,884 27,884 27,884 27,884 27,295
Abr, in. 0.0329 0.0724 0.1120 0.1514 0.1909 0.2341 0.2699 0.3094 0.3489 0.3209
2ndOrder Disp.,in. 8.458 7.929 7.168 6.350 5.394 4.442 3.468 2.572 1.642 0.817
346
Chapter 7
γ = 1 + 0.22
4(G A − GB ) 2 + (G A + 3)(G B + 2) [(G A + 2)(GB + 2) − 1]2 (727)
where GA and GB are the stiffness ratios as defined in Section 7.4.1. The flexibility factor γ has a rather small range of variation (from 1.0 for GA = GB = ∞ , to 1.22 for GA = GB = 0.). For design purposes a conservative average value of γ can be used for the entire frame. Lai and MacGregor(726) suggest an average value of γ = 1.15, while Stevens(710) has proposed an average value of γ = 1.11. To include the CS effect in the previously discussed Pdelta methods, it is sufficient to use γΣP instead of ΣP wherever the term ΣP appears. EXAMPLE 74 For the 10story frame of Example 71, compute the secondorder displacements and story drifts at the first, fifth, and the roof levels by the modified direct Pdelta method. An average value of γ = 1.11 is assumed for all calculations. Using the values listed in Table 74 we have:
• at the roof: γ (ΣP )∆1 (1.11)(180)(0.517) = = 0.024 (ΣV1 )h (30.22)(144) 1 µ= = 1.025 1 − 0.024 ∆ 2 = µ∆1 = (1.025)(0.517) = 0.530 in. • at the fifth level:
γ (ΣP )∆1 (1.11)(1260)(0.914) = = 0.076 (ΣV1 )h (116.21)(144) 1 µ= = 1.082 1 − 0.076 ∆ 2 = µ∆1 = (1.082)(0.914) = 0.989 in.
• and at the first level:
γ (ΣP )∆1 (1.11)(2124)(0.765) = = 0.070 (ΣV1 )h (142.31)(180) 1 µ= = 1.075 1 − 0.070 ∆ 2 = µ∆1 = (1.075)(0.765) = 0.822 in. Comparison of these results with those obtained by the original method reveals an increase of less than 1% in the story drifts due to this modification. 7.4.4
"Exact" PDelta Analysis
Construction of the geometric stiffness matrix is the backbone of any exact secondorder analysis. The same matrix is also essential for any finite element buckling analysis procedure. In this section, the concept of geometric stiffness matrix is introduced, and a general approach to "exact" secondorder structural analysis is discussed. Consider the deformed column shown in Figure 718. For the sake of simplicity, neglect the axial deformation of the member, and the small CS effect. The slope deflection equations for this column can be written as(712)
Mt =
6∆ 6∆ EI 4 θ t + 2θ b − t + b L L L
(728)
Mb =
6∆ 6∆ EI 2 θ t + 4θ b − t + b L L L
(729)
From force equilibrium:
Ft = −
M t + M b P( ∆ t − ∆ b ) − L L
Fb = − Ft
(730)
(731)
7. Design for Drift and Lateral Stability
347
Substituting Equations 728 and 729 into Equation 730:
Ft = −
6 EI EI P ( θt + θb ) + 12 3 − ( ∆ t − ∆ b ) 2 L L L (732)
Now if we rewrite the above equations in a matrix form, we obtain: 4 EI Mt L 2 EI M b L = F − 6 EI t L2 Fb 6 EI L2
2 EI
−
6 EI 2 L
−
6 EI 2 L
L 4 EI L −
6 EI 2 L
6 EI 2 L
P 12 EI − 3 L L −
P 12 EI 3 + L L
θ t 6 EI θ 2 b L P 12 EI − + ∆t 3 L L P 12 EI ∆ b − 3 L L 6 EI 2 L
(733) Since we wrote the equilibrium equations for the deformed shape of the member, this is a secondorder stiffness matrix. Notice that the only difference between this matrix, and a standard firstorder beam stiffness matrix, is the presence of P/L or geometric terms. The stiffness matrix given by Equation 733 can also be written as:
[K ] = [K f ]− [K g ]
(734)
where [Kf] is the standard firstorder stiffness matrix (material matrix) and [Kg] is the geometric stiffness matrix given by:
0 0 0 0
0 0 0 0 0 0 0 + P / L − P / L 0 − P / L + P / L
Inspection of the simple secondorder stiffness matrix given by Equation 733 shows why general secondorder structural analysis
has an iterative nature. The matrix includes P/L terms, but the axial force P is not known before an analysis is performed. For the first analysis cycle, P can be assumed to be zero (standard firstorder analysis). In each subsequent analysis cycle, the member forces obtained from the previous cycle are used to form a new geometric stiffness matrix, and the analysis continues until convergence is achieved. If inelastic material behavior is to be considered, then the material stiffness matrix must also be revised at appropriate steps in the analysis. Substantial research has been performed on the formulation of geometric stiffness matrices and finite element stability analysis of structures(728,736). A complete formulation of the threedimensional geometric stiffness matrix for wide flange beamcolumns has been proposed by Yang and McGuire (736). The common assumption that floor diaphragms are rigid in their own plane, allows condensation of lateral degrees of freedom into three degrees of freedom per floor level: two horizontal translations and a rotation about the vertical axis. This simplification significantly reduces the effort required for an "exact" secondorder analysis. A number of schemes have been developed to permit direct and noniterative inclusion of PDelta effects in the analysis of rigiddiaphragm buildings (737, 738, 739) . The geometric stiffness matrix for a three dimensional rigid diaphragm building is given in Figure 719(737, 738). For a threedimensional building with N floor levels, [Kg] is a 3N × 3N matrix. For planar frames, the matrix reduces to an N × N tridiagonal matrix. The nonzero terms of this matrix are given by:
αi =
(ΣP ) i (ΣP ) i +1 + hi hi +1
(735)
βi =
(ΣT ) i (ΣT ) i +1 + hi hi +1
(736)
348
Chapter 7
ηi = −
(ΣP) i hi
(737)
λi = −
(ΣT ) i hi
(738)
imposed unit rigid body rotation of the floor. Assuming that the dead load is evenly distributed over the floor and that a roughly uniform vertical support system is provided over the plan area of the floor, Equation 742 can be further simplified to
Ti = mRi
g hi
(743)
where mRi is the rotational mass moment of inertia of the ith floor and g is the gravitational acceleration. The approximation involved in the derivation of Equation 743 is usually insignificant(739). Hence, for most practical problems, Equation 743 can be used instead of Equation 742, thereby allowing the direct inclusion of the Pdelta effect in a three dimensional structural analysis. 7.4.5 Figure 718. Geometric stiffness matrix for threedimensional rigid diaphragm buildings.
where hi is the floor height for level i, Pi is weight of the i th level, Ti is the secondorder story torque, and n
(ΣP) i = ∑ Pj
(739)
j=i n
( ΣT ) i = ∑ T j
(740)
j=i
(ΣP)i can also be represented in terms of story mass, mi, and gravitational acceleration, g, as
n (ΣP ) i = ∑ m j × g j=i
(741)
The story torque, Ti, is given by (738)
n θ Ti = ∑ p j d 2j j=i hi
(742)
where pj is the vertical force carried by the jth column, dj is the distance of jth column from the center of rotation of the floor, and θ is an
Choice of Member Stiffnesses for Drift and PDelta Analysis
A common difficulty in seismic analysis of reinforced concrete structures is the selection of a set of rational stiffness values to be used in force and displacement analyses. Should one use gross concrete section properties? Should one use some reduced section properties? Or should the gross concrete properties be used for one type of analysis and reduced section properties be used for another type of analysis? The seismic design codes in the United States are not specific about this matter. Hence, the choice of section properties used in lateral analysis in general, and seismic analysis in particular, varies widely. Contributing to the complexity of this issue, are the following factors: 1. Although elastic material behavior is usually assumed for the sake of simplicity, reinforced concrete is not a homogeneous, linearly elastic material. 2. Stiffness and idealized elastic material properties of a reinforced concrete section vary with the state of behavior of the section (e.g. uncracked, cracked and ultimate states).
7. Design for Drift and Lateral Stability 3. Not all reinforced concrete members in a structure, and not all cross sections along a particular member, are in the same state of behavior at the same time. 4. For many beams and other nonsymmetrically reinforced members, the stiffness properties for positive bending and negative bending are different. 5. Stiffness of reinforced concrete members and structures varies with the time, and with the history of past exposure to wind forces and earthquake ground motions. 6. Stiffness of reinforced concrete members and structures varies with the amplitude of the applied forces. Analytical and experimental studies(740) have indicated that for motions which are within the working stress design limits of members, the measured fundamental periods of concrete structures are generally slightly less than the periods computed using gross concrete section properties. According to Reference 740, in the case of large amplitude motions up to the yield level, the stiffness of the building is usually somewhere between the computed values based on the gross concrete section properties and the cracked section properties. Based on this observation, the same reference suggests that for force analysis, the gross concrete section properties and the clear span dimensions be used and the effect of nonseismic structural and nonstructural elements be considered. For drift calculations, either the lateral displacements determined using the above assumptions should be doubled or the center to center dimensions along with the average of the gross section and the cracked section properties, or one half of the gross section properties should be used. Furthermore, the nonseismic structural and nonstructural elements should be neglected, if they do not create a potential torsional reaction. Similar sets of assumptions have been proposed by research workers who have been concerned about the choice of member stiffnesses to be used in the Pdelta analysis of concrete structures. For example, for secondorder analysis of concrete structures subjected
349 to combinations of gravity and wind loads, MacGregor and Hage(716) recommend using 40% of the gross section moment of inertia for beams and 80% of the gross section moment of inertia for columns. See Chapter 15 for more information on this subject.
7.5
DRIFT DESIGN PROCEDURES
7.5.1
Drift Design of Moment Frames and Framed Tubes
The lateral displacements and story drifts of moment resistant frames and symmetrical framed tubes are caused by bent action, cantilever action, the shear leak effect, and panel zone distortions. With the simplified methods presented in this section, the contribution of each of these actions to the story drift can be estimated separately. The story drifts so obtained are then added to obtain an estimate of the total story drift. Once an estimate of the drift and the extent of the contribution of each of these actions to the total drift are known, proper corrective measures can be adopted to reduce story drifts to an acceptable level. Bent Displacements A significant portion of drift in rigid frames and framed tubes is caused by end rotations of beams and columns (Figure 720). This phenomenon is commonly referred to as bent action (also called frame action, or racking). For most typical low to midrise rigid frames, almost all of the drift is caused by the bent action. However, for taller frames, other actions such as axial deformation of columns (cantilever or chord action) become more significant. For extremely tall frames, the contribution of cantilever action to drift may be several times larger than that of the bent action. In the design of framed tubes, it is usually desirable to limit the bent action drifts to 30 to 40% of the total drift. If a framed tube is also braced, the bent action drifts are usually limited to about 20 to 25% of the total drift(71). The
350
Chapter 7
bent action drift ∆bi for any level i of a frame, may be estimated by(741):
Figure 720. Typical subassemblage used in derivation of the bent action drift equation (741).
Figure 719. Frame deformation caused by the bent action.
(ΣV ) i h 1 1 ∆ bi = + 12 E (ΣK g )i (ΣK c ) i 2 i
(744)
where (ΣV)i is the story shear, hi is the story height b, and (ΣKg)i = summation of Igi / Lgi for all girders (ΣKc)i= summation of Ici / hi for all columns Igi = individual girder moment of inertia Lgi = individual bay length Ici = individual column moment of inertia Equation 744 can be derived by applying the slope deflection equations to the typical subassemblage shown in Figure 721. In the derivation of Equation 744, it is assumed that the points of contraflexure are at the midspan of beams and columns.
b
Depending on the modeling assumption, centertocenter length, clear length, or something in between may be used.
Figure 721. The bent at the 5th floor (Example 75).
Other, but similar, relationships for bent drift design have been proposed(742, 743). Equation 744 can also be used to modify existing beam and column sizes to satisfy a given drift limit. Example 75 illustrates such an application. EXAMPLE 75 For the bent at the 5th floor of the 10story frame of Example 71 (Figure 722), estimate the story drift caused by bent action. Modify member sizes, if necessary, to limit the bent drift ratio to 0.0030. Neglect the Pdelta effect. W14×68 W14×90 W21×50
Ic1 = 723 in4 Ic2 = 999 in4 Ig = 984 in4
I g (3)(984) = 12.30 in 3 (12)( 20) = g
∑ L
7. Design for Drift and Lateral Stability
351
I (2)(723 + 999) ∑ hc = (144) = 23.92 in.3
2. Increasing beam sizes only:
116(144) 2 1 1 ∆ bi = + (12)( 29000) 12.3 23.92 = 0.85 in. δ bi =
1 1 0.432 = 6.912 → Φ g = 3.93 + 12.3 Φ 23 .92 g I g = (3.93)(984)
= 3867 in. 4 → use W30 × 99 : I = 3990 in 4
0.85 = 0.0059 > 0.0030 N.G. 144
Check the new bent drift:
1. Increasing both beam and column sizes:
I g (3)(3990) = = 49.9 in.3 240 g 1 1 ∆ bi = 6.912 + 49.9 23.92 = 0.427 in. < 0.432 in. O.K.
∑ L
∆Limit = (0.0030)(144) = 0.432 in.
∆ Limit =
∆ bi 0.85 or 0.432 = → Φ = 1.97 Φ Φ
Select new beam and column sizes: I c1 = (1.97)(723) = 1424 in 4 → use W14 × 120 : I = 1380 in 4
= 1968 in 4 → use W14 × 176 : I = 2140 in. 4
I g = (1.97)( 984) = 1938 in → use W24 × 76 : I = 2100 in
4
Check the new bent drift:
I g (3)( 2100) = = 26.25 in.3 240 g
∑ L
Ic
∑ h =
Additional member weight required for drift control: W = 3(9950)(20) = 2940 lb
I c 2 = (1.97)(999)
4
W=3 (7650)(20) + 2 (176+1206890)(12) =4872 lb
( 2)(1380 + 2140) = 48.89 in 3 144
1 1 ∆ bi = 6.912 + 26.25 48.89 = 0.405 in. < 0.432 in. O.K. Additional member weight required for drift control:
3. Increasing column sizes only: 1 1 → Φ c < 0. 0.432 = 6.912 + 12 . 3 23 . 92 Φ c
Therefore, bent drift control by increasing column sizes only is not feasible. In this case, drift control by increasing beam sizes only, requires less material. However, in general, one should be careful about increasing beam sizes alone, since it can jeopardize the desirable strong columnweak girder behavior. Cantilever Displacements In tall frames and tubes, there is significant axial deformation in the columns caused by the overturning moments. The distribution of axial forces among the columns due to the overturning moments is very similar to distribution of flexural stresses in a cantilever beam. The overturning moments cause larger axial forces and deformations on the columns which are
352
Chapter 7
farther from the center line of the frame. This action, which causes a lateral deformation that closely resembles the deformation of a cantilever beam (Figure 723), is called the cantilever or chord action. In a properly proportioned framed tube, the cantilever deflections are significantly smaller than a similar rigid frame. As shown in Figure 724, this is due to the participation of some of the columns in the flange frames in resistance to cantilever deformations. The taller the framed tube, the closer the column spacings, and the stronger the spandrel girders, the more significant the tube action becomes.
Figure 723. Tube action in response to lateral loads.
Figure 722. Cantilever or chord deformation.
Cantilever displacements may be estimated by simple application of the momentarea method. The moment of inertia for an equivalent cantilever beam is computed as:
I 0i = ∑ ( Aci d i2 )
Step 1 Compute story moment of inertia Ioi using Equation 745. Step 2 Compute overturning moments Mi. Step 3 Compute Area under the M/EIoi from:
Ai =
( M i + M i +1 )hi 2 EI 0i
(746)
(745)
where Aci is cross sectional area of an individual column and di is its distance from the centerline of the frame. The summation is carried over all the columns of the web frames, and those columns of the flange frames which are believed to participate in resistance to cantilever deflections. The computation of cantilever displacements for each floor level can be summarized in the following steps.
Step 4 Compute xi (see Figure 725) from:
xi =
hi M i + 2 M i +1 3 M i + M i +1
(747)
Step 5 Compute story displacement from: i −1
∆ ci = Ai ( hi − xi ) + ∑ A j ( H i − x j )
(748)
j =1
where Hi is the total height of the ith floor measured from the base of the structure.
7. Design for Drift and Lateral Stability
353
( 20)(18,000 + 24,000) = 28.00 in. 18,000 + 12,000 ( 20)(12,000 + 12,000) x2 = = 26.67 in. 12,000 + 6000 (20)(6000 + 0) x3 = = 20.00 in. 6000 + 0 x1 =
Displacements:
∆1 = 0.01552(60 − 28) = 0.497 in. ∆ 2 = 0.01552(120 − 28) + 0.00931(60 − 26.67) = 1.738 in. ∆ 3 = 0.01552(180 − 28) + 0.00931(120 − 26.67) + 0.00310(60 − 20) = 3.352 in.
Figure 724. Estimating cantilever displacements by the moment area method.
EXAMPLE 76 Use the momentarea method and the procedure explained in this section to compute displacements at points 1, 2 and 3 of the simple cantilever column shown in Figure 726. Assume EI = 58 × 106, kipsin2 Overturning moments: M3 = 0. M2 = (100)(60) = 6000. in.kips M1 = (100)(120) = 12000. in.kips M0 = (100)(180) = 18000. in.kips Area under M/EI curve: A0 = 0.
(18,000 + 12,000)(60) A1 = = 0.01552 ( 2)(58 × 10 6 ) (12,000 + 6000)(60) A2 = = 0.00931 (2)(58 × 106 ) (6000 + 0)(60) A3 = = 0.00310 ( 2)(58 × 10 6 ) xi distances: x0 = 0
Figure 725. Cantilever column of example 76.
Shear Leak Displacements In buildings with closely spaced columns and deep girders, such as framed tubes, the contribution of shearing deformations to the lateral displacements (called the shear leak effect) may be significant. Story drifts due to the shear leak effect at level i, ∆shi, may be estimated as (741)
∆ shi =
ΣVi hi2 G
1 1 ΣA′ L + ΣA′ h ci i gi gi
(749)
354
Chapter 7
where G is the shear modulus and A'gi and A'ci are the shear areas of individual girders and columns at level i. In order to simplify the design process, an effective moment of inertia, Ieff, can be defined where the contributions of both flexural and shearing deformations are considered
I eff =
A′L2 I 24(1 + v ) I + A′L2
(750)
and the panel zone distortions. It also assumes a uniform distribution of shear stress throughout the panel zone. A simple beamcolumn subassemblage and the corresponding force and displacement diagrams, as assumed by this method, are shown in Figure 727. It can be shown that the deformation angle γ and the additional lateral story drift due to panel zone distortion, ∆p, are:
γ= where A' is the shear area, L is span length, I is the moment of Inertia of the section, and v is Poisson's ratio. EXAMPLE 77 For the bent of Example 75, estimate the additional story drift caused by the shear leak effect. We have
2( M c / d g ) − V
∆p =
Gtd c γ(h − d g ) 2
(751) (752)
where Mc is the moment from one column, dg is the girder depth, V is the column shear, G is the shear modulus, t is the panel zone thickness, dc is the column depth, and h is the story height. Hence, (h  dg) is the clear column height.
W14×68: A'= dtw = (14.00)(0.415) = 5.83 in.2 W14×90: A'= dtw = (14.02)(0.440) = 6.17 in.2 W21×50: A'= dtw = (20.83)(0.380) = 7.92 in.2 ΣA'giLi = (3)(7.92)(240) = 5702.4 ΣA'cihi = (2)(6.17 + 5.83)(144) = 3456.0 Using Equation 749: ∆ shi =
116(144) 2 11,200
1 1 + = 0.10 in. 5702 . 4 3456 .0
Panel Zone Distortions When joint shear forces are high, and the beamcolumn panel zones are not adequately stiffened, panel zone distortions can have a measurable impact on the story drift. The panel zone forcedeformation behavior is complex and nonlinear. Currently, there is no real consensus among researchers on appropriateness of various designoriented approaches to this problem. CheongSiatMoy(744) has recommended a simple method based on elastic theory to estimate this effect. The method assumes a linear relationship between the shearing forces
Figure 726. Effect of panel zone deformation(744)
7. Design for Drift and Lateral Stability
355
If the points of contraflexure are assumed to be at midspan of the beams and columns, Equation 751 can be further simplified to:
γ =V
(h / d g ) − 2 Gtd c
(753)
Considering the approximate nature of the above formula, it is not necessary to apply it to each individual column. Instead, it can be used in an average sense (see Example 78). A series of experimental and analytical studies on the behavior of steel beamcolumn panel zones have been conducted by various research institutions (745,746,747,748). In one of these studies(748), conducted at Lehigh University, several beamcolumn subassemblage specimens were subjected to cyclic loads far beyond their elastic limits. Based on these tests a formula, similar to Equation 753, for estimation of panel zone distortions was recommended:
γ=
V Lc L − Gd c t d g h
(754)
where L is the beam span length, Lc is clear column length , G is the shear modulus which is taken as 11,000 ksi, and γ is the panel zone distortion in radians. There is a serious need for further research on the seismic behavior of beamcolumn panel zones. EXAMPLE 78 For the bent of Example 75, estimate the contribution of panel zone distortion to story drift assuming two conditions: a) No doubler plates, and b) 1/4in. doubler plates. d = 14.04 in t = 0.450 in W14 × 68 W14 × 90 d = 14.02 in t = 0.440 in dg = 20.83 in W21 × 50 Using CheongSiat Moy method (Equations 752 and 753), we have
γ =V
h / dg − 2 Gtd c
without doubler plates: Average t = 2
0.450 + 0.440 = 0.445 in. 4
Average V = 116/4 = 29 kips
144 / 20.83 − 2 = 0.0020 (11200)( 0.445)(14.03) h − dg ∆p = γ = 0.0010(144 − 20.83) 2 = 0.123 in. γ = 29
with doubler plates: Average t = 0.445 + 0.25 = 0.695 in
∆ p = 0.0013
144 − 20.83 = 0.080 in. 2
Using Lehigh’s formula (Equation 754):
γ=
V Lc L − Gd c t d g h
Lc= 144  20.83 = 123.17 in L = 12(20) = 240 in without doubler plates: t = 0.445 in
( 29)(123.17 / 20.83 − 240 / 144) = (11000)(14.03)(0.445) 0.00179 rad.
γ =
∆p = (0.00179)(144  20.83)/2 = 0.110 in. with doubler plates: t = 0.695 in.
γ =
(0.00179)(0.445) = 0.00115 rad. 0.695
∆p = (0.110)(0.00115)/(0.00179) = 0.071 in.
356
Chapter 7
Table 76. Calculation of bentaction story drifts and lateral displacements for the 10story unbraced frame Level h, Σ(Ic/h), ∆bi,in. ΣV, Σ(Ig/Lg), in.3 in.3 (Eq. 744) in. kips 10 144 30.22 6.475 12.68 0.420 9 144 52.17 6.475 12.68 0.725 8 144 71.74 10.538 17.56 0.649 7 144 88.93 10.538 17.56 0.805 6 144 103.76 12.300 23.92 0.761 5 144 116.21 12.300 23.92 0.856 4 144 126.30 16.875 29.47 0.701 3 144 134.01 16.875 29.47 0.744 2 144 139.34 16.875 43.61 0.682 0.461 1 180 142.31 16.875 52.33* * Twothirds of the first story height was used in calculation of the bentaction drift. Table 77. Calculation of shearleak story drifts and lateral displacements for the 10story unbraced frame. Level h, Σ(Ac ′h), ∆shi,in. ΣP, Σ(Ag′Lg), in.3 in.3 (Eq. 744) in. kips 10 144 30.22 3516 2550 0.0379 9 144 52.17 3516 2550 0.0653 8 144 71.74 5206 3161 0.0675 7 144 88.93 5206 3161 0.0837 6 144 103.76 5999 3455 0.0893 5 144 116.21 5999 3455 0.1000 4 144 126.30 6703 4267 0.0897 3 144 134.01 6703 4267 0.0951 2 144 139.34 6703 5379 0.0864 1 180 142.31 6703 5379 0.1226
Drift Design of a 10 Story Moment Resistant Frame In this subsection the approximate methods for drift and Pdelta analysis which were explained previously, are put into practice by performing a complete drift design for the 10story moment resistant steel frame introduced in Example 71. The goal is to achieve an economical design that meets the story drift index limitation of 0.0033. The first step is to estimate the lateral displacements and story drifts of the structure. Calculations of story drifts and lateral displacements due to bent action, the shear leak effect, and chord action are presented in Tables 76, 77 and 78 respectively. It was demonstrated in Example 78 that the contribution of panel zone deformations to story drifts for this structure, at the level of forces considered here, is not significant.
Bent Disp., in. 6.802 6.382 5.657 5.001 4.203 3.442 2.588 1.877 1.143 0.461
Bent Disp., In. 0.8377 0.7998 0.7345 0.6670 0.5833 0.4940 0.3939 0.3042 0.2091 0.1226
Therefore, this effect is ignored in subsequent analyses. The total displacements and story drifts are magnified using the direct Pdelta Method. These calculations are shown in Table 79. Notice that in sizing the members for strength, all lateral load related forces and moments should also be multiplied by the corresponding story magnification factors (see µ in Table 79). Once the internal forces are thus magnified, it is rational to design the members using an equivalent length factor of one. Figures 728 and 729 depict the contribution of each action to the total lateral displacement and story drift. The dominance of bent action in the lateral response of this frame can be clearly seen in these figures. As explained previously, if the frame was significantly taller, bent action would be
7. Design for Drift and Lateral Stability
357
Table 78. Calculation of chordacrtion and lateral displacements for the 10story unbraced frame Level h, Ioi, Chord disp. A, x ΣV Mov,a in. inkips in. in. in4 kips 10 144 30.22 4,352 3,672,000 0.294×105 48.00 0.5746 9 144 52.17 11,864 3,672,000 1.096×105 60.88 0.5024 64.72 0.4250 8 144 71.74 22,194 4,619,520 1.830×105 66.63 0.3412 7 144 88.93 35,001 4,619,520 3.074×105 67.78 0.2617 6 144 103.76 49,942 5,947,200 3.546×105 68.56 0.1840 5 144 116.21 66,677 5,947,200 4.868×105 69.12 0.1176 4 144 126.30 84,864 7,168,320 5.249×105 69.55 0.0619 3 144 134.01 104,161 7,168,320 6.547×105 69.89 0.0253 2 144 139.34 124,226 9,639,360 5.882×105 87.20 0.0082 1 180 142.31 149,841 9,639,360 8.824×105 a Overturning moment.
Chord drift, in. 0.0722 0.0774 0.0838 0.0795 0.0777 0.0664 0.0557 0.0366 0.0171 0.0082
Table 79. Calculation of total first and second order story drifts and lateral displacements for the 10story unbraced frame 2nd –Order Level h, µ ∆2= µ ∆1 ΣV, ΣP, ∆1 in. in. Disp.,in. in. kips kips 10 144 30.22 180 0.517 1.022 0.528 8.547 9 144 52.17 396 0.849 1.047 0.889 8.019 8 144 71.74 612 0.773 1.048 0.810 7.130 7 144 88.93 828 0.941 1.065 1.002 6.320 6 144 103.76 1044 0.898 1.067 0.958 5.318 5 144 116.21 1260 0.987 1.080 1.066 4.360 4 144 126.30 1476 0.833 1.073 0.894 3.294 3 144 134.01 1692 0.865 1.082 0.936 2.400 2 144 139.34 1908 0.786 1.081 0.850 1.464 1 180 142.31 2124 0.584 0.614 0.614 0.614
replaced by chord action as the dominant contributor to lateral displacement. The results of this approximate analysis are compared to the results of an exact elastic analysis in Figures 730 and 731, where the good agreement between the two sets of results may be observed. Given the dominance of bent action in this case, a simple drift design strategy based on reducing the bent drift is adopted. The maximum bent drift is about 80% of the maximum total drift. Hence, it would be rational to reduce the bent drift ratios to 80% of the maximum allowable value of 0.0033 (≈ 0.0026). It should be noted that increasing member sizes would further reduce the contribution of chord and shear leak actions to the drift. Assuming that the drift control is to be achieved by increasing both beam and column sizes, the average magnification factors Φ by which the moment of inertia of beams and
columns should be multiplied can be calculated as described in part 1 of Example 75. Based on the average values of Φ, new member sizes for beams and columns are selected. These member sizes are shown in Figure 732, where the computed values of Φ are shown in parenthesis. At this stage, another round of displacement analysis, similar to that performed in Tables 76 to 79, is necessary to make sure that the new design satisfies the drift design criteria. Results of this analysis are shown in Figures 733 and 734, which indicate that the new design satisfies the design drift criteria. This was also confirmed by performing an exact structural analysis (Figures 735 and 736). The last item on the agenda, is to check the satisfaction of the strength criteria by the new design. Codified equivalent static lateral forces, which are based on a predetermined fundamental period for the structure, do not necessarily change with variation of stiffness.
358
Figure 727. Contribution of various actions to the total lateral displacement of the 10 story frame.
Figure 728. Contribution of various actions to the total interstory drift ratios of the 10 story frame.
Chapter 7
Figure 729. Comparison of approximate and “exact” secondorder displacements.
Figure 730. Comparison of approximate and “exact” secondorder interstoy drift ratios.
7. Design for Drift and Lateral Stability
359
Figure 731. Member sections after drift design.
Figure 733. Approximate interstory drift ratios for the 10 story frame after drift design.
Figure 732. Approximate lateral displacements for the 10 story frame after drift design.
Figure 734. “Exact” versus approximate displacements for the 10 story frame after drift design.
360
Chapter 7 to withstand increased inertial forces should be examined. Let us assume that the design ground motion for this example is represented by the design spectrum shown in Figure 737. Application of the Rayleigh method, or a simple dynamic analysis, reveals that the fundamental period of the original design (Figure 714) is about 2.7 seconds. The fundamental period of vibration of the structure after drift design (Figure 732) is about 1.9 seconds. Given the design spectrum of Figure 737, the spectral acceleration corresponding to the first mode of vibration of the structure, is about 0.15g for the original design and 0.20g for the modified design. Hence, the modified design will be expected to withstand about 33% more inertial forces than the original one.
Figure 735. “Exact” versus approximate interstory drift ratios for the 10 story frame after drift design.
7.5.2
Drift Design of Braced Frames
Lateral displacements of braced frames are primarily caused by two actions: deformation of the braces, and axial deformation of the columns (chord action). Several methods are available for estimation of braced frame displacements (744, 749, 750). The contribution of brace deformations to story drift may be estimated by(744):
S br = ∑
∆ br =
Figure 736. Influence of drift design on imposed inertial forces.
In reality, however, increasing member sizes for drift control, increases the stiffness of the structure and reduces its natural periods. In multistory buildings, reduction of natural periods usually implies an increase in the inertial forces exerted on the structure. Therefore, the adequacy of the modified design
Abr E cos 2 α Lbr
ΣV S br
(755)
(756)
where ∆br is story drift due to brace deformations, ΣV is the story shear, Sbr is the sum of stiffnesses of the braces at the level under consideration, E is the modulus of elasticity of brace, Abr and Lbr are the cross sectional area and the length of each brace, and α is the angle that a brace makes with the horizontal axis. The summation is carried out over all braces at the level under consideration. Equation 755 is valid as long as the braces do not yield or buckle.
7. Design for Drift and Lateral Stability For ordinary braced frames, the bent story stiffness is negligible in comparison with the brace stiffness. However, in cases where rigid beamcolumn connections are utilized (such as eccentrically braced frames) the bent stiffness can be significant. In these situations, the bent story stiffness (see Sec. 7.5.1, “Bent Displacements”) should be added to the brace stiffness. The cantilever drifts may be computed via the Moment Area Method as explained in Sec. 7.5.1, “Cantilever Displacements”. Note that in ordinary braced frames, where beams and columns are not joined by moment connections, only some of the columns (those in the vicinity of braces) provide significant resistance to cantilever deflections.
361 beam to column connections are simple. The tributary width of the frame is 30 ft. The gravity load is 100 psf on the roof level and 120 psf on typical floors. Assume that the braces are so proportioned that none of them either yield or buckle under the given loads. We have W8×35
A = 10.3 in2
For braces at typical floors, Lbr = (10) 2 + (12) 2 = 15.62 ft. = 187.44 in. cos α = 10 / 15.62 = 0.6402
S br = ∑ E
Abr cos 2 α Lbr
= 2( 29000)(10.3)(0.6402 ) 2 / 187.44 = 1306.27 kips/in. For braces at the first floor, Lbr = (10) 2 + (15) 2 = 18.03 ft. = 216.33in. cos α = 10/18.03= 0.5547 S br =
Figure 737. Braced frame elevation (Example 79).
EXAMPLE 79 Estimate the first and secondorder lateral displacements and story drifts for the 10story braced steel frame shown in Figure 738. All
2(29000)(10.3)(0.5547) 2 = 849.67 kips/in. 216.33
The brace action story drifts and lateral displacements are calculated in Table 710. To show the accuracy of the above simple procedure, an exact firstorder elastic analysis was also performed, in which large column areas were used to eliminate axial deformation of the columns. Results of the exact and approximate analyses are compared in Figure 739, where good agreement can be observed. The chord action story drifts and lateral displacements are calculated in Table 711. The total drifts are magnified using the direct Pdelta method in Table 712. The extent of contribution of each action to the lateral response of the frame is shown in Figure 740, where the dominance of chord action is evident. The results obtained by the above simple procedure are compared with those obtained by an exact secondorder analysis in Figures 741 and 742.
362
Chapter 7
Table 710. Calculation of braceaction story drifts and lateral displacements for the 10story braced frame of example 79. Lat. disp. Level h, ∆br, ΣV, Sbr kips/in. in. in. in. kips 10 144 30.22 1306 0.0231 0.8279 9 144 52.17 1306 0.0399 0.8048 8 144 71.74 1306 0.0549 0.7649 7 144 88.93 1306 0.0681 0.7100 6 144 103.76 1306 0.0794 0.6419 5 144 116.21 1306 0.0890 0.5625 4 144 126.30 1306 0.0967 0.4735 3 144 134.01 1306 0.1026 0.3768 2 144 139.34 1306 0.1067 0.2742 1 180 142.31 850 0.1675 0.1675
Table 711. Calculation of chordaction story drifts and lateral displacements for the braced frame of Example 79. Ioi, A, Chord disp., Chord Level h, ΣV, Mov, x, in.kips in.4 in.2 in. drift, in. in. kips in. 10 144 30.22 4,352 406,080 2.66×105 48.00 2.958 0.452 9 144 52.17 11,864 406,080 9.92×105 60.88 2.506 0.443 64.72 2.063 0.426 8 144 71.74 22,194 576,000 14.7×105 66.63 1.637 0.397 7 144 88.93 35,001 576,000 24.6×105 67.78 1.240 0.360 6 144 103.76 49,942 763,200 27.6×105 68.56 0.880 0.312 5 144 116.21 66,677 763,200 37.9×105 69.12 0.568 0.256 4 144 126.30 84,864 921,600 40.8×105 69.55 0.312 0.190 3 144 134.01 104,161 921,600 50.9×105 69.89 0.122 0.122 2 144 139.34 124,226 1,344,960 42.2×105 87.20 0.000 0.000 1 180 142.31 149,841 1,344,960 63.2×105
Table 712 Calculation of total firstorder and secondorder story drifts and lateral displacements for the braced frame of example 79. Level h, ΣV, ΣP, ∆1 2ndOrder µ ∆2= µ ∆1, in. kips kips in. Disp.,in. in. 10 144 30.22 180 0.475 1.020 0.485 3.897 9 144 52.17 396 0.483 1.026 0.496 3.412 8 144 71.74 612 0.481 1.029 0.495 2.916 7 144 88.93 828 0.465 1.031 0.479 2.421 6 144 103.76 1044 0.439 1.032 0.453 1.942 5 144 116.21 1260 0.401 1.031 0.413 1.489 4 144 126.30 1476 0.353 1.029 0.363 1.076 3 144 134.01 1692 0.301 1.027 0.309 0.713 2 144 139.34 1908 0.229 1.022 0.234 0.404 1 180 142.31 2124 0.168 1.014 0.170 0.170
7. Design for Drift and Lateral Stability
Figure 738. Lateral displacements caused by brace deformations.
Figure 739. Contribution of various actions to the total lateral displacement of the braced frame of Example 79.
363
Figure 740. “Exact” versus approximate lateral displacements for the braced frame of example 79.
Figure 741. “Exact” versus approximate interstory drift ratios for the braced frame of Example 79.
364
Chapter 7
Figure 742. Design aid for drift design of frameshear wall systems(751) (Sc/Sb=1).
7. Design for Drift and Lateral Stability
Figure 743. Design aid for drift design of frameshear wall systems (751) (Sc/Sb=5).
365
366
Chapter 7
Figure 744. Design aid for drift design of frameshear wall systems(751) (Sc/Sb=10)
7. Design for Drift and Lateral Stability 7.5.3
Drift Design of Frame  Shear Wall Systems
Estimates of the lateral displacements of FrameShear wall systems may be obtained using the charts developed by Khan and Sbarounis(751). Some of these charts, for the case of constant stiffness over the height, are reproduced in Figures 743 to 745. A sample application of the charts is presented in Example 710. In order to utilize the charts, the sum of stiffnesses of beams (Sb), columns (Sc) and shear walls (Ss) should be computed by adding the corresponding EI/L terms. The charts provide the ratio of the lateral deflection of the frameshear wall system to the free deflection (at the top) of the shear wall alone. Note that the ratio of Ss/Sc should be normalized by multiplying it by (10/N)2, where N is the number of stories in the structure. Another method for estimating drift and natural periods of frameshear wall systems, has been developed by Stafford Smith et al.(752, 753) The method has been shown to provide accurate estimates of lateral displacements for a variety of structural systems. It can be easily adapted to programmable calculators. It is rather tedious, however, for hand calculations. EXAMPLE 710 Use the Khan and Sbarounis charts to estimate the lateral displacement at the top of the 30story frameshear wall building shown in Figure 746. Assume a uniform lateral pressure of 30psf. Story heights are 12.5 feet. Use gross concrete section properties and E = 4000 ksi. Column Stiffnesses: Col. Type C1 C2 C3 C4
b, in. 24 28 32 36
h, in. 24 28 32 36
367 Beams: 3 = 2.625 ft4 I = (14)(36) 5 (12)
B1:
Figure 745. Plan of the 30 story frameshear wall building(752).
B2:
(18)(24) 3 I= = 1.000 ft4 5 (12)
B3:
I=
(18)(32) 3 = 2.370 ft4 5 (12)
(4)(2.625) (2)(2.625) + 24 28 (6)(1.00) (1)(2.37) + + 28 28 3 = 0.924 ft
Total I /L =
Walls:
( 2)( 28) 3 = 3658.67 ft 4 12 ( 2)(3658.67) Total I/L = = 585.39 12.5 I=
I, ft4 1.333 2.470 4.214 6.750
Total I / L = 4(0.1067) + 6(0.1976) +4(0.3371) + 2(0.5400) = 4.041 ft3
I / L, ft3 0.1067 0.1976 0.3371 0.5400
2
S s 585.39 10 = = 16.10 Sc 4.041 30 S c 4.041 = = 4.37 S b 0.924
368
Chapter 7
Free deflection of the wall:
w=
30( 4)( 24) = 2.88 kips/ft 1000
wl 4 (2.88)(375) 4 = 8EI (8)(576000)(3658.67)( 2) = 1.69 ft. = 20.28 in.
∆=
Using the curve corresponding to Ss/Sc = 20 from Chart (a) of Figure 744, we have Dtop = (0.22)(20.28) = 4.06 inches, which compares very well with the computed exact displacement of 4.23 inches (see Figure 747).
For buildings in which the locations and relative stiffnesses of the lateral load resisting subsystems (e.g. frames and walls) do not vary significantly along the height, the torsional displacements may be estimated as follows: 1. For buildings which are composed of only one type of lateral load resisting system (moment frames, braced frames, or walls), the torsional rotation at the ith floor, θi, and the corresponding torsional drift of the j th frame at this floor, ∆j, may be estimated as:
θi =
(ΣVi ) ei2 J
∆ j = R j θi
(757) (758)
where ΣVi is the story shear, ei is the eccentricity of the "center of rigidity" from the center of mass, Rj is the closest distance from the jth frame to the center of rigidity, and J is the torsional story stiffness given by
J = ΣK j R 2j
Figure 746. Lateral displacement of the 30 story frameshear wall building.
7.5.4
Torsional Effects
One of the most important tasks in the process of the selection, and the subsequent proportioning, of a structural system, is the minimization of torsional response. In general, this is a rather difficult task, and its success is strongly dependent on the intuition and experience of the designer.
(759)
2. For combination systems (frameshear wall systems, moment frame and braced frame combinations), the process is more complex: – The direct lateral displacements and story drifts of the structure are obtained via the KhanSbarounis charts or any other appropriate method. – The total direct story shear carried by the frames subjected to the above displacements, Vfi, are calculated (see Section 7.5.1, “Bent Displacements). – The shear Vfi is distributed among the various frames according to their relative stiffness in the direction of applied load. – The rest of the story shear (ΣVi  Vfi) is distributed among the various walls (braced frames) according to their relative stiffness in the direction of applied loads. – The shear in each frame or wall, as calculated in the two preceding steps, is used as a measure of rigidity, and the center of rigidity of the entire system is located.
7. Design for Drift and Lateral Stability
369
– The torsional rotation and the corresponding torsional drift of individual frames and walls are calculated using Equations 757 and 758. It may be noticed that the concept of the "center of rigidity" is of significant use in the preliminary evaluation of the torsional response. However, the physical limitations of such a concept when applied to the seismic response of general, three dimensional, multistory structures should be clearly understood. In a three dimensional, multistory structure, if it exhibits significant plan and elevation irregularities, the lateral resistance is provided by a combination of strongly interdependent actions, both within a single story, and among various floors. In general, for such a complex system, centers of rigidity (points of application of forces for a torsionfree response) do not exist. Furthermore, if and when they exist, they must all lie on a single vertical line(754).
7.6
SEISMIC CODE REQUIREMENTS FOR DRIFT AND PDELTA ANALYSIS
7.6.1
UBC97 Provisions
UBC97(757), addresses design for drift and lateral stiffness within the framework of strength design. The reduced lateral displacement calculated by utilizing the reduction factor, R, is called ∆S. The maximum inelastic response displacement is called ∆M and is calculated from
∆ M = 0.7 R∆ S
θ=
Px ∆ Sx V x hsx
(761)
where ∆Sx = story drift based on ∆S acting between levels x and x1 Vx = the design seismic shear force acting between levels x and x1 hsx = the story height below level x Px = the total unfactored vertical design load at and above level x. In seismic zones 3 and 4, Pdelta effects need not be considered when the story drift index does not exceed 0.02/R. UBC97 permitted drift using ∆M is a function of the fundamental period of the structure
∆ Mx ≤ 0.025hsx for T < 0.7 sec . (761) ∆ Mx ≤ 0.020hsx for T ≥ 0.7 sec . where ∆Sx = story drift based on ∆M acting between levels x and x1 The fundamental period used in drift calculations is not subject to lowerbound period formulas of the code (see Chapter 4) and may be based on the Rayleigh formula or other rational calculations such as a detailed computer model of the structure. Furthermore, UBC97 permits these drift limits to be exceeded when the engineer can demonstrate that greater drift can be tolerated by both structural and nonstructural elements whose performance can affect the seismic safety of the structure. Therefore, if local drift is exceeded locally in an area without a serious seismic ramification, it can be tolerated and there is no need for a redesign.
(760)
Alternatively, ∆M may be computed by nonlinear time history analysis. The analysis to determine ∆M must consider Pdelta effects. Pdelta effects, however, may be ignored when the ratio of secondary moments to firstorder moments does not exceed 0.10. This ratio is calculated from
7.6.2
IBC2000 Provisions
The provisions of IBC2000(758) embody a convergence of the efforts initiated by the Applied Technology Council's ATC 306(759) document published in 1978 and its successive modifications by the Federal Emergency Management Agency(760) and that of the UBC
370
Chapter 7
provisions. Therefore, setting aside the difference in the language and vocabulary, IBC2000 and UBC97 drift and Pdelta provisions are very similar (760). Quite rationally, IBC2000 addresses seismic design for drift and lateral stiffness exclusively at the ultimate limit state of building behavior. According to IBC2000 provisions, the design story drift, ∆, is computed as the difference of the deflections, δx, at the top and bottom of the story under consideration in accordance with the following formula
δx =
Cd δ xe IE
(762)
where: Cd = the deflection amplification factor as given in Table 517, δxe.= the deflection determined by an elastic analysis of the forceresisting system, and ΙE = the occupancy importance factor as given in Section 5.4.2. The maximum interstory drift index calculated using Equation 762 should not exceed the corresponding limits described in Section 5.4.15. Furthermore, for structures assigned to seismic design categories C, D, E, or F having plan irregularity types 1a or 1b (see Chapter 5) the design story drift is to be computed as the largest difference of the deflections along any of the edges of the structure at the top and bottom of the story under consideration. To determine whether a Pdelta analysis is required, a stability coefficient is used. This is in fact, the same as the stability index introduced previously in this Chapter. Pdelta effects need not be considered when the stability coefficient, θ as determined from Equation 763 is less than 0.10:
θ=
Px ∆ Vx hsx C d
where ∆ = the design story drift
(763)
Vx = the seismic shear force acting between level x and x1 hsx = the story height below level x, and Px = the total unfactored vertical design load at and above level x. The stability coefficient, θ, should not exceed an upper limit of θmax given as
θ max =
0.5 ≤ 0.25 βC d
(763)
where: β = the ratio of shear demand to shear capacity for the story between level x and x1. If this ratio is not calculated, a value of β = 1 should be used. When θ is greater than 0.10 but less than θmax, IBC2000 permits direct calculation of Pdelta effects in a manner very similar to the direct Pdelta method discussed earlier in this Chapter. That is, the calculated firstorder interstory drifts are to be multiplied by a factor of 1/(1θ)>1. If, however, θ is larger than θmax the structure is potentially unstable and should be redesigned.
REFERENCES 71 Council on Tall Buildings, Committee 16, "Stability," Chapter SB4, Vol. SB of Monograph on Planning and Design of Tall Buildings, ASCE, New York, 1979. 72 Scholl, R.E. (ed), "Effects Prediction Guidelines for Structures Subjected to Ground Motion," Report No. JAB99115, URS/Blume Engineers, San Francisco, 1975. 73 Scholl, R.E., "Brace Dampers: An Alternative Structural System for Improving the Earthquake Performance of Buildings," Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, Vol 5., Prentice Hall, 1984. 74 Ohta, Y., and Omote, S., "An Investigation into Human Psychology and Behavior During an Earthquake," Proceedings of the 6th World Conference on Earthquake Engineering, India, 1977. 75 Mileti, D.S., and Nigg, J.M., "Earthquakes and Human Behavior," Earthquake Spectra, EERI, Vol. 1, No. 1, Feb., 1984.
7. Design for Drift and Lateral Stability 76 Durkin, M.E., "Behavior of Building Occupants in Earthquakes," Earthquake Spectra, EERI, Vol. 1, No. 2, Feb., 1985. 77 Chen, W.F., and Lui, E.M., Structural Stability, Elsevier Science Publishing Company, New York, 1987. 78 Wakabayashi, M., Design of Earthquake Resistant Buildings, MacGraw Hill, New York, 1986. 79 Rosenblueth, E., "Slenderness Effects in Buildings," Journal of the Structural Division, ASCE, Vol. 91, No. ST1, Proc. Paper 4235, January, 1967, pp. 229252. 710 Stevens, L. K., "Elastic Stability of Practical MultiStory Frames," Proceedings, Institution of Civil Engineers, Vol. 36, 1967, pp. 99117. 711 Goldberg, J. E., "Approximate Methods for Stability and Frequency Analysis of Tall Buildings," Proceedings, Regional Conference on Tall Buildings, Madrid, Spain, September, 1973, pp. 123146. 712 Council on Tall Buildings, Committee 23, "Stability," Chapter CB8, Vol. CB of Monograph on Planning and Design of Tall Buildings, ASCE, New York, 1978. 713 Singh, J.P., "Earthquake Ground Motions: Implications for Designing Structures and Reconciling Structural Damage," Earthquake Spectra, EERI, Vol. 1, No. 2, Feb., 1985. 714 Naeim, F., "On Seismic Design Implications of the 1994 Northridge Earthquake Records,” Earthquake Spectra, EERI, Vol. 11, No. 1, Feb. 1995. 715 Anderson, J.C., and Bertero, V.V., "Uncertainties in Establishing Design Earthquakes," Proceedings of a twoday Course from EERI on Strong Ground Motion, San Francisco and Los Angeles, Apr., 1987. 716 MacGregor, J.G, and Hage, S.E., "Stability Analysis and Design of Concrete Frames," Journal of the Structural Division, ASCE, Vol. 103, No. ST10, October, 1977, pp. 19531970. 717 Wynhoven, J.H., and Adams, P.F., "Behavior of Structures Under Loads Causing Torsion," Journal of the Structural Division, ASCE, Vol. 98, No. ST7, July, 1972, pp. 13611376. 718 Salmon, C.G., and Johnson, J.E., Steel Structures, Design and Behavior, 2nd Edition, Harper and Row, 1980, pp. 843851. 719 Johnston, B.G. (ed.), Structural Stability Research Council, Guide to Stability Design Criteria for Metal Structures, 3rd Edition, John Wiley and Sons, New York, 1976. 720 CheongSiatMoy, F., "Frame Design Without Using Effective Column Length," Journal of the Structural Division, ASCE, Vol. 104, No. ST1, Jan., 1978, pp. 2333. 721 CheongSiatMoy, F., "KFactor Paradox," Journal of Structural Engineering, ASCE, Vol. 112, No. 8, Aug., 1986, pp. 17471760.
371 722 Nair, R. S., "A Simple Method of Overall Stability Analysis for Multistory Buildings," Developments in Tall Buildings  1983, Council on Tall Buildings and Urban Habitat, Lynn S. Beedle (EditorinChief), Van Nostrand Reinhold, New York, 1983. 723 Nair, R.S., "Overall Elastic Stability of Multistory Buildings," Journal of the Structural Division, ASCE, Vol. 101, No. ST12, Dec., 1975, pp. 24872503. 724 Springfield, J., and Adams, P.F., "Aspects of Column Design in Tall Steel Buildings," Journal of the Structural Division, ASCE, Vol. 98, No. ST5, May, 1972, pp. 10691083. 725 Wood, B.R., Beaulieu, D., and Adams, P.F., "Column Design by PDelta Method," Journal of the Structural Division, ASCE, Vol. 102, No. ST2, Feb., 1976, pp. 411427. 726 Lai, S.A., and MacGregor, J.G., "Geometric Nonlinearities in Unbraced Multistory Frames," Journal of Structural Engineering, ASCE, Vol. 109, No. 11, Nov., 1983, pp. 25282545. 727 Nixon, D., Beaulieu, D., and Adams, P.F., "Simplified SecondOrder Frame Analysis," Canadian Journal of Civil Engineering, Vol. 2, No. 4, Dec., 1975, pp. 602605. 728 Renton, J.D., "Stability of Space Frames by Computer Analysis," Journal of the Structural Division, ASCE, Vol. 88, No. ST4, Aug., 1962, pp. 81103. 729 Chu, K.H., and Rampetsreiter, R.H., " Large Deflection Buckling of Space Frames," Journal of the Structural Division, ASCE, Vol. 98, No. ST12, Dec., 1972, pp. 27012722. 730 Connor, J.J., Jr., Logcher, R.D., and Chen, S.C., "Nonlinear Analysis of Elastic Frame Structures," Journal of the Structural Division, ASCE, Vol. 94, No. ST6, Jun., 1968, pp. 15251547. 731 Zarghammee, M.S., and Shah, J.M., "Stability of Space Frames," Journal of the Engineering Mechanics Division, ASCE, Vol. 94, No. EM2, Apr., 1968, pp. 371384. 732 Krajcinovic, D., "A Consistent Discrete Elements Technique for ThinWalled Assemblages," International Journal of Solids and Structures, Vol. 5, 1969, pp. 639662. 733 Barsoum, R.S., and Gallagher, R.H., "Finite Element Analysis of Torsional Flexural Stability Problems," International Journal for Numerical Methods in Engineering, Vol. 2, 1970, pp. 335352. 734 Bazant, Z.P., and El Nimeiri, M., "LargeDeflection Buckling of Thin Walled Beams and Frames," Journal of the Engineering Mechanics Division, ASCE, Vol. 79, No. EM6, Dec., 1973, pp. 12591281. 735 Yoo, C.H., "Bimoment Contribution to Stability of ThinWalled Assemblages," Computers and Structures, Vol. 11, 1980, pp. 465471.
372 736 Yang, Y., and McGuire, W., "Stiffness Matrix for Geometric Nonlinear Analysis," Journal of Structural Engineering, ASCE, Vol. 112, No. 4, Apr., 1986, pp. 853877. 737 Naeim, F., An Automated Design Study of the Economics of Earthquake Resistant Structures, Ph.D. Dissertation, Department of Civil Engineering, University of Southern California, Aug., 1982. 738 Neuss, C.F., Maison, B.F., and Bouwkamp, J.G., A Study of Computer Modeling Formulation and Special Analytical Procedures for Earthquake Response of Multistory Buildings, A Report to National Science Foundation," J.G. Bouwkamp, Inc., Berkeley, California, Jan., 1983, pp. 335362. 739 Wilson, E.L., and Habibullah, A., "Static and Dynamic Analysis of MultiStory Buildings Including the PDelta Effects," Earthquake Spectra, EERI, Vol.3, No.2, May, 1987. 740 Freeman, S.A., Czarncki, R.M., and Honda, K.K., "Significance of Stiffness Assumptions on Lateral Force Criteria," in Reinforced Concrete Structures Subjected to Wind and Earthquake Forces, Publication SP63, American Concrete Institute, Detroit, Michigan, 1980. 741 Wong, C.H., El Nimeiri, M.M., and Tang, J.W., "Preliminary Analysis and Member Sizing of Tall Tubular Steel Buildings," AISC Engineering Journal, American Institute of Steel Construction, Second Quarter, 1981, pp. 3347. 742 Council on Tall Buildings, Committee 14, "Elastic Analysis and Design," Chapter SB2, Vol. SB of Monograph on Planning and Design of Tall Buildings, ASCE, New York, 1979. 743 CheongSiatMoy, F., "Multistory Frame Design Using Story Stiffness Concept," Journal of the Structural Division, ASCE, Vol. 102, No. ST6, Jun., 1976, pp. 11971212. 744 CheongSiatMoy, F., "Consideration of Secondary Effects in Frame Design," Journal of the Structural Division, ASCE, Vol. 103, No. ST10, Oct., 1977, pp. 20052019. 745 Krawinkler, H., Bertero, V.V., and Popov, E.P., Inelastic Behavior of Steel BeamtoColumn Subassemblages, Earthquake Engineering Research Center, University of California, Berkeley, Report No. EERC 717, Oct., 1971. 746 Becker, E.R., Panel Zone Effect on the Strength and Stiffness of Rigid Steel Frames, Structural Mechanics Laboratory Report, University of Southern California, Jun., 1971. 747 Richards, R.M., and Pettijohn, D.R., Analytical Study of Panel Zone Behavior in BeamColumn Connections, University of Arizona, Nov., 1981. 748 Slutter, R.G., Tests of Panel Zone Behavior in BeamColumn Connections, Fritz Engineering Laboratory, Lehigh University, Report No. 200814031, 1981.
Chapter 7 749 Teal, E.J., Practical Design of Eccentric Braced Frames to Resist Seismic Forces, Structural Steel Educational Council. 750 White, R.N., and Salmon, C.G., (eds), Building Structural Design Handbook, John Wiley and Sons, New York, 1987. 751 Khan, F.R., and Sbarounis, J.A., " Interaction of Shear Walls and Frames," Journal of the Structural Division, ASCE, Vol. 90, No. ST3, Jun., 1964, pp. 285335. 752 Stafford Smith, B., Kuster, M., and HoenderKamp, J.C.D., "Generalized Method for Estimating Drift in HighRise Structures," Journal of Structural Engineering, ASCE, Vol. 110, No. 7, Jul., 1984, pp. 15491562. 753 Stafford Smith, B., and Crowe, E.,"Estimating Periods of Vibration of Tall Buildings," Journal of Structural Engineering, ASCE, Vol. 112, No. 5, May, 1986, pp. 10051018. 754 Riddell, R., and Vasques, J.,"Existence of Centers of Resistance and Torsional Uncoupling of Earthquake Response of Buildings," Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, Vol 5., Prentice Hall, 1984. 755 International Conference of Building Officials, Uniform Building Code 1997, Whittier, California, 1997. 756 International Code Council, International Building Code 2000, Falls Church, Virginia, 2000. 757 Applied Technology Council, Tentative Provisions for the Development of Seismic Regulations for Buildings, Publication ATC306, 1978. 758 Federal Emergency Management Agency, 1997 Edition of NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, 1997.
8. Seismic Design of Floor Diaphragms
373
Chapter 8 Seismic Design of Floor Diaphragms
Farzad Naeim, Ph.D.,S.E. Vice President and Director of Research and Development, John A. Martin & Associates, Los Angeles, California.
R. Rao Boppana, Ph.D.,S.E. President, Sato & Boppana, Los Angeles, California.
Key words:
Design, Diaphragm, Earthquake, Flexible Diaphragms, IBC2000, Reinforced Concrete, Seismic, Structural Steel, Rigid Diaphragms, Timber, UBC97.
Abstract:
This chapter surveys the seismic behavior and design of floor and roof diaphragms. Following some introductory remarks, a classification of diaphragm behavior is presented in Section 8.2, and a discussion on the determination of diaphragm rigidity in Section 8.3. Potential diaphragm problems are explained in Section 8.4 where examples are provided to clarify the subject. Provisions of major United States building codes for seismic design of diaphragms are summarized in Section 8.5. Finally, in Section 8.6, the current standard procedures for design of diaphragms are presented via their application in a number of realistic design examples
373
374
Chapter 8
8. Seismic Design of Floor Diaphragms
8.1
INTRODUCTION
The primary function of floor and roof systems is to support gravity loads and to transfer these loads to other structural members such as columns and walls. Furthermore, they play a central role in the distribution of wind and seismic forces to the vertical elements of the lateral load resisting system (such as frames and structural walls). The behavior of the floor/roof systems under the influence of gravity loads is well established and guidelines for use in structural design have been adopted (81,82) . In the earthquake resistant design of building structures, the building is designed and detailed to act as a single unit under the action of seismic forces. Design of a building as a single unit helps to increase the redundancy and the integrity of the building. The horizontal forces generated by earthquake excitations are transferred to the ground by the vertical systems of the building which are designed for lateral load resistance (e.g. frames, bracing, and walls). These vertical systems are generally tied together as a unit by means of the building floors and roof. In this sense, the floor/roof structural systems, used primarily to create enclosures and resist gravity (or out of plane) loads are also designed as horizontal diaphragms to resist and to transfer horizontal (or inplane) loads to the appropriate vertical elements. The analysis and design of a floor or roof deck under the influence of horizontal loads is performed assuming that the floor or roof deck behaves as a horizontal continuous beam supported by the vertical lateral load resisting elements (hereafter referred to as VLLR elements). The floor deck is assumed to act as the web of the continuous beam and the beams at the floor periphery are assumed to act as the flanges of the continuous beam (see Figure 81). Accurate determination of the inplane shears and bending moments acting on a floor diaphragm, and the corresponding horizontal force distribution among various VLLR
375 elements requires a three dimensional analysis that accounts for the relative rigidity of the various elements including the floor diaphragms. Increasingly, this type of analysis is being performed for design and rehabilitation of major buildings that feature significant plan irregularities. In general, however, some assumptions are made on the horizontal diaphragm rigidity and a relatively simple analysis is performed to determine distribution of lateral forces. Obviously, the accuracy of the results obtained depends on the validity of the assumptions made. In addition, the behavior of certain floor systems such as plywood, metal deck, and precast concrete diaphragms are difficult to model analytically due to their various attachments. In some cases testing may be required to establish the strength and stiffness properties of such systems.
Figure 81. Design forces on a diaphragm
376
Chapter 8
While for the great majority of structures, simplified analysis procedures result in a safe design, studies indicate that neglecting the real behavior of floor diaphragms can sometimes lead to serious errors in assessing the required lateral load resistance capacities of the VLLR elements(83, 84, 85). This chapter addresses the major issues of seismic behavior and design of diaphragms. It starts by classification of diaphragm behavior in Section 8.2, and a discussion on the determination of diaphragm rigidity in Section 8.3. Potential diaphragm problems are explained in Section 8.4 where examples are provided to clarify the subject. Provisions of major United States building codes for seismic design of diaphragms are summarized in Section 8.5. Finally, in Section 8.6, the current standard procedures for design of diaphragms are presented via their application in a number of realistic design examples.
8.2
be made as to a diaphragm's rigidity or flexibility in order to simplify the analysis. If the diaphragm deflection and the deflection of the VLLR elements are of the same order of magnitude, then the diaphragm can not reasonably be assumed as either rigid or flexible. Such a diaphragm is classified as semirigid.
CLASSIFICATION OF DIAPHRAGM BEHAVIOR
The distribution of horizontal forces by the horizontal diaphragm to the various VLLR elements depends on the relative rigidity of the horizontal diaphragm and the VLLR elements. Diaphragms are classified as "rigid", "flexible", and "semirigid" based on this relative rigidity. A diaphragm is classified as rigid if it can distribute the horizontal forces to the VLLR elements in proportion to their relative stiffness. In the case of rigid diaphragms, the diaphragm deflection when compared to that of the VLLR elements will be insignificant. A diaphragm is called flexible if the distribution of horizontal forces to the vertical lateral load resisting elements is independent of their relative stiffness. In the case of a flexible diaphragm, the diaphragm deflection as compared to that of the VLLR elements will be significantly large. A flexible diaphragm distributes lateral loads to the VLLR elements as a series of simple beams spanning between these elements. No diaphragm is perfectly rigid or perfectly flexible. Reasonable assumptions, however, can
Figure 82. Diaphragm behavior. (a) Loading and building proportions. (b) Rigid diaphragm behavior. (c) Flexible diaphragm behavior, (d) Semi rigid diaphragm behavior
Exact analysis of structural systems containing semirigid diaphragms is complex, since any such analysis should account for the relative rigidity of all structural elements including the diaphragm. The horizontal load distribution of a semirigid diaphragm may be approximated as that of a continuous beam supported on elastic supports. In most cases consisting of semirigid diaphragms,
8. Seismic Design of Floor Diaphragms assumptions can be made to bound the exact solution without resorting to a complex analysis. The absolute size and stiffness of a diaphragm, while important, are not the final determining factors whether or not a diaphragm will behave as rigid, flexible, or semirigid(83). Consider the onestory concrete shear wall building shown in Figure 82a. Keeping the width and the thickness of walls and slabs constant, it is possible to simulate rigid, flexible and semirigid diaphragms as the wall heights and diaphragm spans are varied. The wall stiffness decreases with an increase in the floor height (H). Similarly, the diaphragm stiffness decreases with an increase in span (L). The dashed line in Figure 82b indicates the deflection of the system under the influence of horizontal forces when the diaphragm is rigid. This can be accomplished by increasing H and decreasing L so that the stiffness of the diaphragm relative to the wall is significantly larger. In such a situation, the deflection of the diaphragm under horizontal loads is insignificant when compared to the deflections of the walls. The diaphragm will move as a rigid body and will force the walls to move together accordingly. The force distribution among the walls will depend only on the relative stiffness of the walls. In Figure 82b it is assumed that the applied load and the wall stiffness are symmetric. If this is not the case, in addition to the rigid body translation, the diaphragm will experience rigid body rotation. Figure 82c shows the deflection of the system under the influence of horizontal forces when the diaphragm is flexible. This can be accomplished by decreasing H and increasing L such that the stiffness of the diaphragm when compared to the walls is small. In such a situation, the diaphragm segments between the walls act as a series of simply supported beams and the load distribution to the walls can be determined based on the tributary area of the diaphragm to the wall. Obviously, a flexible diaphragm can not experience the rotation or torsion that occurs due to the rigid body rotation of a rigid diaphragm.
377 The dashed line in Figure 82d indicates the deflection pattern of a semirigid diaphragm under the influence of lateral forces. Here the stiffness of the walls and the diaphragm are of the same order. Both wall deflections and diaphragm deflections do contribute to the total system deflection. Determination of exact load distribution among the walls requires a three dimensional analysis of the entire system (including the diaphragm).
8.3
DETERMINATION OF DIAPHRAGM RIGIDITY
In order to estimate the diaphragm rigidity, it is necessary to predict the deflection of the diaphragm under the influence of lateral loads. The various floor and roof systems that have evolved primarily for the purpose of supporting gravity loads do not lend themselves easily to analytical calculation of lateral deflections. Some of the more common floor systems in use today are: (1) castinplace concrete; (2) precast planks or Tees with or without concrete topping; (3) metal deck with or without concrete fill and; (4) wood framing with plywood sheathing. With the single exception of castinplace concrete floor system which is a monolithic construction, all the other floor systems mentioned above consist of different units joined together with some kind of connections. In precast concrete construction, adjacent units are generally connected together by welding embedded plates or reinforcing bars. This will help the units to deflect vertically without separation while providing some diaphragm action. The strength and rigidity of such a diaphragm will depend to a great extent on the type and spacing of connections. Analytical computation of deflections and stiffness of such a diaphragm is complex. As an alternative, a bonded topping slab on precast floor or roof can be provided with sufficient reinforcement to ensure continuity and resistance for shear transfer mechanism. In floor systems consisting of metal decks, the deck is welded intermittently to the supports below. Adjacent
378
Chapter 8
units of the deck are connected together by means of button punching or welding. Here again, the diaphragm stiffness is directly related to the spacing and type of connections. In the wood construction, the plywood sheathing is nailed directly to the framing members. Again, strength and stiffness depends on the spacing of the nails and whether or not the diaphragm is blocked. It is general practice to consider the diaphragms made of cast in place concrete, precast with concrete topping, and metal deck with concrete fill as rigid while the diaphragms consisting of precast planks without concrete topping, metal deck without concrete fill, and plywood sheathing as flexible. This classification is valid for most cases. Gross errors in force distribution, however, can occur if the above assumption is used without paying attention to the relative rigidity of the VLLR elements and the diaphragm(83, 84, 85). Metal deck manufacturers have established test programs to provide strength and deflection characteristics of various metal decks and various connection patterns(86, 87). Similarly, the Uniform Building Code provides an empirical formula to compute plywood diaphragm deflections and tables to establish the strength of such diaphragms.
8.4
SIGNIFICANT FACTORS AFFECTING DIAPHRAGM BEHAVIOR
Identifying every situation where special attention should be given to the design and detailing of floor diaphragms requires substantial experience and a good amount of engineering judgement. Certain cases, however, more often than not, require special attention and in this section guidelines for identification of such cases are provided. In general, lowrise buildings and buildings with very stiff vertical elements such as shear walls are more susceptible to floor diaphragm flexibility problems than taller structures.
In buildings with long and narrow plans, if seismic resistance is provided either by the end walls alone, or if the shear walls are spaced far away from each other, floor diaphragms may exhibit the socalled bow action (see Figure 83). The bow action subjects the end walls to torsional deformation and stresses. If sufficient bond is not provided between the walls and the diaphragm, the two will be separated from each other starting at the wall corners. This separation results in a dramatic increase in the wall torsion and might lead to collapse.
Figure 83 A plan showing how the socalled bow action subjects the end walls to torsion
The Arvin High School Administrative Building in California which suffered extensive damage during the Kern County earthquake of July 21, 1952 is a good example in this regard. Schematic plans and elevations of this building are shown in Figure 84. An analytical study of this building by Jain(88) indicated that the two lowest natural frequencies of the building were close to the fundamental frequencies of the floor and roof diaphragms modeled as simply supported beams. When an analytical model of the building was subjected to a 0.20g constant spectral acceleration, with four translational modes considered, the two diaphragm modes represented 74 percent of the sum of the modal base shears. As documented by Steinburgge (89) diaphragm deflections caused a separation between the roof diaphragm and the wall corners at the second story wall located at the west end of the building. This action subjected the wall to significant torsional stresses beyond its capacity.
8. Seismic Design of Floor Diaphragms
Figure 84. Plan and elevation of the Arvin High School Administrative Building (88)
Another potential problem in diaphragms can be due to any abrupt and significant changes in a wall stiffness below and above a diaphragm level, or any such changes in the relative stiffness of adjacent walls in passing through one floor level to another (Figure 85). This can cause high shear stresses in the floor diaphragm and/or a redistribution of shear forces among the walls.
Figure 85. Abrupt changes in stiffness and location of VLLR elements can cause drastic redistribution of forces
379 As an example consider the three story concrete shear wall building shown in Figure 86. The concrete floor diaphragms are eight inches thick. A set of static lateral forces of 24 kips, 48 kips and 73 kips are applied at the center of mass of the first, second, and third levels, respectively. The base of the building is assumed to be fixed and the reported results are based on an elastic analysis. An analysis based on a rigiddiaphragm assumption and a finite element analysis considering the uncracked diaphragm stiffness, yield very close results. However, if we make a simple change in the elevation of the building by moving the opening at the second level, from the wall on line A to the wall on line B (Figure 87), the results of the two methods will be markedly different (see Figure 88). For example, the rigid diaphragm assumption suggests that the shear force in wall A is reduced from 94.3 kips above the first floor diaphragm to 26 kips below this level, while the finite element model of the building, shows that such a large portion of the shear force is not transferred away from this wall by the floor diaphragm.
Figure 86. Plan and elevation of a simple three story shear wall building (Note the uniform stiffness along the height of walls on lines A and B.)
380
Chapter 8
Figure 87. Altered plan and elevation of the three story shear wall building (Note the abrupt change of stiffness along the height of walls on lines A and B.)
In buildings with significant plan irregularities, such as multiwing plans, Lshape, Hshape, Vshape plans, etc. (Figure 89) particular attention should be paid to accurately access the inplane diaphragm stress at the joints of the wings and to design for them. In this type of buildings, the fanlike deformations in the wings of diaphragm can lead to a stress concentration at the junction of the diaphragms (see Figure 810). If these stress concentrations are not accounted for, serious problems can arise. For the case of reinforced concrete diaphragms, it is recommended to limit the maximum compressive stresses to 0.2f′c. Alternatively, special transverse reinforcement can be provided. In some cases the diaphragm stresses at the junctions may be so excessive that a feasible diaphragm thickness and reinforcement can not be accommodated. In these cases the wings should be separated by seismic joints. One example for this type of problems was provided by the West Anchorage High School Building in Anchorage, Alaska, which suffered severe damage during the
Figure 88. Computed shears of walls on lines A and B
8. Seismic Design of Floor Diaphragms
381
Figure 89. Typical plan Irregularities
Alaskan earthquake of March 27, 1964 (see Figure 615). Other classes of buildings deserving special attention to diaphragm design include those with relatively large openings in one or more of the floor decks (Figure 811) and tall buildings resting on a significantly larger lowrise part (Figure 812). In the later case, the action of the lowrise portion as the shear base and the corresponding redistribution of shear forces (kickbacks) may subject the diaphragm located at the junction of the lowrise and highrise parts (and sometimes a number of floor diaphragms above and below the junction) to some significant inplane shear deformations.
Figure 810. Fanlike deformation of wings causes stress concentration at the junction
Figure 811. Significant floor openings are cause for concern
382
Chapter 8 Ft + F px =
n
∑F
i
i= x
n
∑W
w px
(81)
i
i= x
The minimum value of Fpx to be used in analysis is 0.5CaIwpx. However, it need not exceed 1.0CaIwpx where:
Figure 812. Elevation of towers on an expanded lowrise base
8.5
CODE PROVISIONS FOR DIAPHRAGM DESIGN
8.5.1
UBC97, ASCE 795, and IBC2000 Provisions
Diaphragm design provisions contained in the UBC97, ASCE 795 and IBC2000 are similar but vary in the degree of detailed information they provide. All these model codes contain a clause limiting the inplane deflection of the floor diaphragms as follows: The deflection in the plane of the diaphragm, as determined by engineering analysis, shall not exceed the permissible deflection of attached elements. Permissible deflection shall be that deflection which will permit the attached element to maintain its structural integrity under the individual loading and continue to support the prescribed loads. UBC97 requires the roof and floor diaphragms to be designed to resist the forces determined in accordance with:
Ca = seismic coefficient (see section 5.3) I = Importance factor (see Section 5.3) i = Index identifying the ith level above the base x = Floor level under design consideration W = Total seismic dead load of the building Fi = the lateral force applied to level i. Ft = that portion of the base shear, V, considered concentrated at the top of the structure in addition to Fn Wi = the portion of W at level i. wpx = the weight of the diaphragm and the elements tributary thereto at level x, including 25% of the floor live load in storage and warehouse occupancies. UBC97 makes an exception for buildings of no more than three stories in height excluding basements, with lightframe construction and for other buildings not more than two stories in height excluding basements, diaphragm design forces may be estimated using a simplified procedure as follows: F px =
3.0C a w px R
(82)
where R is the numerical coefficient representative of the inherent overstrength and global ductility of the lateralforce–resisting system as described in Chapter 5. In the above equation, Fpx should not be less than 0.5Cawpx and need not exceed Cawpx. ASCE 795 requires the floor and roof diaphragms to be designed for a minimum seismic force equivalent to 50% of the seismic coefficient Ca times the weight of the
8. Seismic Design of Floor Diaphragms
383
diaphragm. Diaphragm connections can be positive connections, mechanical or welded. IBC2000 requires the roof and floor diaphragm to be designed to resist the force Fp as follows:
F p = 0.2 I E S DS w p + V px
(83)
where: Fp = The seismic force induced by the parts. IE = Occupancy importance factor (see Section 5.4.2). SDS = The short period site design spectral response acceleration coefficient (see Section 5.4.6). wp= The weight of the diaphragm and other elements of the structure attached to. Vpx = The portion of the seismic shear force at the level of diaphragm, required to be transferred to the VLLR elements because of the offsets or changes in stiffness of the VLLR elements above or below the diaphragm. Notice that vertical distribution of lateral forces in IBC2000 takes place in accordance with Equations 525 and 526 (see Section 5.4.13) which do not necessarily conform with the distributions obtained according to the UBC97 formulas. IBC2000 provisions also require that diaphragms be designed to resist both shear and bending stresses resulting from these forces. Ties or struts should be provided to distribute the wall anchorage forces. Obviously, the floor or roof diaphragm at every level need to be designed to span horizontally between the VLLR elements and to transfer the force Fpx to these elements (see Figure 813a). All contemporary model codes require the diaphragms to be designed to transfer lateral forces from the vertical lateral load resisting elements above the diaphragm to the other VLLR elements below the diaphragm due to offsets in the placement of VLLR elements or due to changes in stiffness of these elements. For example, in Figure 813b, the
force P1 has to be transferred by the diaphragm to the VLLR elements below the diaphragm since the VLLR element above the diaphragm has been discontinued at this level. In addition, the force P2 from the other VLLR element above, has to be redistributed among the VLLR elements below the diaphragm. The diaphragm must be designed to transfer these additional loads.
Figure 813.
Code provisions for diaphragm design
As per UBC97, additional requirements for the design of diaphragms are as follows: Diaphragms supporting concrete or masonry walls should be designed with continuous ties between diaphragm chords to distribute the anchorage forces into the diaphragm. Added chords of subdiaphragms may be used to form subdiaphragms to transmit the anchorage forces to the main continuous crossties. The length to width ratio of the wood structural subdiaphragms should not exceed 2½ to 1. Diaphragm deformations should also be considered in the design of supported walls. Furthermore, in design of wood diaphragms providing lateral support for concrete or
384
Chapter 8
masonry walls in seismic zones 2, 3, and 4, anchorage should not be accomplished by use of toenails or nails subjected to withdrawal. In addition, wood framing should not be used in crossgrain bending or tension. For structures in Seismic Zones 3 and 4 having a plan irregularity of type 2 in Table 510, diaphragm chords and drag members should be designed considering independent movement of the projecting wings of the structure. Each of these diaphragm elements should be designed for the more severe of the following two conditions: 1. Motion of the projecting wings in the same direction; and 2. Motion of the projecting wings in opposing directions. This requirement is considered satisfied if a threedimensional dynamic analysis according to the code provisions is performed. As a requirement for flexible diaphragms, the design seismic forces providing lateral support for walls or frames of masonry or concrete are to be based on Equation 81 and determined with the value of the response modification factor, R, not exceeding 4.0. 8.5.2
ACI 31895 Provisions
The thickness of concrete slabs and composite topping slabs serving as structural diaphragms used to transmit earthquake forces cannot be less than 2 inches. This requirement reflects current usage in joist and waffle systems and composite topping slabs on precast floor and roof systems. Thicker slabs are required when the topping slab does not act compositely with the precast system to resist the design seismic forces. A composite castinplace concrete topping slab on precast units is permitted to be used as a structural diaphragm provided the topping slab is reinforced and its connections are proportioned and detailed for complete transfer of forces to the elements of the lateral force resisting system. A bonded topping slab is required so that the floor or roof system can provide restraint against slab buckling.
Reinforcement is required to ensure the continuity of the shear transfer across precast joints. The connection requirements are to promote provisions of a complete system with necessary shear transfers. Obviously, the castinplace topping on a precast floor or roof system can be used without the composite action provided that the topping alone is proportioned and detailed to resist the design forces. In this case, a thicker topping slab has to be provided. The shear strength requirements are the same as those for slender structural walls (see Chapter 10). The term Acv in the equation for calculating the nominal shear strength refers to the thickness times the width of the diaphragm.
8.6
DESIGN EXAMPLES
As discussed in Chapter 6, it is desirable from the structural point of view to have regular buildings with minimal offset in the location of VLLR elements and without sudden changes in stiffness from floor to floor. Quite often, however, other requirements of the project (such as architectural considerations) control these parameters and the structural engineer is faced with buildings that are considered irregular in terms of seismic behavior and design. Diaphragm design consists primarily of the following tasks: 1. Determining the lateral force distribution on the diaphragm and computing diaphragm shears and moments at different locations. 2. Providing adequate inplane shear capacity in the diaphragm to transfer lateral forces to the VLLR elements. 3. Providing suitable connection between the diaphragm and the VLLR elements. 4. Design of boundary members or reinforcement to develop chord forces, and 5. Computing diaphragm deflections, when necessary, to ascertain that the diaphragm is stiff enough to support the curtain walls, etc. without excessive deflections.
8. Seismic Design of Floor Diaphragms In addition, the diaphragm must be designed and detailed for local effects caused by various openings such as those caused by the elevator shafts. Parking structure diaphragms with ramps are a special case of diaphragms with openings. The effect of the ramp attachment to floors above and below the ramp should be considered in lateral force distribution, especially for nonshear wall buildings. In this section, the current design procedures for seismic design of floor diaphragms are demonstrated by means of four design examples which are worked out in detail. In the first example, a concrete floor diaphragm at the top of a parking level under a two story wood framed apartment building is designed. The second example explains diaphragm design for a four story concrete parking structure, which has setbacks in elevation of the building and the shear walls. In the third example, the metaldeck diaphragm of a three story steel framed office building is designed. Finally, the fourth example, explains the wood diaphragm design
385 for a typical one story neighborhood shopping center. EXAMPLE 81 It is proposed to build a two story wood framed apartment building on top of one story concrete parking. The building will be located in a zone of high seismicity. The concrete floor supporting the wood construction (see Figure 814) will be a 14 inch thick, hard rock concrete, flat plate (fc′ = 4000 lb/in2). The lateral force resisting system for the concrete parking structure consists of concrete block masonry walls (fm′ = 3000 lb/in2). Given that the superimposed dead load from the two story wood framing above is 65 pounds per square foot, design the concrete diaphragm per typical requirements of the modern model codes. Floor to floor height is 10 feet. Assume that the structural analysis of the building has produced a seismic base shear coefficient of 0.293 for strength design purposes (V=0.293W).
Figure 814. Second floor framing plan (Example 81)
386
Chapter 8
SOLUTION •
Dead loads and seismic shears:
Superimposed dead load from wood framing above = 65 lb/ft2 Concrete slab at 150 lb/ft3 = (14/12)(150) = 175 lb/ft2 Miscellaneous (M + E + top half of column weights) = 10 lb/ft2 Total floor weight = (175)(89.66)(65+175+10) = 3922.6 kips
one half of the height of a wall above and below the diaphragm will contribute to the mass of each floor. The parameters needed for determination of the center of mass of the walls are calculated in Table 81. Therefore, the center of mass of the walls is located at:
∑ xW = 12,703.0 = 88.31 ft ∑W 143.85 yW 8,564.1 =∑ = = 59.53 ft ∑W 143.85
x1 = y1
NS walls: 12in walls at 124 lb/ft2 = 4(5)(17.33)(0.124)= 43 kips EW walls: 8" wall at 78 lb/ft2 = (5)(175)(0.078) = 68.25 kips 12" walls at 124 lb/ft2 = (5)(17.33+35.33)(0.124) = 32.65 kips Figure 815. Locations of centers of mass and rigidity.
The weight of the walls parallel to the applied seismic force does not contribute to the diaphragm shears. However, in general, they are included conservatively in the design of concrete floor diaphragms. In this example, the weight of the walls parallel to the applied seismic force is not included in calculating diaphragm shears.
Since the slab is of uniform thickness, the center of mass of the floor coincides with its geometric centroid: x2 = 87.50 ft y2 = 44.83 ft Location of the combined center of mass:
EW weight = Wx = 3922.6 + 43 = 3965.6 kips NS weight = Wy = 3922.6 +68.25 + 32.65 = 4023.5 kips •
143.9(88.31) + 3922.6(87.5) 143.9 + 3922.6 = 87.53 ft
xm =
Base shears:
143.9(59.53) + 3922.6( 44.83) 143.9 + 3922.6 = 45.35 ft
ym = FPy =0.293(3965.6)=1161.9 kips (in y direction) FPx =0.293(4023.5)=1178.9 kips (in x direction) •
Center of mass (see Figure 815):
In computing the location of the center of mass of the walls it is generally assumed that
•
Center of rigidity:
For a cantilever wall (see Figure 816):
8. Seismic Design of Floor Diaphragms
387
∆=
4 P ( h / L) 3 3P ( h / L) + Et Et
The relative wall rigidities, R = 1/D, may be computed assuming a constant value of P, say P=1,000,000 pounds. Using the parameters generated in Tables 82 and 83, the location of the center of rigidity is established as:
xr =
∑ xR ∑R
y
=
4886.0 = 87.50 ft 55.84
=
6506.93 = 55.23 ft 117.8
y
Figure 816. Deformation of a cantliever wall panel
∆=
yr =
3
Ph 1.2 Ph + 3EI AG
∑ yR ∑R
x
x
•
Denoting wall thickness by t and assuming G = 0.40E for masonry, this relation may be rewritten as:
Torsional eccentricity:
ex = xr – xm = 87.5  87.53 ≈ 0 ft ey = yr – ym = 55.23  45.35 = 9.88 ft
Table 81 Center of Mass Calculations for Example 81 Length, Area, Weight, Wall Weight, ft ft2 Kips No. Lb/ft2 1 2 3 4 5 6 7
124 124 124 124 78 124 124
17.33 17.33 17.33 17.33 175.00 17.33 35.33
86.65 86.65 86.65 86.65 875.00 86.65 176.70
Σ
10.74 10.74 10.74 10.74 68.25 10.74 21.90
Dir.
x, ft
xW, ftkips
y, ft
yW ftkips
y y y y x x x
0.50 0.50 174.50 174.50 87.50 55.84 110.16
5.37 5.37 1,874.10 1,874.10 5,971.88 559.72 2,412.50
66.00 33.67 66.00 33.67 89.33 10.00 10.00
708.84 361.62 708.84 361.62 6,096.78 107.40 219.00
143.85
Table 82. Relative Rigidity of the Walls Wall Height, Length, No. ft ft 1 2 3 4 5 6 7
10 10 10 10 10 10 10
17.33 17.33 17.33 17.33 175.00 17.33 35.33
12,703.
8,564.
H/L
E, lb/in2
t, in.
∆
R = 1/∆
0.5770 0.5770 0.5770 0.5770 0.0571 0.5770 0.2830
3,000,000 3,000,000 3,000,000 3,000,000 1,500,000 3,000,000 3,000,000
11.625 11.625 11.625 11.625 7.625 11.625 11.625
0.0716 0.0716 0.0716 0.0716 0.0150 0.0716 0.0269
13.96 13.96 13.96 13.96 66.67 13.96 37.17
388
Chapter 8
Table 83. CenterofRigidity Calculations for Example 81 Wall No. Dir. x y 1 2 3 4 5 6 7
y y y y x x x
0.50 0.50 174.50 174.50 
89.33 10.00 10.00
Σ
Table 84. Wall Shear for Seismic forces in the NS Direction Wall Rx Ry dx, ft dy, ft Rd Rd2 No 1 2 3 4 5 6 7
0 0 0 0 66.67 13.96 37.17
13.96 13.96 13.96 13.96 0 0 0
87.00 87.00 87.00 87.00 34.10  45.23  45.23
1214.52 1214.52 1214.52 1214.52 2273.45 631.41 168.20
105,663 105,663 105,663 105,663 77,524 28,559 76,041
Rx
Ry
xRy
66.67 13.96 37.17
13.96 13.96 13.96 13.96 
6.98 6.98 2,436.02 2,436.02 
5,995.63 139.60 371.70
117.80
55.84
4,886.00
6,506.93
Fv, kips
Ft1, kips
Ft2, kips
Ftotal1 kips
Ftotal2 kips
Fdesign kips
294.70 294.70 294.70 294.70 0.00 0.00 0.00
20.70 20.70 20.70 20.70 38.80 10.80 28.70
20.70 20.70 20.70 20.70 38.80 10.80 28.70
274.00 274.00 315.40 315.40 38.80 10.80 28.70
315.40 315.40 274.00 274.00 38.80 10.80 28.70
315.40 315.40 315.40 315.40 38.80 10.80 28.70
Σ
1179.50
Modern codes generally require shifting of the center of mass of each level of the building a minimum of 5% of the building dimension at that perpendicular to the direction of force in addition to the actual eccentricity: ex = 0.05(175) = ± 8.75 ft ey = 9.88 ± 0.05(89.67) = 14.36 ft or 5.4 ft •
yRx
Fvy = V y
Ry ∑ Ry
and the inplane wall forces due to torsion are computed from
Ftx = Tx
Rd ∑ Rd
2
Torsional Moments:
Ty = FPy ex = 1178.9(±8.75) = ±10315.4 ftk Tx+ = FPx ey+ = 1161.9(14.36) =16,684.9 ftk Tx = FPx ey =1161.9( 5.40) = 6,274.2 ftk Inplane forces in the walls due to direct shear are computed from
Fvx = V x
Rx ∑ Rx
Fty = T y
Rd ∑ Rd
2
where d is the distance of each wall from the center of rigidity. Using these formulas, the wall forces for seismic force acting in the NS and EW directions are calculated and reported in Tables 84 and 85, respectively. Note that the contribution of torsion, if it reduces the magnitude of the design wall shears, is ignored. The design shear forces are summarized in Table 86.
8. Seismic Design of Floor Diaphragms
389
Table 85. Wall Shear for Seismic forces in the EW Direction dx, dy, Wall Ry Rd Rd2 Rx ft ft No 1 2 3 4 5 6 7
0 0 0 0 66.67 13.96 37.17
13.96 13.96 13.96 13.96 0 0 0
87.00 87.00 87.00 87.00 34.10  45.23  45.23
1214.52 1214.52 1214.52 1214.52 2273.45 631.41 168.20
Fv, kips
105,663 105,663 105,663 105,663 77,524 28,559 76,041
0.00 0.00 0.00 0.00 657.60 137.70 366.60
Ft1, kips
Ft2, kips
33.52 12.60 33.52 12.60 33.52 12.60 33.52 12.60 594.85 23.60 155.10 6.60 413.00 17.50
Ftotal1 kips
Ftotal2 kips
Fdesign Kips
33.52 12.60 33.52 33.52 12.60 33.52 33.52 12.60 33.52 33.52 12.60 33.52 594.85 634.00 634.00 155.10 144.30 155.10 413.00 384.10 413.00
Σ
1,162.95
or Table 86. Shear Design Forces (kips) Wall Wall L EW NS No ft. Load Load 1 17.33 33.52 315.40 2 17.33 33.52 315.40 3 17.33 33.52 315.40 4 17.33 33.52 315.40 5 175.00 634.00 38.80 6 17.33 155.10 10.80 7 35.33 413.00 28.70
•
VL + 2VR = 22.24
Max Load 315.40 315.40 315.40 315.40 634.00 155.10 413.00
( II )
Solving equations I and II for VL and VR yields:
VL = 4.72 k/ft, and VR = 8.76 k/ft. The midspan diaphragm moment1 (Figure 818) is: M = 548(87.5) – 19.4(79.66) – 4.72(87.5)(58.33)/2 – 6.74(87.5)(29.17)/2 = 25,758 ftkips
Diaphragm design for seismic force in the NS direction:
Check slab shear stress along walls 1 and 2: The wall forces and the assumed direction of torque due to the eccentricity are shown in Figure 817. Using this information, the distribution of the applied force on the diaphragm may be calculated. Denoting the left and right diaphragm reactions per unit length by VL and VR, from force equilibrium (see Figure 818),
VL
L = 17.33 ft,
t = 14 inches
Slab capacity without shear reinforcement
φVc = φ (2) f ' bt c =
175 175 + VR = 1179.5 Kips 2 2
0.85(2) 4000 (14)(17.33)(12) 1000
or
VL + VR = 13.48
(I )
from moment equilibrium:
175 175 175 175 VL + 2 VR = 3 2 3 2 1179.5(96.25)
1
The midspan moment has been used in this example to demonstrate the chord design procedures. This moment, however, is not necessarily the maximum moment. In a real design situation the maximum moment should be calculated and used for the chord design.
390
Chapter 8
Figure 817. Design wall forces for seismic load in the N  S direction 1 1.0(25,758) Tu = M = = 301kips d (89.66 − 4.0)
As =
T u = 301 = 5.57 in 2 φf 0.9(60) y
Provide 6#9 chord bars (As = 6.0 in2) along the slab edges at the North and South sides of the building. Here, we have assumed that the chord bars will be placed over a 4 ft. strip of the slab. •
Figure 818 Force distribution and diaphragm moments for seismic load in the NS direction.
= 313 kips ≈ 315.4 O.K. Chord Design:
Diaphragm design for seismic force in the NS direction:
A sketch of the wall forces indicating the assumed direction of the torque due to eccentricity is shown in Figure 819. Similar to the NS direction, the force and moment equilibrium equations may be used to obtain the distribution of lateral force on the diaphragm: 1
Arguably, strict conformity with the UBC97 would require this moment to be multiplied by a factor of 1.1 (UBC97 Sec. 1612.2.1 Exception 2). No such requirement exists, however, in the IBC2000 which replaces UBC97.
8. Seismic Design of Floor Diaphragms
391
φVc =
0.85(2) 4000 (14)(175)(12) 1000
=3,161 kips > 634 O.K.
1 4 .3 6 ’ o r 5 .4 ’
Figure 819. Design wall forces for seismic load in the EW direction
VL
89.66 89.66 + VR = 1162.95 Kips 2 2
or
VL + VR = 25.95
( III )
and
VL
89.66 89.66 (59.77) ( 29.89) + VR 2 2
= 1162.95(45.35) or
VL + 2VR = 39.36
( IV )
solving equations III and IV for VL and VR: VL = 12.54 k/ft and VR = 13.41 k/ft The midspan diaphragm moment (Figure 820): M = 568(34.83) + 33.52(175) – 12.55(44.83)(29.83)/2 – 12.98(44.83)(14.94)/2 = 12,916 ftkips Similarly, diaphragm moments at other locations, including the cantilever portion of the diaphragm can be calculated. •
Check diaphragm shear capacity:
along wall 5: L = 175 ft, t = 14 in.
Figure 820. Force distribution and diaphragm moments for seismic load in the EW direction
along wall 6: L = 17.33 ft, t = 14 in.
φVc =
0.85(2) 4000 (14)(17.3)(12) 1000
= 313 kips > 155 O.K. along wall 7: L = 35.33 ft, t = 14 in.
φVc =
0.85(2) 4000 (14)(35.33)(12) 1000
= 638 kips > 413 O.K. Chord Design:
392
Chapter 8
12,916 Tu = M = = 74.23 kips d (175.0 − 1.0) As =
T u = 74.23 = 1.37 in2 φf 0.9(60) y
Provide 4#6 chord bars (As = 1.76 in2) along the slab edges at the East and West sides of the building where the maximum chord force occurs.
inches thick posttensioned slabs spanning to 36 in. deep posttensioned beams. Typical floor dead load for purposes of seismic design is estimated at 150 pounds per square foot. This includes contributing wall and column weights. Typical floor to floor height is 10 feet. This building is irregular and therefore needs to be analyzed using the dynamic response procedures. Furthermore, the redundancy factor for the building needs to be calculated and applied. For preliminary design purposes only, however, use the UBC97 static lateral force procedure and ignore accidental torsion. Soil profile type is SD, I = 1.0, Na = Nv =1.0. Use fc′ = 5,000 lb/in2 and Fy = 60,000 lb/in2.
Figure 821.Ground floor framing plan (Example 82).
EXAMPLE 82 Perform a preliminary design the third floor diaphragm of the four story parking structure shown in Figures 821 through 825. The building is to be located in southern California (UBC seismic zone 4). Access to each floor will be provided from an adjacent parking structure that will be separated by a seismic joint. Typical floor and roof framing consists of a 5½
Figure 822.Second and third floor framing plan (Example 82)
SOLUTION •
Weight Computations:
Roof Weight = (68')(185')(0.15 k/ft2) = 1887 kips
8. Seismic Design of Floor Diaphragms
393
4th Floor Weight = (85')(185')(0.15 k/ft2) = 2359 kips 3rd Floor Weight = (104')(185')(0.15 k/ft2) = 2886 kips 2nd Floor Weight = (104')(185')(0.15 k/ft2) = 2886 kips Total Weight = 1887 + 2359 + 2(2886) = 10018 kips
= 0.447(W ) > (0.11Ca I )W = 0.048W ZN v I > 0.8 (W ) = 0.07W R > 2.5
Ca I (W ) = 0.244W R
∴ V = 0.244 W = 2444.4 kips
Fx = (V − Ft )
Fpx =
W x hx ∑ Wx hx
Ft + ∑ Fi
∑W
w px
i
T = 0.318 Sec. < 0.7 Sec. ⇒ Ft = 0
Figure 823. Fourth floor framing plan (Example 82)
•
Design Lateral Forces
T = Ct (hn)3/4 Take Ct =0.02 ∴ T = 0.02(40)3/4 = 0.318 Sec.
C I v (W ) RT 0.64(1.0) (W ) = 4.5(0.318)
Base Shear (V ) =
Figure 824. Roof framing plan (Example 82)
394
Chapter 8
Figure 825. A section through the building (Example 82)
Values of Fpx for various floors are calculated in Table 87. Concrete diaphragm is assumed to be rigid. The seismic shear forces acting on the walls were obtained by a computer analysis and are shown in Figures 826 and 827.
Figure 827. Forces on the third floor diaphragm due to EW seismic loading (Wall shears above the diaphragm are shown with solid arrows while wall shears below the diaphragm are indicated by dashed lines.)
•
Diaphragm Design in the NS Direction:
Net shear forces acting on the walls and the corresponding diaphragm load, shear and moment diagrams are shown in Figure 828. Check 8" thick slab shear capacity along the walls on grid lines B and C: Maximum slab shear = 283.75 kips Slab capacity without shear reinforcement =
0.85(2) 5000 (5.5)(37)(12) 1000 = 294 > 283.75 kips O.K.
φVc = φ 2 f c' =
Figure 826. Forces on the third floor diaphragm due to NS seismic loading (Wall shears above the diaphragm are shown with solid arrows while wall shears below the diaphragm are indicated by dashed lines.)
Therefore, no shear reinforcement seems to be required by the code. Chord Design:
8. Seismic Design of Floor Diaphragms
395
Table 87. Calculation of Diaphragm Design Forces for Example 82 Level
hx, ft
Wx, Kips
Wx.hx
Wx.hx ΣWihi
Fx, Kips
Roof 4th 3rd 2nd
40 30 20 10
1,887 2,359 2,886 2,886
75,480 70,770 57,720 28,860
0.324 0.304 0.248 0.124
792.4 743.0 606.0 303.0
10,018
232,830
1.00
2444.4
Σ
•
ΣFx Kips 792.4 1,535.4 2,141.4 2,444.4
Diaphragm Direction:
ΣWi, Kips 1,887 4,246 7,132 10,018
ΣFi Σwi 0.420 0.362 0.300 0.244
Design
in
Fpx, Kips 792.4 853.1 866.5 704.2
the
EW
Net shear forces acting on the walls and the corresponding diaphragm load, shear and moment diagrams are shown in Figure 829. Moment Calculations: at Section AA:
M A− A = 1,401(25.4) −
8.53(25.4) 2 2
= 32,833 ft  kips at Section BB: M B−B = 1,401(50.8) − 590.6( 4.5) −
8.53(50.8) 2 2
= 57,505 ft  kips at Section CC:
M C −C = 56(25.4) −
Figure 828. Diaphragm loading, shear, and moment diagrams for seismic load in the NS direction
8,586 Tu = M = = 85.4 kips d (101.58 − 1.0) T As = u = 85.4 = 1.58 in2 φf 0.9(60) y Therefore provide 3 #7 chord bars (As = 1.8 in2) along slab edges on the North and South sides of the building.
8.53(25.4) 2 2
+ 16.1 (1031)(63.5) 37 = 27,158 ft  kips ∴Estimated maximum moment1 = 57,505 ftk Chord Design:
57,505 Tu = M = = 315 kips d (184.5 − 2.0)
1
A more accurate value of the maximum moment may be obtained by reading the moment diagram plotted to a larger scale.
396
Chapter 8 for L = 184.5 ft, slab capacity without shear reinforcement is:
0.85(2) 5000 (5.5)(184.5)(12) = 1000 = 1465 kips > 1401 O.K.
φVc =
Check the capacity of 30 foot long slab with #4 bars @ 18 inches, at the top and bottom of the slab:
φVc = 238 kips #4 @ 18" As = 0.13 in2/ ft
φVs = (0.85)(2×0.13)(60)(30 ft) = 398 kips φVn = 398 + 238 = 636 kips < 1401 kips Drag struts are needed to transfer the difference (1401  636 = 765 kips). •
Design of Drag Struts (see Figure 830):
Figure 829. Diaphragm loading, shear, and moment diagrams for seismic load in the EW direction
As =
T u = 315 = 5.83 in 2 φf 0.9(60) y
Therefore provide 6 #9 chord bars (As= 6.0 in2) along slab edges on the east and west sides of the building
Cu = Tu Compression Cu to be resisted by edge beam and concrete slab. Check 5½in.thick slab shear capacity along the wall on line 1: For L = 30 ft, slab capacity without shear Reinforcement is:
0.85(2) 5000 (5.5)(30)(12) 1000 = 238 kips < 1401 N.G.
φVc =
Figure 830. Diaphragm chord, drag, and shear reinforcement
The two beams along the Grid line 1 may be designed to transfer the slab shear into the walls:
8. Seismic Design of Floor Diaphragms
As =
(7652 ) (0.9)(60)
φVn = 294 + 490 =784 kips > 515.5
= 7.08 in 2
∴ Provide 8 #9 bars (As = 8.0 in2) in the beams for seismic shear transfer. Drag strut length provided = 2(77.3) = 154.6 ft Capacity of slab along drag strut
=
397
0.85(2) 5000 (5.5)(154.6)(12) 1000 = 1228 kips > 693 O.K.
Check shear capacity of 5½in. thick slab at the wall on grid line 4 to carry 590.8/2 = 295.4 kips of shear (notice that slab occurs on both sides of the wall):
0.85(2) 5000 (5.5)(21)(12) 1000 = 167 kips < 295.4 N.G.
φVc =
Therefore Shear reinforcement is required. Using #4 bars @ 18 inches at the top and bottom of the slab:
φVs = (0.85)(2×0.13)(60)(21) = 278 kips φVn = 167 + 278 = 445 kips > 295.4 O.K. Therefore drag struts are not required. It can be realized by observation that the slab shear capacity along the walls on the grid line 7 is sufficient. Check the shear capacity of the slab along the cross walls on grid lines B and C. Here again, slab occurs on both sides of the wall:
0.85(2) 5000 (5.5)(37)(12) 1000 1031 = 294 kips < = 515.5 N.G. 2
φVc =
Therefore shear Reinforcement is required. Try #4 bars @ 18 inches at the top and bottom of the slab:
φVs = (0.85)(0.13×2)(60)(37) = 490 kips
Therefore drag struts are not required. EXAMPLE 83 Design the roof diaphragm of the three story steel framed building shown in Figure 831. The building is supported on the top of a one story subterranean concrete parking structure. The parking structure deck may be considered as the shear base for the steel structure. The lateral load resisting system for the steel building consists of moment resisting frames in both directions. Beams and columns which are not part of the lateral system are not shown in Figure 831. The floor construction consists of 3 1/4 inches of lightweight concrete on the top of a 3 inch deep, 20 gage, metal deck. The maximum spacing of floor purlins is 10 feet. Mechanical equipment is located on the roof, west of grid line D. The roof construction west of grid line D consists of 4 1/2 inches of hard rock concrete on the top of a 3 inch deep, 18 gage, metal deck. The maximum spacing of the roof purlins is 8 feet. The roof construction east of grid line D is similar to the typical floor construction. The estimated total dead loads for seismic design are 100 psf at the typical floors, 200 psf at the mechanical areas of the roof, and 70 psf elsewhere on the roof. The building is located in area of high seismicity. A three dimensional computer analysis of the building has resulted in a working stress level (WSD) roof diaphragm design force of 364.8 kips in the NS and EW directions. The distribution of the roof diaphragm shear among the momentreistant steel frames are shown in Figures 832 and 833. SOLUTION •
Diaphragm Design in the EW Direction
398
Chapter 8
Figure 831. Typical floor framing plan for building of Example 83 (Opening shown exist on second and third floors only)
The design lateral force of 3604.8 kips is distributed along the roof in the same proportion as the mass distribution at this level. This loading pattern and the corresponding diaphragm shear diagram are shown in Figure 834. The maximum diaphragm shear per linear foot occurs at grid line 10 and is equal to:
Figure 833. Frame shears for NS seismic loading
v=
Figure 832. Frame shears for EW seismic loading
29.9 kips = 1.44 k/ft (3.8 + 14.5 + 2.5) ft
This value, has to be compared with the allowable shear values supplied by the metal deck manufacturer. For example, if a Verco 20 gage, W3 Formlok deck with 3 1/4 lightweight
8. Seismic Design of Floor Diaphragms
399
Figure 834. Diaphragm loading and shear diagrams for the EW seismic loading
concrete fill and puddle welds in every flute is used, the allowable shear would be 1.74 kips compared to the required value of 1.44 kips (see Figure 835). Check diaphragm chord requirements: As mentioned earlier in this Chapter, the frame beams at the perimeter of the building will act as chord members or flanges of the diaphragm. To get a handle on the magnitude of
the chord forces, diaphragm moments are computed at various sections. The transverse shear forces (in the NS frames) are small and hence, are ignored in this analysis. Moment at grid line 13 = 29.9(60) 0.38(11)(57)  0.57(9)(47) 0.90(10.75)(37.125) – 1.15(31.75)2/2 = 375.8 kipsft
400
Chapter 8
Figure 835. A Verco Formlok diaphragm design table (reproduced with permission of Verco Manufacturing Company, Benicia, California)
8. Seismic Design of Floor Diaphragms
401
Figure. 835 (continued)
Chord force at grid line 13 = 375.8/57.58 = 6.52 kips Moment at grid line 16 = 29.9(120) – 0.38(11)(137) – 0.57(9)(107) – 0.90(10.75)(97.125) – 1.15(87.92)(47.76) – 4.24(3.8)2/2 + 68.5(60) = 777.2 kft Chord force at grid line 16 = 777.2/57.58 = 13.5 kips
Similarly, diaphragm moments and chord forces can be computed at other locations. In design of beams and the beamcolumn connections, these chord forces must be considered. The metal deckbeam welds must be verified to be able to develop the chord forces in addition to their shear transfer capability. •
Diaphragm Design in the NS Direction
402
Chapter 8
Here again, the applied lateral force of 364.8 kips is distributed in proportion to the mass distribution (see Figure 836). Diaphragm shears and moments at any location can be computed similar to the eastwest seismic analysis. For example,
calculations, we compute moment at grid line D:
the
diaphragm
diaphragm moment at grid line D
= 99.3(58.92 ) −
3.01(60.67 ) 2
2
= 311 ft  kips Chord force at grid line D = 311.05/52.92 = 5.87 kips To complete this design, diaphragm moments should be computed at a few other locations on the diaphragm, in order to establish the maximum moment, and the corresponding maximum chord force. The beams along grids 16 and 18, near grid line D may be designed to carry these chord forces. EXAMPLE 84
Figure 836. Diaphragm loading diagrams for the NS seismic loading
diaphragm shear at grid line G.1
=
99.3  3.01(1.75) = 1.59 kips/ft 59.25
diaphragm shear at grid line D
=
3.01(60.67 ) − 99.3 = 1.40 kips/ft 59.25
Both of the above computed diaphragm shears are less than the allowable shear value of 3.07 kips per linear foot for a Verco 18 gage, W3 Formlok deck with puddle welds in all flutes. As an example of diaphragm moment
The ground floor and roof plans of a one story neighborhood shopping center which is being planned for a city in a zone of high seimsicity are shown in Figure 837. The roof framing consists of plywood panelized roof with glue laminated beams and purlins. The roof dead load for the purposes of seismic design calculations is estimated to be 16 pounds per square foot. In addition to the framing weight, this includes allowances for composition roof, insulation, acoustic tile ceiling and a miscellaneous load of 1.5 pounds per square foot. Design the roof diaphragm in accordance with the UBC97 requirements (IBC2000 diaphragm design process is virtually the same). Assume Z = 0.40, I =1.0, Na = Nv = 1.0, and the SB soil type. •
Dead load and base shear in the NS direction
north wall at 75 lb/ft2 = 75(14/2 + 2)(180) = 121,500 lb
8. Seismic Design of Floor Diaphragms
403
Figure 837. Floor plans for building of Example 84
pilasters in North wall = 75(14/2)(1.33×8) = 5,600 lb
roof at 16 lb/ft2 = 16(180)(56.67) = 163,210 lb
pilasters in South piers = 75(14/2)(1.33 ×10) = 7,000 lb
total dead load = 121,500 + 5,600 + 7000 + 40500+16,200 + 163,210 = 354,010 lb
south piers at 75 lb/ft2 = 75(14/2+2)(10×6) = 40500 lb glass window at 15 lb/ft2 = 15(14/2 + 2)(7×14 + 2×11) = 16,200 lb
Because this is a one story lightweight structure, we can use the simplified method according to UBC97 section 1629.8.2. Notice that for flexible diaphragms providing lateral
404
Chapter 8
support for masonry, an R value of 4.0 must be used (UBC97 section 1633.2.9.3)
Base Shear (V ) = Fpx =
3.0C a W R
3.0C a W px R
3.0(0.4) Fpx = W px = 0.30W px 4.0 = 0.30(354,010) = 106,203 lb in N − S direction This value, however, is intended for strength design purposes. To convert it to the corresponding working stress design value, we divide it by a load factor of 1.4.
Fpx (WSD ) = •
106,203 = 75,859 lb 1.40
Diaphragm design in the NS direction (see Figure 838):
75,859 lb 180 ft = 421 lb/ft
N − S diaphragm load =
East wall: diaphragm shear = 421(80/2) = 16,840 lb diaphragm unit shear = 16,840/56.67 = 297 lb/ft force in the drag strut = 297(32.67) = 9,703 lb Center Wall: east side shear = 421(80/2) = 16,840 lb diaphragm unit shear = 16,840/56.67 = 297 lb/ft west side shear = 421(100/2) = 21,050 lb diaphragm unit shear = 21,050/56.67 = 372 lb/ft force in the drag strut = (297 + 372)(32.67) =21,856 lb West Wall: diaphragm shear = 421(100/2) = 21,050 lb diaphragm unit shear = 21,050/56.67 = 372 lb/ft force in the drag strut = 372(32.67) = 12, 153 lb
Figure 838. Chord forces for the NS seismic loading based on flexible diaphragm assumption.
The diaphragm is assumed to be flexible. Therefore, in both directions, the wall loads will be based on the tributary diaphragm areas.
Diaphragm plywood requirements: Per UBC97 Table 23IIH (or similarly from IBC2000 Table 2306.3.1), use 3/8in. Structural 1 wood panel diaphragm, blocked, 8d nails at 21/2in. on center at the boundaries and continuos panel edges, 8d nails at 4 in. on center at other panel edges, and 12 in. on center on intermediate framing members. Allowable diaphragm shear is 530/1.4= 378 lb/ft which is greater than the maximum demand of 372 lb/ft. Chord Design (see Figure 838): for the 100 ft span:
8. Seismic Design of Floor Diaphragms
421(100) = 526,250 ft  lb 8 2
M =
d = 56.67 − C or T =
8 = 56.0 ft 12
405 means of the steel angle shown in Figure 840. The steel angle is welded to the steel beam and bolted to the wall. A wood ledger is used to transfer the diaphragm shear from the plywood to the wall, and to attach purlins to the wall.
536,250 = 9,397 lb 56.0
for the 80 ft span:
421(80) M = = 336,800 ft  lb 8 2
d = 56.67 − C or T =
8 = 56.0 ft 12
Figure 839. Typical detail for transfer of shear from plywood to the drag strut
336,800 = 6,014 lb 56.0
Provide horizontal reinforcement as chord reinforcement in the North wall at the roof level. The maximum required area of steel is:
As =
9,397 = 0.30 in 2 1.33(24,000)
Therefore a #5 continuous horizontal bar may be used typically (AS = 0.31 in2). A chord member is also required on the south side of the diaphragm. Alternatively, a timber chord member may be designed and used. Since the required chord area is small, one can design the edge purlin to act as a chord. Bolt purlin to the piers and provide metal strap across the beams for continuity of the chord. Design of drag struts: The steel beams may be designed to act as drag struts to transfer the drag force from the steel beam to the block walls (see Figure 838). Diaphragm shear is transferred from plywood to the drag strut by means of the nailer as shown in Figure 839. The nailer is bolted to the drag strut. The plywood sheathing is nailed to the nailer. The drag strut force is transferred to the wall by
Figure 840. Typical detail for transfer of force from drag struts to a block shear wall
•
Dead load and base shear in the EW direction:
east and West walls at 75 psf = 75(14/2+2)(2)(24) + 75(14/2)(24) = 45,000 lb pilasters at 75 psf = 75(14/2)(16/12)(3) = 2,100 lb glass windows at 15 psf = 15(14/2 + 2)(2)(32.67) = 8,821 lb roof at 16 psf = 16(180)(56.67) = 163,210 lb total dead load = 45,000 + 2,100 + 8,821 + 163,210 = 219,131 lb
406
Chapter 8
Fpx = 0.30W px = 0.3(219,131) = 65,739 lb Fpx (WSD ) = •
65,739 = 46,957 lb 1.40
diaphragm design in the EW direction (see Figure 841):
M = 829(56.67)2/8 = 332,791 ftlb d = 180  8/12 = 179.33 ft C or T = 332,791/179.33 = 1,856 lb The chord force is small. Hence, the steel beam and the horizontal reinforcement in the block wall will work as chord members. •
North wall:
46,957 lb 56.67 ft = 829 lb/ft 56.67 diaphragm shear = 829 × = 23,490 lb 2 E − W diaphragm load =
effective length of diaphragm = 180 ft
23,490 180 =131 lb/ft < 378 lb/ft
Diaphragm deflections:
The span to width ratio of the diaphragm in both directions is less than 4. Therefore, deflection is not expected to be a problem. However, if a deflection check is necessary, a simple procedure described in the Timber Construction Manual(814) or formula 231 of the IBC2000 may be used to estimate diaphragm deflections.
diaphragm unit shear =
Therefore plywood requirements specified for NS seismic is adequate along this wall. South wall:
diaphragm shear = 829 ×
56.67 2
= 23,490 lb Length of diaphragm in direct contact with the wall is 10×6ft = 60 ft. However, the southside edge purlins, which were also designed and detailed as the chord for NS seismic, will act as drag members along the south wall. Therefore, diaphragm shear = 23,490/180=131 < 378 lb/ft. Hence, previously specified plywood detailing will be adequate. Push or pull at the wall in a typical drag strut is T = (131 lb/ft)(14/2 ft) = 917 lb. The edge purlin and its bolting to the wall must be verified for the above force. Chord design: diaphragm span = 56.67 ft
Figure 841. Chord forces for EW seismic loading
8. Seismic Design of Floor Diaphragms
REFERENCES 81 International Conference of Building Officials (1997), The Uniform Building Code –1997 Edition, Whittier, California. 82 International Code Council (2000), International Building Code 2000, Virginia. 83 Boppana, R.R., and Naeim, F., "Modeling of Floor Diaphragms in Concrete Shear Wall Buildings," Concrete International, Design & Construction, ACI, July, 1985. 84 Roper, S.C., and Iding, R.H., "Appropriateness of the Rigid Floor Assumption for Buildings with Irregular Features," Proceedings of 8th World Conference on Earthquake Engineering, San Francisco, California, 1984. 85 Mendes, S., "Wood Diaphragms: Rigid Versus Flexible Inappropriate Assumptions Can Cause ShearWall Failures," Proceedings of the 56th Annual Convention, Structural Engineers Association of California, San Diego, California, 1987. 86 S.B. Barnes and Associates, "Report on Use of H.H. Robertson Steel Roof and Floor Decks as Horizontal Diaphragms," prepared for H.H. Robertson Company by S.B. Barnes and Associates, Los Angeles, California, July, 1963. 87 American Iron and Steel Institute, "Design of Light Gage Steel Diaphragms," American Iron and Steel Institute, New York, 1982. 88 Jain, S.K., "Analytical Models for the Dynamics of Buildings," Earthquake Engineering Research Laboratory, Report No. 8302, California Institute of Technology, Passadena, May, 1983. 89 Steinburgge, K.V., Manning, J.H, and Dagenkolb, H.J., "Building Damage in Anchorage," in The Prince Williams Sound, Alaska, Earthquake of 1964, and Aftershocks, F.J. Wood (EditorinChief), U.S. Department of Commerce, Washington, D.C., 1967. 810 Buildings Officials and Code Administrators International, "The BOCA Basic Building Code," Homewood, Illinois, 1987. 811 American National Standards Institute, "American National Standards Building Code Requirements for Minimum Design Loads in Buildings and Other Structures," ANSI A58.11982, New York, 1982. 812 Federal Emergency Management Agency, "1997 Edition of NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings,", 1997 813 Seismology Committee of Structural Engineers Association Of California, "Tentative Lateral Force Requirements," NovemberDecember, 1986. 814 American Institute of Timber Construction, "Timber Construction Manual," 2nd edition, American Institute of Timber Construction, Englewood, Colorado, 1974.
407 815 American Concrete Institute (1995), Building Code Requirements for Reinforced Concrete—ACI 31895, Detroit, Michigan. 816 Building Seismic Safety Council, “NEHRP Recommended Provisions for Seismic Regulations for New Buildings and other Structures, (NEHRP— National Earthquake Hazards Reduction Program), 1997 Edition, Part 1: Provisions, Part 2: Commentary, Washington, DC 20005. 817 American Society of Civil Engineers (1996), Minimum Design Loads for Buildings and other Structures (ASCE 795), ASCE, New York, 1996
408
Chapter 8
Chapter 9 Seismic Design of Steel Structures
ChiaMing Uang, Ph.D. Professor of Structural Engineering, University of California, San Diego
Michel Bruneau, Ph.D., P.Eng. Professor of Civil Engineering, State University of New York at Buffalo
Andrew S. Whittaker, Ph.D., S.E. Associate Professor of Civil Engineering, State University of New York at Buffalo
KeyChyuan Tsai, Ph.D., S.E. Professor of Civil Engineering, National Taiwan University
Key words:
Seismic Design, Steel Structures, NEHRP Recommended Seismic Provisions, AISC Seismic Provisions, R Factor, Ductility, System Overstrength, Capacity Design, 1994 Northridge Earthquake, MomentResisting Frames, Brittle Fracture, Moment Connections, Concentrically Braced Frames, Buckling, Braces, Eccentrically Braced Frames, Links.
Abstract:
Seismic design of steel building structures has undergone significant changes since the Northridge, California earthquake in 1994. Steel structures, thought to be ductile for earthquake resistance, experienced brittle fracture in welded moment connections. The latest AISC Seismic Provisions reflect the significant research findings that resulted from the Northridge earthquake. This chapter first starts with a description of the seismic design philosophy, the concept of system parameters (R, Cd, and Ωo) and capacity design. Background information for the seismic requirements in the AISC Seismic Provisions of Moment Frames, Concentrically Braced Frames, and Eccentrically Braced Frames are then presented. Design examples are provided for each of the three structural systems.
409
410
Chapter 9
9. Seismic Design of Steel Structures
9.1
Introduction
9.1.1
General
Steel is one of the most widely used materials for building construction in North America. The inherent strength and toughness of steel are characteristics that are well suited to a variety of applications, and its high ductility is ideal for seismic design. To utilize these advantages for seismic applications, the design engineer has to be familiar with the relevant steel design provisions and their intent and must ensure that the construction is properly executed. This is especially important when welding is involved. The seismic design of building structures presented in this chapter is based on the NEHRP Recommended Provisions for the Development of Seismic Regulation for New Buildings (BSSC 1997). For seismic steel design, the NEHRP Recommended Provisions incorporate by reference the AISC Seismic Provisions for Structural Steel Buildings (1997b). 9.1.2
NEHRP Seismic Design Concept
The NEHRP Recommended Provisions are based on the Rfactor design procedure. In this procedure, certain structural components are designated as the structural fuses and are specially detailed to respond in the inelastic range to dissipate energy during a major earthquake. Since these components are expected to experience significant damage, their locations are often selected such that the damage of these components would not impair the gravity loadcarrying capacity of the system. Aside from these energy dissipating components, all other structural components including connections are then proportioned following the capacity design concept to remain in the elastic range. Consider a structural response envelope shown in Figure 91, where the abscissa and ordinate represent the story drift and base shear
411 ratio, respectively. If the structure is designed to respond elastically during a major earthquake, the required elastic base shear ratio, Ceu, would be high. For economical reasons, the NEHRP Recommended Provisions take advantage of the structure's inherent energy dissipation capacity by specifying a design seismic force level, Cs, which is reduced significantly from Ceu by a response modification factor, R: Cs =
C eu R
(91)
The Cs design force level is the first significant yield level of the structure, which corresponds to the force level beyond which the structural response starts to deviate significantly from the elastic response. Idealizing the actual response envelope by a linearly elasticperfectly plastic response shown in Figure 91, it can be shown that the R factor is composed of two contributing factors (Uang 1991): R = Rµ Ω o
(92)
The ductility reduction factor, Rµ, accounts for the reduction of seismic forces from Ceu to Cy, Such a force reduction is possible because ductility, which is measured by the ductility factor µ (= δs/δy), is provided by the energydissipating components in the structural system. The system overstrength factor, Ωo, in Eq. 92 accounts for the reserve strength between the force levels Cy and Cs. Several factors contribute to this overstrength factor. These include structural redundancy, story drift limits, material overstrength, member oversize, nonseismic load combinations, and so on. The Rfactor design approach greatly simplifies the design process because the design engineer only has to perform an elastic structural analysis even though the structure is expected to deform well into the inelastic range during a major earthquake. After the elastic story drift, δe, is computed from a structural analysis, the NEHRP Recommended Provisions then specify a deflection amplification factor,
412
Chapter 9
Figure 91. General structural response envelope
Cd, to estimate the Design Story Drift, δs, in Figure 91: δs =
Cd δe I
(93)
where I is the Occupancy Importance Factor. The story drift thus computed cannot exceed the allowable drift specified in the NEHRP Recommended Provisions. Depending on the Seismic Use Group, the allowable drift for steel buildings varies from 1.5% to 2.5% of the story height. Note that the ultimate strength of the structure (Cy in Figure 91) is not known if only an elastic analysis is performed at the Cs design force level. Nevertheless, the ultimate strength of the structure is required in capacity design to estimate, for example, the axial force in the columns when a yield mechanism forms in the structure. For this purpose, the NEHRP Recommended Provisions specify Ωo values to simplify the design process. Therefore, in addition to the load combinations prescribed in
the AISC LRFD Specification (1993), the AISC Seismic Provisions require that the columns be checked for two additional special load combinations using the amplified horizontal earthquake load effects, ΩoE: 1 .2 D + 0 .5 L + 0 .2 S + Ω o E
(94)
0.9 D − Ω o E
(95)
The amplified seismic load effects are to be applied without consideration of any concurrent bending moment on the columns. In addition, the required strengths determined from these two load combinations need not exceed either (1) the maximum load transferred to the column considering 1.1Ry times the nominal strengths of the connecting beam or brace elements of the frame, or (2) the limit as determined by the resistance of the foundation to uplift. Refer to the next section for the factor Ry. The R, Cd, and Ωo values specified in the NEHRP Recommended Provisions for different types of steel framing systems are listed in
9. Seismic Design of Steel Structures
413
Table 91. Steel framing systems and design parameters (NEHRP 1997) Frame System Bearing Wall Systems Ordinary Concentrically Braced Frames (OCBFs) Building Frame Systems Eccentrically Braced Frames (EBFs) • Moment connections at columns away from links • Nonmoment connections at columns away from links Special Concentrically Braced Frames (SCBFs) Ordinary Concentrically Braced Frames(OCBFs) Moment Resisting Frame Systems Special Moment Frames (SMFs) Intermediate Moment Frames (IMFs) Ordinary Moment Frames (OMFs) Special Truss Moment Frames (STMFs) Dual Systems with SMFs Capable of Resisting at Least 25% of Prescribed Seismic Forces Eccentrically Braced Frames (EBFs) • Moment connections at columns away from links • Nonmoment connections at columns away from links Special Concentrically Braced frames (SCBFs) Ordinary Concentrically Braced Frames (OCBFs)
Table 91. Seismic design of three widely used systems (momentresisting frames, concentrically braced frames, and eccentrically braced frames) that are presented later in this chapter makes use of these parameters. 9.1.3
Structural Steel Materials
The ductility of steel generally reduces with an increase of the yield stress. Therefore, the AISC Seismic Provisions permit only the following grades of steel for seismic design: ASTM A36, A53, A500 (Grades B and C), A501, A572 (Grades 42 or 50), A588, A913 (Grade 50 or 65), or A992. Further, for those structural members that are designed to yield under load combinations involving Ωo times the design seismic forces, the specified minimum yield strength, Fy, shall not exceed 50 ksi unless the suitability of the material is determined by testing or other rational criteria. This limitation does not apply to columns of A588 or A913
R
Ωo
Cd
4
2
3½
8 7 6 5
2 2 2 2
4 4 5 4½
8 6 4 7
3 3 3 3
5½ 5 3½ 5½
8 7 8 6
2½ 2½ 2½ 2½
4 4 6½ 5
Grade 65 steel for which the only expected inelastic behavior is yielding at the column base. The specified minimum yield strength is used to design the structural components that are expected to yield during the design earthquake. However, to estimate the force demand these components would impose on other structural components (including connections) that are expected to remain elastic, the expected yield strength, Fye, of the energy dissipating components needs to be used for capacity design: F ye = R y F y
(96)
For rolled shapes and bars, the AISC Seismic Provisions stipulate that Ry shall be taken as 1.5 for A36 and 1.3 for A572 Grade 42. For rolled shapes and bars of other grades of steel and for plates, Ry shall be taken as 1.1 (SSPC 1995).
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Chapter 9
Figure 92. (a) Geometry considering finite dimensions of members, (b) Typical moment diagram under lateral loading, and (c) Corresponding member forces on beams, columns, and panel zones
9. Seismic Design of Steel Structures
9.2
Behavior and Design of MomentResisting Frames
9.2.1
Introduction
Steel momentresisting frames (SMFs) are rectilinear assemblies of columns and beams that are typically joined by welding or highstrength bolting or both. Resistance to lateral loads is provided by flexural and shearing actions in the beams and the columns. Lateral stiffness is provided by the flexural stiffness of the beams and columns; the flexibility of the beamcolumn connections are often ignored although such flexibility may substantially increase deflections in a momentresisting frame. Components of an SMF together with sample internal actions are shown in Figure. 92. The AISC Seismic Provisions define three types of seismic steel momentresisting frames: Ordinary Moment Frames, Intermediate Moment Frames, and Special Moment Frames. All three framing systems are designed assuming ductile behavior of varying degrees, for earthquake forces that are reduced from the elastic forces by a response modification factor, R (see Table 91 for values of R). SMFs are considered to be the most ductile of the three types of moment frames considered by AISC. For this reason, and due to their architectural versatility, SMFs have been the most popular seismic framing system in high seismic regions in the United States. SMFs are designed for earthquake loads calculated using a value of R equal to 8. Stringent requirements are placed on the design of beams, columns, beamtocolumn connections, and panel zones. Beamtocolumn connections in SMFs are required to have a minimum inelastic rotation capacity of 0.03 radian. Intermediate Moment Frames (IMFs) are assumed to be less ductile than SMFs but are expected to withstand moderate inelastic deformations in the design earthquake. IMFs are designed using a value of R equal to 6; fully restrained (FR) or partially restrained (PR)
415 connections can be used in such frames. Beamtocolumn connections in IMFs are required to have an inelastic rotation capacity of 0.02 radian. Other requirements are listed in the AISC Seismic Provisions (1997b). Ordinary moment frames (OMFs) are less ductile than IMFs, and are expected to sustain only limited inelastic deformations in their components and connections in the design earthquake. Beamtocolumn connections in OMFs are required to have an inelastic rotation capacity of 0.01 radian. FR and PR connections can be used in OMFs. Because OMFs are less ductile than IMFs, an OMF must be designed for higher seismic forces than an IMF; an OMF is designed for earthquake loads calculated using a value of R equal to 4. The remainder of this section addresses issues associated with the design, detailing, and testing of special moment frames and components. The design philosophy for such frames is to dissipate earthquakeinduced energy in plastic hinging zones that typically form in the beams and panel zones of the frame. Columns and beamtocolumn connections are typically designed to remain elastic using capacity design procedures. 9.2.2
Analysis and Detailing of Special Moment Frames
Because the SMF is a flexible framing system, beam and column sizes in SMFs are often selected to satisfy story drift requirements. As such, the nominal structural strength of an SMF can substantially exceed the minimum base shear force required by the NEHRP Recommended Provisions. When analyzing SMFs, all sources of deformation should be considered in the mathematical model. NEHRP stipulates that panel zone deformations must be included in the calculation of story drift. The AISC Seismic Provisions prescribe general requirements for materials and connections that are particularly relevant to SMF construction:
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Chapter 9
1. Steel in SMF construction must comply with the requirements described in Section 9.1.3. In addition, a minimum Charpy Vnotch toughness of 20 ftlbs at 70°F is required for thick materials in SMFs: ASTM A6 Group 3 shapes with flanges 1½ inches or thicker, ASTM A6 Groups 4 and 5 shapes, and plates that are 1½ inches or greater in thickness in builtup members. 2. Calculation of maximum component strengths (e.g., for strong columnweak beam calculations) for capacity design must be based on the expected yield strength, Fye (see Eq. 96). 3. To prevent brittle fractures at the welds, AISC prescribes that welded joints be performed in accordance with an approved Welding Procedure Specifications and that all welds used in primary members and connections in the seismic force resisting system be made with a filler metal that has a minimum Charpy Vnotch toughness of 20 ftlbs at minus 20°F. 9.2.3
Beam Design
A beam in a steel SMF is assumed to be able to develop its full plastic moment (Mp) calculated as M p = Z b Fy
(97)
where Z b is the plastic section modulus. In order to prevent premature beam flange or web local buckling, and to maintain this moment for large plastic deformations, the widththickness ratios of the web and flange elements should be limited to the values of λ ps given in Table 92. (The λ p values are for nonseismic design.) In addition, both flanges of the beam must be laterally braced near potential plastic hinges; the unbraced length of the beam must not exceed 2500 ry /Fy, where ry is the radius of
gyration about the weak axis for outofplane buckling. 9.2.4
BeamtoColumn Connections
Introduction For discussion purposes, a beamtocolumn connection includes the beamcolumn panel zone and the beamtocolumn joints. Connections in an SMF need to satisfy three criteria: (1) a sufficient strength to develop the full plastic moment of the beam, (2) a sufficient stiffness to satisfy the assumption of a fully rigid (FR) connection, and (3) a large postyield deformation capacity without significant loss of strength. Prior to the 1994 Northridge, California earthquake, the welded flangebolted web steel moment connections were assumed by design professionals to easily satisfy all three criteria. Unfortunately, many momentresisting connections suffered extensive damage during this earthquake. In addition to brittle fracture in the groove welded connections (mostly in the beam bottom flange), other types of fracture that were seldom observed in laboratory testing prior to the Northridge earthquake were also reported. Figure 93a shows cracks extending into the column panel zone, and Figure 93b presents a “divot” pullout from the column flange. The causes of failure are discussed in Bruneau et al. (1997). The poor performance of welded momentframe connections in more than 200 multistory buildings in the Northridge earthquake led to the development of a national program, funded by the Federal Emergency Management Agency (FEMA), to investigate the causes of failure and to develop alternative connections for repair, rehabilitation, and new construction. Part of the FEMA program involved fullscale testing of largesize steel beamcolumn connections (SAC 1996). The laboratory testing of the preNorthridge prequalified welded flangebolted web connection replicated many of the failure modes observed in the field after the earthquake. The mean value of beam plastic
9. Seismic Design of Steel Structures
417
Table 92. Limiting widththickness ratios Description of Widthλp Element Thickness Ratio Flanges of Ishaped b/t 65 / Fy beams and channels in flexure Webs of Ishaped for Pu/φbPy ≤ 0.125: h/tw beams in combined flexure and axial 640 2.75Pu 1− compression φb Py Fy for Pu/φbPy > 0.125: 191 P 2.33 − u φb Py Fy Round HHS in axial compression or flexure Rectangular HHS in axial compression or flexure
253 ≥ Fy
λps
52 / Fy for Pu/φbPy
≤ 0.125:
520 (1 − 1.54Pu φb Py Fy for Pu/φbPy > 0.125: 191 (2.33 − Pu φb Py Fy
253 ≥ Fy
D/t
2070 Fy
1300 Fy
b/t
190
110
Fy
Fy
rotation capacity from all of the tests of the preNorthridge connection detail was 0.004 radian (Whittaker et al. 1998), which was significantly less than the target value of 0.03 radian. In response to these findings, the 1997 AISC Seismic Provisions require that (1) the design of beamtocolumn joints and connections in SMFs must be based on qualifying tests of at least two specimens, and (2) each connection must develop a plastic rotation of 0.03 radian.
welded connections are considered in this section. These connection details fall in one of the two categories: weakening the beam crosssection away from the face of the column, or reinforcing the beam crosssection at the column face. Only nonproprietary moment connections are discussed.
BeamtoColumn Connection Details
A variety of reinforced connections have been developed since the Northridge earthquake. Some reinforced connection details are shown in Figure. 94: cover plates, welded flange plates, triangular haunches, straight haunches, and vertical plate ribs. Note that these connection details would not only increase the beam plastic hinge rotation demand but also increase the maximum moment demand at the face of the column, which could require a stronger panel zone or a larger section for the column to maintain the strong columnweak beam system (SAC 1995). Typical design practice for reinforced connections is to keep the reinforced component in the elastic range for moments associated with substantial strain
Shortly after the 1994 earthquake, the prequalified welded flangebolted web connection was deleted from most building codes and replaced by general provisions that required the design professional to demonstrate the adequacy of the connection by either fullscale testing or calculations supported by test data. In response to this action, design professionals have proposed new types of momentresisting connections for steel buildings. Some of these proposals are discussed below. In all cases, the proposed connection details relocate the beam plastic hinge away from the face of the column. Only
Reinforced Connections
418 hardening in the beam beyond the reinforcement. Although it may be tempting to assume a linear distribution of bending moment along the length of the beam to size the reinforcement, the effects of gravity load on the beam bending moment diagram, if significant, must be carefully considered. For all of the connection details described below, notchtoughness rated weld filler metal, qualified welders, and high quality inspection should be specified. Immediately after the Northridge earthquake, cover plates (see Figure 94a) have been one of the more popular strategies for reinforcing beamtocolumn connections. Testing has been completed at a number of laboratories and significant data are available (e.g., Engelhardt and Sabol 1996, and SAC 1996). In most cases, the bottom cover plate is rectangular and wider than the beam bottom flange, and the top cover plate is tapered and narrower than the beam top flange. This configuration permits the bottom cover plate to be used as an erection seat, and facilitates downhand welding in the field. Welded, not bolted, web connections are recommended as an effective way of reducing the thickness of the cover plates. Although a significant number of cover plated connection specimens have achieved beam plastic rotations exceeding 0.03 radian, Hamburger (1996) reported a failure rate of approximately 20 percent for coverplated connections in laboratory tests. Another concern with the coverplate connection is that the seam between the flange cover plate and the beam flange acts as a notch at the column face that may lead to cracks propagating into the column flange and beyond. Further information is available in SAC (1997). The welded flangeplate connection (see Figure 94b) is closely related to the coverplate connection, with the major difference being that only the flange plates are groove welded to the column (Jokerst and Soyer 1996, Noel and Uang 1996). As such, flange plates of the welded flangeplate connection are thicker than the comparable cover plates shown in Figure 94a. There is no notch effect associated with the
Chapter 9 welded flangeplate connection because the beam flanges are not welded to the column flange. The bottom welded flange plate can be shop welded to the column, thereby eliminating one field groove weld, and providing an erection seat for the beam. Welded triangular and straight haunch reinforced connections (see Figures 94c and d) underwent extensive laboratory testing following the Northridge earthquake (e.g., SAC 1996, Gross et al. 1998) because both reinforcements could be used for seismic repair and retrofit. Most of the haunch connection tests conducted to date incorporated a haunch to the bottom flange, although the addition of haunches to both the top and bottom flanges was also considered. Of the different types of haunch details tested to date, the triangular Tshaped haunches appear to be the most effective (Yu et al. 2000). Large plastic rotations were achieved with this type of connection. Vertical rib plates (see Figure 94e) can also be used to reduce the stress demand in the welded joint (Chi and Uang 2000). Reduced Beam Sections An alternative to relocating the plastic hinge away from the face of the column is to reduce the plastic moment of the beam at a short distance from the column face. Beam sections can be reduced by tapering the flanges, or by radiuscutting the flanges as shown in Figure 95. The latter approach appears to be the most promising because the reentrant corners of the tapered flange profile tend to promote premature fracture in the beam flanges. Originally proposed and tested by Plumier (1990), the use of the reduced beam section (RBS), also termed the dogbone by many design professionals, has seen broad support from engineers, steel producers, and fabricators. Both reducedbeamsection profiles have achieved plastic rotations in excess of 0.03 radian. Additional information is provided in Iwankiw and Carter (1996), Chen et al. (1996), Engelhardt et al. (1996), and Zekioglu et al. (1996).
9. Seismic Design of Steel Structures
419
(a) Beam bottom flange weld fracture propagating through column flange and web
(b) Beam bottom flange weld fracture causing a column divot fracture Figure 93. Examples of brittle fracture of steel moment frame connections (courtesy of David P. O’sullivan, EQE International, San Francisco)
420
Chapter 9
Figure 94. Reinforced moment connections: (a) cover plates, (b) welded flange plates, (c) triangular haunches, (d) straight haunch, (e) rib plates
9. Seismic Design of Steel Structures
421 First, the lateraltorsional buckling amplitude of the beam tends to be larger when the RBS is used. Second, the stress in the column produced by warping torsion is highly dependent on the ratio (d c − t cf ) / t cf3 . For example, this ratio is
(a) Tapered Flange Profile
(b) Circular Flange Profile Figure 95. Moment connection with reduced beam section
Reducing the width of the beam flange serves to delay flange local buckling but increases the likelihood of web local buckling and lateraltorsional buckling because the inplane stiffness of the flanges is significantly reduced. The reduced beam section usually experiences web local buckling first, followed by lateraltorsional buckling and flange local buckling. The stability of RBS beams was studied as part of the SAC Joint Venture (Uang and Fan 2000). It was found from a statistical study that web local buckling is the governing mode of buckling. While the λps values presented in Section 9.2.3 for flange local buckling and lateraltorsional buckling still can be used for RBS design, the λps value for web local buckling needs to be reduced from 520/ Fy to 418/ Fy (SAC 2000). The study also showed that additional lateral bracing near the RBS is generally unnecessary. Design engineers frequently use deep columns in a moment frame to control drift. When the deep section wideflange columns are used, however, an experimental study showed that significant torsion leading to the twisting of the column could result (Gilton et al. 2000). Two factors contribute to the column twisting.
equal to 0.671/in2 for a W14×398 section (Ix = 6000 in4). If the designer chooses a deep section W27×161 for a comparable moment of inertia (Ix = 6280 in4) to control drift, the ratio is drastically increased to 21.04/in2, implying that this section is susceptible to column twisting. Lateral bracing near the RBS region then may be required to minimize the twisting. A procedure to check if column twisting is a concern has been developed (Gilton et al. 2000). 9.2.5
Beamto Column Panel Zones
Introduction A beamtocolumn panel zone is a flexible component of a steel momentresisting frame that is geometrically defined by the flanges of the column and the beam (see Figure 96). Although seismic building codes require the consideration of panel zone deformations in the story drift computations, panel zones are rarely modelled explicitly in mathematical models of steel momentresisting frames. Mathematical representations of momentresisting frames are generally composed of beams and columns modelled as line elements spanning between the beamcolumn intersection points. Such a representation will underestimate the elastic flexibility of a momentresisting frame. An approximate analysis procedure that includes the flexibility of panel zones for drift computations have been proposed (Tsai and Popov 1990). This procedure will be demonstrated in an SMF design in Section 9.5.2. Typical internal forces on a panel zone are shown in Figure 96a; axial, shearing, and flexural forces are typically present in a panel zone. In this figure, continuity plates are shown in the column at the level of the beam flanges and the moments M 1 and M 2 represent
422 earthquake actions. Assuming that the flanges resist 100 percent of the moment and that the distance between the centroids of the flanges is 95 percent of the beam depth, compression and tension flange forces as shown in Figure 96b can replace the beam moments.
Chapter 9 restrained joints, such as continuity plates, induces residual stress in steel members. In addition to the normal variation of material properties in the column, the process of mill rotary straightening of the Wshaped member alters the mechanical properties by cold working in the “k” area. (The “k” area is defined by AISC as the region extending from about the midpoint of the radius of the fillet into the web approximately 1 to 1.5 in. beyond the point of tangency between the fillet and web.) As a result, a reduction in ductility and toughness in the “k” area may occur. In some cases, values of Charpy Vnotch toughness less than 5 ftlb at 70° F have been reported. Since welding in the “k” area may increase the likelihood of fracture, a recent AISC Advisory (1997a) has suggested that welds for the continuity plates be stopped short of the “k” area. Fillet welds and/or partial joint penetration welds, proportioned to transfer the calculated forces, are preferred to complete joint penetration welds. Required Shear Strength
Figure 96. Internal forces acting on a panel zone of a momentresisting frame subjected to lateral loading
Using the information presented in Figure 96b, and taking a freebody diagram immediately below the upper continuity plate, the horizontal shearing force in the panel zone ( V pz ) can be calculated as V pz =
The continuity plates shown in Figure 96 serve to prevent column flange distortion and column web yielding and crippling. If such plates are not provided in a column with thin flanges, and the beam flange imposes a tensile force on the column flange, inelastic strains across the groove weld of the beam flange are much higher opposite the column web than they are at the flange tips. Thus, weld cracks and fractures may result. Because the design of beamtocolumn joints and connections is based upon qualifying cyclic tests, AISC (1997) requires that continuity plates of the size used in the qualifying tests be provided in the connection. However, welding of the highly
M1 M2 + − Vc 0.95d b1 0.95d b 2
(98)
where all terms are defined above and in the figure, and Vc is the shearing force in the column immediately above the panel zone. Because Vc reduces the shearing force in the panel zone, and its magnitude is substantially smaller than the first two terms on the right hand side of this equation, Vc can be ignored conservatively in the calculation of the maximum shearing force. Therefore, for beams of equal depth, V pz ≈
∆M db
(99)
9. Seismic Design of Steel Structures
423
where ∆M = ( M 1 + M 2 ) is the unbalanced beam moment. Prior to the publication of the 1988 Uniform Building Code (ICBO 1988), panel zones were designed to remain elastic for M 1 = M 2 = M p , where M p is the nominal plastic moment of the beam under consideration. The strength of the panel zone at first yield was computed as 0.55Fyc Awc , where Fyc is the nominal yield strength of the column and Awc is the area of the column web (= d c t cw ). This design procedure was intended to produce strong panel zones such that yielding in the momentresisting frame was minimized in the panel zone region. Both the 1988 Uniform Building Code and the 1992 AISC Seismic Provisions relaxed the design provisions for panel zone regions and permitted intermediate strength panel zones and minimum strength panel zones. Previous studies by Krawinkler et al. (1975) had shown that panel zone yielding could dissipate a large amount of energy in a stable manner. Intermediate and minimum strength panel zones were introduced to encourage panel zone yielding. According to the 1992 AISC Seismic Provisions, intermediate strength panel zones were designed for ∆M = ∑ M p − 2M g
(910)
where Mg is the gravity moment for one beam. If the gravity moment is taken to be 20 percent of the plastic moment, the above equation gives ∆M = 0.8ΣM p . Minimum strength panel zones were allowed for a value ∆ M = Σ M E ≤ 0 .8 Σ M p , of where the unbalanced beam moment produced by the prescribed design seismic forces is ΣM E = ( M E1 + M E 2 ) . It has been shown (Tsai and Popov 1988) that steel moment frames with intermediate or minimumstrength panel zones are likely to have a substantially smaller overstrength factor, Ω o , than those with strong panel zones. In addition, the lateral stiffness of
an intermediate or minimumstrength panelzone frame can be significantly smaller than that computed using a mathematical model based on centerline dimensions. Current AISC provisions (AISC 1997) require the use of Ω o equal to 3.0 (see Table 91) for beam moments induced by the design earthquake loads. It also replaces the nominal plastic moment by the expected plastic moment and prescribes that the required strength of a panel zone need not exceed the shear force determined from 0.8∑ M *pb , where ∑ M *pb is the sum of the beam moment(s) at the intersection of the beam and column centrelines. ( ∑ M *pb is determined by summing the projections of the expected beam flexural strength(s) at the plastic hinge location(s) to the column centreline.) That is, the panel zone shall be designed for the following unbalanced moment: ∆M = Ω o ΣM E ≤ 0.8ΣM *pb
(911)
Substituting Eq. (911) into Eq. (99) would give the required shear strength in the panel zone. PostYield Strength Requirements
and
Detailing
The 1992 AISC equation for calculating the design shear strength of a panel zone ( φ v V n , where φv =0.75) was based on the work of Krawinkler et al. (1975): 3bcf tcf2 φvVn = φ v 0.60 Fyc d c t p 1 + d b d ct p
(912)
where d c is the depth of the column, t p is the total thickness of the panel zone, including doubler plates ( t p = t cw if no doubler plates are present), bcf is the width of the column flange, t cf is the thickness of the column flange, and d c is the depth of the column. The second term
424
Chapter 9
in the parentheses represents the contribution of column flanges (assumed to be linearly elastic) to the shear strength of the panel zone. The equation used to calculate Vn assumes a level of shear strain of 4 γ y in the panel zone, where γ y is the yield shearing strain. A panel zone must also be checked for a minimum thickness ( t ) to prevent premature local buckling under large inelastic shear deformations: t=
(d z + w z ) 90
(913)
In this empirical equation, d z is the depth of the panel zone between the continuity plates, and wz is the width of the panel zone between the column flanges. If doubler plates are used to satisfy this equation for t , the plates must be plug welded to the column web such that the plates do not buckle independently of the web. If used, doubler plates must be welded to the column flanges using either a complete joint penetration groove weld or a fillet weld that develops the design shear strength of the full doubler plate thickness. When such plates are welded directly to the column web and extend beyond the panel zone, minimum weld size can be used to connect the top and bottom edges to the column web. However, because of the cold working due to the rotary straightening practice and the resulting variations of material properties exhibited in the column ”k” areas, the AISC Advisory (1997) suggested that, as an interim measure, the design engineer increase the column size to avoid the use of doubler plates. 9.2.6
Column Design
The column of an SMF must be designed per the LRFD Specifications (1997) as a beamcolumn to avoid axial yielding, buckling, and flexural yielding. Columns are routinely spliced by groove welding. Such connections are required to have sufficient strength to resist the imposed axial, shearing, and flexural forces
calculated using the specified load combinations. In addition, the column axial strength should be sufficient to resist the axial forces produced by the special load combinations of Eqs. 94 and 95. Additional strength is required if either the welds are partial penetration groove welds or the welds are subjected to net tension forces. Column splices using fillet welds or partial joint penetration groove welds shall not be located within 4 feet or onehalf the column clear height of beamtocolumn connections, which is less. Special moment frames are designed using the strong columnweak beam philosophy because such an approach improves the energy dissipation capacity of the frame, promotes plastic hinge formation in the beams, increases the seismic resistance of the frame, and ostensibly prevents the formation of a soft story mechanism. Seismic regulations seek to achieve a strong columnweak beam system by ensuring that, at a beamtocolumn connection, the sum of the column plastic moments exceeds the sum of the beam plastic moments. With few exceptions, AISC (1997) requires that:
∑ M *pc ∑ M *pb where
> 1.0
∑ M *pc
(914)
is the sum of the moment
capacities in the columns above and below the joint at the intersection of the beam and column centerlines, and ∑ M *pb is the sum of the moment demands in the beams at the intersection of the beam and column centerlines. The value of ∑ M *pc is determined by summing the projections of the nominal flexural strength of the columns above and below the connection to the beam centerline, with a reduction for the axial force in the column. Σ M *pc can be conservatively approximated as
∑ Z c ( Fyc − Puc / Ag ) ,
where
Ag is the gross area of the column, Puc is the required column compressive strength, Zc is the
9. Seismic Design of Steel Structures plastic section modulus of the column, and Fyc is the minimum specified yield strength of column. The value of ∑ M *pb is calculated by summing the projection of the expected beam flexural strengths at the plastic hinge locations to the column. ∑ M *pb can be approximated as
∑ (1.1R y Fy Z + M v ) ,
where Z is the plastic
modulus of the beam section at the potential plastic hinge location, and Mv accounts for the additional moment due to shear amplification from the location of the plastic hinge to the column centerline. As illustrated in Figure 97, for reinforced connections using haunches or vertical ribs, SAC (1996) suggests that plastic hinges be assumed to be located at a distance sh = d/3 from the toe of haunch or ribs. For cover plated connections, SAC recommends that the plastic hinge be located at a distance sh = d/4 beyond the end of cover plate. When the ratio in Eq. 914 is no greater than 1.25, the widthtothickness ratios of the flange and web
425 elements of the column section shall be limited to the λps values in Table 92 because plastic hinge formation in the column may occur due to the shift of inflection point during an earthquake. Otherwise, columns shall comply with the limiting values of λ p in the same table.
9.3
Behavior and Design of Concentrically Braced Frames
9.3.1
Design Philosophy
Concentrically braced frames are frequently used to provide lateral strength and stiffness to low and midrise buildings to resist wind and earthquake forces. Although some architects favor the less intrusive moment frames, others have found architectural expression in exposing braced frames which the public intuitively
Figure 97. Assumed beam plastic hinge locations (Adapted from Interim Guidelines Advisory No. 1, SAC 1997)
426
Chapter 9
associates with seismic safety in some earthquakeprone regions. However, for those frames to provide adequate earthquake resistance, they must be designed for appropriate strength and ductility. This is possible for many of the concentrically braced frame (CBF) configurations shown in Figure 98, but not all, as described in this section. In a manner consistent with the earthquakeresistant design philosophy presented elsewhere in this chapter, modern concentrically braced frames are expected to undergo inelastic response during infrequent, yet large earthquakes. Specially designed diagonal braces in these frames can sustain plastic deformations and dissipate hysteretic energy in a stable manner through successive cycles of buckling in compression and yielding in tension. The preferred design strategy is, therefore, to ensure that plastic deformations only occur in the braces, leaving the columns, beams, and
connections undamaged, thus allowing the structure to survive strong earthquakes without losing gravityload resistance. Past earthquakes have demonstrated that this idealized behavior may not be realized if the braced frame and its connections are not properly designed. Numerous examples of poor seismic performance have been reported (Tremblay et al. 1995, 1996; AIJ 1995). As shown in Figure 99, braces with bolted connections have fractured through their net section at bolt holes, beams and columns have suffered damage, and welded and bolted connections have fractured. Collapses have occurred as a consequence of such uncontrolled inelastic behavior. The design requirements necessary to achieve adequate strength and ductility in concentrically braced frames are presented in this section. Two types of systems are permitted by the AISC Seismic Provisions: Special
Figure 98. Typical brace configuration
9. Seismic Design of Steel Structures Concentrically Braced Frames (SCBs) and Ordinary Concentrically Braced Frames (OCBFs). The emphasis herein is on the SCBF, which is designed for stable inelastic performance using a response modification factor, R, of 6. Some of the more stringent ductile detailing requirements are relaxed for the OCBFs because it is assumed that these frames will be subjected to smaller inelastic deformation demands due to the use of a smaller response modification factor. However, if an earthquake greater than that considered for design occurs, SCBFs are expected to perform better than OCBFs because of their substantially improved deformation capacity. 9.3.2
Hysteretic Energy Dissipation Capacity of Braces
Given that diagonal braces are the structural members chosen to plastically dissipate seismic energy, an examination of the physical behavior of a single brace subjected to axial load reversal is useful. It is customary to express the inelastic behavior of axially loaded members in terms of the axial force, P, versus the axial elongation, δ. According to convention, tension forces and elongations are expressed with positive values. A schematic representation of such a hysteretic curve is shown in Figure 910. Note that the transverse member deflection at midspan is represented by ∆. A full cycle of inelastic deformations can be described as follows. Starting from an initially unloaded condition (point O in Figure 910), the member is first compressed axially in an elastic manner. Buckling occurs at point A. Slender members will experience elastic buckling along plateau AB, for which the applied axial force can be sustained while the member deflects laterally. Up to that point, the brace behavior has remained elastic and unloading would proceed along the line BAO if the axial compressive was removed. During buckling, flexural moments develop along the member, equal to the product of the axial force and lateral deflection, with the largest value reached at the point of maximum
427 deflection, ∆, at midspan. Eventually, the plastic moment of the member, reduced by the axial load, is reached at midspan, and a plastic hinge starts to develop there (point B in Figure 910). The interaction of flexure and axial force on the plastic moment must be taken into account to determine the actual value of ∆ corresponding to point B. Along segment BC, further increases in ∆ result in greater plastic hinge rotations at midspan (i.e., the member develops a “plastic kink”) and a corresponding drop in axial resistance. The relationship between P and δ is nonlinear, partly as a result of the plastic interaction between flexure and axial force. Upon unloading (starting at point C in Figure 910), the applied compression force is removed in an elastic manner. After unloading, the member retains a large residual axial deformation as well as a large lateral deflection. When loading the member in tension, behavior is first elastic, up to point D. Then, at point D, the product of the axial force, P, and the midspan transverse deformation, ∆, equals the member reduced plastic moment and a plastic hinge forms at midspan. However, this time, along segment DE, plastic hinge rotations act in the reverse sense to those along segment BC, and the transverse deflection reduces. As a result, progressively larger axial forces can be applied. The bracing member cannot be brought back to a perfectly straight position before the member yields in tension. Consequently, when unloaded and reloaded in compression, the brace behaves as a member with an initial deformation and its buckling capacity, Pcr′ , is typically lower that the corresponding buckling capacity upon first loading, Pcr. Upon further cycles of loading, the value of Pcr′ rapidly stabilizes to a relatively constant value. Typically, the ratio of Pcr′ /Pcr depends on the member slenderness ratio, KL/r, and expressions have been proposed to capture this relationship (Bruneau et al. 1997). For simplicity, a constant value of Pcr′ = 0.8Pcr is specified in the AISC Seismic Provisions (1992) and must be considered whenever it gives a more critical design condition.
428
Chapter 9
(a) Net section fracture at bolt holes
(b) Severe distortion of beam without lateral support at location of chevron braces Figure 99. Examples of damage to nonductile braced frames
429
Chapter 9
(c) Fracture of welded connection and web tearout in brace
(d) Weld fracture Figure 99 Examples of damage to NonDuctile braced frames (continued)
430
Chapter 9
Figure 910. Hysteresis of a brace under cyclic axial loading
Beyond this difference, the hysteretic curve repeats itself in each subsequent cycle of axial loading and inelastic deformations, with a shape similar to the OABCDEF of Figure 910. 9.3.3
dissipation. The energy absorption capability of a brace in compression depends on its slenderness ratio (KL/r) and its resistance to local buckling during repeated cycles of inelastic deformation.
Design Requirements Limits on Effective Slenderness Ratio
Concentrically braced frames exhibit their best seismic performance when both yielding in tension and inelastic buckling in compression of their diagonal members contribute significantly to the total hysteretic energy
As can be deduced from Figure 910, slenderness has a major impact on the ability of a brace to dissipate hysteretic energy. For a very slender brace, segment OA is short while
Figure 911. Brace Hysteresis loops by experimentation. (Nakashima and Wakabayashi 1992, referring to a figure by Shibata et al. 1973, with permission from CRC Press, Boca Raton, Florida)
9. Seismic Design of Steel Structures
431
Figure 912. Schematic hysteretic behavior of braces of short, long, and intermediate slenderness (Nakashima and Wakabayashi 1992, with permission from CRC Press, Boca Raton, Florida).
segment AB is long, resulting in poor energy dissipation capacity in compression. For stocky braces, the reverse is true, and segment AB (i.e., elastic buckling) may not exist. Slenderness has no impact on the energy dissipation capability of braces in tension. Typical hysteretic loops obtained experimentally for axially loaded members of intermediate and large slenderness ratios are shown in Figure 911, where the parameter λ (= Kl /( rπ) Fy / E ) is a nondimensional slenderness ratio (Nakashima and Wakabayashi 1992). Schematic illustrations of simplified hysteresis loops for short, intermediate and long braces are shown in Figure 912. Very slender brace members (such as bars or plates) can result from a practice called tensiononly design, often used prior to the promulgation of modern seismic provisions for steel buildings, and still used in nonseismic regions. In that design approach, the tension brace is sized to resist all the lateral loads, and the contribution of the buckled compression brace is ignored. While tensiononly design may be acceptable for wind resistance, it is not permissible for earthquake resistance. As shown in Figure 913, braced frames with very slender members must progressively drift
further and further to be able to dissipate the same amount of energy at each cycle, perhaps leading to collapse due to secondorder effects. Seismic detailing provisions typically limit brace slenderness to prevent the above problem and to ensure good energy dissipation capacity. Many seismic codes require: KL 720 ≤ r Fy
(915)
where Fy is in ksi. For ASTM A992 or A572 Grade 50 steel, this corresponds to an effective slenderness ratio of 102. Recently, the AISC Seismic Provisions (1997) have relaxed this limit to: KL 1000 ≤ r Fy
(916)
for bracing members in SCBFs, but kept the more stringent limit of Eq. 915 for OCBFs. Nevertheless, the authors recommend the use of Eq. 915 for both SCBFs and OCBFs. Limits on WidthtoThickness Ratio
432
Chapter 9
Figure 913. Hysteretic Behavior of SingleStory braced frame having very slender braces
The plastic hinge that forms at midspan of a buckled brace may develop large plastic rotations that could lead to local buckling and rapid loss of compressive capacity and energy dissipation during repeated cycles of inelastic deformations. Past earthquakes and tests have shown that locally buckled braces can also
suffer lowcycle fatigue and fracture after a few cycles of severe inelastic deformations (especially when braces are coldformed rectangular hollow sections). For these reasons, braces in SCBFs must satisfy the widthtothickness ratio limits for compact sections. For OCBFs, braces can be compact or non
9. Seismic Design of Steel Structures compact, but not slender, i.e., b / t ≤ λr per LRFD Specification. Based on experimental evidence, more stringent limits are specified for some types of structural shapes. In particular, the widthtothickness ratio of angles (b/t), the outside diameter to wall thickness ratio of unstiffened circular hollow sections (D/t), and the outside width to wall thickness ratio of unstiffened rectangular sections must not exceed 52/ F y , 1300/Fy, and 110/ F y , respectively (see Table 92). Note that the AISC Seismic Provisions (1997) define b for rectangular hollow sections as the “outtoout width”, not the flatwidth (= b−3t) as defined in the AISC Specifications (AISC 1994). Redundancy Energy dissipation by tension yielding of braces is more reliable than buckling of braces in compression. To provide structural redundancy and a good balance of energy dissipation between compression and tension members, structural configurations that depend predominantly on the compression resistance of braces should be avoided. Examples of poor braced frames layout are shown in Figure 914, together with recommended alternatives. Four braces in compression and only one brace in tension resist the load applied on the 5bay braced frame shown in Figure 914a. All braces in the bracedcore of Figure 914c are in compression to resist the torsional moment resulting from seismicallyinduced inertial force acting at the center of mass. (For simplicity, columns resisting only gravity loads are not shown in that figure.) Better designs are shown in Figures 914b and 914d for each of these cases, respectively. Seismic design codes attempt to prevent the use of nonredundant structural systems by requiring that braces in a given line be deployed such that at least 30% of the total lateral horizontal force acting along that line is resisted by tension braces, and at least 30% by compression braces. Although the wording of such clauses does not cover the case shown in
433 Figure 914c, the intent does. Codes generally waive this requirement if nearly elastic response is expected during earthquakes, something achieved in the AISC Seismic Provisions by the special load conditions described in Section 9.1. Note that in calculating the strength of an OCBF, the AISC Seismic Provisions also require that φcPcr (= 0.9φcPcr) be used instead of φcPcr, for the reasons described in the previous section. There is no such requirement for SCBFs, but the authors prefer to observe this requirement for both OCBFs and SCBFs, recognizing, however, that the tension brace may have sufficient strength to accommodate the strength degradation of the compression brace upon repeated cycling, and that such a force redistribution may be considered when calculating the strength of the braced panel using φcPcr. This approach is not recommended for V and invertedVtypes of OCBF. 9.3.4
Bracing Connections Design Requirements
When a brace is in tension, net section fracture and block shear rupture at the end of the brace must be avoided. Likewise, the brace connections to beams and columns must be stronger than the braces themselves. Using capacity design, calculation of brace strength must recognize that the expected yield strength of the brace, Fye, will typically exceed its specified minimum yield strength, Fy (see Eq. 96). Thus, connections must be designed to resist an axial force equal to RyFyAg. However, when plastic analysis is used to demonstrate that braces are unlikely to yield, connections may be designed for the maximum force obtained from such an analysis. Connections must also be able to resist the forces due to buckling of the brace. If strong connections permit the development of a plastic hinge at each end of a brace, they should be designed to resist a moment equal to 1.1RyMp of the brace in the direction of buckling. Otherwise, the connecting elements will themselves yield in flexure (such as gussets out
434
Chapter 9
Figure 914. Brace configurations to ensure structural redundancy and balanced energy dissipation between compression and tension members: (a and c) poor configurations; (b and d) acceptable configurations
9. Seismic Design of Steel Structures of their plane); these must then be designed to resist the maximum brace compression force in a stable manner while undergoing the large plastic rotations that result from brace buckling. AstanehAsl et al. (1986) suggested providing a clear distance of twice the plate thickness between the end of the brace and the assumed line of restraint for the gusset plate to permit plastic rotations and to preclude plate buckling (see Figure 915).
435 with the further specification that the maximum axial tension forces in columns need not be taken larger than the value corresponding to foundation uplift. For SCBFs, the Provisions also require that columns satisfy the same widthtothickness ratio limits as braces (i.e., λ ps in Table 92). Partial penetration groove welds in column splices have been observed to fail in a brittle manner (Bruneau and Mahin 1990). When a welded column splice is expected to be in tension under the loading combination shown in Eq. 95, the AISC Seismic Provisions mandate that the partial joint penetration groove welded joints in SCBFs be designed to resist 200% of the strength required by elastic analysis using codespecified forces. Column splices also need to be designed to develop at least the nominal shear strength of the smaller connected member and 50% of the nominal flexural strength of the smaller connected section. 9.3.6
Figure 915. Bracetogusset connection detail to permit ductile outofplane brace buckling (AISC 1997, with permission from American Institute of Steel Construction, Chicago, Illinois)
9.3.5
Columns and Beams
Beams and columns in braced frames must be designed to remain elastic when all braces have reached their maximum tension or compression capacity (1.1Ry times the nominal strength) to eliminate inelastic response in all components except for the braces. This requirement could be too severe for columns, however, as the braces along the height of a multistory frame do not necessarily reach their capacity simultaneously during an earthquake. Statistical approaches have been proposed to evaluate the maximum likely column load (Redwood and Channagiri 1991). The AISC Seismic Provisions address this issue using special load conditions described in Section 9.1,
Special Bracing Configuration Requirements
Special requirements apply to the design of Vtype and inverted Vtype braced frames (also known as chevron braced frames). Because braces meet at the midspan of beams in these frames, the vertical force resulting from the unequal compression and tension strengths of these braces can have a considerable impact on the cyclic behavior of the frame. That vertical force introduces flexure in the beam, and possibly a plastic hinge in the beam, producing the plastic collapse mechanism shown in Figure. 916. Therefore, it is imperative that beams in chevron braced frames be continuous between columns. It has also been observed that once a yielding mechanism develops in a chevrontype brace at a particular story, damage tends to concentrate at that story. A comprehensive discussion of the seismic behavior of chevron braced frames under seismic loading is beyond the scope of this chapter, and is presented elsewhere (Bruneau et al. 1997).
436
Figure 916. Plastic collapse mechanism of chevron braced frame having plastic hinge in beam
Figure 917. Plastic collapse mechanism of KBraced frame with plastic hinge in column
Seismic provisions usually require that beams in chevron braced frames be capable of resisting their tributary gravity loads neglecting the presence of the braces. The AISC Seismic Provisions also require that each beam in an SCBF be designed to resist a maximum unbalanced vertical load calculated using full yield strength for the brace in tension, and 30% of the brace buckling strength in compression. In OCBFs, this latter provision need not be considered. However, braces in OCBFs must be designed to have 1.5 times the strength required by load combinations that include seismic forces, which is equivalent to designing chevron braced frames for a smaller value of R to compensate for their smaller ductility. Finally, to prevent instability of a beam bottom flange at the intersection point of the braces in a chevron braced frame, in a manner similar to that shown in Figure 99b, the top and bottom flanges of beams in SCBFs and OCBFs must be designed to resist a lateral force equal to 2% of the nominal beam flange strength (i.e., 0.02AfFy). This requirement is best met by the addition of a beam perpendicular to the chevron braced frame. The above concepts also explain why a number of braced frame configurations are undesirable in seismic regions. For example, in
Chapter 9 a Ktype braced frame (see Figure 917), the unequal buckling and tensionyielding strengths of the braces would create an unbalanced horizontal load at the midheight of the columns, jeopardizing the ability of the column to carry gravity loads if a plastic hinge forms at the midheight of the column.
9.4
Behavior and Design of Eccentrically Braced Frames
9.4.1
Introduction
While a properly designed and constructed steel moment frame can behave in a very ductile manner, moment frames are very flexible and their design is usually dictated by the drift limitations. Concentrically braced frames, on the other hand, have a large lateral stiffness, but their energy dissipation capacity is affected by brace buckling. In the early 1970s, an innovative steel system called the Eccentrically Braced Frame (EBF) that combines the advantages of both the steel moment frame and braced frame was proposed in Japan (Fujimoto et al. 1972, Tanabashi et al. 1974). The EBF dissipates energy by controlled yielding of shear or moment links. In the United Sates, the EBF system was first studied by Roeder and Popov (1978). This attractive system rapidly gained acceptance by the design profession (Teal 1979, Libby 1981, Merovich et al. 1982), some being constructed well before detailed design provisions were developed in the United States. In the 1980s, numerous studies on link behavior provided insight into the cyclic response of EBFs (Manheim and Popov 1983, Hjelmnstad and Popov 1983, 1984, Malley and Popov 1984, Kasai and Popov 1986a and 1986b, Ricles and Popov 1989, Engelhardt and Popov 1989). EBF design provisions were first promulgated in the 1988 Uniform Building Code. Experimental verifications of EBF response at the system level were also conducted in the mid to late1980s (Yang 1985, Roeder et al. 1987, Whittaker et al. 1989).
9. Seismic Design of Steel Structures
437 are proportioned following capacity design provisions to remain essentially elastic during the design earthquake. Elastic Stiffness
Figure 918. Typical EBF configurations
9.4.2
Basic Concept and EBF Behavior
An eccentrically braced frame is a framing system in which the axial force induced in the braces are transferred either to a column or another brace through shear and bending in a small segment of the beam. Typical EBF geometries are shown in Figure 918. The critical beam segment is called a “link” and is designated by its length, e. Links in EBFs act as structural fuses to dissipate the earthquake induced energy in a building in a stable manner. To serve its intended purpose, a link needs to be properly detailed to have adequate strength and stable energy dissipation. All the other structural components (beam segments outside of the link, braces, columns, and connections)
The variations of the lateral stiffness of a simple EBF with respect to the link length is shown in Figure 919 (Hjelmstad and Popov 1984). Note that e/L ratios of 0.0 and 1.0 correspond to a concentrically braced frame and a moment frame, respectively. The figure clearly shows the advantage of using a short link for drift control. Link Deformation Consider the idealized split Vtype EBF in Figure 918b. Once the links have yielded in shear, the plastic mechanism shown in Figure 920a will form. Applying simple plastic theory, the kinematics of the plastic mechanism require that: γp =
L θp e
Figure 919. Variations of lateral stiffness with respect to link length (Hjelmstad and Popov 1994)
(917)
438
Chapter 9
Figure 920. Kinematic mechanism and link plastic angle of a Ktype EBF
where θp is the plastic drift angle (or plastic story drift ratio), and γp is the plastic deformation of the link. Based on Eq. 917, the variation of γp with respect to the link length is shown in Figure 920b. Because the elastic component of the total drift angle is generally small, the plastic story drift angle, θp, can be conservatively estimated as the total story drift divided by the story height, h: θp ≈
∆ s Cd ∆ e = h h
(918)
where ∆ e is the story drift produced by the prescribed design earthquake force, and Cd (= 4) is the deflection amplification factor. To ensure that the deformation capacity of the link is not exceeded, it is obvious from Eq. 917 that the link length cannot be too short. Note that the kink that forms between the link and the beam outside the link also implies damage of the concrete slab at the ends of the link.
Ultimate Strength Unless architectural considerations dictate otherwise, a short link is usually used so that the link will yield primarily in shear. The lateral strength of such an EBF can then be calculated conveniently using simple plastic theory. Assuming that the link behaves in an elasticperfectly plastic manner, the lateral strength, Pu, of the simple onestory split Vshaped EBF frame can be computed by equating the external work to the internal work: External work = Pu (hθp) Internal work =
e
∫0 V p γ p dx = eV p γ p
(919a) (919b)
where Vp is the shear strength of the link. Substituting Eq. 917 into Eq. 919b, the resulting ultimate strength of the EBF frame is Pu =
Vp L h
(920)
9. Seismic Design of Steel Structures As long as the link yields in shear, the above equation shows that the ultimate strength is independent of the link length. This simple plastic theory can also be applied to multistory frames (Kasai and Popov 1985). Once the link length exceeds a threshold value, flexure and shear dominates the link strength. The ultimate strength of the frame then decreases with an increase in link length. Figure 921 illustrates the strength variations. This figure also indicates that the ultimate strength of an EBF with short links is significantly larger than that of a moment frame (i.e., e/L = 1.0).
439 moment, Mp. A shear hinge is said to form when the shear reaches Vp. The plastic moment and shear capacities are respectively computed as follows: Mp = Fy Z
(921a)
Vp = 0.6 F y (d − 2t f )t w
(921b)
Figure 922. Link deformation and freebody diagram
A balanced yielding condition corresponds to the simultaneous formation of flexural hinges and a shear hinge. The corresponding link length is e0 =
2M p Vp
(922)
In a short link ( e ≤ e0 ), a shear hinge will form. When e > e0 , a flexural (or moment) hinge forms at both ends of the link, and the corresponding shear force is: Figure 921. Variations of EBF ultimate strength with e/L (Kasai and Popov 1985)
9.4.3
Link Behavior
Critical Length for Shear Link Figure 922 shows the freebody diagram of a link. Ignoring the effects of axial force and the interaction between moment and shear in the link, flexural hinges form at two ends of the link when both MA and MB reach the plastic
V=
2M p e
(923)
Based on plastic theory, Eq. 922 can be modified slightly to include the effect of interaction between M and V. Nevertheless, experimental results indicated that the interaction is weak and that such interaction can be ignored (Kasai and Popov 196b). Test results also showed that a properly stiffened short link can strain harden and develop a shear strength equal to 1.5Vp. The end moments of a link that has yielded in shear can continue to increase
440
Chapter 9
due to strain hardening and, therefore, flexural hinges can develop. To avoid lowcycle fatigue failure of the link flanges due to high strains, these end moments are limited to 1.2Mp, and the maximum length (e0 in Eq. 922) for a shear link was modified as follows (Kasai and Popov 1986b): e0 =
2(1.2 M p ) 1.5V p
=
1.6 M p Vp
Based on experimental results, the link deformation capacity, γa, as given by the AISC Seismic Provisions is shown in Figure 924. The calculated rotation angle, γp, cannot exceed γa.
(924)
Longer Links Experimental results have shown that the inelastic deformation capacity of an EBF can be greatly reduced when long links ( e > e0 ) are used. Following the above logic, it can be shown that flexural hinges dominate the link response when e is larger than 2.6 M p / V p . In the transition region where 1.6 M p / V p < e < 2.6 M p / V p , the link undergoes simultaneous shear and flexural yielding (Engelhardt and Popov 1989). Figure 923 classifies links in EBFs. Note that when longer links are used in the Dtype or Vtype EBF (see Figure 918), the welded connection between the link and the column is subjected to high moments and it could be vulnerable to brittle fracture if detailed similar to the connections that failed during the Northridge earthquake (see Section 9.2).
Figure 924. Allowable link angles per AISC Seismic Provisions (1997)
Effect of Axial Force The presence of an axial force in a link reduces not only the flexural and shear capacities but also its inelastic deformation capacity (Kasai and Popov 1986b). When the axial force, Pu, exceeds 15% of the yield force, Py (= Ag Fy ) , the PM interaction formula for plastic design (AISC 1989) can be used to compute the reduced plastic moment, Mpa: P M pa = 1.18M p 1 − u Py
(925)
The reduced shear capacity is (Manheim and Popov 1983): V pa = V p 1 − ( Pu / Py ) 2
(926)
Replacing Mp and Vp in Eq. 924 by Mpa and Vpa, the reduced value of e0 when ρ′Aw / Ag ≥ 0.3 can be approximated as follows (Kasai and Popov 1986b): Figure 923. Classification of links
9. Seismic Design of Steel Structures A 1.6 M p e0 = 1.15 − 0.5ρ′ w Ag V p
441 (927)
where ρ′ = P / V , and Aw = (d − 2t f )t w . The correction is unnecessary if ρ′Aw / Ag < 0.3 , in which case the AISC Seismic Provisions (1997) require that the link length shall not exceed that given by Eq. 924. Effect of Concrete Slab Research conducted on composite links showed that composite action can significantly increase the link shear capacity during the first cycles of large inelastic deformations. However, composite action deteriorates rapidly in subsequent cycles due to local concrete floor damage at both ends of the link (Ricles and Popov 1989). For design purposes, it is conservative to ignore the contribution of composite action for calculating the link shear strength. But the overstrength produced by the composite slab effect needs to be considered when estimating the maximum forces that the shear link imposes to other structural components (Whittaker et al. 1989). Link Detailing Fulldepth web stiffeners must be placed symmetrically on both sides of the link web at the diagonal brace ends of the link. These end stiffeners are required to have a combined width not less than (bf −2tw) and a thickness not less than 0.75tw nor 3/8 inch, whichever is larger. The link section needs to satisfy the same compactness requirement as the beam section for special moment frames. Further, the link needs to be stiffened in order to delay the onset of web buckling and to prevent flange local buckling. The stiffening requirement is dependent on the length of link (see Figure 923). For a shear link with e ≤ 1.6 M p / V p , a relationship among the link web deformation angle, the web panel aspect ratio as well as the beam web slenderness ratio was developed
(Kasai and Popov 1986a). A conservative approximation of the relationship follows: a = CB tw −
d 5
(928)
where a = stiffener spacing, d = link depth, tw = link web thickness, and CB = 56, 38, and 29 for γ p = 0.03, 0.06, and 0.09 radian, respectively. These CB values are slightly modified and are adopted in the AISC Seismic Provisions (1997) as follows: (1) When e ≤ 1.6M p / V p , intermediate stiffeners are needed per Eq. 928, but the coefficient CB is a function of the deformation demand; the relationship between CB and γ p implied by the AISC Seismic Provisions is shown in Figure 925. 2.6 M p / V p ≤ e ≤ 5M p / V p , (2) When intermediate stiffeners shall be provided at a distance 1.5bf from each end of the link to control flange local buckling. (3) When 1.6 M p / V p ≤ e ≤ 2.6M p / V p , intermediate stiffeners satisfying the requirements of both Cases 1 and 2 are needed. (4) When e > 5M p / V p , intermediate stiffeners are not required. Intermediate link web stiffeners must be full depth. While twosided stiffeners are required at the end of the link where the diagonal brace intersects the link, intermediate stiffeners placed on one side of the link web are sufficient for links less than 25 inches in depth. Fillet welds connecting a link stiffener
Figure 925. Variation of CB
442
Chapter 9
to the link web shall have a design strength to resist a force of AstFy, where Ast is the stiffener area. The design strength of fillet welds fastening the stiffener to the flanges shall be adequate to resist a force of AstFy/4. Lateral Bracing of Link To ensure stable hysteresis, a link must be laterally braced at each end to avoid outofplane twisting. Lateral bracing also stabilizes the eccentric bracing and the beam segment outside the link. The concrete slab alone cannot be relied upon to provide lateral bracing. Therefore, both top and bottom flanges of the link beam must be braced. The bracing should be designed for 6 percent of the expected link flange strength, RyFybf tf. 9.4.4
At the connection between the diagonal brace and the beam, the intersection of the brace and beam centerlines shall be at the end of the link or within the length of the link (see Figure 926a). If the intersection point lies outside the link length, the eccentricity together with the brace axial force produces additional moments in the beam and brace. The diagonal bracetobeam connection at the link end of the brace shall also be designed to develop the expected strength of the brace. No part of this connection shall extend over the link length to reduce the link length, e. If the connection is designed as a pin (see Figure. 926b), the gusset plate needs to be properly stiffened at the free edge to avoid local buckling (Roeder et al. 1989).
Capacity Design of Other Structural Components
Links in an EBF are designated as structural fuses and are sized for codespecified design seismic forces. All other elements (beam segments outside the link, braces, columns, and connections) are then designed for the forces generated by the actual (or expected) capacity of the links rather than the codespecified design seismic forces. The capacity design concept thus requires that the computation of the link strength not only be based on the expected yield strength of the steel but also includes the consideration of strainhardening and overstrength due to composite action of the slab. Diagonal Brace The required axial and flexural strength of the diagonal brace shall be those generated by the expected shear strength of the link RyVn increased by 125 percent to account for strainhardening. The nominal shear capacity, Vn, is the lesser of Vp or 2Mp/e. Although braces are not expected to experience buckling, the AISC Seismic Provisions take a conservative approach by requiring that a compact section ( λ < λ p ) be used for the brace.
Figure 926. EBF link and connection details 1997)
(AISC
9. Seismic Design of Steel Structures
LinktoColumn Connections Of the common EBF configurations shown in Figure 918, it is highly desirable to use the split Vbraced EBF in order to avoid the moment connection between the link and column. Prior to the 1994 Northridge earthquake, test results showed that the fully restrained welded connection between the column and the link (especially longer links) is vulnerable to brittle fracture similar to those found in the beamtocolumn moment connections after the Northridge earthquake. Therefore, the AISC Seismic Provisions (1997) require that the deformation capacity of the linktocolumn connections be verified by qualifying cyclic tests. Test results shall demonstrate that the link inelastic rotation capacity is at least 20 percent greater than that calculated by Eq. 917. When reinforcements like cover plates are used to reinforce the linktocolumn connection, the link over the reinforced length may not yield. Under such circumstances, the link is defined as the segment between the end of the reinforcement and the brace connection. Cyclic testing is not needed when (1) the shortened link length does not exceed eo in Eq. 924, and (2) the design strength of the reinforced connection is equal to or greater than the force produced by a shear force of 1.25 RyVn in the link. Tests also demonstrated that the welded connections of links to the weakaxis of a column were vulnerable to brittle fracture (Engelhardt and Popov 1989); this type of connection should be avoided. BeamtoColumn Connection For the preferred EBF configuration where the link is not adjacent to a column, a simple framing connection between the beam and the column is considered adequate if it provides some restraint against torsion in the beam. The AISC Seismic Provisions stipulate that the magnitude of this torsion be calculated by
443 considering perpendicular forces equal to 2 percent of the beam flange nominal strength, Fybf tf, applied in opposite directions on each flange. Beam Outside of Link The link end moment is distributed between the brace and the beam outside of the link according to their relative stiffness. In preliminary design, it is conservative to assume that all the link end moment is resisted by the beam. The link end moment shall be calculated using 1.1 times the expected nominal shear strength (RyVn) of the link. Because a continuous member is generally used for both the link and the beam outside the link, it is too conservative to use the expected yield strength (RyFy) for estimating the force demand produced by the link while the beam strength is based on the nominal yield strength (Fy). Therefore, the AISC Seismic Provisions allow designers to increase the design strength of the beam by a factor Ry. The horizontal component of the brace produces a significant axial force in the beam, particularly if the angle between the diagonal brace and the beam is small. Therefore, the beam outside the link needs to be designed as a beamcolumn. When lateral bracing is used to increase the capacity of the beamcolumn, this bracing must be designed to resist 2 percent of the beam flange nominal strength, Fybf tf. Column Using a capacity design approach, columns in braced bays shall have a sufficient strength to resist the sum of gravityload actions and the moments and axial forces generated by 1.1 times the expected nominal strength (RyVn) of the link. This procedure assumes that all links will yield and reach their maximum strengths simultaneously. Nevertheless, available multistory EBF test results showed that this preferred yielding mechanism is difficult to develop. For example, shaking table testing of a 6story reduced scale EBF model showed that
444
Chapter 9
links in the bottom two stories dissipated most of the energy (Whittaker et al. 1989). Therefore, this design procedure may be appropriate for lowrise buildings and the upper stories of medium and highrise buildings but may be too conservative in other instances. The alternative design procedure permitted by the AISC Seismic Provisions is to amplify the design seismic axial forces and moments in columns by the overstrength factor, Ωo (= 2.0, see Table 91). See Eqs. 94 and 95 for the load combinations. The computed column
forces need not exceed those computed by the first procedure. Therefore, the first design procedure will generally produce a more conservative design for columns.
9.5
Design Examples
9.5.1
General
A sixstory office building having the floor plan shown in Figure 927 is used to
Figure 927. A sixstory office building
9. Seismic Design of Steel Structures
445
demonstrate the seismic design procedures. The design follows the AISC Seismic Provisions (1997) and the Load and Resistance Factor Design Specification for Structural Steel Buildings (1993). Special MomentResisting Frames (SMFs) are used in the EW direction, and their design is presented in Section 9.5.2. Braced frames provide lateral loadresistance in the NS direction; these are designed as Special Concentrically Braced Frames (SCBFs) in Section 9.5.3 and Eccentrically Braced Frames (EBFs) in Section 9.5.4, respectively. The design gravity loads are listed in Table 93. The NEHRP Recommended Provisions (1997) are the basis for computing the design seismic forces. It is assumed that the building is located in a high seismic region with the following design parameters: SS = 1.5 g S1 = 0.6 g Site Class = B I = 1.0 (Seismic User Group I) Seismic Design Category = D
Figure 928. Elastic design response spectrum
The design response spectrum is shown in Figure 928. The design follows the Equivalent Lateral Force Procedure of the NEHRP Recommended Provisions. The design base shear ratio, Cs, is computed as follows: Cs =
S S D1 ≤ DS T (R / I ) (R / I )
(929)
where S D1 (= 0.4 g) and S DS (= 1.0 g) are the design spectral response accelerations at a period of 1.0 second and in the short period range, respectively. The values of R for the three framing systems considered in this example are listed in Table 94. The NEHRP empirical period formula is used to compute the approximate fundamental period, Ta:
Ta = CT hn3 / 4
where hn (ft) is the building height, and the coefficient CT is equal to 0.035, 0.030, and 0.02 for SMFs, EBFs, and SCBFs, respectively. Alternatively, the value of T obtained from a dynamic analysis can be used in design, but the period thus obtained cannot be taken larger than CuTa for the calculation of required structural strengths, where Cu = 1.2
(930)
for this design example. To establish seismic forces for story drift computations, however, this upper limit is waived by the NEHRP Recommended Provisions. Recognizing that the analytically predicted period of a multistory SMF is generally larger than CuTa, this upper bound value is used to compute the design base shear ratio for preliminary design. Based on Eq. 91, the design base shear ratios for the three types of frames are listed in Table 94. The following two load combinations are to be considered: 1.2D + 0.5L + 1.0E
(931)
0.9D – 1.0E
(932)
where E = ρ QE + 0.2 SDSD. The Redundancy Factor, ρ , is ρ = 2−
20 rmax A
= 2−
20 0.25 8549
= 1.13 (933)
(See the NEHRP Recommended Provisions on ρ.) Therefore, the above two load combinations can be expressed as
446
Chapter 9
Table 93. Design gravity loads Load
Live Load +(psf)
Dead Load (psf)
Roof
70
20
Floor
90*
50
Cladding
20

*80 psf for computing reactive weight. + Use L = L0 0.25 + 15 / AI for live load reduction (ASCE 1998)
(
)
Table 94. System parameters and design base shears Framing R Ωo Cd System SMF 8 3 5½ SCBF 6 2 5
Ta (sec) 1.07 0.51
Cs
EBF
8 2 4 0.76 *Values have been increased by 5% to account for accidental eccentricity.
1.2D + 0.5L + 1.0( ρ QE + 0.2D) = 1.4D + 0.5L + ρ QE
(934)
0.9D − 1.0(ρQE + 0.2D) = 0.7D − ρQE (935) The design base shear, VB (= CsW, where W = building reactive weight), for computing the seismic effect (QE) is distributed to each floor level as follows: Wx hxk Fx = VB ∑ Wi hik
(936)
where the values of k listed in Table 94 are used to consider the highermode effect. Based on Eq. 936, the design story shears for each example frame are summarized in Table 95. 9.5.2
The story shear distribution of the SMF listed in Table 95 is for strength computations. To compute story drift, however, it is permissible to use the actual fundamental period, T, of the structure. The actual period of this 6story SMF is expected to be larger than
k 1.285
0.131
305
1.0
0.066
156
1.13
the approximate period, Ta (= 1.07 seconds), determined from Eq. 930. There exists many approaches to the preliminary design of SMF. The one followed in this section has been proposed by Becker (1997). First, the fundamental period can be estimated using a simplified Rayleigh method (Teal 1975): T = 0.25
∆r C1
(937)
where T = fundamental period, ∆ r = lateral deflection at the top of the building under the lateral load V, C1 = V/W, V = lateral force producing deflection, and W = building reactive weight. The story drift requirement is:
Special Moment Frames (SMF)
Story Shear Distribution
0.047
VB * (kip) 111
δx =
Cd ∆ < 0.02 H , I
0.02 HI 0.02 × 75 × 12 = = 3.27 in 5 .5 Cd Assuming conservatively that the total deflection is about 60% of the allowable value, ∆ r = 0.60(3.27) = 1.96 in ∆r <
9. Seismic Design of Steel Structures
Table 95. Design story shears Floor Wi (kips) R 322 6 387 5 387 4 387 3 387 2 392 *ρ (= 1.13) is included.
447
hi (ft) 75 63 51 39 27 15
0.4 0.059 = T (R / I ) T (where 1.05 accounts for torsion)
C1 = 1.05ρ(C s ) = 1.05(1.13)
T = 0.25
1.96 = 1.44 T 0.059 / T
Solving the above equation gives a value of T equal to 2.0 seconds. For this value, however, S D1 0.4 Cs = = T ( R / I ) 2(8 / 1.0) , = 0.025 < 0.044 S DS and C s = 0.044 controls. That is, the minimum seismic base shear for drift computations is V = 1.05 × ρ × 0.044W = 0.052W Since the base shear ratios for strength and drift designs are 0.047 and 0.044, respectively, a scaling factor of 0.94 (= 0.044/0.047) can be used to reduce the story shears listed in Table 95 for drift computations.
SMF 34 67 92 109 120 125
Story Shear* (kips) SCBF 84 168 236 288 324 345
EBF 45 90 125 150 167 176
Member Proportions For brevity in this design example, detailed calculations are presented only for the beams on the fourth floor and the columns above and below that floor (see Figure 929). The portal method is used for preliminary design. Assuming that the point of inflection occurs at the midlength of each member: 2F1 + 3F2 = 0.94(109) = 102.5 kips F1 = F2/2,
F2 = 25.6 kips
Consider the interior beamcolumn assembly shown in Figure 929. Summing the moments at the point of inflection at point P, the beam shear, F3, is calculated to be: 12F2 = 25F3,
F3 = 12.3 kips
The story drift due to column and girder deformations is:
Figure 929. Typical shear force distributions in beams and columns
448
Chapter 9 ∆ = ∆c + ∆ g =
F2 h 3 F Lh 2 + 2 12 EI c 12 EI g
(938)
where ∆ = story drift, ∆ c = drift produced by column deformation, ∆ g = drift produced by beam deformations, F2 = column shear, h = story height, L = beam length between points of inflection, Ic = moment of inertia of column, and Ig = moment of inertia of beam. Eq. 938 uses centerline dimensions and ignores the shear and axial deformations of the beams and column. In equating Eq. 938 to the allowable drift, it is assumed that the panel zone deformation will contribute 15% to the story drift; the actual contribution of the panel zone deformation will be verified later. Cd ∆ ≤ (0.85)0.02h, I ∆ ≤ 0.0031h = 0.45 in
quick check of this requirement for the flexural strength of both the beams and columns is worthwhile before the moment connections are designed. It is assumed that the column axial stress ( Pu / Ag ) is equal to 0.15Fy. Beams are designed using the reduced beam section strategy in this example. Assuming that (1) the reduced beam plastic sectional modulus (ZRBS) is 70% of the beam plastic sectional modulus (ZBM), and (2) the moment gradient (Mv) from the plastic hinge location to the column centerline is 15% of the design plastic moment at the plastic hinge location: ∑ M *pc = 2[ Z c ( Fy − Pu /Ag )] ≈ 2 (0.85 Ζ C Fy )
Ζ RBS ≈ 0.7 Z BM , M v ≈ 0.15(1.1R y Z RBS F y ) ∑ M *pb = 2[1.1R y Ζ RBS F y + M v ]
(
δx =
F2 h 2 12 E
h + L ≤ 0.45 Ic I g
≈ 2 1.1R y Ζ RBS F y × 1.15
)
≈ 2(1.1×1.1× 0.7 ×1.15Z BM F y ) = 2(0.97)Z BM F y
(939)
The above relationship dictates the stiffness required for both the beams and columns in order to meet the story drift requirement. By setting I = I c = I as a first attempt, Eq. 939 gives a required I = 1532 in4. Using A992 steel for both the columns and the beams, it is possible to select W14 × 132 columns ( I c = 1530 in4) and W24 × 62 beams ( I g = 1532 in4). In addition to satisfying the story drift requirements, the strength of the columns and beams also need to be checked for the forces produced by the normal seismic load combinations (Eqs. 934 and 935). However, beam and column sizes of this 6story SMF are generally governed by the story drift and strongcolumn weakgirder requirements. Therefore, the strength evaluations of these members are not presented here. A formal check of the strong columnweak beam requirement will be performed later. A
To satisfy Eq. 914: ∑ M *pc ∑
M b*
=
2(0.85Ζ C ) ≥ 1.0 2(0.97 Ζ BM )
ΖC ≥ 1.15 Ζ BM
For the beam and column sizes selected: ZC 234 = = 1.53 ≥ 1.15 Z BM 153
(OK)
Both W14×132 and W24×62 satisfy the λ ps requirements given in Table 92. Since the RBS is to be used, additional check of the beam web compactness is required: 418 h = 50.1 < = 59.1 tw Fy
(OK)
9. Seismic Design of Steel Structures
449
Figure 930. Reduced beam section and the welded beamcolumn connection details.
BeamtoColumn Connection Design Reduced beam section details employing radiuscut (Figure 930) is the most promising beamtocolumn connection detail. The key dimensions of the radius cut include the distance from the face of the column (dimension a), the length of the cut (dimension b), and the depth of the cut (dimension c). To minimize the moment gradient between the narrowest section and the face of column, the dimensions a and b should be kept small. However, making these dimensions too short may result in high strains either at the face of column or within the reduced beam sections. It has been recommended that (Engelhardt et al. 1996): a ≈ (0.5 to 0.75)bf b ≈ (0.65 to 0.85)d
c≥
Z α(L − a − 0.5b ) 1− ≤ 0.25b f 2t f (d − t f ) 1.1L
where L (= 142.7 in) is the distance from the face of the column to the point of inflection in the beam, bf, tf and d are flange width, flange thickness and beam depth, respectively. To determine the maximum cut dimension, c, it is assumed that the strainhardened plastic moment developed at the narrowest beam section is equal to 1.1 times the plastic moment of the reduced section (ZRBSFye). The factor 1.1 accounts for strain hardening. The factor α limits the beam moment ( α Mp) developed at the face of column. The maximum value of α should range between 0.85 and 1.0. Based on an α equal to 0.90, a = 4.0 in = 0.57bf b = 16 in = 0.67d c = 1.375 in ≈ 0.20bf
450
Chapter 9 R=
4c 2 + b 2 = 24 in 8c
Following the SAC Interim Guidelines (SAC 1997), other features of the connection include the use of notchtoughness weld metal, the use of a welded web connection, and the use of continuity plates. Lateral supports capable of resisting a minimum of 2% of the unreduced flange force should be provided such that the unbraced length is no larger than the following (see Figure 931): Lb =
2500 2500 ry = (1.38) = 69 in = 5.75 ft Fy 50
Additional bracing near the RBS is unnecessary because deep section is not used for the column.
+(25×12)(0.02) = 125 kips and the live load axial force, including live load reduction, is L= L D 0.25 +
= 22.5 psf 4(4)(25 × 14) 15
PL= 3(25×14)(0.0225)+(25×14)(0.02×0.45) = 26.8 kips Therefore, the factored axial load is Pu = 1.4(125)+0.5(26.8) = 188 kips The column moment capacity is
∑ M ∗pc = ∑ Z c ( F yc − Puc
Ag )
= 2(234)(50188/38.8) = 21132 kipin The plastic sectional modulus of the RBS is Z RBS = Z BM − 2ct f (d − t f ) = 153 − 2(1.375)(23.74 − 0.59) = 115 in 4 The design plastic moment capacity of the reduced beam section is M pd = 1.1R y Z RBS F y = 1.1× 1.1×115 × 50 = 6958 kip − in
and the corresponding beam shear is Vpd = Figure 931. Lateral support for the beam
Strong ColumnWeak Beam Criterion The axial force in interior columns in a moment frame, produced by seismic loading, can be ignored generally. The axial force due to dead load on the upper floors, roof, and cladding is: PD = roof + (4 to 6)floors + cladding = (25×14)(0.07)+3(25×14)(0.09)
1.1R y Z RBS F y
[0.5(25 × 12) − d c
2 − a − b 2]
= 53 kips After extrapolating the beam moment at the plastic hinge location to the column centerline, the beam moment demand is
∑ M ∗pb = ∑ (M pd + M v ) = ∑ [ M pd + V pd (d c / 2 + a + b / 2)] 14.66 = 2 26958 + 53 + 4 + 8 2 = 15965 kipin
9. Seismic Design of Steel Structures
∑ M ∗pc ∑ M ∗pb
= 1.32 > 1.0
451 t(req’d) =
(OK)
Therefore, the strong columnweak beam condition is satisfied. Panel Zone Design The unbalanced beam moment, ∆ M, for the panel zone design is determined from the special load combination in Eq. 94, where the beam moment at the column face produced by Ω o (ρQ E ) is
Since both the thicknesses of column web and doubler plate are larger than the required thickness, plug welds are not required to connect the doubler plate to the column web. See Figure 9.30 for the connection details. The component of story drift produced by the panel zone deformation is computed as follows (Tsai and Popov 1990): γp = =
= 3.0(12.3/0.94)142.7 = 5602 kipin
But the above moment need not be greater than 0.8 ∑ M pb . Extrapolating the beam moment at the plastic hinge location to the column face, M pb is computed as follows: b M *pd = M pd + V pd a + 2 = 6958+ 53(4.0 + 8.0) = 7594 kip − in
where G is the shear modulus. The story drift due to the panel zone deformation, ∆ P , is: ∆ P = 0.00070 × 12 × 12 = 0.10 in
The total story drift produced by the column, beam, and panel zone deformations is:
∆M ∆M Vu = − = 497 − 78 = 419 kips 0.95d b h
The shear capacity of the panel zone is φVn = 0.75(0.6 )(50.0)(14.66)t p 3(14.725)(1.03)2 × 1 + (23.74)(14.66)t p
Equating Vu and φVn to solve for the required panel zone thickness gives tp = 1.14 in. Since the column web thickness is 0.645 in, use a 1/2 in thick doubler plate. (The column size needs to be increased if the designer prefers not to use doubler plates.) Check Eq. 913 for local buckling of the doubler plate:
F2 h(h − d b ) 12 EI c
2
∆c + ∆ g + ∆P =
0.8∑ M *pb = 0.8(2)(7594) = 12150 kipin Therefore, the shear in the panel zone is
V dct pG
419 × 0.94 / Ω o 14.66(0.645 + 0.50)29000 / 2.6 = 0.00070 rad
M1 = M2 = Ω o (F3/0.94)L ∆M = M 1 + M 2 = 11204 kipin
d z + wz = 0.39 in 90
+
F2 h 2 (L − d c ) + ∆P 12 EI g
= 0.10+0.281+0.10 = 0.48 < 0.52 in (OK) Note that the clear lengths are used to compute the deformations of the beams and column in the above equation. 9.5.3
Special Concentrically Braced Frames (SCBFs)
The sixstory invertedV braced frame shown in Figure 932 is analyzed for the loads specified earlier. The service dead load, live load, and seismic member forces, calculated taking into account loadpaths and live load reduction, and maximum forces resulting from the critical load combination, are presented in Tables 96 and 97, where the axial forces and moments are expressed in kips and kipft,
452
Chapter 9
Figure 932. Concentrically braced frame elevation
respectively. Cladding panels are assumed connected at the columns. Note that the load combination 1.2D+0.5L+1.0E governs for the design of all members. In the first phase of design (called “strength design” hereafter), members are sized without attention paid to special seismic detailing requirements, as normally done in nonseismic applications, and results are also presented in Tables 96 and 97. Members are selected per a minimum weight criterion, with beams and braces constrained to be wideflanges sections of same width, and columns constrained to be W14 shapes continuous over two stories. ASTM A992 steel is used for all members, and the effective length factors, K, of 1.0 were respectively used in calculating the inplane and outofplane buckling strength of braces. Additional information on the effects of endfixity on the inelastic nonlinear behavior of
braces is presented elsewhere (Bruneau et al. 1997). Note that this frame geometry leads to substantial foundation uplift forces. Although not done here, increasing the number of braced bays will reduce the uplift forces. In the second phase of design, (hereafter called “ductile design”), the seismic requirements are checked, and design is modified as necessary. The special ductile detailing requirements of braces are first checked. Here, all braces are found to have a slenderness ratio in excess of the permissible limit (Eq. 915), and some also violate the specified flange widthtothickness ratio limit. For example, for the fifth story braces (W8×31), the slenderness ratio is: (KL/r)y = (1.0)(19.21)(12)/2.02 = 114.1 > 720 / 50 = 102
(NG)
9. Seismic Design of Steel Structures
453
Table 96. Strength design results for columns and Wshape braces (axial force in kips) Story
PL
PD
PLr
Pua
PQE or TQE
Tub
Member
Columns 37 W14×30 132 W14×30 261 25 W14×53 416 90 W14×53 593 177 W14×90 783 276 W14×90 Braces 6 5.9 1.7 48 62 49.6 W8×24 5 7.7 4.2 95 120 102 W8×31 4 7.7 4.2 133 163 145 W8×35 3 7.7 4.2 164 198 180 W8×48 2 7.7 4.2 182 218 200 W8×48 1 6.9 3.8 218 257 240 W8×67 a from load combination 1.2D + 0.5L + 1.0E (see Eq. 934), where E = ρQE. b from load combination 0.9D  1.0E (see Eq. 935). 6 5 4 3 2 1
26.6 66.7 108 149 191 232
10.5 18.9 26.2 33.0 39.6
4.20 4.86 4.86 4.86 4.86 4.86
0 29.8 89 172 275 388
φcPn
KL/r
bf/2tf
h/tw
190 190 439 439 1008 947
96.6 96.6 75.0 75.0 38.9 48.6
8.7 8.7 6.1 6.1 10.2 10.2
45.4 45.4 30.8 30.8 25.9 25.9
73.7 149 170 244 244 292
143.2 114.1 113.6 110.8 110.8 120.1
8.1 9.2 8.1 5.9 5.9 4.4
25.8 22.2 20.4 15.8 15.8 11.1
Table 97. Strength design results for beams (axial force in kips, moment in kipin) Level
PD
MD
PL
ML
PLr
MLr
PQbE
Pua
M ua
Section
11.1 2.0 37.2 42 16.5 W8×21 4.6 14.7 5.0 0.8 74.2 81 30.3 W8×24 6.0 14.8 2.1 3.3 104 113 30.4 W8×31 6.0 15.2 1.4 3.1 128 141 31 W8×31 6.1 14.9 1.2 2.9 142 156 30.6 W8×31 5.9 15.6 1.2 2.9 154 170 31.7 W8×31 a from load combination 1.2D + 0.5L + 1.0E (see Eq. 934), where E = ρQE.
Roof 6 5 4 3 2
φcPn
φbMn
KL/r
bf /2tf
h/tw
64.4 120 217 217 217 217
76.5 87.0 114 114 114 114
142.9 111.8 89.1 89.1 89.1 89.1
6.6 8.1 9.2 9.2 9.2 9.2
27.5 25.8 22.2 22.2 22.2 22.2
b. M = 0. E
and the widthtothickness ratio is: b/t = bf /2tf = 9.2 > 52/ 50 = 7.35
b/t = 10/0.25 = 40 >110/ 46 = 16.22 (NG) (NG)
These braces, therefore, have insufficient capacity to dissipate seismic energy through repeated cycles of yielding and inelastic buckling. Coldformed square structural tubes with a specified yield strength of 46 ksi under ASTM A500 Grade B are first selected to replace the wideflange brace sections. As shown in Table 98, a strength design using such hollow shapes effectively reduces brace slenderness, but does not necessarily satisfy the stringent widthtothickness ratio limits prescribed for seismic design. For example, for the first story braces (TS10×10×1/4), the widthtothickness ratio is:
Consequently, new brace sections are selected to comply with both the widthtothickness and member slenderness ratio limits. These are presented in Table 99. At each story, the reduced compression strength 0.8( φc Pn ) is then considered. Here, the tension brace at each level has sufficient reserve strength to compensate for the loss in compression resistance upon repeated cyclic loading, and the chosen braces are thus adequate. For example, for the TS6×6×5/8 braces at the third story, Factored design forces:
Pu = 198 kips Tu = 180 kips
454
Chapter 9
Table 98. Strength design results for TSshape braces (axial force in kips) Story
Pua
Tub
Member
φcPn
6 62 49.6 TS6×6×3/16 88 5 120 102 TS8×8×3/16 157 4 163 145 TS9×9×3/16 178 3 198 180 TS8×8×1/4 207 2 218 200 TS9×9×1/4 253 1 257 240 TS10×10×1/4 284 a from load combination 1.2D + 0.5L + 1.0E (see Eq. 934), where E = ρQE. b from load combination 0.9D  1.0E (see Eq. 935).
KL/r
b/t
97.7 72.5 64.2 73.2 64.8 64.3
32.0 42.7 48.0 32.0 36.0 40.0
KL/r
b/t
101 101 104 107 88 84
16.0 16.0 10.0 9.6 14.0 16.0
Table 99. Ductility design results for TSshape braces (axial force in kips) Story
Pua
Tub
Member
φcPn
6 62 49.6 TS6×6×3/8 157 5 120 102 TS6×6×3/8 157 4 163 145 TS6×6×1/2 196 3 198 180 TS6×6×5/8 224 2 218 200 TS7×7×1/2 288 1 257 240 TS8×8×1/2 351 a from load combination 1.2D + 0.5L + 1.0E (see Eq. 934), where E = ρQE. b from load combination 0.9D − 1.0E (see Eq. 935).
Design strengths: φc Pn
= 224 kips, φ t Tn = φ t Ag Fy = 513 kips
Reduced compression design strength: 0.8( φc Pn ) = 0.8(224) = 179 kips
Finally, the redundancy requirement is satisfied by checking that members in tension carry at least 30% but no more than 70% of the story shear. Note that for bays with the same number of compression and tension braces, satisfying the above member slenderness limits, this is usually not a concern. For example, check the first story brace as follows: Tu / cos θ 240 / 0.707 = = 0.56 305 VB
which is between 0.3 and 0.7.
Design Forces in Connections Connections are designed to resist their expected brace tension yield force of RyAgFy. For example, for the braces in the first story, this would correspond to a force of (1.1)(14.4)(46) = 729 kips. The brace gusset used with tubular braces usually permits outofplane buckling and needs to be detailed per Figure 915 to resist the applied axial force while undergoing large plastic rotation. Design Forces in Columns When Pu/ φc Pn in columns is greater than 0.4 (as is the case here), the AISC Seismic Provisions require that columns also be designed to resist forces calculated according to the special load combinations in Eqs. 94 and 95. However, these forces need not exceed those calculated considering 1.1Ry Tn and 1.1Ry Pn of the braces. Members designed to satisfy this requirement are presented in Table 910.
9. Seismic Design of Steel Structures
455
Table 910. Ductility design results for columns (axial force in kips) Story
Pua
Tub
∑ Pnc
∑ Tnd
6 34 34 5 146 225 221 4 314 80 420 465 3 538 211 647 790 2 796 378 886 1150 1 1078 568 1196 1543 a from load combination 1.2D + 0.5L ± Ωo QE (see Eq. 94), where Ωo =2.0. b from load combination 0.9D ± ΩoQE (see Eq. 95). c 1.2D + 0.5L + Σ(1.1RyPn) , where Pn is the brace nominal compressive strength. d 1.2D + 0.5L + Σ(1.1RyTn) , where Tn is the brace nominal tensile strength. Cases b and d are used to check column splices and foundation uplift.
Member
φcPn
W14×30 W14×30 W14×61 W14×61 W14×109 W14×109
190 190 591 591 1220 1147
Table 911. Ductility design results for beamsa (force in kips, moment in kipin) Unbalanced Force Level Tn 0.3φcPn Vertical Horizontal Mux Pu Section φbMnx 6 372 47.3 203 1522 127 254 W30×148 1789 5 372 47.3 203 1522 127 254 W30×148 1789 4 479 58.7 263 1973 164 328 W30×173 2164 3 524 63.0 288 2160 180 360 W30×191 2398 2 570 86.4 302 2265 189 378 W30×191 2398 a Ductility design not required at top story of a chevron braced frame per AISC Seismic provisions.
Note that columns splices would have to be designed to resist the significant uplift forces shown in this table, although the AISC Seismic Provisions indicate that the tension forces calculated in Table 910 need not exceed the value corresponding the uplift resistance of the foundation. Design Forces in Beams Finally, beams are checked for compliance with the special requirements presented in Section 9.3. Here, all beams are continuous between columns, and are braced laterally at the ends and midspan. W30 shapes were chosen to limit beam depth. Beams are, therefore, redesigned to resist the unbalanced vertical force induced when the compression braces are buckled and the tension braces are yielded. In this example, this substantial force governs the design. The corresponding moments and axial forces acting on the beams are shown in Table 911, along with the resulting new beam sizes. Note that the
φcPn 1172 1172 1765 1956 1956
Ratio 0.90 0.90 0.96 0.94 0.99
adequacy of these beams is checked using the AISC (1993) beamcolumn interaction equations. For example, for the W30×191 beam on the second floor: M ux 189 2265 Pu = + + 2φ c Pn φb M nx 2 (1956) 2398 = 0.05 + 0.94 = 0.99 < 1.0 (OK)
Incidentally, note that this section is a compact section. 9.5.4
Eccentrically Braced Frames (EBFs)
The configuration of the splitVbraced EBF is shown in Figure. 933, and the design seismic forces are listed in Table 95. The geometry is chosen such that the link length is about 10% of the bay width, and the inclined angle of the braces is between 35 to 60 degrees: e = 0.1L = 3 ft = 36 in
456
Chapter 9 θ = tan −1 (15 / 13.5) = 48°
(first story)
θ = tan −1 (12 / 13.5) = 42°
(other stories)
In this example, detailed design calculations are only presented for members at the first story to illustrate the procedure. Unless indicated otherwise, ASTM A992 steel is used.
Assuming that the braces are rigidly connected to the link, that the beam can resist 95% of the link end moment, and that the beam flexural capacity is reduced by 30% due to the presence of an axial force: (0.7) R y (φ b M p ) ≥ 0.85(1.1) R yVn (e / 2) or 1.35Mp/Vn ≥ e For shear links, the above requirement for the maximum link length is more stringent than 1.6Mp/Vp. The required strengths for the link on the second floor are Vu = 1.4 D + 0.5 L + E = 1.4(1.1) + 0.5(0.4) + 98.0 = 100 kips M u = 1.4 D + 0.5L + E = 1.4(8.0) + 0.5(3.0) + 98(3.0 / 2) = 160 kipft Note that there is no axial force acting on the shear links (i.e., Pu = 0 kip). Illustrating this procedure for the shear link on the second floor: 1.35M p / V p ≥ 36 in ⇒ M p / V p ≥ 26.7 in Vu = 100 kips ≤ φV n = φV p = 0.9(0.6)(50)t w (d − 2t f ) ⇒ t w (d − 2t f ) ≥ 3.70 in 2
(
Vu = 100 kips ≤ φVn = φ 2 M p / e
)
= 2(0.9)(50Z x ) / 36 ⇒ Z x ≥ 40.0 in 3 Based on the above three requirements, select a W12×45 section for the link:
Figure 933. Eccentrically braced frame elevation
Link Design Shear links with e ≤ 1.6 Mp/Vp are used to achieve higher structural stiffness and strength. The AISC Seismic Provisions stipulate that the beam outside the link shall be able to resist the forces generated by at least 1.1 times the expected nominal shear strength of the link.
Z x = 64.7 in 3 > 40.0 in 3
(OK)
t w (d − 2t f ) = 3.87 in 2 > 3.70 in 2
(OK)
Mp
(OK)
Vp
=
Zx = 27.9 in > 26.7 in 0.6(d − 2t f )tw
9. Seismic Design of Steel Structures bf 2t f
= 7.0 <
52 Fy
= 7.4
457 (OK)
φ b M p = 243, BF = 5.07, C b ≈ 1.67
[
(
φ b M n = C b φ b M p − BF Lb − L p
520 h = 29.0 < = 73.5 tw Fy
)]
= 350 ≥ φ b M p
(OK)
Use φ b M n = 243 kipft
Vn = min {V p , 2 M p e}
= min {116, 180} = 116 kips
(OK)
π 2 EI x
Pe1 =
( KL x ) 2
= 3817 kips
Cm = 0.85 Beam Outside of Link
B1 =
The moment at both ends of the link is: M u = 1.1( R yVn e / 2)
Use B1 = 1.0
= 1.1(1.1 × 116 × 3.0/2) = 211 kipin This moment is resisted by both the rigidly connected brace and the beam outside the link. Assuming that the beam resists 85% of the link moment, the beam end moment including the gravity load effect (MD = 8 kipft, ML = 3 kipft) is Mu = 0.85(211) + 1.2(8.0) + 0.5(3.0) = 190 kipft
= 0.97 < 1.0
(OK)
1.25R y Vn (e 2) Lb
= 18 kips
Therefore, the brace force including the gravity load effect (VD = 5.7 kips, VL = 2.2 kips) is Pu = (1.25R yV n + Vb + 1.2V D + 0.5V L ) / sin(θ)
(OK)
Checking the strength of the beam segment as a beamcolumn: = 6.9 ft < [Lb = 13.5 ft ]< [Lr = 20.3 ft ]
1.25RyVn (e 2) = 240 kip  ft Vb =
253 ≥ Fy
h λ= = 16.7 < λ ps tw
]
Pu 8 B1 M u + = 0.34 + 0.63 R y (φ c Pn ) 9 R y (φ b M n )
1.25 R yVn = 160 kips
Checking the beam web local buckling (see Table 92):
p
φ c Pn = 337 kips
To compute the beam shear, Vb, assume the beam moment at the column end is zero.
Pu 126 = = 0.212 > 0.125 φ b Py 0.9 Ag F y
[L
( KL) y = 13.5 ft,
Diagonal Brace
The axial force ratio in the beam is
P 191 2.33 − u λ ps = φ b Py F y = 57.2
Cm 0.85 = = 0.88 < 1.0 Pu 126 1− 1− 3817 Pe1
= 250 kips
The brace length is 20.2 ft. Selecting a square tubular section TS8 × 8 × 1/2 (A500 Grade B steel): φ c Pn = 366 kips > 250 kips
(OK)
458
Chapter 9
b/t = 6.5/0.5 = 13 < λ p = 190 / F y = 28 (OK) Once the brace size is determined, it is possible to determine the link end moment based on the relative stiffness (I/L) of the brace and the beam segment outside the link. The moment distribution factor is
(DF )br
Therefore, the moment at the end of brace is Mu = 240 × (DF)br = 48 kipft The brace capacity is checked as a beamcolumn: φ b M n = φ b ( Z x F y ) = 137 kipft
(OK)
t = max {tw , 3/8} = 3/8 in The required stiffener spacing, a, is based on Eq. 928, where CB is (see Figure 925):
The axial force produced by the design seismic force in the first story is P = 88 / cos(θ) = 132 kips The axial deformation of the brace is PL 132(20.2 × 12) = = 0.077 in EA 29000(14.4)
The elastic story drift is ∆ = 0.115 in cos (θ)
and the design story drift is δ s = C d δ e I = 4.0( 0.115 ) / 1.0 = 0.46 in
Therefore, the link rotation is Lδ γ p = s e h
Lateral bracing similar to that shown in Figure 931 is needed for the links, except that the bracing needs to be designed for 6% of the expected link flange force, RyFybftf.
Onesided intermediate stiffeners are permitted because the link depth is less than 25 inches. The required thickness is
Link Rotation
δe =
t = max {0.75tw, 3/8} = 3/8 in
Link Stiffeners
Pu 8 Mu + = 0.68 + 0.31 φ c Pn 9 φ b M n
∆=
Lateral Bracing Fulldepth stiffeners of A36 steel are to be used in pairs at each end of the links. The required thickness of these stiffeners is
I br /Lbr = = 0.20 I br /Lbr + I b /Lb
= 0.99 < 1
The link rotation capacity is 0.08 rad because the link length (= 36 in) is smaller than 1.6 Mp/Vp (= 44.6 in). Thus, the link deformation capacity is sufficient.
30.0 0.46 = = 0.026 rad 3.0 15 × 12
C B = 59.3 − 367 γ P = 50.1
a = CBtw −
d = 14.4 in 5
Therefore, three intermediate stiffeners are provided. The weld between the stiffener and the link web should be designed to resist the following force: F = Ast Fy = (3.75)(0.375)(36) = 51 kips The required total design force between the stiffener and the flanges is F = Ast Fy/4 = 12.8 kips A minimum fillet weld size of ¼ in. satisfies the above force requirement. Columns
9. Seismic Design of Steel Structures
Table 912. Summary of member sizes and column axial loads Floor Link Σ1.1RyVn ΣPD Level Size (kips) (kips) R W10×45 113 30 6 W10×45 226 71 5 W10×45 339 112 4 W10×45 452 153 3 W12×45 592 194 2 W12×45 732 235
Columns must be designed to satisfy the special load combination presented in Eq. 94, where Ω o E is replaced by the seismic force generated by 1.1 times the expected nominal strength (RyVn) of the links. The column axial load produced by both gravity loads and seismic forces are listed in Table 912. The required axial compressive strength is Pu = 1.2(235) + 0.5(42) + 732 = 1035 kips A W12×106 column, with a design axial load capacity of 1040 kips, is chosen for the lowest two stories. The column splice must be designed for the tensile force determined from the load combination in Eq. 8.5: Pu = 0.9D − Ω o Q E = 0.9(235) − 732 = −521 kips As stated in the SCBF design example, using more than one braced bay in the bottom stories may reduce the tensile force in the columns and increase the overturning resistance of the building.
459
ΣPL (kips)
Column Size
Brace Size
5
W12×40
TS8×8×½
14
W12×40
TS8×8×½
22
W12×72
TS8×8×½
29
W12×72
TS8×8×½
35
W12×106
TS8×8×½
42
W12×106
TS8×8×½
460
REFERENCES 91 AIJ, Performance of Steel Buildings during the 1995 HyogokenNanbu Earthquake (in Japanese with English summary), Architectural Inst. of Japan, Tokyo, Japan, 1995. 92 AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, IL, 1993. 93 AISC, Specifications for Structural Steel Buildings, AISC, Chicago, IL, 1989. 94 AISC, Near the Fillet of Wide Flange Shapes and Interim Recommendations, Advisory Statement on Mechanical Properties, AISC, Chicago, IL, 1997a. 95 AISC, Seismic Provisions for Structural Steel Buildings, with Supplement No. 1 (1999), AISC, Chicago, IL, 1992 and 1997b. 96 ASCE, Minimum Design Loads for Buildings and Other Structures, ASCE Standard 798, ASCE, New York, NY, 1998. 97 AstanehAsl, A., Goel, S.C., Hanson, R.D., “Earthquakeresistant Design of Double Angle Bracings,” Engrg. J., Vol. 23, No. 4, pp. 133147, AISC, 1981. 98 Becker, R., “Seismic Design of Steel Buildings Using LRFD,” in Steel Design Handbook (editor: A. R. Tamboli), McGrawHill, 1997. 99 Bruneau, M., Mahin, S.A., “Ultimate Behavior of Heavy Steel Section Welded Splices and Design Implications,” J. Struct. Engrg., Vol. 116, No. 8, pp. 22142235, ASCE, 1990. 910 Bruneau, M., Uang, C.M, Whittaker, A., Ductile Design of Steel Structures, McGraw Hill, New York, NY, 1997. 911 BSSC, NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings, Federal Emergency Management Agency, Washington, DC, 1997. 912 Chen, S.J., Yeh, C.H. and Chu, J.M., “Ductile Steel BeamColumn Connections for Seismic Resistance,” J. Struct. Engrg., Vol. 122, No. 11, pp. 12921299. ASCE, 1996. 913 Chi, B. and Uang, C.M., “Seismic Retrofit Study on Steel Moment Connections for the Los Angeles Department of Public Works Headquarters Building,” Report No. TR2000/14, Department of Structural Engineering, University of California, San Diego, La Jolla, CA, 2000. 914 Engelhardt, M. D. and Popov, E. P., "On Design of Eccentrically Braced Frames," Earthquake Spectra, Vol. 5, No. 3, pp. 495511, EERI, 1989. 915 Engelhardt, M.D. and Sabol, T., “Reinforcing of Steel Moment Connections with Cover Plates: Benefits and Limitations,” Engrg. J., Vol. 20, Nos. 46, pp. 510520, 1998.
Chapter 9 916 Engelhardt, M.D., Winneburger, T., Zekany, A.J., and Potyraj, T.J., “The Dogbone Connection: Part II,” Modern Steel Construct., Vol. 36. No. 8, pp. 4655, AISC, 1996. 917 Fujimoto, M., Aoyagi, T., Ukai, K., Wada, A., and K. Saito, "Structural Characteristics of Eccentric KBraced Frames," Trans., No. 195, pp. 3949, AIJ, May 1972. 918 Gilton, C., Chi, B., and Uang, C.M., "Cyclic Response of RBS Moment Connections: WeakAxis Configuration and Deep Column Effects," Report No. SSRP2000/03, Department of Structural Engineering, University of California, San Diego, La Jolla, CA, 2000. 919 Gross, J. L., Engelhardt, M. D., Uang, C.M., Kasai, K., and N. Iwankiw, Modification of Existing Welded Steel Moment Frame Connections for Seismic Resistance, Steel Design Guide Series 12, AISC and NIST, 1998. 920 Hamburger, R. “More on Welded Connections,” SEAONC News, Structural Engineers Association of Northern California, San Francisco, CA, April, 1996. 921 Hjelmstad, K. D. and Popov, E. P., "Characteristics of Eccentrically Braced Frames," J. Struct. Engrg., Vol. 110, No. 2, pp. 340353, ASCE, 1984. 922 Hjelmstad, K. D. and Popov, E. P., "Cyclic Behavior and Design of Link Beams," J. Struct. Engrg., Vol. 109, No. 10, pp. 23872403, ASCE, 1983. 923 ICBO, Uniform Building Code, Int. Conf. of Building Officials, Whittier, CA, 1988. 924 Iwankiw, R.N. and Carter, C.J., “The Dogbone: A New Idea to Chew On,” Modern Steel Construct., Vol. 36. No. 4, pp. 1823, AISC, Chicago IL, 1996. 925 Jokerst, M. S. and Soyer, C., "San Francisco Civic Center Complex: Performance Based Design with Passive Dampers and Welded Steel Frames," Proc., 65th Annual Convention, pp. 119134, SEAOC, 1996. 926 Kasai, K. and Popov, E. P., "On Seismic Design of Eccentrically Braced Frames," Proc., 8th World Conf. Earthquake Engrg., Vol. V, pp. 387394, IAEE, San Francisco, 1985. 927 Kasai, K. and Popov, E. P., "Cyclic Web Buckling Control for Shear Link Beams," J. Struct. Engrg., Vol. 112, No. 3, pp. 505523, ASCE, 1986a. 928 Kasai, K. and Popov, E. P., "General Behavior of WF Steel Shear Link Beams," J. Struct. Engrg., Vol. 112, No. 2, pp. 362382, ASCE, 1986b. 929 Krawinkler, H., Bertero, V.V., and Popov, E.P., “Inelastic Behavior of Steel BeamColumn Subassemblages,” Report No. UCB/EERC71/7, Univ. of Calif., Berkeley, Berkeley, CA, 1971. 930 Krawinkler, H., Bertero, V.V., and Popov, E.P., “Shear Behavior of Steel Frame Joints,” J. Struct. Div., Vol. 101, ST11, pp. 23172338. ASCE, 1975.
9. Seismic Design of Steel Structures 931 Libby, J. R., "Eccentrically Braced Frame Construction−A Case Study," Engrg. J., Vol. 18, No. 4, pp. 149153, AISC, 1981. 932 Malley, J. O. and Popov, E. P., "Shear Links in Eccentrically Braced Frames," J. Struct. Engrg., Vol. 110, No. 9, pp. 22752295, ASCE, 1984. 933 Manheim, D. N. and Popov, E. P., "Plastic Shear Hinges in Steel Frames," J. Struct. Engrg., Vol. 109, No. 10, pp. 24042419, ASCE, 1983. 934 Merovich, A., Nicoletti, J. P., and Hartle, E., "Eccentric Bracing in Tall Buildings," J. Struct. Div., Vol. 108, No. ST9, pp. 20662080, ASCE, 1982. 935 Nakashima, M. And Wakabayashi, M., “Analysis and Design of Steel Braces and Braced Frames,” in Stability and Ductility of Steel Structures under Cyclic Loading, pp. 309322, CRC Press, 1992. 936 Noel, S. and Uang, C.M., “Cyclic Testing of Steel Moment Connections for the San Francisco Civic Center Complex,” Report No. TR96/07, Univ. of California, San Diego, La Jolla, CA, 1996. 937 Plumier, A., “New Idea for Safe Structures in Seismic Zones,” Proc., Symposium of Mixed Structures Including New Materials, pp. 431436, IABSE, Brussels, Belgium, 1990. 938 Redwood, R.G., and Channagiri, V.S., “EarthquakeResistant Design of Concentrically Braced Steel Frames,” Canadian J. Civil Engrg., Vol. 18, No. 5, pp. 839850, 1991. 939 Ricles, J. M. and Popov, E. P., "Composite Action in Eccentrically Braced Frames," J. Struct. Engrg., Vol. 115, No. 8, pp. 20462065, ASCE, 1989. 940 Roeder, C. W. and Popov, E. P., "Eccentrically Braced Steel Frames For Earthquakes," J. Struct. Div., Vol. 104, No. ST3, pp. 391411, ASCE, 1978. 941 Roeder, C. W., Foutch, D. A., and Goel, S. C., "Seismic Testing of FullScale Steel Building−Part II," J. Struct. Engrg., Vol. 113, No. 11, pp. 21302145, ASCE, 1987. 942 SAC, “Interim Guidelines: Evaluation, Repair, Modification, and Design of Welded Steel Moment Frame Structures,” Report FEMA No. 267/SAC9502, SAC Joint Venture, Sacramento, CA, 1995. 943 SAC, “Interim Guidelines Advisory No. 1, Supplement to FEMA 267,” Report No. FEMA 267A/SAC9603, SAC Joint Venture, Sacramento, CA, 1997. 944 SAC, “Technical Report: Experimental Investigations of BeamColumn Subassemblages, Parts 1 and 2,” Report No. SAC9601, SAC Joint Venture, Sacramento, CA, 1996. 945 SAC, “Recommended Seismic Design Criteria for New MomentResisting Steel Frame Structures,” Report No. FEMA 350, SAC Joint Venture, Sacramento, CA, 2000. 946 Shibata, M., Nakamura T., Yoshida, N., Morino, S., Nonaka, T., and Wakabayashi, M., “ElasticPlastic Behavior of Steel Braces under Repeated Axial
461 Loading,” Proc., 5th World Conf. Earthquake Engrg., Vol. 1, pp. 845848, IAEE, Rome, Italy, 1973. 947 SSPC, Statistical Analysis of Tensile Data for WideFlange Structural Shapes, Chaparral Steel Co., Midlothian, TX, 1994. 948 Tanabashi, R., Naneta, K., and Ishida, T., "On the Rigidity and Ductility of Steel Bracing Assemblage," Proc., 5th World Conf. Earthquake Engrg., Vol. 1, pp. 834840, IAEE, Rome, Italy, 1974. 949 Teal, E. J., "Seismic Drift Control Criteria," Engrg. J., Vol. 12, No. 2, pp. 5667, AISC, 1975. 950 Teal, E., Practical Design of Eccentric Braced Frames to Resist Seismic Forces, Struct. Steel Res. Council, CA, 1979. 951 Tremblay, R., Bruneau, M., Nakashima, M., Prion, H.G.L., Filiatrault, M., DeVall, R., “Seismic Design of Steel Buildings: Lessons from the 1995 Hyogoken Nanbu Earthquake”, Canadian J. Civil Engrg., Vol. 23, No. 3, pp. 757770, 1996. 952 Tremblay, R., Timler, P., Bruneau, M., Filiatrault, A., “Performance of Steel Structures during the January 17, 1994, Northridge Earthquake,” Canadian J. Civil Engrg., Vol. 22, No. 2, pp. 338360, 1995. 953 Tsai, K.C., and Popov, E.P., “Seismic Panel Zone Design Effects on Elastic Story Drift of Steel Frames,” J. Struct. Engrg., Vol. 116, No. 12, pp. 32853301, ASCE, 1990. 954 Tsai, K.C., and Popov, E.P., “Steel BeamColumn Joints in Seismic MomentResisting Frames,” Report No. UCB/EERC88/19, Univ. of Calif., Berkeley, 1988. 955 Yu, Q.S., Uang, C.M., and Gross, J., “Seismic Rehabilitation Design of Steel Moment Connection with Welded Haunch,” Journal of Structural Engineering, Vol. 126, No. 1, pp. 6978, ASCE, 2000. 956 Uang, C.M. "Establishing R (or Rw) and Cd Factors for Building Seismic Provisions," J. Struct. Engrg., Vol. 117, No. 1, pp. 1928, ASCE, 1991. 957 Uang, C.M. and Noel, S., “Cyclic Testing of Strongand WeakAxis Steel Moment Connection with Reduced Beam Flanges,” Report No. TR96/01, Univ. of Calif., San Diego, CA, 1996. 958 Uang, C.M and Fan, C.C, “Cyclic Stability of Moment Connections with Reduced Beam Section,” Report No. SSRP99/21, Department of Structural Engineering, University of California, San Diego, La Jolla, CA, 1999. 959 Whittaker, A. S., Uang, C.M., and Bertero, V. V., "Seismic Testing of Eccentrically Braced Dual Steel Frames," Earthquake Spectra, Vol. 5, No. 2, pp. 429449, EERI, 1989. 960 Whittaker, A.S. and Gilani, A., “Cyclic Testing of Steel BeamColumn Connections,” Report No. EERCSTI/9604, Univ. of Calif., Berkeley, 1996. 961 Whittaker, A.S., Bertero V.V., and Gilani, A.S. “Evaluation of PreNorthridge Steel Moment
462 Resisting Joints.” Struct. Design of Tall Buildings. Vol. 7, No. 4, pp. 263283, 1998. 962 Yang, M.S., "Shaking Table Studies of an Eccentrically Braced Steel Structure," Proc., 8th World Conf. Earthquake Engrg., Vol. IV, pp. 257264, IAEE, San Francisco, CA, 1985. 963 Zekioglu, A., Mozaffarian, H., Chang, K.L., Uang, C.M., and Noel, S., “Designing after Northridge”, Modern Steel Construct., Vol. 37. No. 3, pp. 3642, AISC, 1997.
Chapter 9
Chapter 10 Seismic Design of Reinforced Concrete Structures
Arnaldo T. Derecho, Ph.D. Consulting Strucutral Engineer, Mount Prospect, Illinois
M. Reza Kianoush, Ph.D. Professor, Ryerson Polytechnic University, Toronto, Ontario, Canada
Key words:
Seismic, Reinforced Concrete, Earthquake, Design, Flexure, Shear, Torsion, Wall, Frame, WallFrame, Building, HiRise, Demand, Capacity, Detailing, Code Provisions, IBC2000, UBC97, ACI318
Abstract:
This chapter covers various aspects of seismic design of reinforced concrete structures with an emphasis on design for regions of high seismicity. Because the requirement for greater ductility in earthquakeresistant buildings represents the principal departure from the conventional design for gravity and wind loading, the major part of the discussion in this chapter will be devoted to considerations associated with providing ductility in members and structures. The discussion in this chapter will be confined to monolithically cast reinforcedconcrete buildings. The concepts of seismic demand and capacity are introduced and elaborated on. Specific provisions for design of seismic resistant reinforced concrete members and systems are presented in detail. Appropriate seismic detailing considerations are discussed. Finally, a numerical example is presented where these principles are applied. Provisions of ACI318/95 and IBC2000 codes are identified and commented on throughout the chapter.
463
464
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10. Seismic Design of Reinforced Concrete Structures
10.1
INTRODUCTION
10.1.1
The Basic Problem
The problem of designing earthquakeresistant reinforced concrete buildings, like the design of structures (whether of concrete, steel, or other material) for other loading conditions, is basically one of defining the anticipated forces and/or deformations in a preliminary design and providing for these by proper proportioning and detailing of members and their connections. Designing a structure to resist the expected loading(s) is generally aimed at satisfying established or prescribed safety and serviceability criteria. This is the general approach to engineering design. The process thus consists of determining the expected demands and providing the necessary capacity to meet these demands for a specific structure. Adjustments to the preliminary design may likely be indicated on the basis of results of the analysisdesignevaluation sequence characterizing the iterative process that eventually converges to the final design. Successful experience with similar structures should increase the efficiency of the design process. In earthquakeresistant design, the problem is complicated somewhat by the greater uncertainty surrounding the estimation of the appropriate design loads as well as the capacities of structural elements and connections. However, information accumulated during the last three decades from analytical and experimental studies, as well as evaluations of structural behavior during recent earthquakes, has provided a strong basis for dealing with this particular problem in a more rational manner. As with other developing fields of knowledge, refinements in design approach can be expected as more information is accumulated on earthquakes and on the response of particular structural configurations to earthquaketype loadings. As in design for other loading conditions, attention in design is generally focused on those areas in a structure which analysis and
465
experience indicate are or will likely be subjected to the most severe demands. Special emphasis is placed on those regions whose failure can affect the integrity and stability of a significant portion of the structure. 10.1.2
Design for Inertial Effects
Earthquakeresistant design of buildings is intended primarily to provide for the inertial effects associated with the waves of distortion that characterize dynamic response to ground shaking. These effects account for most of the damage resulting from earthquakes. In a few cases, significant damage has resulted from conditions where inertial effects in the structure were negligible. Examples of these latter cases occurred in the excessive tilting of several multistory buildings in Niigata, Japan, during the earthquake of June 16, 1964, as a result of the liquefaction of the sand on which the buildings were founded, and the loss of a number of residences due to large landslides in the Turnagain Heights area in Anchorage, Alaska, during the March 28, 1964 earthquake. Both of the above effects, which result from ground motions due to the passage of seismic waves, are usually referred to as secondary effects. They are distinguished from socalled primary effects, which are due directly to the causative process, such as faulting (or volcanic action, in the case of earthquakes of volcanic origin). 10.1.3
Estimates of Demand
Estimates of force and deformation demands in critical regions of structures have been based on dynamic analyses—first, of simple systems, and second, on inelastic analyses of more complex structural configurations. The latter approach has allowed estimation of force and deformation demands in local regions of specific structural models. Dynamic inelastic analyses of models of representative structures have been used to generate information on the variation of demand with major structural as well as groundmotion parameters. Such an effort involves consideration of the practical
466
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range of values of the principal structural parameters as well as the expected range of variation of the groundmotion parameters. Structural parameters include the structure fundamental period, principal member yield levels, and force—displacement characteristics; input motions of reasonable duration and varying intensity and frequency characteristics normally have to be considered. A major source of uncertainty in the process of estimating demands is the characterization of the design earthquake in terms of intensity, frequency characteristics, and duration of largeamplitude pulses. Estimates of the intensity of ground shaking that can be expected at particular sites have generally been based on historical records. Variations in frequency characteristics and duration can be included in an analysis by considering an ensemble of representative input motions. Useful information on demands has also been obtained from tests on specimens subjected to simulated earthquake motions using shaking tables and, the pseudodynamic method of testing. The latter method is a combination of the socalled quasistatic, or slowly reversed, loading test and the dynamic shakingtable test. In this method, the specimen is subjected to essentially statically applied increments of deformation at discrete points, the magnitudes of which are calculated on the basis of predetermined earthquake input and the measured stiffness and estimated damping of the structure. Each increment of load after the initial increment is based on the measured stiffness of the structure during its response to the imposed loading of the preceding increment. 10.1.4
Estimates of Capacity
Proportioning and detailing of critical regions in earthquakeresistant structures have mainly been based on results of tests on laboratory specimens tested by the quasistatic method, i.e., under slowly reversed cycles of loading. Data from shakingtable tests and from pseudodynamic tests have also contributed to the general understanding of structural behavior
under earthquaketype loading. Design and detailing practice, as it has evolved over the last two or three decades, has also benefited from observations of the performance of structures subjected to actual destructive earthquakes. Earthquakeresistant design has tended to be viewed as a special field of study, not only because many engineers do not have to be concerned with it, but also because it involves additional requirements not normally dealt with in designing for wind. Thus, while it is generally sufficient to provide adequate stiffness and strength in designing buildings for wind, in the case of earthquakeresistant design, a third basic requirement, that of ductility or inelastic deformation capacity, must be considered. This third requirement arises because it is generally uneconomical to design most buildings to respond elastically to moderatetostrong earthquakes. To survive such earthquakes, codes require that structures possess adequate ductility to allow them to dissipate most of the energy from the ground motions through inelastic deformations. However, deformations in the seismic force resisting system must be controlled to protect elements of the structure that are not part of the lateral force resisting system. The fact is that many elements of the structure that are not intended as a part of the lateral force resisting system and are not detailed for ductility will participate in the lateral force resistant mechanism and can become severely damaged as a result. In the case of wind, structures are generally expected to respond to the design wind within their “elastic” range of stresses. When wind loading governs the design (drift or strength), the structure still should comply with the appropriate seismic detailing requirements. This is required in order to provide a ductile system to resist earthquake forces. Figure 101 attempts to depict the interrelationships between the various considerations involved in earthquakeresistant design.
10. Seismic Design of Reinforced Concrete Structures
Figure 10 1. Components of and considerations in earthquakeresistant building design
10.1.5
The Need for a Good Design Concept and Proper Detailing
Because of the appreciable forces and deformations that can be expected in critical regions of structures subjected to strong ground motions and a basic uncertainty concerning the intensity and character of the ground motions at a particular site, a good design concept is essential at the start. A good design concept implies a structure with a configuration that behaves well under earthquake excitation and designed in a manner that allows it to respond to strong ground motions according to a predetermined pattern or sequence of yielding. The need to start with a sound structural configuration that minimizes “incidental” and often substantial increases in member forces resulting from torsion due to asymmetry or force concentrations associated with discontinuities cannot be overemphasized. Although this idea may not be met with favor by some architects, clear (mainly economic) benefits can be derived from structural configurations emphasizing symmetry, regularity, and the avoidance of severe discontinuities in mass, geometry, stiffness, or strength. A direct path for the lateral (inertial) forces from the superstructure to an appropriately designed foundation is very desirable. On numerous occasions, failure to take account of the increase in forces and deformations in certain elements due to torsion or discontinuities has led to severe structural
467
distress and even collapse. The provision of relative strengths in the various types of elements making up a structure with the aim of controlling the sequence of yielding in such elements has been recognized as desirable from the standpoint of structural safety as well as minimizing postearthquake repair work. An important characteristic of a good design concept and one intimately tied to the idea of ductility is structural redundancy. Since yielding at critically stressed regions and subsequent redistribution of forces to less stressed regions is central to the ductile performance of a structure, good practice suggests providing as much redundancy as possible in a structure. In monolithically cast reinforced concrete structures, redundancy is normally achieved by continuity between momentresisting elements. In addition to continuity, redundancy or the provision of multiple load paths may also be accomplished by using several types of lateralloadresisting systems in a building so that a “backup system” can absorb some of the load from a primary lateralloadresisting system in the event of a partial loss of capacity in the latter. Just as important as a good design concept is the proper detailing of members and their connections to achieve the requisite strength and ductility. Such detailing should aim at preventing nonductile failures, such as those associated with shear and with bond anchorage. In addition, a deliberate effort should be made to securely tie all parts of a structure that are intended to act as a unit together. Because dynamic response to strong earthquakes, characterized by repeated and reversed cycles of largeamplitude deformations in critical elements, tends to concentrate deformation demands in highly stressed portions of yielding members, the importance of proper detailing of potential hinging regions should command as much attention as the development of a good design concept. As with most designs but more so in design for earthquake resistance, where the relatively large repeated deformations tend to “seek and expose,” in a manner of speaking, weaknesses in a structure—the proper field implementation of engineering drawings
468
Chapter 10
ultimately determines how well a structure performs under the design loading. Experience and observation have shown that properly designed, detailed, and constructed reinforcedconcrete buildings can provide the necessary strength, stiffness, and inelastic deformation capacity to perform satisfactorily under severe earthquake loading. 10.1.6
Accent on Design for Strong Earthquakes
The focus in the following discussion will be on the design of buildings for moderatetostrong earthquake motions. These cases correspond roughly to buildings located in seismic zones 2, 3 and 4 as defined in the Uniform Building Code (UBC97).(101) By emphasizing design for strong ground motions, it is hoped that the reader will gain an appreciation of the special considerations involved in this most important loading case. Adjustments for buildings located in regions of lesser seismic risk will generally involve relaxation of some of the requirements associated with highly seismic areas. Because the requirement for greater ductility in earthquakeresistant buildings represents the principal departure from the conventional design for gravity and wind loading, the major part of the discussion in this chapter will be devoted to considerations associated with providing ductility in members and structures. The discussion in this chapter will be confined to monolithically cast reinforcedconcrete buildings.
10.2
DUCTILITY IN EARTHQUAKERESISTANT DESIGN
10.2.1
Design Objective
In general, the design of economical earthquake resistant structures should aim at providing the appropriate dynamic and structural characteristics so that acceptable
levels of response result under the design earthquake. The magnitude of the maximum acceptable deformation will vary depending upon the type of structure and/or its function. In some structures, such as slender, freestanding towers or smokestacks or suspensiontype buildings consisting of a centrally located corewall from which floor slabs are suspended by means of peripheral hangers, the stability of the structure is dependent on the stiffness and integrity of the single major element making up the structure. For such cases, significant yielding in the principal element cannot be tolerated and the design has to be based on an essentially elastic response. For most buildings, however, and particularly those consisting of rigidly connected frame members and other multiply redundant structures, economy is achieved by allowing yielding to take place in some critically stressed elements under moderatetostrong earthquakes. This means designing a building for force levels significantly lower than would be required to ensure a linearly elastic response. Analysis and experience have shown that structures having adequate structural redundancy can be designed safely to withstand strong ground motions even if yielding is allowed to take place in some elements. As a consequence of allowing inelastic deformations to take place under strong earthquakes in structures designed to such reduced force levels, an additional requirement has resulted and this is the need to insure that yielding elements be capable of sustaining adequate inelastic deformations without significant loss of strength, i.e., they must possess sufficient ductility. Thus, where the strength (or yield level) of a structure is less than that which would insure a linearly elastic response, sufficient ductility has to be built in. 10.2.2
Ductility vs. Yield Level
As a general observation, it can be stated that for a given earthquake intensity and structure period, the ductility demand increases as the strength or yield level of a structure decreases. To illustrate this point, consider two
10. Seismic Design of Reinforced Concrete Structures vertical cantilever walls having the same initial fundamental period. For the same mass and mass distribution, this would imply the same stiffness properties. This is shown in Figure 102, where idealized forcedeformation curves for the two structures are marked (1) and (2). Analyses(102, 103) have shown that the maximum lateral displacements of structures with the same initial fundamental period and reasonable properties are approximately the same when subjected to the same input motion. This phenomenon is largely attributable to the reduction in local accelerations, and hence displacements, associated with reductions in stiffness due to yielding in critically stressed portions of a structure. Since in a vertical cantilever the rotation at the base determines to a large extent the displacements of points above the base, the same observation concerning approximate equality of maximum lateral displacements can be made with respect to maximum rotations in the hinging region at the bases of the walls. This can be seen in Figure 103, from Reference 103, which shows results of dynamic analysis of isolated structural walls having the same fundamental period (T1 = 1.4 sec) but different yield levels My. The structures were subjected to the first 10 sec of the east— west component of the 1940 El Centro record with intensity normalized to 1.5 times that of the north—south component of the same
469
record. It is seen in Figure 103a that, except for the structure with a very low yield level (My = 500,000 in.kips), the maximum displacements for the different structures are about the same. The corresponding ductility demands, expressed as the ratio of the maximum hinge rotations, θmax to the corresponding rotations at first yield, θy, are shown in Figure 103b. The increase in ductility demand with decreasing yield level is apparent in the figure.
Figure 102. Decrease in ductility ratio demand with increase in yield level or strength of a structure.
Figure 103. Effect of yield level on ductility demand. Note approximately equal maximum displacements for structures with reasonable yield levels. (From Ref. 103.)
470 A plot showing the variation of rotational ductility demand at the base of an isolated structural wall with both the flexural yield level and the initial fundamental period is shown in Figure 104.(104) The results shown in Figure 104 were obtained from dynamic inelastic analysis of models representing 20story isolated structural walls subjected to six input motions of 10sec duration having different frequency characteristics and an intensity normalized to 1.5 times that of the north—south component of the 1940 El Centro record. Again, note the increase in ductility demand with decreasing yield level; also the decrease in ductility demand with increasing fundamental period of the structure.
Chapter 10 The abovenoted relationship between strength or yield level and ductility is the basis for code provisions requiring greater strength (by specifying higher design lateral forces) for materials or systems that are deemed to have less available ductility. 10.2.3
Some Remarks about Ductility
One should note the distinction between inelastic deformation demand expressed as a ductility ratio, µ (as it usually is) on one hand, and in terms of absolute rotation on the other. An observation made with respect to one quantity may not apply to the other. As an example, Figure 105, from Reference 103,
Figure 104. Rotational ductility demand as a function of initial fundamental period and yield level of 20story structural walls. (From Ref. 104.)
10. Seismic Design of Reinforced Concrete Structures shows results of dynamic analysis of two isolated structural walls having the same yield level (My = 500,000 in.kips) but different stiffnesses, as reflected in the lower initial fundamental period T1 of the stiffer structure. Both structures were subjected to the E—W component of the 1940 El Centro record. Even though the maximum rotation for the flexible structure (with T1 = 2.0 sec) is 3.3 times that of the stiff structure, the ductility ratio for the stiff structure is 1.5 times that of the flexible structure. The latter result is, of course, partly due to the lower yield rotation of the stiffer structure.
rotation per unit length. This is discussed in detail later in this Chapter. Another important distinction worth noting with respect to ductility is the difference between displacement ductility and rotational ductility. The term displacement ductility refers to the ratio of the maximum horizontal (or transverse) displacement of a structure to the corresponding displacement at first yield. In a rigid frame or even a single cantilever structure responding inelastically to earthquake excitation, the lateral displacement of the structure is achieved by flexural yielding at local critically stressed regions. Because of this, it is reasonable to expect—and results of analyses bear this out(102, 103, 105)—that rotational ductilities at these critical regions are generally higher than the associated displacement ductility. Thus, overall displacement ductility ratios of 3 to 6 may imply local rotational ductility demands of 6 to 12 or more in the critically stressed regions of a structure. 10.2.4
The term “curvature ductility” is also a commonly used term which is defined as
Results of a Recent Study on Cantilever Walls
In a recent study by Priestley and Kowalsky on isolated cantilever walls, it has been shown that the yield curvature is not directly proportional to the yield moment; this is in contrast to that shown in Figure 102 which in their opinions leads to significant errors. In fact, they have shown that yield curvature is a function of the wall length alone, for a given steel yield stress as indicated in Figure 106. The strength and stiffness of the wall vary proportionally as the strength of the section is changed by varying the amount of flexural reinforcement and/or the level of axial load. This implies that the yield curvature, not the section stiffness, should be considered the fundamental section property. Since wall yield curvature is inversely proportional to wall length, structures containing walls of different length cannot be designed such that they yield simultaneously. In addition, it is stated that wall design should be proportioned to the square of (106)
Figure 105. Rotational ductility ratio versus maximum absolute rotation as measures of inelastic deformation.
471
472
Chapter 10
wall length, L2, rather than the current design assumption, which is based on L3 . It should be noted that the above findings apply to cantilever walls only. Further research in this area in various aspects is currently underway at several institutions.
M1
M
In certain members, such as conventionally reinforced short walls—with heighttowidth ratios of 2 to 3 or less—the very nature of the principal resisting mechanism would make a sheartype failure difficult to avoid. Diagonal reinforcement, in conjunction with horizontal and vertical reinforcement, has been shown to improve the performance of such members (107). 10.3.2
M2 M3
y
Figure 106. Influence of strength on momentcurvature relationship (From Ref. 106).
10.3
BEHAVIOR OF CONCRETE MEMBERS UNDER EARTHQUAKETYPE LOADING
10.3.1
General Objectives of Member Design
A general objective in the design of reinforced concrete members is to so proportion such elements that they not only possess adequate stiffness and strength but so that the strength is, to the extent possible, governed by flexure rather than by shear or bond/anchorage. Code design requirements are framed with the intent of allowing members to develop their flexural or axial load capacity before shear or bond/anchorage failure occurs. This desirable feature in conventional reinforced concrete design becomes imperative in design for earthquake motions where significant ductility is required.
Types of Loading Used in Experiments
The bulk of information on behavior of reinforcedconcrete members under load has ‘generally been obtained from tests of fullsize or nearfullsize specimens. The loadings used in these tests fall under four broad categories, namely: 1. Static monotonic loading—where load in one direction only is applied in increments until failure or excessive deformation occurs. Data which form the basis for the design of reinforced concrete members under gravity and wind loading have been obtained mainly from this type of test. Results of this test can serve as bases for comparison with results obtained from other types of test that are more representative of earthquake loading. 2. Slowly reversed cyclic (“quasistatic”) loading—where the specimen is subjected to (force or deformation) loading cycles of predetermined amplitude. In most cases, the load amplitude is progressively increased until failure occurs. This is shown schematically in Figure 107a. As mentioned earlier, much of the data upon which current design procedures for earthquake resistance are based have been obtained from tests of this type. In a few cases, a loading program patterned after analytically determined dynamic response(108) has been used. The latter, which is depicted in Figure 107b, is usually characterized by largeamplitude load cycles early in the test, which can produce early deterioration of the strength of a specimen.(109) In both of the above cases, the load application points are fixed so that the moments and shears are always in phase—a condition, incidentally, that does not always occur in dynamic response.
10. Seismic Design of Reinforced Concrete Structures
473
Figure 107. Two types of loading program used in quasistatic tests.
This type of test provides the reversing character of the loading that distinguishes dynamic response from response to unidirectional static loading. In addition, the relatively slow application of the load allows close observation of the specimen as the test progresses. However, questions concerning the effects of the sequence of loading as well as the phase relationship between moment and shear associated with this type of test as it is normally conducted need to be explored further. 3. Pseudodynamic tests. In this type of test, the specimen base is fixed to the test floor while timevarying displacements determined by an online computer are applied to selected points on the structure. By coupling loading rams with a computer that carries out an incremental dynamic analysis of the specimen response to a preselected input motion, using measured stiffness data from the preceding loading increment and prescribed data on specimen mass and damping, a more realistic distribution of horizontal displacements in the test structure is achieved. The relatively slow rate at which the loading is imposed allows convenient inspection of the condition of the structure during the progress of the test. This type of test, which has been used mainly for testing structures, rather than members or structural elements, requires a fairly large reaction block to take the thrust from the many loading rams normally used.
4. Dynamic tests using shaking tables (earthquake simulators). The most realistic test conditions are achieved in this setup, where a specimen is subjected to a properly scaled input motion while fastened to a test bed impelled by computercontrolled actuators. Most current earthquake simulators are capable of imparting controlled motions in one horizontal direction and in the vertical direction. The relatively rapid rate at which the loading is imposed in a typical dynamic test generally does not allow close inspection of the specimen while the test is in progress, although photographic records can be viewed after the test. Most currently available earthquake simulators are limited in their capacity to smallscale models of multistory structures or nearfullscale models of segments of a structure of two or three stories. The difficulty of viewing the progress of damage in a specimen as the loading is applied and the limited capacity of available (and costly) earthquake simulators has tended to favor the recently developed pseudodynamic test as a basic research tool for testing structural systems. The effect of progressively increasing lateral displacements on actual structures has been studied in a few isolated cases by means of forcedvibration testing. These tests have usually been carried out on buildings or portions of buildings intended for demolition.
474 10.3.3
Chapter 10 Effects of Different Variables on the Ductility of Reinforced Concrete Members
Figure 108 shows typical stress—strain curves of concrete having different compressive strengths. The steeper downward slope beyond the point of maximum stress of curves corresponding to the higher strength concrete is worth noting. The greater ductility of the lowerstrength concrete is apparent in the figure. Typical stressstrain curves for the commonly available grades of reinforcing steel, with nominal yield strengths of 60 ksi and 40 ksi, are shown in Figure 109. Note in the figure that the ultimate stress is significantly higher than the yield stress. Since strains well into the strainhardening range can occur in hinging regions of flexural members, stresses in excess of the nominal yield stress (normally used in conventional design as the limiting stress in steel) can develop in the reinforcement at these locations.
Figure 108. Typical stressstrain curves for concrete of varying compressive strengths.
Rate of Loading An increase in the strain rate of loading is generally accompanied by an increase in the strength of concrete or the yield stress of steel. The greater rate of loading associated with earthquake response, as compared with static loading, results in a slight increase in the strength of reinforced concrete members, due primarily to the increase in the
yield strength of the reinforcement. The calculation of the strength of reinforced concrete members in earthquakeresistant structures on the basis of material properties obtained by static tests (i.e., normal strain rates of loading) is thus reasonable and conservative.
Figure 109. Typical stressstrain curves for ordinary reinforcing steel.
Confinement Reinforcement The American Concrete Institute Building Code Requirements for Reinforced Concrete, ACI 31895(1010) (hereafter referred to as the ACI Code), specifies a maximum usable compressive strain in concrete, εcu of 0.003. Lateral confinement, whether from active forces such as transverse compressive loads, or passive restraints from other framing members or lateral reinforcement, tends to increase the value of εcu. Tests have shown that εcu, can range from 0.0025 for unconfined concrete to about 0.01 for concrete confined by lateral reinforcement subjected to predominantly axial (concentric) load. Under eccentric loading, values of εcu for confined concrete of 0.05 and more have been observed.(1011, 1012,1013) Effective lateral confinement of concrete increases its compressive strength and deformation capacity in the longitudinal direction, whether such longitudinal stress represents a purely axial load or the compressive component of a bending couple.
10. Seismic Design of Reinforced Concrete Structures In reinforced concrete members, the confinement commonly takes the form of lateral ties or spiral reinforcement covered by a thin shell of concrete. The passive confining effect of the lateral reinforcement is not mobilized until the concrete undergoes sufficient lateral expansion under the action of compressive forces in the longitudinal direction. At this stage, the outer shell of concrete usually has reached its useful load limit and starts to spall. Because of this, the net increase in strength of the section due to the confined core may not amount to much in view of the loss in capacity of the spalled concrete cover. In many cases, the total strength of the confined core may be slightly less than that of the original section. The increase in ductility due to effective confining reinforcement, however, is significant. The confining action of rectangular hoops mainly involves reactive forces at the corners, with only minor restraint provided along the straight unsupported sides. Because of this, rectangular hoops are generally not as effective as circular spiral reinforcement in confining the concrete core of members subjected to compressive loads. However, confinement in rectangular sections can be improved using additional transverse ties. Square spirals, because of their continuity, are slightly better
475
than separate rectangular hoops. The stress—strain characteristics of concrete, as represented by the maximum usable compressive strain εcu is important in designing for ductility of reinforced concrete members. However, other factors also influence the ductility of a section: factors which may increase or diminish the effect of confinement on the ductility of concrete. Note the distinction between the ductility of concrete as affected by confinement and the ductility of a reinforced concrete section (i.e., sectional ductility) as influenced by the ductility of the concrete as well as other factors. Sectional Ductility A convenient measure of the ductility of a section subjected to flexure or combined flexure and axial load is the ratio µ of the ultimate curvature attainable without significant loss of strength, φu , to the curvature corresponding to first yield of the tension reinforcement, φy. Thus Sectional ductility, µ =
φu φy
Figure 1010, which shows the strains and resultant forces on a typical reinforced concrete section under flexure, corresponds to the condition when the maximum usable compressive strain in concrete, εcu is reached. The corresponding curvature is denoted as the
Figure 1010. Strains and stresses in a typical reinforced concrete section under flexure at ultimate condition.
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Chapter 10
ultimate curvature, φu.. It will be seen in the figure that
φu =
ε cu ku d
where kud is the distance from the extreme compression fiber to the neutral axis. The variables affecting sectional ductility may be classified under three groups, namely: (i) material variables, such as the maximum usable compressive strain in concrete, particularly as this is affected by confinement, and grade of reinforcement; (ii) geometric variables, such as the amount of tension and compression reinforcement, and the shape of the section; (iii) loading variables, such as the level of the axial load and accompanying shear. As is apparent from the above expression for ultimate curvature, factors that tend to increase εcu or decrease kud tend to increase sectional ductility. As mentioned earlier, a major factor affecting the value of εcu is lateral confinement. Tests have also indicated that εcu increases as the distance to the neutral axis decreases, that is, as the strain gradient across the section increases(1014, 1015) and as the moment gradient along the span of the member increases or as the shear span decreases.(1016, 1017) (For a given maximum moment, the moment gradient increases as the distance from the point of zero moment to the section considered decreases.) The presence of compressive reinforcement and the use of concrete with a high compressive strength,a as well as the use of flanged sections, tend to reduce the required depth of the compressive block, kud, and hence to increase the ultimate curvature φu. In addition, the compressive reinforcement also helps confine the concrete compression zone and, in combination with adequate transverse reinforcement, allows the spread of the inelastic action in a hinging region over a longer length than would otherwise occur, thus improving the a
The lower ductility of the higherstrength (f′c >5000 psi ), however, has been shown to result in a decrease in sectional ductility, particularly for sections with low reinforcement indexes. (1018)
ductility of the member.(1019) On the other hand, compressive axial loads and large amounts of tensile reinforcement, especially tensile reinforcement with a high yield stress, tend to increase the required kud and thus decrease the ultimate curvature φu. Figure 1011 shows axialload—momentstrength interaction curves for a reinforcedconcrete section subjected to a compressive axial load and bending about the horizontal axis. Both confined and unconfined conditions are assumed. The interaction curve provides a convenient way of displaying the combinations of bending moment M and axial load P which a given section can carry. A point on the interaction curve is obtained by calculating the forces M and P associated with an assumed linear strain distribution across the section, account being taken of the appropriate stress— strain relationships for concrete and steel. For an ultimate load curve, the concrete strain at the extreme compressive fiber, εc is assumed to be at the maximum usable strain, εcu while the strain in the tensile reinforcement, εs, varies. A loading combination represented by a point on or inside the interaction curve can be safely resisted by the section. The balance point in the interaction curve corresponds to the condition in which the tensile reinforcement is stressed to its yield point at the same time that the extreme concrete fiber reaches its useful limit of compressive strain. Points on the interaction curve above the balance point represent conditions in which the strain in the tensile reinforcement is less than its yield strain εy, so that the strength of the section in this range is governed by failure of the concrete compressive zone. For those points on the curve below the balance point, εs > εy. Hence, the strength of the section in this range is governed by rupture of the tensile reinforcement. Figure 1011 also shows the variation of the ultimate curvature φu (in units of 1/h) with the axial load P. It is important to note the greater ultimate curvature (being a measure of sectional ductility) associated with values of P less than that corresponding to the balance condition, for both unconfined and confined cases. The significant increase in ultimate curvature
10. Seismic Design of Reinforced Concrete Structures
477
Figure 1011. Axial loadmoment interaction and loadcurvature curves for a typical reinforced concrete section with unconfined and confined cores.
resulting from confinement is also worth noting in Figure 1011b. In the preceding, the flexural deformation capacity of the hinging region in members was examined in terms of the curvature at a section, φ, and hence the sectional or curvature ductility. Using this simple model, it was possible to arrive at important conclusions concerning the effects of various parameters on the ductility of reinforced concrete members. In the hinging region of members, however, the curvature can vary widely in value over the length of the “plastic hinge.” Because of this, the total rotation over the plastic hinge, θ, provides a more meaningful measure of the inelastic flexural deformation in the hinging regions of members and one that can be related directly to experimental measurements. (One can, of course, speak of average curvature over the hinging region, i.e., total rotation divided by length of the plastic hinge.)
Shear The level of shear present can have a major effect on the ductility of flexural hinging regions. To study the effect of this variable, controlled tests of laboratory specimens have been conducted. This will be discussed further in the following section. 10.3.4
Some Results of Experimental and Analytical Studies on the Behavior of Reinforced Concrete Members under EarthquakeType Loading and Related Code Provisions
Experimental studies of the behavior of structural elements under earthquaketype loading have been concerned mainly with identifying and/or quantifying the effects of variables that influence the ability of critically stressed regions in such specimens to perform properly. Proper performance means primarily possessing adequate ductility. In terms of the
478 quasistatic test that has been the most widely used for this purpose, proper performance would logically require that these critical regions be capable of sustaining a minimum number of deformation cycles of specified amplitude without significant loss of strength. In the United States, there is at present no standard set of performance requirements corresponding to designated areas of seismic risk that can be used in connection with the quasistatic test. Such requirements would have to specify not only the minimum amplitude (i.e., ductility ratio) and number of deformation cycles, but also the sequence of application of the largeamplitude cycles in relation to any smallamplitude cycles and the permissible reduction in strength at the end of the loading. As mentioned earlier, the bulk of experimental information on the behavior of elements under earthquaketype loading has been obtained by quasistatic tests using loading cycles of progressively increasing amplitude, such as is shown schematically in Figure 107a. Adequacy with respect to ductility for regions of high seismicity has usually been inferred when displacement ductility ratios of anywhere from 4 to 6 or greater were achieved without appreciable loss of strength. In New Zealand,(1020) moment resisting frames are designed for a maximum ductility, µ, of 6 and shear walls are designed for a maximum ductility of between 2.5 to 5. Adequate ductile capacity is considered to be present if all primary that are required to resist earthquakeinduced forces are accordingly designed and detailed. In the following, some results of tests and analyses of typical reinforcedconcrete members will be briefly reviewed. Where appropriate, related code provisions, mainly those in Chapter 21 of the ACI Code(1010) are also discussed. Beams Under earthquake loading, beams will generally be most critically stressed at and near their intersections with the supporting columns. An exception may be where a heavy concentrated load is carried at some intermediate point on the span. As a result, the focus of attention in the design of beams is on
Chapter 10 these critical regions where plastic hinging can take place. At potential hinging regions, the need to develop and maintain the strength and ductility of the member through a number of cycles of reversed inelastic deformation calls for special attention in design. This special attention relates mainly to the lateral reinforcement, which takes the form of closed hoops or spirals. As might be expected, the requirements governing the design of lateral reinforcement for potential hinging regions are more stringent than those for members designed for gravity and wind loads, or the less critically stressed parts of members in earthquakeresistant structures. The lateral reinforcement in hinging regions of beams is designed to provide (i) confinement of the concrete core, (ii) support for the longitudinal compressive reinforcement against inelastic buckling, and (iii) resistance, in conjunction with the confined concrete, against transverse shear. In addition to confirming the results of sectional analyses regarding the influence of such variables as concrete strength, confinement of concrete, and amounts and yield strengths of tensile and compressive reinforcement and compression flanges mentioned earlier, tests, both monotonic and reversed cyclic, have shown that the flexural ductility of hinging regions in beams is significantly affected by the level of shear present. A review of test results by Bertero(1021) indicates that when the nominal shear stress exceeds about 3 f c′
, members designed
according to the present seismic codes can expect to suffer some reduction in ductility as well as stiffness when subjected to loading associated with strong earthquake response. When the shear accompanying flexural hinging is of the order of 5 f c′ or higher, very significant strength and stiffness degradation has been observed to occur under cyclic reversed loading. The behavior of a segment at the support region of a typical reinforcedconcrete beam subjected to reversed cycles of inelastic deformation in the presence of high shear(1022,
10. Seismic Design of Reinforced Concrete Structures 1023)
is shown schematically in Figure 1012. In Figure 1012a, yielding of the top longitudinal steel under a downward movement of the beam end causes flexure—shear cracks to form at the top. A reversal of the load and subsequent yielding of the bottom longitudinal steel is also accompanied by cracking at the bottom of the beam (see Figure 10l2b). If the area of the bottom steel is at least equal to that of the top steel, the top cracks remain open during the early stages of the load reversal until the top steel yields in compression, allowing the top crack to close and the concrete to carry some compression. Otherwise, as in the more typical case where the top steel has greater area than the bottom steel, the top steel does not yield in compression (and we assume it does not buckle), so that the top crack remains open during the reversal of the load (directed upward). Even in the former case, complete closure of the crack at the top may be prevented by loose particles of concrete that may fall into the open cracks. With a crack traversing the entire depth of the beam, the resisting flexural couple consists of the forces in the tensile and compressive steel areas, while the shear along the throughdepth crack is resisted primarily by dowel action of the longitudinal steel. With subsequent reversals of the load and progressive deterioration of the concrete in the hinging region (Figure 1012c), the throughdepth crack widens. The resulting increase in total length of the member due to the opening of throughdepth cracks under repeated load reversals is sometimes referred to as growth of the member. Where the shear accompanying the moment is high, sliding along the throughdepth crack(s) can occur. This sliding shear displacement, which is resisted mainly by dowel action of the longitudinal reinforcement, is reflected in a pinching of the associated load—deflection curve near the origin, as indicated in Figure 1013. Since the area under the load—deflection curve is a measure of the energydissipation capacity of the member, the pinching in this curve due to sliding shear represents a degradation not only of the strength but also the energydissipation capacity of the hinging
479
region. Where the longitudinal steel is not adequately restrained by lateral reinforcement, inelastic buckling of the compressive reinforcement followed by a rapid loss of flexural strength can occur.
Figure 1012. Plastic hinging in beam under high shear. (Adapted from Ref. 1031.)
Figure 1013. Pinching in loaddisplacement hysteresis loop due to mainly to sliding shear
Because of the significant effect that shear can have on the ductility of hinging regions, it has been suggested(1024) that when two or more load reversals at a displacement ductility of 4 or more are expected, the nominal shear stress in critical regions reinforced according to normal
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U.S. code requirements for earthquakeresistant design should be limited to 6
f c′ . Results of
tests reported in Reference 1024 have shown that the use of crossing diagonal or inclined web reinforcement, in combination with vertical ties, as shown in Figure 1014, can effectively minimize the degradation of stiffness associated with sliding shear. Relatively stable hysteretic force— displacement loops, with minimal or no pinching, were observed. Tests reported in Reference 1025 also indicate the effectiveness of intermediate longitudinal shear reinforcement, shown in Figure 1015, in reducing pinching of the force—displacement loops of specimens subjected to moderate levels of shear stresses, i.e., between 3 6
to be equal to 1.25fy and using a strength reduction factor φ equal to 1.0 (instead of 0.9). This is illustrated in Figure 1016 for the case of uniformly distributed beam. The use of the factor 1.25 to be applied to fy is intended to take account of the likelihood of the actual yield stress in the steel being greater (tests indicate it to be commonly 10 to 25% greater) than the specified nominal yield stress, and also in recognition of the strong possibility of strain hardening developing in the reinforcement when plastic hinging occurs at the beam ends.
f c′ and
f c′ .
Figure 1015. Intermediate longitudinal web reinforcement for hinging regions under moderate levels of shear.
Figure 1014. Crossing diagonal web reinforcement in combination with vertical web steel for hinging regions under high shear. (Adapted from Ref. 1024)
As mentioned earlier, a major objective in the design of reinforced concrete members is to have the strength controlled by flexure rather than shear or other less ductile failure mechanisms. To insure that beams develop their full strength in flexure before failing in shear, ACI Chapter 21 requires that the design for shear in beams be based not on the factored shears obtained from a lateralload analysis but rather on the shears corresponding to the maximum probable flexural strength, Mpr, that can be developed at the beam ends. Such a probable flexural strength is calculated by assuming the stress in the tensile reinforcement
VcA =
M prA
l + M prB
+
Wu l 2
Wl − u l 2 based on f s = 1.25 f y and φ = 1.0 VcB =
M pr
M prA + M prB
Figure 1016. Loading cases for shear design of beams uniformly distributed gravity loads
10. Seismic Design of Reinforced Concrete Structures ACI Chapter 21 requires that when the earthquakeinduced shear force calculated on the basis of the maximum probable flexural strength at the beam ends is equal to or more than onehalf the total design shear, the contribution of the concrete in resisting shear, Vc, be neglected if the factored axial compressive force including earthquake effects is less than Ag f c′ /20, where Ag is the gross area of the member crosssection. In the 1995 New Zealand Code,(1026) the concrete contribution is to be entirely neglected and web reinforcement provided to carry the total shear force in plastichinging regions. It should be pointed out that the New Zealand seismic design code appears to be generally more conservative than comparable U.S. codes. This will be discussed further in subsequent sections. Columns The current approach to the design of earthquakeresistant reinforced concrete rigid (i.e., momentresisting) frames is to have most of the significant inelastic action or plastic hinging occur in the beams rather than in the columns. This is referred to as the “strong columnweak beam” concept and is intended to help insure the stability of the frame while undergoing large lateral displacements under earthquake excitation. Plastic hinging at both ends of most of the columns in a story can precipitate a storysidesway mechanism leading to collapse of the structure at and above the story. ACI Chapter 21 requires that the sum of the flexural strengths of the columns meeting at a joint, under the most unfavorable axial load, be at least equal to 1.2 times the sum of the design flexural strengths of the girders in the same plane framing into the joint. The most unfavorable axial load is the factored axial force resulting in the lowest corresponding flexural strength in the column and which is consistent with the direction of the lateral forces considered. Where this requirement is satisfied, closely spaced transverse reinforcement need be provided only over a short distance near the ends of the columns where potential hinging can occur. Otherwise, closely spaced transverse reinforcement is required over the full height of the columns.
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The requirements associated with the strong columnweak beam concept, however, do not insure that plastic hinging will not occur in the columns. As pointed out in Reference 105, a bendingmoment distribution among frame members such as is shown in Figure 1017, characterized by points of inflection located away from the midheight of columns, is not uncommon. This condition, which has been observed even under static lateral loading, occurs when the flexural mode of deformation (as contrasted with the shear—beam component of deformation) in tall frame structures becomes significant and may also arise as a result of highermode response under dynamic loading. As Figure 1017 shows, a major portion of the girder moments at a joint is resisted (assuming the columns remain elastic) by one column segment, rather than being shared about equally (as when the points of inflection are located at midheight of the columns) by the column sections above and below a joint. In extreme cases, such as might result from substantial differences in the stiffnesses of adjoining column segments in a column stack, the point of contraflexure can be outside the column height. In such cases, the moment resisted by a column segment may exceed the sum of the girder moments. In recognition of this, and the likelihood of the hinging region spreading over a longer length than would normally occur, most building codes require confinement reinforcement to be provided over the full height of the column. Tests on beamcolumn specimens incorporating slabs,(1027, 1028) as in normal monolithic construction, have shown that slabs significantly increase the effective flexural strength of the beams and hence reduce the columntobeam flexural strength ratio, if the beam strength is based on the bare beam section. Reference 1027 recommends consideration of the slab reinforcement over a width equal to at least the width of the beam on each side of the member when calculating the flexural strength of the beam.
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Figure 1017. Distribution of bending moments in columns at a joint when the point of inflection is located away from midheight.
Another phenomenon that may lead to plastic hinging in the columns occurs in twoway (threedimensional rigid) frames subjected to ground motions along a direction inclined with respect to the principal axes of the structure. In such cases, the resultant moment from girders lying in perpendicular planes framing into a column will generally be greater than that corresponding to either girder considered separately.(105) ( except for certain categories of structures and those with certain irregularities, codes allow consideration of design earthquake loads along each principal axes of a structure separately, as nonconcurrent loadings.) Furthermore, the biaxial moment capacity of a reinforcedconcrete column under skew bending will generally be less than the larger uniaxial moment capacity. Tests reported in Reference 1028 indicate that where bidirectional loading occurs in rectangular columns, the decrease in strength of the column due to spalling of concrete cover, and bond deterioration along the column longitudinal bars at and near the corner can be large enough to shift the hinging from the beams to the columns. Thus, under concurrent bidirectional loading, columns in twoway frames designed according to the strong columnweak beam
Chapter 10 concept mentioned above can either yield before the framing girders or start yielding immediately following yielding of the girders. It is worth noting that the 1985 report of ACIASCE Committee 352 on beamcolumn joints in monolithic reinforced concrete structures(1029) recommends a minimum overstrength factor of 1.4, instead of the 1.2 given in ACI 31895, for the flexural strength of columns relative to that of beams meeting at a joint when the beam strength is based only on the bare beam section (excluding slab). A design procedure (capacity design), based on the work of Paulay,(1013,1030) that attempts to minimize the possibility of yielding in the columns of a typical frame due to the factors described in the preceding paragraph has been adopted in New Zealand.(1026) The avowed purpose of capacity design is to limit inelastic action, as well as the formation of plastic hinges, to selected elements of the primary lateralforceresisting system. In the case of frames, the ideal location for plastic hinges would be the beams and the bases of the first or lowest story columns. Other elements, such as columns, are intended to remain essentially elastic under the design earthquake by designing them with sufficient overstrength relative to the yielding members. Thus elements intended to remain elastic are designed to have strengths in the plastic hinges. For all elements, and particularly regions designed to develop plastic hinges, undesirable modes of failure, such as shear or bond/anchorage failures, are precluded by proper design/detailing. The general philosophy of capacity design is no different from that underlying the current approach to earthquakeresistant design found in ACI Chapter 21, UBC97 and IBC2000. The principle difference lies in the details of implementation and particularly in the recommended overstrength factors. For example, the procedure prescribes overstrength factors of 1.5 or greater(1013,1032) for determining the flexural strength of columns relative to beams. This compares with the 1.2 factor specified in ACI Chapter 21. In capacity design, the flexural strength of T or invertedL beams is to be determined by considering the
10. Seismic Design of Reinforced Concrete Structures slab reinforcement over the specified width (depending upon column location) beyond the column faces as effective in resisting negative moments. It is clear from the above that the New Zealand capacity design requirements call for greater relative column strength than is currently required in U.S. practice. A similar approach has also been adopted in the Canadian Concrete Code of Practice, CSA Standard A23.394.(1033) Reference 1013 gives detailed recommendations, including worked out examples, relating to the application of capacity design to both frames and structural wall systems. To safeguard against strength degradation due to hinging in the columns of a frame, codes generally require lateral reinforcement for both confinement and shear in regions of potential plastic hinging. As in potential hinging regions of beams, the closely spaced transverse reinforcement in critically stressed regions of columns is intended to provide confinement for the concrete core, lateral support of the longitudinal column reinforcement against buckling and resistance (in conjunction with the confined core) against transverse shear. The transverse reinforcement can take the form of spirals, circular hoops, or rectangular hoops, the last with crossties as needed. Early tests(1034) of reinforced concrete columns subjected to large shear reversals had indicated the need to provide adequate transverse reinforcement not only to confine the concrete but also to carry most, if not all, of the shear in the hinging regions of columns. The beneficial effect of axial load—a maximum axial load of onehalf the balance load was used in the tests—in delaying the degradation of shear strength in the hinging region was also noted in these tests. An increase in column strength due to improved confinement by longitudinal reinforcement uniformly distributed along the periphery of the column section was noted in tests reported in Reference 1035. Tests cited in Reference 1032 have indicated that under high axial load, the plastic hinging region in columns with confinement reinforcement provided over the usually assumed hinging length (i.e., the longer section
483
dimension in rectangular columns or the diameter in circular columns) tends to spread beyond the confined region. To prevent flexural failure in the less heavily confined regions of columns, the New Zealand Code(1020) requires that confining steel be extended to 2 to 3 times the usual assumed plastichinge length when the axial load exceeds 0.25φ f c′ Ag, where φ = 0.85 and Ag is the gross area of the column section. The basic intent of the ACI Code provisions relating to confinement reinforcement in potential hinging regions of columns is to preserve the axialloadcarrying capacity of the column after spalling of the cover concrete has occurred. This is similar to the intent underlying the column design provisions for gravity and wind loading. The amount of confinement reinforcement required by these provisions is independent of the level of axial load. Design for shear is to be based on the largest nominal moment strengths at the column ends consistent with the factored design axial compressive load. Some investigators,(105) however, have suggested that an approach that recognizes the potential for hinging in critically stressed regions of columns should aim primarily at achieving a minimum ductility in these regions. Studies by Park and associates, based on sectional analyses(1032) as well as tests,(1036, 1037) indicate that although the ACI Code provisions based on maintaining the loadcarrying capacity of a column after spalling of the cover concrete has occurred are conservative for low axial loads, they can be unconservative for high axial loads, with particular regard to attaining adequate ductility. Results of these studies indicate the desirability of varying the confinement requirements for the hinging regions in columns according to the magnitude of the axial load, more confinement being called for in the case of high axial loads. ACI Chapter 21 limits the spacing of confinement reinforcement to 1/4 the minimum member dimension or 4 in., with no limitation related to the longitudinal bar diameter. The New Zealand Code requires that the maximum spacing of transverse reinforcement in the potential plastic hinge regions not exceed the
484 least of 1/4 the minimum column dimension or 6 times the diameter of the longitudinal reinforcement. The second limitation is intended to relate the maximum allowable spacing to the need to prevent premature buckling of the longitudinal reinforcement. In terms of shear reinforcement, ACI Chapter 21 requires that the design shear force be based on the maximum flexural strength, Mpr , at each end of the column associated with the range of factored axial loads. However, at each column end, the moments to be used in calculating the design shear will be limited by the probable moment strengths of the beams (the negative moment strength on one side and the positive moment strength on the other side of a joint) framing into the column. The larger amount of transverse reinforcement required for either confinement or shear is to be used. One should note the significant economy, particularly with respect to volume of lateral reinforcement, to be derived from the use of spirally reinforced columns.(1032) The saving in the required amount of lateral reinforcement, relative to a tied column of the same nominal capacity, which has also been observed in designs for gravity and wind loading, acquires greater importance in earthquakeresistant design in view of the superior ductile performance of the spirally reinforced column. Figure 1018b, from Reference 1038, shows one of the spirally reinforced columns in the first story of the Olive View Hospital building in California following the February 9, 1971 San Fernando earthquake. A tied corner column in the first story of the same building is shown in Figure 1018c. The upper floors in the fourstory building, which were stiffened by shear walls that were discontinued below the secondfloor level, shifted approximately 2 ft. horizontally relative to the base of the firststory columns, as indicated in Figure 1018a. Beam—Column Joints Beamcolumn joints are critical elements in frame structures. These elements can be subjected to high shear and bondslip deformations under earthquake loading. Beamcolumn joints have to be
Chapter 10 designed so that the connected elements can perform properly. This requires that the joints be proportioned and detailed to allow the columns and beams framing into them to develop and maintain their strength as well as stiffness while undergoing large inelastic deformations. A loss in strength or stiffness in a frame resulting from deterioration in the joints can lead to a substantial increase in lateral displacements of the frame, including possible instability due to Pdelta effects. The design of beamcolumn joints is primarily aimed at (i) preserving the integrity of the joint so that the strength and deformation capacity of the connected beams and columns can be developed and substantially maintained, and (ii) preventing significant degradation of the joint stiffness due to cracking of the joint and loss of bond between concrete and the longitudinal column and beam reinforcement or anchorage failure of beam reinforcement. Of major concern here is the disruption of the joint core as a result of high shear reversals. As in the hinging regions of beams and columns, measures aimed at insuring proper performance of beamcolumn joints have focused on providing adequate confinement as well as shear resistance to the joint. The forces acting on a typical interior beamcolumn joint in a frame undergoing lateral displacement are shown in Figure 1019a. It is worth noting in Figure 1019a that each of the longitudinal beam and column bars is subjected to a pull on one side and a push on the other side of the joint. This combination of forces tends to push the bars through the joint, a condition that leads to slippage of the bars and even a complete pull through in some test specimens. Slippage resulting from bond degradation under repeated yielding of the beam reinforcement is reflected in a reduction in the beamend fixity and thus increased beam rotations at the column faces. This loss in beam stiffness can lead to increased lateral displacements of the frame and potential instability.
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485
(a)
(b)
(c)
Figure 1018. Damage to columns of the 4story Olive View Hospital building during the February 9, 1971 San Fernando, California, earthquake. (From Ref. 1038.) (a) A wing of the building showing approximately 2 ft drift in its first story. (b) Spirally reinforced concrete column in first story. (c) Tied rectangular corner column in first story.
486
Figure 1019. Forces and postulated shearresisting mechanisms in a typical interior beamcolumn joint. (Adapted from Ref. 1032.) (a) Forces acting on beamcolumn joint. (b) Diagonal strut mechanism. (c) Truss mechanism.
Two basic mechanisms have been postulated as contributing to the shear resistance of beam—column joints. These are the diagonal strut and the joint truss (or diagonal compression field) mechanisms, shown in Figure 1019b and c, respectively. After several cycles of inelastic deformation in the beams framing into a joint, the effectiveness of the diagonal strut mechanism tends to diminish as throughdepth cracks start to open
Chapter 10 between the faces of the column and the framing beams and as yielding in the beam bars penetrates into the joint core. The joint truss mechanism develops as a result of the interaction between confining horizontal and vertical reinforcement and a diagonal compression field acting on the elements of the confined concrete core between diagonal cracks. Ideally, truss action to resist horizontal and vertical shears would require both horizontal confining steel and intermediate vertical column bars (between column corner bars). Tests cited in Reference 1039 indicate that where no intermediate vertical bars are provided, the performance of the joint is worse than where such bars are provided. Tests of beamcolumn joints(1027,1040,1041) in which the framing beams were subjected to large inelastic displacement cycles have indicated that the presence of transverse beams (perpendicular to the plane of the loaded beams) considerably improves joint behavior. Results reported in Reference 1027 show that the effect of an increase in joint lateral reinforcement becomes more pronounced in the absence of transverse beams. However, the same tests indicated that slippage of column reinforcement through the joint occurred with or without transverse beams. The use of smallerdiameter longitudinal bars has been suggested (1039) as a means of minimizing bar slippage. Another suggestion has been to force the plastic hinge in the beam to form away from the column face, thus preventing high longitudinal steel strains from developing in the immediate vicinity of the joint. This can be accomplished by suitably strengthening the segment of beam close to the column (usually a distance equal to the total depth of the beam) using appropriate details. Some of the details proposed include a combination of heavy vertical reinforcement with crossties (see Figure 1014), intermediate longitudinal shear reinforcement (see Figure 1015),(1042) and supplementary flexural reinforcement and haunches, as shown in Figure 1020.(1032) The current approach to beam—column joint design in the United States, as contained in ACI Chapter 21, is based on providing
10. Seismic Design of Reinforced Concrete Structures sufficient horizontal joint crosssectional area that is adequately confined to resist the shear stresses in the joint. The approach is based mainly on results of a study by Meinheit and Jirsa(1041) and subsequent studies by Jirsa and associates. The parametric study reported in Reference 1041 identified the horizontal crosssectional area of the joint as the most significant variable affecting the shear strength of beam—column connections. Although recognizing the role of the diagonal strut and joint truss mechanisms, the current approach defines the shear strength of a joint simply in terms of its horizontal crosssectional area. The approach presumes the provision of confinement reinforcement in the joint. In the ACI Chapter 21 method, shear resistance calculated as a function of the horizontal crosssectional area at midheight of the joint is compared with the total horizontal shear across the same midheight section. Figure 1021 shows the forces involved in calculating the shear at midheight of a typical joint. Note that the stress in the yielded longitudinal beam bars is to be taken equal to 1.25 times the specified nominal yield strength fy of the reinforcement. The ACIASCE Committee 352 have added a Recommendations(1029) requirement relating to the uniform distribution of the longitudinal column reinforcement around the perimeter of the column core, with a maximum spacing between perimeter bars of 8 in. or onethird the column diameter or the crosssection dimension. The lateral confinement, whether from steel hoops or beams, and the distributed vertical column reinforcement, in conjunction with the confined concrete core, provide the necessary elements for the development of an effective truss mechanism to resist the horizontal and vertical shears acting on a beam—column joint. Results of recent tests on bidirectionally loaded beam—column joint specimens(1028) confirm the strong correlation between joint shear strength and the horizontal crosssectional area noted by Meinheit and Jirsa.(1041) Some investigators(1013, 1032, 1039) have suggested that the ACI Chapter 21 approach does not fully reflect the effect of the different
487
variables influencing the mechanisms of resistance operating in a beamcolumn joint and have proposed alternative expressions based on idealizations of the strut and joint truss mechanisms.
Figure 1020. Proposed details for forcing beam hinging away from column face(1026). See also Fig. 1015. (a) Supplementary flexural reinforcement. (b) Haunch. (c) Special reinforcement detail.
To limit slippage of beam bars through interior beamcolumn joints, the ACIASCE Committee 352 Recommendations call for a minimum column dimension equal to 20 times the diameter of beam bars passing through the joint. For exterior joints, where beam bars terminate in the joint, the maximum size of beam bar allowed is a No. 11 bar.
488
Figure 1021. Shear force at midheight of beamcolumn joint ACI Chapter 21 design practice.
When the depth of an exterior column is not sufficient to accommodate the required development length for beam bars, a beam stub at the far (exterior) side of the column,(1032) such as is shown in Figure 1022, can be used. Embedding the 90o beam bar hooks outside of the heavily stressed joint region reduces the stiffness degradation due to slippage and improves the overall performance of the connection.
Figure 1022. Exterior beam stub for anchoring beam bars
Slab—Column Connections By omitting consideration of the reinforced concrete flat plate in its provisions governing the design of structures in highseismicrisk areas, ACI Chapter 21 essentially excludes the use of such a system as part of a ductile frame resisting
Chapter 10 seismic loads in such areas. Twoway slabs without beams, i.e., flat plates, are, however, allowed in areas of moderate seismic risk. The flat plate structure is an economical and widely used form of construction in nonseismic areas, especially for multistory residential construction. Its weakest feature, as is well known, is its vulnerability to a punching shear failure at the slabcolumn junctions. The collapse of a number of buildings using such a system during the 1964 Anchorage, Alaska and the 1967 Caracas, Venezuela earthquakes, as well as several buildings using waffle slabs during the September 1985 Mexican earthquake,(1043, 1044) clearly dramatized this vulnerability. Although a flat plate may be designed to carry vertical loads only, with structural walls taking the lateral loads, significant shears may still be induced at the slabcolumn junctions as the structure displaces laterally during earthquake response. Tests on slab—column connections subjected to reversed cyclic loading(1045, 1046) indicate that the ductility of flatslab—column connections can be significantly increased through the use of stirrups enclosing bands of flexural slab reinforcement passing through the columns. Such shearreinforced bands essentially function as shallow beams connecting the columns. Structural Walls Reinforced concrete structural walls (commonly referred to as shear walls), when properly designed, represent economical and effective lateral stiffening elements that can be used to reduce potentially damaging interstory displacements in multistory structures during strong earthquakes. The structural wall, like the vertical steel truss in steel buildings, has had a long history of use for stiffening buildings laterally against wind forces. The effectiveness of properly designed structural walls in reducing earthquake damage in multistory buildings has been well demonstrated in a number of recent earthquakes. In earthquakeresistant design, the appreciable lateral stiffness of structural walls can be particularly well utilized in combination with properly proportioned coupling beams in
10. Seismic Design of Reinforced Concrete Structures coupled wall systems. Such systems allow considerable inelastic energy dissipation to take place in the coupling beams (which are relatively easy to repair) at critical levels, sometimes even before yielding occurs at the bases of the walls. Attention in the following discussion will be focused on slender structural walls, i.e., walls with a heighttowidth ratio greater than about 2.0, such as are used in multistory buildings. These walls generally behave like vertical cantilever beams. Short or squat walls, on the other hand, resist horizontal forces in their plane by a predominantly trusstype mechanism, with the concrete providing the diagonal compressive strut(s) and the steel reinforcement the equilibrating vertical and horizontal ties. Tests on lowrise walls subjected to slowly reversed horizontal loading(1047) indicate that for walls with heighttowidth ratios of about 1.0 , horizontal and vertical reinforcement are equally effective. As the heighttowidth ratio of a wall becomes smaller, the vertical reinforcement becomes more effective in resisting shear than the horizontal steel.(1048) In the following discussion, it will be assumed that the isolated structural wall is loaded by a resultant horizontal force acting at some distance above the base. Under such a loading, flexural hinging will occur at the base of the wall. Where the wall is designed and loaded so that it yields in flexure at the base, as might be expected under strong earthquakes, its behavior becomes a function primarily of the magnitude of the shear force that accompanies such flexural hinging as well as the reinforcement details used in the hinging region near the base. Thus, if the horizontal force acts high above the base (long shear arm), it will take a lesser magnitude of the force to produce flexural hinging at the base than when the point of application of the load is close to the base (short shear arm). For the same value of the base yield moment, the momenttoshear ratio in the former case is high and the magnitude of the applied force (or shear) is low, while in the latter case the momenttoshear ratio is low and the applied shear is high. In both cases, the
489
magnitude of the applied shear is limited by the flexural yield strength at the base of the wall. In this connection, it is of interest to note that dynamic inelastic analyses of isolated walls(104) covering a wide range of structural and ground motion parameters have indicated that the maximum calculated shear at the base of walls can be from 1.5 to 3.5 times greater than the shear necessary to produce flexural yielding at the base, when such shear is distributed in a triangular manner over the height of the wall, as is prescribed for design in most codes. This is shown in Figure 1023, which gives the ratio of the calculated maximum dynamic shear, Vdynmax, to the resultant of the triangularly distributed shear necessary to produce flexural yielding at the base, VT, as a function of the fundamental period T1 and the available rotational ductility µar . The input accelerograms used in the analyses had different frequency characteristics and were normalized with respect to intensity so that their spectrum intensity (i.e., the area under the corresponding 5%damped velocity response spectrum, between periods 0.1 and 3.0 sec) was 1.5 times that of the NS component of the 1940 El Centro record. The results shown in Figure 1023 indicate that a resultant shear force equal to the calculated maximum dynamic shear need not be applied as high as twothirds the height of the wall above the base to produce yielding at the base. Figure 1024, also from Reference 104, shows the distance (expressed dyn ) from the base at which as the ratio M y / Vmax the resultant dynamic force would have to act to produce yielding at the base, as a function of the fundamental period and the available rotational ductility of the wall. The ordinate on the right side of the figure gives the distance above the base as a fraction of the wall height. Note that for all cases, the resultant dynamic force lies below the approximate twothirds point associated with the triangular loading specified in codes.
490
Chapter 10 specified forces by a flexural overstrength factor and a "dynamic shear magnification factor”. The flexural overstrength factor in this case represents the ratio of flexural overstrength (accounting for upward deviations from the nominal strength of materials and other factors) to the moment due to the codespecified forces, with a typical value of about 1.39 or higher. Recommended values for the dynamic shear magnification factor range from 1.0 for a onestory high wall to a maximum of 1.8 for walls 6stories or more in height.
dyn
Figure 1023. Ratio Vmax /VT as a function of T1 and
µ ra . 20 story isolated structural walls. (From Ref. 104.) These analytical results suggest not only that under strong earthquakes the maximum dynamic shear can be substantially greater than that associated with the lateral loads used to design the flexural strength of the base of the wall, but also, as a corollary, that the momenttoshear ratio obtained under dynamic conditions is significantly less than that implied by the codespecified distribution of design lateral loads. These results are important because unlike beams in frames, where the design shear can be based on the maximum probable flexural strengths at the ends of the member as required by statics (see Figure 1016), in cantilever walls it is not possible to determine a similar design shear as a function of the flexural strength at the base of the wall using statics alone, unless an assumption is made concerning the height of the applied resultant horizontal force. In the capacity design method adopted in New Zealand as applied to structural walls,(1013,1049) the design base shear at the base of a wall is obtained by multiplying the shear at the base corresponding to the code
dyn
Figure 1024. Ratio Y = My/ Vmax as a function of T1 and
µ ra  20 story isolated structural walls. (From Ref. 104.)
Tests on isolated structural walls(1050,1051) have shown that the hinging region, i.e., the region where most of the inelastic deformation occurs, extends a distance above the base roughly equal to the width of the wall. The ductility of the hinging region at the base of a wall, like the hinging region in beams and columns, is heavily dependent on the reinforcing details used to prevent early disruption of critically stressed areas within the region. As observed in beams and columns, tests of structural walls have confirmed the
10. Seismic Design of Reinforced Concrete Structures
491
Figure 1025. Momentcurvature curves for statically loaded rectangular walls as a function of reinforcement distribution.(1052)
effectiveness of adequate confinement in maintaining the strength of the hinging region through cycles of reversed inelastic deformation. The adverse effects of high shears, acting simultaneously with the yield moment, on the deformation capacity of the hinging region of walls has also been noted in tests. Early tests of slender structural walls under static monotonic loading(1052) have indicated that the concentration of wellconfined longitudinal reinforcement at the ends of the wall section can significantly increase the ductility of the wall. This is shown in Figure 1025 from Reference 1052. This improvement in behavior resulting from a concentration of wellconfined longitudinal reinforcement at the ends of a wall section has also been observed in
tests of isolated walls under cyclic reversed loading.(1050, 1051) Plain rectangular walls, not having relatively stiff confined boundary elements, are prone to lateral buckling of the compression edge under large horizontal displacements.(1050, 1052) Figure 1026 shows a sketch of the region at the base of a wall with boundary elements after a few cycles of lateral loading. Several modes of failure have been observed in the laboratory. Failure of the section can occur in flexure by rupture of the longitudinal reinforcement or by a combination of crushing and sliding in a weakened compression flange. Alternatively, failure, i.e., loss of lateralloadresisting capacity, can occur by sliding along a nearhorizontal plane near the base (in rectangular
492 section walls especially) or by crushing of the web concrete at the junction of the diagonal struts and the compression flange (in walls with thin webs and/or heavy boundary elements).
Chapter 10 stresses), the compression steel in members subjected to reversed cycles of inelastic loading tends to buckle earlier than in comparable monotonically loaded specimens. As in beams and columns, degradation of strength and ductility of the hinging region of walls is strongly influenced by the magnitude of the shear that accompanies flexural yielding. High shears ( > 6
Figure 1026. Momentcurvature curves for statically loaded rectangular walls as a function of reinforcement distribution.(1054)
Since walls are generally designed to be underreinforced, crushing in the usual sense associated with monotonic loading does not occur. However, when the flanges are inadequately confined, i.e., with the longitudinal and lateral reinforcement spaced far apart, concrete fragments within the cores of the flanges that had cracked in flexure under earlier cycles of loading can be lost in subsequent loading cycles. The longitudinal bars can buckle under compression and when subsequently stretched on reversal of the loading can rupture in lowcycle fatigue. It is also worth noting that because of the Bauschinger effect (i.e., the early yielding, reflected in the rounding of the stress—strain curve of steel, that occurs during load reversals in the inelastic range and the consequent reduction in the tangent modulus of the steel reinforcement at relatively low compressive
f c′ ), when acting on a web
area traversed by crisscrossing diagonal cracks, can precipitate failure of the wall by crushing of the diagonal web struts or a combined compression—sliding failure of the compression flange near the base. Shear in the hinging region is resisted by several mechanisms, namely, shearfriction along a nearhorizontal plane across the width of the wall, dowel action of the tensile reinforcement and to a major extent (as in beams) by shear across the compression flange. After several cycles of load reversals and for moderate momenttoshear ratios, the flexural cracks become wide enough to reduce the amount of shear carried by shear friction. As suggested by Figure 1026, the truss action that develops in the hinging region involves a horizontal (shear) component of the diagonal strut that acts on the segment of the compression flange close to the base. If the compression flange is relatively slender and inadequately confined, the loss of core concrete under load reversals results in a loss of stiffness of this segment of the compression flange. The loss of stiffness and strength in the compression flange or its inability to support the combined horizontal (shear) component of the diagonal strut and the flexural compressive force can lead to failure of the wall. Thus confinement of the flanges of walls, and especially those in the hinging region, is necessary not only to increase the compressive strain capacity of the core concrete but also to delay inelastic bar buckling and, together with the longitudinal reinforcement, prevent loss of the core concrete during load reversals (the socalled “basketing effect”). By maintaining the strength and stiffness of the flanges, confinement reinforcement improves the shear transfer capacity of the hinging region through
10. Seismic Design of Reinforced Concrete Structures the socalled “dowel action” of the compression flange, in addition to serving as shear reinforcement. As in beams, the diagonal tension cracking that occurs in walls and the associated truss action that develops induces tensile stresses in the horizontal web reinforcement. This suggests the need for proper anchorage of the horizontal reinforcement in the flanges. Where high shears are involved, properly anchored crossing diagonal reinforcement in the hinging regions of walls, just as in beams, provides an efficient means of resisting shear and particularly the tendency toward sliding along cracked and weakened planes. A series of tests of isolated structural wall specimens at the Portland Cement Association(1050, 1051) have provided some indication of the effect of several important variables on the behavior of walls subjected to slowly reversed cycles of inelastic deformations. Some results of this investigation have already been mentioned in the preceding. Three different wall crosssections were considered in the study, namely, plain rectangular sections, barbell sections with heavy flanges (columns) at the ends, and flanged sections with the flanges having about the same thickness as the web. In the following, results for some of the parameters considered will be presented briefly. 1. Monotonic vs. reversed cyclic loading. In an initial set of two nominally identical specimens designed to explore the effect of load reversals, a 15% decrease in flexural strength was observed for a specimen loaded by cycles of progressively increasing amplitude of displacement when compared with a specimen that was loaded monotonically. Figures 1027a and 1028a show the corresponding load— deflection curves for the specimens. A comparison of these figures shows not only a reduction in strength but also that the maximum deflection of the wall subjected to reversed loading was only 8 in., compared to about 12 in. for the monotonically load specimen, indicating a reduction in deflection capacity of about 30%. Figure 1028b, when compared
493
with Figure 1027b, shows the more severe cracking that results from load reversals. 2.Level of shear stress. Figure 1029 shows a plot of the variation of the maximum rotational ductility with the maximum nominal shear stress in isolated structural wall specimens reported in References 1050 and 1051. The decrease in rotational ductility with increasing values of the maximum shear stress will be noted. The maximum rotation used in determining ductility was taken as that for the last cycle in which at least 80% of the previous maximum observed load was sustained throughout the cycle. The yield rotation was defined as the rotation associated with the yielding of all of the tensile reinforcement in one of the boundary elements. The presence of axial loads—of the order of 10% of the compressive strength of the walls— increased the ductility of specimens subjected to high shears. In Figure 1029, the specimens subjected to axial loads are denoted by open symbols. The principal effect of the axial load was to reduce the shear distortions and hence increase the shear stiffness of the hinging region. It may be of interest to note that for walls loaded monotonically,(1052) axial compressive stress was observed to increase moment capacity and reduce ultimate curvature, results consistent with analytical results from sectional analysis. 3. Section shape. As mentioned earlier, the use of wall sections having stiff and wellconfined flanges or boundary elements, as against plain rectangular walls, not only allows development of substantial flexural capacity (in addition to being less susceptible to lateral buckling), but also improves the shear resistance and ductility of the wall. In walls with relatively stiff and wellconfined boundary elements, some amount of web crushing can occur without necessarily limiting the flexural capacity of the wall. Corley et al.(1053) point out that trying to avoid shear failure in walls, particularly walls with stiff and wellconfined boundary elements, may be a questionable design objective.
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Chapter 10
(a)
(b) Figure 1027. (a) Loaddeflection curve of monotonically loaded specimen. (b) view of specimen at +12 in. top deflection.(1053)
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Figure 1028. (a) Loaddeflection curve of specimen subjected to load cycles of progressively increasing amplitude. (b) (1053) View of specimen at +8 in. top deflection.
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Chapter 10
Figure 1029. Variation of rotational ductility with maximum average shear stress in PCA isolated wall tests(1051).
Thus, although ACI Chapter 21 limits the maximum average shear stress in walls to 10
f c′ (a value based on monotonic tests)
with the intent of preventing web crushing, web crushing occurred in some specimens subjected to shear stresses only slightly greater than 7
f c′ . However, those specimens where web
crushing failure occurred were able to develop deformations well beyond the yield deformation prior to loss of capacity. 4. Sequence of largeamplitude load cycles. Dynamic inelastic analyses of isolated walls(108) have indicated that in a majority of cases, the maximum or a nearmaximum response to earthquakes occurs early, with perhaps only one elastic response cycle preceding it. This contrasts with the loading program commonly used in quasistatic tests, which consists of load cycles of progressively increasing amplitude. To examine the effect of imposing largeamplitude load cycles early in the test, two nominally identical isolated wall specimens were tested. One specimen was subjected to load cycles of progressively increasing amplitude, as were most of the specimens in this series. Figure 1030a indicates that specimen B7 was able to sustain a rotational ductility of slightly greater than 5 through three
repeated loading cycles. The second specimen (B9) was tested using a modified loading program similar to that shown in Figure 107b, in which the maximum load amplitude was imposed on the specimen after only one elastic load cycle. The maximum load amplitude corresponded to a rotational ductility of 5. As indicated in Figure 1030b, the specimen failed before completing the second load cycle. Although results from this pair of specimens cannot be considered conclusive, they suggest that tests using load cycles of progressively increasing amplitude may overestimate the ductility that can be developed under what may be considered more realistic earthquake response conditions. The results do tend to confirm the reasonable expectation that an extensively cracked and “softened” specimen subjected to several previous load cycles of lesser amplitude can better accommodate large reversed lateral deflections than a virtually uncracked specimen that is loaded to nearcapacity early in the test. From this standpoint, the greater severity of the modified loading program, compared to the commonly used progressively increasingamplitude loading program, appears obvious. 5. Reinforcement detailing. On the basis of the tests on isolated walls reported in References 1050 and 1051, Oesterle et al.(1054) proposed the following detailing requirements for the hinging regions of walls: • The maximum spacing of transverse reinforcement in boundary elements should be 5db, where db is the diameter of the longitudinal reinforcement. • Transverse reinforcement in the boundary element should be designed for a shear Vnb = Mnb/1.5 lb , where Mnb = nominal moment strength of boundary element lb =width of boundary element (in the plane of the wall)
10. Seismic Design of Reinforced Concrete Structures
497
(a)
(b) Figure 1030. Comparison of behavior of isolated walls subjected to different loading histories. (1053) (a) specimen subjected to progressively increasing load amplitudes (see Fig. 107a). (b) Specimen subjected to loading history characterized by largeamplitude cycles early in loading (see Fig. 107b).
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Chapter 10
• No lap splices should be used for crossties in segments of boundary elements within the hinging region. • A recommendation on anchoring horizontal web reinforcement in the boundary elements, such as is shown in Figure 1031a, has been adopted by ACI Chapter 21. For levels of shear in the range of 5
f c′ to 10
f c′ , the study
indicates that alternate 90° and 135° hooks, as shown in Figure 1031b, can be used.
Figure 1031. Alternative details for anchorage of horizontal web reinforcement in boundary elements.(1054) (a) detail for walls subjected to low –tomoderate stress levels. (b) Detail for walls subjected to high shear stress levels.
The specimens tested in this series had special confinement reinforcement only over a length near the base equal to the width of the wall, i.e., the approximate length of the hinging region. Strain readings as well as observations of the general condition of the walls after failure showed that significant inelasticity and damage were generally confined to the hinging region. In view of this, it has been suggested that special confinement reinforcement for boundary elements need be provided only over the lengths of potential hinging regions. These are most likely to occur at the base and at points
along the height of the wall where discontinuities, associated with abrupt and significant changes in geometry, strength, or stiffness, occur. Coupled Walls As mentioned earlier, a desirable characteristic in an earthquakeresistant structure is the ability to respond to strong ground motion by progressively mobilizing the energydissipative capacities of an ascending hierarchy of elements making up the structure. In terms of their importance to the general stability and safety of a building, the components of a structure may be grouped into primary and secondary elements. Primary elements are those upon the integrity of which depend the stability and safety of the entire structure or a major part of it. In this category fall most of the vertical or nearvertical elements supporting gravity loads, such as columns and structural walls, as well as longspan horizontal elements. Secondary elements are those components whose failure would affect only limited areas or portions of a structure. The strong columnweak beam design concept discussed earlier in relation to momentresisting frames is an example of an attempt to control the sequence of yielding in a structure. The “capacity design” approach adopted in New Zealand which, by using even greater conservatism in the design of columns relative to beams, seeks to insure that no yielding occurs in the columns (except at their bases)— is yet another effort to achieve a controlled response in relation to inelastic action. By deliberately building in greater flexural strength in the primary elements (the columns), these design approaches force yielding and inelastic energy dissipation to take place in the secondary elements (the beams). When properly proportioned, the coupledwall system can be viewed as a further extension of the above design concept. By combining the considerable lateral stiffness of structural walls with properly proportioned coupling beams that can provide most of the energydissipative mechanism during response
10. Seismic Design of Reinforced Concrete Structures to strong ground motions, a betterperforming structural system is obtained. The stiffness of the structural wall makes it a desirable primary element from the standpoint of damage control (by restricting interstory distortions), while the more conveniently repairable coupling beams provide the energydissipating secondary elements. Figure 1032a shows a twowall coupledwall system and the forces acting at the base and on a typical coupling beam. The total overturning moment at the base of the coupled wall = M1 + M2 + TL. A typical distribution of the elastic shear force in the coupling beams along the height of the structure due to a statically applied lateral load is shown in Figure 1032b. Note that the accumulated shears at each end of the coupling beams, summed over the height of the structure, are each equal to the axial force (T) at the base of the corresponding wall. The height to the most critically stressed coupling beam tends to move downward as the couplingbeam stiffness (i.e., the degree of coupling between the two walls) increases.
499
energy through inelastic action. These requirements call for fairly stiff and strong beams. Furthermore, the desire for greater lateralloadresisting efficiency in the system would favor stiff and strong coupling beams. However, the beams should not be so stiff or strong flexurally that they induce appreciable tension in the walls, since a net tension would reduce not only the yield moment but also the shear resistance of the wall (recall that a moderate amount of compression improves the shear resistance and ductility of isolated walls). This in turn can lead to early flexural yielding and shearrelated inelastic action at the base of the tension wall. Dynamic inelastic analyses of coupledwall systems(1056) have shown, and tests on coupledwall systems under cyclic reversed loading(1057) have indicated, that when the coupling beams have appreciable stiffness and strength, so that significant net tension is induced in the “tension wall”, a major part of the total base shear is resisted by the “compression wall” (i.e., the wall subjected to axial compression for the direction of loading considered), a situation not unlike that which occurs in a beam. The design of a coupledwall system would then involve adjusting the walltocoupling beam strength and stiffness ratios so as to strike a balance between these conflicting requirements. A basis for choosing an appropriate beamtowall strength ratio, developed from dynamic inelastic response data on coupledwall systems, is indicated in Reference 1058. The Canadian Code for Concrete, CSA Standard A23.394(1033), recommends that in order to classify as a fully effective coupled wall system, the ratio TL must be greater than 2/3. Those M 1 + M 2 + TL
Figure 1032. Laterally loaded coupled wall system. (a) Forces on walls at base. (b) Typical distribution of shears in coupling beams over height of structure.
In a properly designed earthquakeresistant coupledwall system, the critically stressed coupling beams should yield first—before the bases of the walls. In addition, they must be capable of dissipating a significant amount of
with lower ratios are classified as partially coupled wall system in which the coupled wall system are to be designed for higher seismic design forces (14% greater) due to their lower amount of energy dissipation capacity due to reduced coupling action. Once the appropriate relative strengths and stiffness have been established, details to insure adequate ductility in potential hinging regions can be addressed.
500 Because of the relatively large shears that develop in deep coupling beams and the likelihood of sliding shear failures under reversed loading, the use of diagonal reinforcement in such elements has been suggested (see Figure 1033). Tests by Paulay and Binney(1059) on diagonally reinforced coupling beams having spantodepth ratios in the range of 1 to 1½ have shown that this arrangement of reinforcement is very effective in resisting reversed cycles of high shear. The specimens exhibited very stable force— deflection hysteresis loops with significantly higher cumulative ductility than comparable conventionally reinforced beams. Tests by Barney et al.(1060) on diagonally reinforced beams with spantodepth ratios in the range of 2.5 to 5.0 also indicated that diagonal reinforcement can be effective even for these larger spantodepth ratios.
Figure 1033. Diagonally reinforced coupling beam. (Adapted from Ref. 1059.)
In the diagonally reinforced couplings beams reported in Reference 1060, no significant flexural reinforcement was used. The diagonal bars are designed to resist both shear and bending and assumed to function at their yield stress in both tension and compression. To prevent early buckling of the diagonal bars, Paulay and Binney recommend the use of closely spaced ties or spiral binding to confine the concrete within each bundle of diagonal bars. A minimum amount of “basketing reinforcement,” consisting of two layers of smalldiameter horizontal and vertical
Chapter 10 bars, is recommended. The grid should provide a reinforcement ratio of at least 0.0025 in each direction, with a maximum spacing of 12 in. between bars.
10.4
CODE PROVISIONS FOR EARTHQUAKERESISTANT DESIGN
10.4.1
Performance Criteria
In recent years, the performance criteria reflected in some building code provisions such as IBC2000(1061) have become more explicit than before. Although these provisions explicitly require design for only a single level of ground motion, it is expected that buildings designed and constructed in accordance with these requirements will generally be able to meet a number of performance criteria, when subjected to earthquake ground motions of differing severity. The major framework of the performance criteria is discussed in the report by the Structural Association of California Vision 2000 (SEAOC, 1995).(1062) In this report, four performance levels are defined and each performance level is expressed as the desired maximum level of damage to a building when subjected to a specific seismic ground motion. Categories of performance are defined as follows: 1. fully operational 2. operational 3. lifesafe 4. near collapse For each of the performance levels, there is a range of damage that corresponds to the building’s functional status following a specified earthquake design level. These earthquake design levels represent a range of earthquake excitation that have defined probabilities of occurrence over the life of the building. SEAOC Vision 2000 performance level definition includes descriptions of structural and nonstructural damage, egress systems and overall building state. Also included in the performance level descriptions
10. Seismic Design of Reinforced Concrete Structures is the level of both transient and permanent drift in the structure. Drift is defined as the ratio of interstory deflection to the story height. The fully operational level represents the least level of damage to the building. Except for very low levels of ground motion, it is generally not practical to design buildings to meet this performance level. Operational performance level is one in which overall building damage is light. Negligible damage to vertical load carrying elements as well as light damage to the lateral load carrying element is expected. The lateral load carrying system retains almost all of its original stiffness and strength, with minor cracking in the elements of the structure is expected. Transient drift are less than 0.5% and there is inappreciable permanent drift. Building occupancy continues unhampered. Lifesafe performance level guidelines include descriptions of damage to contents, as well as structural and nonstructural elements. Overall, the building damage is described as moderate. Lateral stiffness has been reduced as well as the capacity for additional loads, while some margin against building collapse remains. Some cracking and crushing of concrete due to flexure and shear is expected. Vertical load carrying elements have substantial capacity to resist gravity loads. Falling debris is limited to minor events. Levels of transient drift are to be below 1.5% and permanent drift is less than 0.5%. Near collapse performance includes severe overall damage to the building, moderate to heavy damage of the vertical load carrying elements and negligible stiffness and strength in the lateral load carrying elements. Collapse is prevented although egress may be inhibited. Permissible levels of transient and permanent drift are less than 2.5%. Repair of a building following this level of performance may not be practical, resulting in a permanent loss of building occupancy. In the IBC2000 provisions, the expected performance of buildings under the various earthquakes that can affect them are controlled by assignment of each building to one of the three seismic use groups. These seismic use
501
groups are categorized based on the type of occupancy and importance of the building. For example, buildings such as hospitals, power plants and fire stations are considered as essential facilities also known as postdisaster buildings and are assigned as seismic use group III. These provisions specify progressively more conservative strength, drift control, system selection, and detailing requirements for buildings contained in the three groups, in order to attain minimum levels of earthquake performance suitable to the individual occupancies. 10.4.2
CodeSpecified Design Lateral Forces
The availability of dynamic analysis programs (see References 1063 to 1068) has made possible the analytical estimation of earthquakeinduced forces and deformations in reasonably realistic models of most structures. However, except perhaps for the relatively simple analysis by modal superposition using response spectra, such dynamic analyses, which can range from a linearly elastic timehistory analysis for a single earthquake record to nonlinear analyses using a representative ensemble of accelerograms, are costly and may be economically justifiable as a design tool only for a few large and important structures. At present, when dynamic timehistory analyses of a particular building are undertaken for the purpose of design, linear elastic response is generally assumed. Nonlinear (inelastic) timehistory analyses are carried out mainly in research work. However, nonlinear pushover static analysis can be used as a design tool to evaluate the performance of the structure in the postyield range of response. Pushover analysis is used to develop the capacity curve, illustrated generally as a base shear versus top story displacement curve. The pushover test shows the sequence of element cracking and yielding as a function of the top story displacement and the base shear. Also, it exposes the elements within the structure subjected to the greatest amount of inelastic deformation. The force displacement relationship shows the strength of
502 the structure and the maximum base shear that can be developed. Pushover analysis, which is relatively a new technology, should be carried out with caution. For example, when the response of a structure is dominated by modes other than the first mode, the results may not represent the actual behavior. For the design of most buildings, reliance will usually have to be placed on the simplified prescriptions found in most codes(101) Although necessarily approximate in characterin view of the need for simplicity and ease of applicationthe provisions of such codes and the philosophy behind them gain in reliability as design guides with continued application and modification to reflect the latest research findings and lessons derived from observations of structural behavior during earthquakes. Code provisions must, however, be viewed in the proper perspective, that is, as minimum requirements covering a broad class of structures of more or less conventional configuration. Unusual structures must still be designed with special care and may call for procedures beyond those normally required by codes. The basic form of modern code provisions on earthquakeresistant design has evolved from rather simplified concepts of the dynamic behavior of structures and has been greatly influenced by observations of the performance of structures subjected to actual earthquakes.(1069) It has been noted, for instance, that many structures built in the 1930s and designed on the basis of more or less arbitrarily chosen lateral forces have successfully withstood severe earthquakes. The satisfactory performance of such structures has been attributed to one or more of the following(1070, 1071) : (i) yielding in critical sections of members (yielding not only may have increased the period of vibration of such structures to values beyond the damaging range of the ground motions, but may have allowed them to dissipate a sizable portion of the input energy from an earthquake); (ii) the greater actual strength of such structures resulting from socalled nonstructural elements which are generally ignored in analysis, and the significant energydissipation capacity that
Chapter 10 cracking in such elements represented; and (iii) the reduced response of the structure due to yielding of the foundation. The distribution of the codespecified design lateral forces along the height of a structure is generally similar to that indicated by the envelope of maximum horizontal forces obtained by elastic dynamic analysis. These forces are considered service loads, i.e., to be resisted within a structure’s elastic range of stresses. However, the magnitudes of these code forces are substantially smaller than those which would be developed in a structure subjected to an earthquake of moderatetostrong intensity, such as that recorded at El Centro in 1940, if the structure were to respond elastically to such ground excitation. Thus, buildings designed under the present codes would be expected to undergo fairly large deformations (four to six times the lateral displacements resulting from the codespecified forces) when subjected to an earthquake with the intensity of the 1940 El Centro.(102) These large deformations will be accompanied by yielding in many members of the structure, and, in fact, such is the intent of the codes. The acceptance of the fact that it is economically unwarranted to design buildings to resist major earthquakes elastically, and the recognition of the capacity of structures possessing adequate strength and ductility to withstand major earthquakes by responding inelastically to them, lies behind the relatively low forces specified by the codes. These reduced forces are coupled with detailing requirements designed to insure adequate inelastic deformation capacity, i.e., ductility. The capacity of an indeterminate structure to deform in a ductile manner, that is to deform well beyond the yield limit without significant loss of strength, allows such a structure to dissipate a major portion of the energy from an earthquake without serious damage.
10. Seismic Design of Reinforced Concrete Structures 10.4.3
Principal EarthquakeDesign Provisions of ASCE 795, IBC2000, UBC97, and ACI Chapter 21 Relating to Reinforced Concrete
The principal steps involved in the design of earthquakeresistant castinplace reinforced concrete buildings, with particular reference to the application of the provisions of nationally accepted model codes or standards, will be discussed below. The minimum design loads specified in ASCE 795, Minimum design Loads for Buildings and Other Structures(1072) and the design and detailing provisions contained in Chapter 21 of ACI 31895, Building Code Requirements for Reinforced Concrete,(1010) will be used as bases for the discussion. Emphasis will be placed on those provisions relating to the proportioning and detailing of reinforced concrete elements, the subject of the determination of earthquake design forces having been treated in Chapters 4 and 5. Where appropriate, reference will be made to differences between the provisions of these model codes and those of related codes. Among the more important of these is the IBC2000(1061) which is primarily a descendant of ATC 306(1073) and the latest edition of the Recommended Lateral Force Requirements of the Structural Engineers Association of California (SEAOC96).(1074) The ASCE 795 provisions relating to earthquake design loads are basically similar to those found in the 1997 Edition of the Uniform Building Code (UBC97)(101). The current UBC97 earthquake design load requirements are based on the 1996 SEAOC Recommendations (SEAOC96). Except for minor modifications, the design and detailing requirements for reinforced concrete members found in UBC97 (SEAOC96) and IBC2000 are essentially those of ACI Chapter 21. Although the various codeformulating bodies in the United States tend to differ in what they consider the most appropriate form in which to cast specific provisions and in their judgment of the adequacy of certain design requirements, there has been a tendency for the different codes and model codes to gradually
503
take certain common general features. And while many questions await answers, it can generally be said that the main features of the earthquakeresistant design provisions in most current regional and national codes have good basis in theoretical and experimental studies as well as field observations. As such, they should provide reasonable assurance of attainment of the stated objectives of earthquakeresistant design. The continual refinement and updating of provisions in the major codes to reflect the latest findings of research and field observations(1075) should inspire increasing confidence in the soundness of their recommendations. The following discussion will focus on the provisions of ASCE 795 and ACI Chapter 21, with occasional references to parallel provisions of IBC2000 and UBC97 (SEAOC96). The design earthquake forces specified in ASCE 795 is intended as equivalent static loads. As its title indicates, ASCE 795 is primarily a load standard, defining minimum loads for structures but otherwise leaving out material and member detailing requirements. ACI Chapter 21 on the other hand, does not specify the manner in which earthquake loads are to be determined, but sets down the requirements by which to proportion and detail monolithic castinplace reinforced concrete members in structures that are expected to undergo inelastic deformations during earthquakes. Principal Design Steps Design of a reinforced concrete building in accordance with the equivalent static force procedure found in current U.S. seismic codes involves the following principal steps: 1. Determination of design “earthquake” forces: • Calculation of base shear corresponding to the computed or estimated fundamental period of vibration of the structure. (A preliminary design of the structure is assumed here.) • Distribution of the base shear over the height of the building.
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2. Analysis of the structure under the (static) lateral forces calculated in step (1), as well as under gravity and wind loads, to obtain member design forces and story drift ratios. The lateral load analysis, of course, can be carried out most conveniently by using a computer program for analysis. For certain class of structures having plan or vertical irregularities, or structure over 240 feet in height, most building codes require dynamic analysis to be performed. In this case, ASCE 795 and IBC2000 require that the design parameters including story shears, moments, drifts and deflections determined from dynamic analysis to be adjusted. Where the design value for base shear obtained from dynamic analysis (Vt) is less than the calculated base shear (V) determined using the step 1 above, these design parameters is to be increased by a factor of V/Vt. 3. Designing members and joints for the most unfavorable combination of gravity and lateral loads. The emphasis here is on the design and detailing of members and their connections to insure their ductile behavior. The above steps are to be carried out in each principal (plan) direction of the building. Most building codes allow the design of a structure in each principal direction independently of the other direction on the assumption that the design lateral forces act nonconcurrently in each principal direction. However, for certain building categories which may be sensitive to torsional oscillations or characterized by significant irregularities and for columns forming part of two or more intersecting lateralforceresisting systems, orthogonal effects need to be considered. For these cases, the codes consider the orthogonal effects requirement satisfied if the design is based on the more severe combination of 100 percent of the prescribed seismic forces in one direction plus 30 percent of the forces in the perpendicular direction. Changes in section dimensions of some members may be indicated in the design phase under step (3) above. However, unless the required changes in dimensions are such as to
Chapter 10 materially affect the overall distribution of forces in the structure, a reanalysis of the structure using the new member dimensions need not be undertaken. Uncertainties in the actual magnitude and distribution of the seismic forces as well as the effects of yielding in redistributing forces in the structure would make such refinement unwarranted. It is, however, most important to design and detail the reinforcement in members and their connections to insure their ductile behavior and thus allow the structure to sustain without collapse the severe distortions that may occur during a major earthquake. The code provisions intended to insure adequate ductility in structural elements represent the major difference between the design requirements for conventional, nonearthquakeresistant structures and those located in regions of high earthquake risk. Load Factors, Strength Reduction Factors, and Loading Combinations Used as Bases for Design Codes generally require that the strength or loadresisting capacity of a structure and its component elements be at least equal to or greater than the forces due to any of a number of loading combinations that may reasonably be expected to act on it during its life. In the United States, concrete structures are commonly designed using the ultimatestrengthb method. In this approach, structures are proportioned so that their (ultimate) capacity is equal to or greater than the required (ultimate) strength. The required strength is based on the most critical combination of factored loads, that is, specified service loads multiplied by appropriate load factors. The capacity of an element, on the other hand, is obtained by applying a strengthreduction factor φ to the nominal resistance of the element as determined by codeprescribed expressions or procedures or from basic mechanics. Load factors are intended to take account of the variability in the magnitude of the specified b
Since ACI 31871, the term “ultimate” has been dropped, so that what used to be referred to as “ultimatestrength design” is now simply called “strength design.”
10. Seismic Design of Reinforced Concrete Structures loads, lower load factors being used for types of loads that are less likely to vary significantly from the specified values. To allow for the lesser likelihood of certain types of loads occurring simultaneously, reduced load factors are specified for some loads when considered in combination with other loads. ACI 31895 requires that structures, their components, and their foundations be designed to have strengths not less than the most severe of the following combinations of loads:
1.4D + 1.7L 0.75[l.4D + 1.7L ± (1.7W or l.87E)] 0.9D ± (1.3Wor 1.43E) U = 1.4D + 1.7L + (1.7H or l.4F) 0.9D + (1.7H or 1.4F) 0.75(1.4D + 1.7L + 1.4T) (10  1) where U = required strength to resist the factored loads D = dead load L = live load W = wind load E = earthquake load F = load due to fluids with and maximum heights welldefined pressures H = load due to soil pressure T = load due to effects of temperature, shrinkage, expansion of shrinkage compensating concrete, creep, differential settlement, or combinations thereof. ASCE 795 specifies slightly different load factors for some load combinations, as follows: 1.4 D 1.2(D + F + T) + 1.6(L + H) + 0.5(L or S or R) r 1.2D + 1.6(L r or S or R) + (0.5L or 0.8 W) U = 1.2 D + 1.3W + 0.5L + 0.5(L r or S or R) 1.2 D + 1.0 E + 0.5 L + 0.2 S 0.9 D + ( 1.3 W or 1.0 E)
(102)
505
where Lr = roof live load S = snow load R = rain load For garages, places of public assembly, and all areas where the live load is greater than 100 lb/ft2, the load factor on L in the third, fourth, and fifth combinations in Equation 102 is to be taken equal to 1.0. For the design of earthquakeresistant structures, UBC97 uses basically the same load combinations specified by ASCE 795 as shown in Equation 102. IBC2000 requires that the load combinations to be the same as those specified by ASCE 795 as shown in Equation 102. However, the effect of seismic load, E, is defined as follows: E = ρ QE + 0.2 SDS D E = ρ QE  0.2 SDS D
(103)
where E = the effect of horizontal and vertical earthquakeinduced forces, SDS = the design spectral response acceleration at short periods D = the effect of dead load ρ = the reliability factor QE = the effect of horizontal seismic forces To consider the extent of structural redundancy inherent in the lateralforceresisting system, the reliability factor, ρ, is introduced for buildings located in areas of moderate to high seismicity. This is basically a penalty factor for buildings in which the lateral resistance is limited to only few members in the structure. The maximum value of ρ is limited to 1.5. The factor 0.2 SDS in Equation (103) is placed on the dead load to account for the effects of vertical acceleration. For situations where failure of an isolated, individual, brittle element can result in the loss of a complete lateralforceresisting system or in instability and collapse, IBC2000 has a specific requirement to determine the seismic design forces. These elements are referred to as collector elements. Columns supporting
506
Chapter 10
discontinuous lateralloadresisting elements such as walls also fall under this category. The seismic loads are as follows: E = Ωo QE + 0.2 SDS D E = Ωo QE  0.2 SDS D
(104)
where Ωo is the system overstrength factor which is defined as the ratio of the ultimate lateral force the structure is capable of resisting to the design strength. The value of Ωo varies between 2 to 3 depending on the type of lateral force resisting system. As mentioned earlier, the capacity of a structural element is calculated by applying a strength reduction factor φ to the nominal strength of the element. The factor φ is intended to take account of variations in material strength and uncertainties in the estimation of the nominal member strength, the nature of the expected failure mode, and the importance of a member to the overall safety of the structure. For conventional reinforced concrete structures, ACI 31895 specifies the following values of the strength reduction factor φ: 0.90 for flexure, with or without axial tension 0.90
for axial tension
0.75
for spirally reinforced members subjected to axial compression, with or without flexure
0.70
for other reinforced members (tied columns) subjected to axial compression, with or without flexure (an increase in the φ value for members subjected to combined axial load and flexure is allowed as the loading condition approaches the case of pure flexure)
0.85 0.70
for shear and torsion for bearing on concrete
ACI Chapter 21 specifies the following exception to the above values of the strength
reduction factor as given in the main body of the ACI Code: For structural members other than joints, a value φ = 0.60 is to be used for shear when the nominal shear strength of a member is less than the shear corresponding to the development of the nominal flexural strength of the member. For shear in joints, φ = 0.85. The above exception applies mainly to lowrise walls or portions of walls between openings. Code Provisions Designed to Insure Ductility in Reinforced Concrete Members The principal provisions of ACI Chapter 21 will be discussed below. As indicated earlier, the requirements for proportioning and detailing reinforced concrete members found in UBC97 (SEAOC96) and IBC2000 are essentially those of ACI Chapter 21. Modifications to the ACI Chapter 21 provisions found in UBC97 and IBC2000 will be referred to where appropriate. Special provisions governing the design of earthquakeresistant structures first appeared in the 1971 edition of the ACI Code. The provisions Chapter 21 supplement or supersede those in the earlier chapters of the code and deal with the design of ductile momentresisting space frames and shear walls of castinplace reinforced concrete. ACI 31895 does not specify the magnitude of the earthquake forces to be used in design. The Commentary to Chapter 21 states that the provisions are intended to result in structures capable of sustaining a series of oscillations in the inelastic range without critical loss in strength. It is generally accepted that the intensity of shaking envisioned by the provisions of the first seven sections of ACI Chapter 21 correspond to those of UBC seismic zones 3 and 4. In the 1983 edition of the ACI Code, a section (Section A.9; now section 21.8) was added to cover the design of frames located in areas of moderate seismic risk, roughly corresponding to UBC seismic zone 2. For structures located in areas of low seismic risk (corresponding to UBC seismic zones 0 and 1)
10. Seismic Design of Reinforced Concrete Structures and designed for the specified earthquake forces, very little inelastic deformation may be expected. In these cases, the ductility provided by designing to the provisions contained in the first 20 Chapters of the code will generally be sufficient. A major objective of the design provisions in ACI Chapter 21, as well as in the earlier chapters of the code, is to have the strength of a structure governed by a ductile type of flexural failure mechanism. Stated another way, the provisions are aimed at preventing the brittle or abrupt types of failure associated with inadequately reinforced and overreinforced members failing in flexure, as well as with shear (i.e., diagonal tension) and anchorage or bond failures. The main difference between Chapter 21 and the earlier chapters of the ACI Code lies in the greater range of deformation, with yielding actually expected at critical locations, and hence the greater ductility required in designs for resistance to major earthquakes. The need for greater ductility follows from the design philosophy that uses reduced forces in proportioning members and provides for the inelastic deformations that are expected under severe earthquakes by special ductility requirements. A provision unique to earthquakeresistant design of frames is the socalled strong columnweak beam requirement. As discussed in Section 10.3.4 under “Beam—Column Joints,” this requirement calls for the sum of the flexural strengths of columns meeting at a frame joint to be at least 1.2 times that of the beams framing into the joint. This is intended to force yielding in such frames to occur in the beams rather than in the columns and thus preclude possible instability due to plastic hinges forming in the columns. As pointed out earlier, this requirement may not guarantee nondevelopment of plastic hinges in the columns. The strong columnweak beam requirement often results in column sizes that are larger than would otherwise be required, particularly in the upper floors of multistory buildings with appreciable beam spans.
507
1. Limitations on material strengths. ACI Chapter 21 requires a minimum specified concrete strength f c′ of 3000 lb/in.2 and a maximum specified yield strength of reinforcement, fy of 60,000 lb/in.2. These limits are imposed with a view to restricting the unfavorable effects that material properties beyond these limits can have on the sectional ductility of members. ACI Chapter 21 requires that reinforcement for resisting flexure and axial forces in frame members and wall boundary elements be ASTM 706 grade 60 lowalloy steel intended for applications where welding or bending, or both, are important. However, ASTM 615 billet steel bars of grade 40 or 60 may be used provided the following two conditions are satisfied: (actual fy) ≤ (specified fy) ± 18,000 lb/in.2
actual ultimate tensile stress ≥ 1.25 actual f y The first requirement helps to limit the increase in magnitude of the actual shears that can develop in a flexural member beyond that computed on the basis of the specified yield stress when plastic hinges form at the ends of a beam. The second requirement is intended to insure reinforcement with a sufficiently long yield plateau. In the “strong columnweak beam” frame intended by the code, the relationship between the moment capacities of columns and beams may be upset if the beams turn out to have much greater moment capacity than intended by the designer. Thus, the substitution of 60ksi steel of the same area for specified 40ksi steel in beams can be detrimental. The shear strength of beams and columns, which is generally based on the condition of plastic hinges forming (i.e., My acting) at the member ends, may become inadequate if the actual moment capacities at the member ends are greater than intended as a result of the steel having a substantially greater yield strength than specified.
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Chapter 10
2.Flexural members (beams). These include members having a clear span greater than four times the effective depth that are subject to a factored axial compressive force not exceeding Ag f c′ /10, where Ag is the gross crosssectional area. Significant provisions relating to flexural members of structures in regions of high seismic risk are discussed below. (a) Limitations on section dimensions width/depth ≥ 0.3 ≥ 10 in. width ≤ width of supporting column + 1.5 × (depth of beam)
(b) Limitations on flexural reinforcement ratio (see also Figure 1034):
ρ min
200/f y two continuous bars at both top = and bottom of member 3 f ' c f y
ρmax = 0.025 The minimum steel required can be waived if the area of tensile reinforcement at every section is at least onethird greater than required by analysis. (c) Moment capacity requirements: At beam ends My+ ≥ 0.5MyAt any section in beam span My+ or My ≥ 0.25 (Mymax at beam ends)
Figure 1034. Longitudinal reinforcement requirements for flexural members
10. Seismic Design of Reinforced Concrete Structures (d) Restrictions on lap splices: Lap splices shall not be used (1) within joints, (2) within 2h from face of support, where h is total depth of beam, (3) at locations of potential plastic hinging. Lap splices, where used, are to be confined by hoops or spiral reinforcement with a maximum spacing or pitch of d/4 or 4 in. (e) Restrictions on welding of longitudinal reinforcement: Welded splices and mechanical connectors may be used provided: (1) they are used only on alternate bars in each layer at any section; (2) the distance between splices of adjacent bars is ≥ 24 in. (3) Except as noted above, welding of reinforcement required to resist load combinations including earthquake effects is not permitted. Also, the welding of stirrups, ties, inserts, or other similar elements to longitudinal bars is prohibited (f) Development length requirements for longitudinal bars in tension: (1) For bar sizes 3 through 11 with a standard 90° hook (as shown in Figure 1035) in normal weight concrete, the development length
f y db 65 f c ' ldh ≥ 8d b 6 in. (db is bar diameter). (2) When bars are embedded in lightweightaggregate concrete, the development length is to be at least equal to the greater of 10db, 7.5 in. or 1.25 times the values indicated above. (3) The 90° hook shall be located within the confined core of a column or boundary element.
509 (4) For straight bars of sizes 3 through 11, the development length, ld ≥ 2.5 x (ldh for bars with 90° hooks) , when the depth of concrete cast in one lift beneath the bar is ≤ 12 in., or ld ≥ 3.5 × (ldh for bars with 90° hooks) if the above mentioned depth is > 12 in.
Figure 1035. Development length for beam bars with 90o hooks.
(5) If a bar is not anchored by means of a 90° hook within the confined column core, the portion of the required straight development length not located within the confined core shall be increased by a factor of 1.6. (6) When epoxycoated bars are used, the development lengths calculated above to be increased by a factor of 1.2. However, for straight bars, with covers less than 3db or clear spacing less than 6db, a factor of 1.5 to be used. (g) Transverse reinforcement requirements for confinement and shear: Transverse reinforcement in beams must satisfy requirements associated with their dual function as confinement reinforcement and shear reinforcement (see Figure 1036). (1) Confinement reinforcement in the form of hoops is required:
510
Chapter 10 (i)
(ii)
over a distance 2d from faces of support (where d is the effective depth of the member); over distances 2d on both sides of sections within the span where flexural yielding may occur due to earthquake loading.
(2) Hoop spacing: (iii) First hoop at 2 in. from face of support. (iv) Maximum spacing d / 4 8 × (diameter of smallest longitudinal bar ) ≤ 24 × (diameter of hoop bars ) 12in.
(4) Where hoops are not required, stirrups with seismic hooks at both ends with a spacing of not more than d/2 to be provided throughout the length of the member. (5) Shear reinforcement—to be provided so as to preclude shear failure prior to development of plastic hinges at beam ends. Design shears for determining shear reinforcement are to be based on a condition where plastic hinges occur at beam ends due to the combined effects of lateral displacement and factored gravity loads (see Figure 1016). The probable flexural strength, Mpr associated with a plastic hinge is to be computed using a strength reduction factor φ = 1.0 and assuming a stress in the tensile reinforcement fs = 1.25fy. (6) In determining the required shear reinforcement, the contribution of the concrete, Vc, is to be neglected if the shear associated with the probable flexural strengths at the beam ends is equal to or greater than onehalf the total design shear and the factored axial compressive force including earthquake effects is less than Ag f c′ /20. (7) The transverse reinforcement provided must satisfy the requirements for confinement or shear, whichever is more stringent. Discussion:
Figure 1036. Transverse reinforcement limitations for flexural members. Minimum bar size #3
(3) Lateral support for perimeter longitudinal bars where hoops are required: Every corner and alternate longitudinal bar shall be supported by the corner of a hoop with an included angle 135°, with no longitudinal bar farther than 6 in. along the tie from such a laterally supported bar. Where the longitudinal perimeter bars are arranged in a circle, a circular hoop may be used.
(a) Limitations on section dimensions: These limitations have been guided by experience with test specimens subjected to cyclic inelastic loading. (b) Flexural reinforcement limitations: Because the ductility of a member decreases with increasing tensile reinforcement ratio, ACI Chapter 21 limits the maximum reinforcement ratio to 0.025. The use of a limiting ratio based on the “balanced condition” as given in the earlier chapters of the code, while applicable to members loaded monotonically, fails to describe conditions in flexural members subjected to
10. Seismic Design of Reinforced Concrete Structures reversals of inelastic deformation. The limiting ratio of 0.025 is based mainly on considerations of steel congestion and also on limiting shear stresses in beams of typical proportions. From a practical standpoint, low steel ratios should be used whenever possible. The requirements of at least two continuous bars top and bottom, refers to construction rather than behavioral requirements. The selection of the size, number, and arrangement of flexural reinforcement should be made with full consideration of construction requirements. This is particularly important in relation to beamcolumn connections, where construction difficulties can arise as a result of reinforcement congestion. The preparation of largescale drawings of the connections, showing all beam, column, and joint reinforcements, will help eliminate unanticipated problems in the field. Such largescale drawings will pay dividends in terms of lower bid prices and a smoothrunning construction job. Reference 1076 provides further recommendations on reinforcement detailing. (c) Positive moment capacity at beam ends: To allow for the possibility of the positive moment at the end of a beam due to earthquakeinduced lateral displacements exceeding the negative moment due to the gravity loads, the code requires a minimum positive moment capacity at beam ends equal to 50% of the corresponding negative moment capacity. (d) Lap splices: Lap splices of flexural reinforcement are not allowed in regions of potential plastic hinging since such splices are not considered to be reliable under reversed inelastic cycles of deformation. Hoops are mandatory for confinement of lap splices at any location because of the likelihood of loss of the concrete cover. (e) Welded splices and mechanical connectors: Welded splices and mechanical connectors are to conform to the requirements given in Chapter 12 of the ACI 31895. A major requirement is that the splices develop at
511
least 125% of the specified yield strength of the bar. As mentioned earlier, the welding of stirrups, ties, inserts, or other similar elements to longitudinal bars is not permitted. (f) Development length: The expression for ldh given above already includes the coefficients 0.7 (for concrete cover) and 0.80 (for ties) that are normally applied to the basic development length, ldb. This is so because ACI Chapter 21 requires that hooks be embedded in the confined core of a column or boundary element. The expression for ldh also includes a factor of about 1.4, representing an increase over the development length required for conventional structures, to provide for the effect of load reversals. Except in very large columns, it is usually not possible to develop the yield strength of a reinforcing bar from the framing beam within the width of a column unless a hook is used. Where beam reinforcement can extend through a column, its capacity is developed by embedment in the column and within the compression zone of the beam on the far side of the connection (see Figure 1034). Where no beam is present on the opposite side of a column, such as in exterior columns, the flexural reinforcement in a framing beam has to be developed within the confined region of the column. This is usually done by means of a standard 90° hook plus whatever extension is necessary to develop the bar, the development length being measured from the near face of the column, as indicated in Figure 1035. The use of a beam stub at the far (exterior) side of a column may also be considered (see Figure 1022). ACI Chapter 21 makes no provision for the use of size 14 and 18 bars because of lack of sufficient information on the behavior at anchorage locations of such bars when subjected to load reversals simulating earthquake effects. (g) Transverse reinforcement: Because the ductile behavior of earthquakeresistant
512
Chapter 10 frames designed to current codes is premised on the ability of the beams to develop plastic hinges with adequate rotational capacity, it is essential to insure that shear failure does not occur before the flexural capacity of the beams has been developed. Transverse reinforcement is required for two related functions: (i) to provide sufficient shear strength so that the full flexural capacity of the member can be developed, and (ii) to insure adequate rotation capacity in plastichinging regions by confining the concrete in the compression zones and by providing lateral support to the compression steel. To be equally effective with respect to both functions under load reversals, the transverse reinforcement should be placed perpendicular to the longitudinal reinforcement. Shear reinforcement in the form of stirrups or stirrup ties is to be designed for the shear due to factored gravity loads and the shear corresponding to plastic hinges forming at both ends of a beam. Plastic end moments associated with lateral displacement in either direction should be considered (Figure 1016). It is important to note that the required shear strength in beams (as in columns) is determined by the flexural strength of the frame member (as well as the factored loads acting on the member), rather than by the factored shear force calculated from a lateral load analysis. The use of the factor 1.25 on fy for calculating the probable moment strength is intended to allow for the actual steel strength exceeding the specified minimum and also recognizes that the strain in reinforcement of sections undergoing large rotations can enter the strainhardening range. To allow for load combinations not accounted for in design, a minimum amount of web reinforcement is required throughout the length of all flexural members. Within regions of potential hinging, stirrup ties or hoops are required.
A hoop may be made of two pieces of reinforcement: a stirrup having 135° hooks with 6diameter extensions anchored in the confined core and a crosstie to close the hoop (see Figure 1037). Consecutive ties are to have their 90° hooks on opposite sides of the flexural member.
Figure 1037. Single and twopiece hoops
3.Frame members subjected to axial load and bending. ACI Chapter 21 makes the distinction between columns or beam— columns and flexural members on the basis of the magnitude of the factored axial load acting on the member. Thus, if the factored axial load does not exceed Ag f c′ /10, the member falls under the category of flexural members, the principal design requirements for which were discussed in the preceding section. When the factored axial force on a member exceeds Ag f c′ /10, the member is considered a beam— column. Major requirements governing the design of such members in structures located in areas of high seismic risk are given below. (a) Limitations on section dimensions: shortest crosssectional dimension ≥ 12 in. (measured on line passing through geometric centroid);
shortest dimension ≥ 0.4 perpendicular dimension
10. Seismic Design of Reinforced Concrete Structures (b) Limitations on longitudinal reinforcement: ρmin = 0.01,
ρmax = 0.06
(c) Flexural strength of columns relative to beams framing into a joint (the socalled “strong columnweak beam” provision):
∑M
e
≥
6 5
∑M
g
(105)
where ∑Me = sum of the design flexural strengths of the columns framing into joint. Column flexural strength to be calculated for the factored axial force, consistent with the direction of the lateral loading considered, that results in the lowest flexural strength ∑Mg =sum of design flexural strengths of beams framing into joint The lateral strength and stiffness of columns not satisfying the above requirement are to be ignored in determining the lateral strength and stiffness of the structure. Such columns have to be designed in accordance with the provisions governing members not proportioned to resist earthquakeinduced forces, as contained in the ACI section 21.7. However, as the commentary to the Code cautions, any negative effect on the building behavior of such nonconforming columns should not be ignored. The potential increase in the base shear or of torsional effects due to the stiffness of such columns should be allowed for. (d) Restriction on use of lap splices: Lap splices are to be used only within the middle half of the column height and are to be designed as tension splices. (e) Welded splices or mechanical connectors for longitudinal reinforcement: Welded splices or mechanical connectors may be used at any section of a column, provided that: (1) they are used only on alternate longitudinal bars at a section;
513
(2) the distance between splices along the longitudinal axis of the reinforcement is ≥ 24 in. (f) Transverse reinforcement for confinement and shear: As in beams, transverse reinforcement in columns must provide confinement to the concrete core and lateral support for the longitudinal bars as well as shear resistance. In columns, however, the transverse reinforcement must all be in the form of closed hoops or continuous spiral reinforcement. Sufficient reinforcement should be provided to satisfy the requirements for confinement or shear, whichever is larger. (1) Confinement requirements (see Figure 1038): – Volumetric ratio of spiral or circular hoop reinforcement: fc ' 0.12 f yh ρs ≥ 0.45 Ag − 1 f c ' A f ch yh
(106)
fyh = specified yield strength of transverse reinforcement, in lb/in.2 Ach = core area of column section, measured to the outside of transverse reinforcement, in in.2 – Rectangular hoop reinforcement, total crosssectional area, within spacings:
A sh
f 'c 0.09 shc f yh ≥ A 0.3sh g − 1 f c ' c f Ach yh
(107)
where hc = crosssectional dimension of column core, measured centertocenter of confining reinforcement s = spacing of transverse reinforcement
514
Chapter 10
Figure 1038. Confinement requirements for column ends.
measured along axis of member, in in. smax = min ¼(smallest crosssectional dimension of member),4 in. maximum permissible spacing in plane of crosssection between legs of overlapping hoops or cross ties is 14 in. (2) Confinement reinforcement is to be provided over a length l0 from each joint face or over distances l0 on both sides of any section where flexural yielding may occur in connection with lateral displacements of the frame, where
depth d of member l0 ≥ 1 / 6( clear span of member ) 18 in. UBC97 further requires that confinement reinforcement be provided at any section of a column where the nominal axial strength, φ Pn is less than the sum of the shears corresponding to
the probable flexural strengths of the beams (i.e., based on fs = 1.25fy and φ = 1.0) framing into the column above the level considered. (3) over segments of a column not provided with transverse reinforcement in accordance with Eqs. (106) and (107) and the related requirements described above, spiral or hoop reinforcement is to be provided, with spacing not exceeding 6 × (diameter of longitudinal column bars) or 6 in., whichever is less. (4) Transverse reinforcement for shear in columns is to be based on the shear associated with the maximum probable moment strength, Mpr, at the column ends (using fs = 1.25 fy and φ = 1.0) corresponding to the range of factored axial forces acting on the column. The calculated end moments of columns meeting at a joint need not exceed the sum of the probable moment strengths of the girders framing into the joint. However, in no case should the design
10. Seismic Design of Reinforced Concrete Structures
515
Figure 1039. Columns supporting discontinued wall.
shear be less than the factored shear determined by analysis of the structure. (g) Column supporting discontinued walls: Columns supporting discontinued shear walls or similar stiff elements are to be provided with transverse reinforcement over their full height below the discontinuity (see Figure 1039) when the axial compressive force due to earthquake effects exceeds Ag f c′ /10. The transverse reinforcement in columns supporting discontinued walls be extended above the discontinuity by at least the development length of the largest vertical bar and below the base by the same amount where the column rests on a wall. Where the column terminates in a footing or mat, the transverse reinforcement is to be extended below the top of the footing or mat a distance of at least 12 in.
Discussion: (b) Reinforcement ratio limitation: ACI Chapter 21 specifies a reduced upper limit for the reinforcement ratio in columns from the 8% of Chapter 10 of the code to 6%. However, construction considerations will in most cases place the practical upper limit on the reinforcement ratio ρ near 4%. Convenience in detailing and placing reinforcement in beamcolumn connections makes it desirable to keep the column reinforcement low. The minimum reinforcement ratio is intended to provide for the effects of timedependent deformations in concrete under axial loads as well as maintain a sizable difference between cracking and yield moments.
516
Chapter 10
6 (M ctpr + M cb (M blpr + M brpr ) pr ) ≥ 5
Figure 1040. Strong columnweak beam frame requirements.
(c) Relative columntobeam flexural strength requirement: To insure the stability of a frame and maintain its verticalloadcarrying capacity while undergoing large lateral displacements, ACI Chapter 21 requires that inelastic deformations be generally restricted to the beams. This is the intent of Equation 105 (see Figure 1040). As mentioned, formation of plastic hinges at both ends of most columns in a story can precipitate a sidesway mechanism leading to collapse of the story and the structure above it. Also, as pointed out in Section 10.3.4 under “Beam—Column Joints,” compliance with this provision does not insure that plastic hinging will not occur in the columns. If Equation 105 is not satisfied at a joint, columns supporting reactions from such a joint are to be provided with transverse reinforcement over their full height. Columns not satisfying Equation 105 are to be ignored in calculating the strength and stiffness of the structure. However, since such columns contribute to the stiffness of the structure before they suffer severe loss of strength due to plastic hinging, they should not be ignored if neglecting them results in unconservative estimates of design forces. This may occur in determining the design base shear or in calculating the effects of torsion in a structure. Columns not satisfying Equation 105 should satisfy the minimum requirements for members not
proportioned to resist earthquakeinduced forces, discussed under item 6 below. (f) Transverse reinforcement for confinement and shear: Sufficient transverse reinforcement in the form of rectangular hoops or spirals should be provided to satisfy the larger requirement for either confinement or shear. Circular spirals represent the most efficient form of confinement reinforcement. The extension of such spirals into the beam—column joint, however, may cause some construction difficulties. Rectangular hoops, when used in place of spirals, are less effective with respect to confinement of the concrete core. Their effectiveness may be increased, however, with the use of supplementary crossties. The crossties have to be of the same size and spacing as the hoops and have to engage a peripheral longitudinal bar at each end. Consecutive crossties are to be alternated end for end along the longitudinal reinforcement and are to be spaced no further than 14 in. in the plane of the column crosssection (see Figure 1041). The requirement of having the crossties engage a longitudinal bar at each end would almost preclude placing them before the longitudinal bars are threaded through.
Figure 1041. Rectangular transverse reinforcement in columns.
10. Seismic Design of Reinforced Concrete Structures
Vt = Vb =
517
M tpr + M bpr h
Figure 1042. Loading cases for design of shear reinforcement for columns.
In addition to confinement requirements, the transverse reinforcement in columns must resist the maximum shear associated with the formation of plastic hinges at the column ends. Although the strong columnweak beam provision governing relative moment strengths of beams and columns meeting at a joint is intended to have most of the inelastic deformation occur in the beams of a frame, the code recognizes that hinging can occur in the columns. Thus, the shear reinforcement in columns is to be based on the shear corresponding to the development of the probable moment strengths at the ends of the columns, i.e., using fs = 1.25 fy and φ = 1.0. The values of these end moments —obtained from the PM interaction diagram for the particular column section considered—are to be the
maximum consistent with the range of possible factored axial forces on the column. Moments associated with lateral displacements of the frame in both directions, as indicated in Figure 1042, should be considered. The axial load corresponding to the maximum moment capacity should then be used in computing the permissible shear in concrete, Vc. (g) Columns supporting discontinued walls: Columns supporting discontinued shear walls tend to be subjected to large shears and compressive forces, and can be expected to develop large inelastic deformations during strong earthquakes; hence the requirement for transverse reinforcement throughout the height of such columns according to equations (106) and (107) if the factored axial force exceeds Ag f c′ /10
518 4. Beamcolumn connections. In conventional reinforcedconcrete buildings, the beamcolumn connections usually are not designed by the structural engineer. Detailing of reinforcement within the joints is normally relegated to a draftsman or detailer. In earthquake resistant frames, however, the design of beamcolumn connections requires as much attention as the design of the members themselves, since the integrity of the frame may well depend on the proper performance of such connections. Because of the congestion that may result from too many bars converging within the limited space of the joint, the requirements for the beam—column connections have to be considered when proportioning the columns of a frame. To minimize placement difficulties, an effort should be made to keep the amount of longitudinal reinforcement in the frame members on the low side of the permissible range. The provisions of ACI Chapter 21 dealing with beamcolumn joints relate mainly to: (a) Transverse reinforcement for confinement: Minimum confinement reinforcement, as required for potential hinging regions in columns and defined by Equations 106 and 107, must be provided in beamcolumn joints. For joints confined on all four sides by framing beams, a 50% reduction in the required amount of confinement reinforcement is allowed, the required amount to be placed within the depth of the shallowest framing member. In this case, the reinforcement spacing is not to exceed onequarter of the minimum member dimension nor 6 in. (instead of 4 in. for nonconfined joints). A framing beam is considered to provide confinement to a joint if it has a width equal to at least threequarters of the width of the column into which it frames. (b) Transverse reinforcement for shear: The horizontal shear force in a joint is to be calculated by assuming the stress in the tensile reinforcement of framing beams equal to 1.25fy (see Figure 1021). The
Chapter 10 shear strength of the connection is to be computed (for normalweight concrete) as φ 20 f c ' A j for joints confined on all four sides φ15 f c ' A j φVc = for joints confined on three sides or on two opposite sides φ12 f c ' A j for all other cases
where φ = 0.85 (for shear) Aj = effective (horizontal) crosssectional area of joint in a plane parallel to the beam reinforcement generating the shear forces (see Figure 1043)
Figure 1043. Beamcolumn panel zone.
As illustrated in Fig. 1043, the effective area, Aj, is the product of the joint depth and the effective width of the joint. The joint depth is taken as the overall depth of the column (parallel to the direction of the shear considered), while the effective width of the joint is to be taken equal to the width of the
10. Seismic Design of Reinforced Concrete Structures column if the beam and the column are of the same width, or, where the column is wider than the framing beam, is not to exceed the smaller of: – beam width plus the joint depth, and – beam width plus twice the least column projection beyond the beam side, i.e. the distance x in Fig. 1043. For lightweight concrete, Vc is to be taken as threefourths the value given above for normalweight concrete. (c) Anchorage of longitudinal beam reinforcement terminated in a column must be extended to the far face of the confined column core and anchored in accordance with the requirements given earlier for development lengths of longitudinal bars in tension and according to the relevant ACI Chapter 12 requirements for bars in compression. Where longitudinal beam bars extend through a joint ACI Chapter 21 requires that the column depth in the direction of loading be not less than 20 times the diameter of the largest longitudinal beam bar. For lightweight concrete, the dimension shall be not less than 26 times the bar diameter.
519
(b) Results of tests reported in Reference 1041 indicate that the shear strength of joints is not too sensitive to the amount of transverse (shear) reinforcement. Based on these results, ACI Chapter 21 defines the shear strength of beamcolumn connections as a function only of the crosssectional area of the joint, (Aj) and f c′ (see Section 10.3.4 under “BeamColumn Joints”). When the design shear in the joint exceeds the shear strength of the concrete, the designer may either increase the column size or increase the depth of the beams. The former will increase the shear capacity of the joint section, while the latter will tend to reduce the required amount of flexural reinforcement in the beams, with accompanying decrease in the shear transmitted to the joint. Yet another alternative is to keep the longitudinal beam bars from yielding at the faces of the columns by detailing the beams so that plastic hinging occurs away from the column faces. (c) The anchorage or developmentlength requirements for longitudinal beam reinforcement in tension have been discussed earlier under flexural members. Note that lap splicing of main flexural reinforcement is not permitted within the joint.
Discussion: (a) Transverse reinforcement for confinement: The transverse reinforcement in a beamcolumn connection helps maintain the verticalloadcarrying capacity of the joint even after spalling of the outer shell. It also helps resist the shear force transmitted by the framing members and improves the bond between steel and concrete within the joint. The minimum amount of transverse reinforcement, as given by Equations 106 and 107, must be provided through the joint regardless of the magnitude of the calculated shear force in the joint. The 50% reduction in the amount of confinement reinforcement allowed for joints having beams framing into all four sides recognizes the beneficial confining effect provided by these members.
5. Shear Walls. When properly proportioned so that they possess adequate lateral stiffness to reduce interstory distortions due to earthquakeinduced motions, shear walls or structural walls reduce the likelihood of damage to the nonstructural elements of a building. When used with rigid frames, walls form a system that combines the gravityloadcarrying efficiency of the rigid frame with the lateralloadresisting efficiency of the structural wall. In the form of coupled walls linked by appropriately proportioned coupling beams (see Section 10.3.4 under “Coupled Walls”), alone or in combination with rigid frames, structural walls provide a laterally stiff structural system that allows significant energy dissipation to take place
520
Chapter 10 in the more easily repairable coupling beams. Observations of the comparative performance of rigidframe buildings and buildings stiffened by structural walls during earthquakes(1077) have pointed to the consistently better performance of the latter. The performance of buildings stiffened by properly designed structural walls has been better with respect to both life safety and damage control. The need to insure that critical facilities remain operational after a major tremor and the need to reduce economic losses from structural and nonstructural damage, in addition to the primary requirement of life safety (i.e., no collapse), has focused attention on the desirability of introducing greater lateral stiffness in earthquakeresistant multistory buildings. Where accelerationsensitive equipment is to be housed in a structure, the greater horizontal accelerations that may be expected in laterally stiffer structures should be allowed or provided for.
The principal provisions of ACI Chapter 21 relating to structural walls and diaphragms are as follows (see Figure 1044): (a) Walls (and diaphragms) are to be provided with shear reinforcement in two orthogonal directions in the plane of the wall. The minimum reinforcement ratio for both longitudinal and transverse directions is ρ
v
=
A sv = ρ A cv
n
≥ 0 . 0025
where the reinforcement is to be continuous and distributed uniformly across the shear area, and Acv = net area of concrete section, i.e., product of thickness and width of wall section Asv = projection on Acv of area of shear reinforcement crossing the plane of Acv
Figure 1044. Structural wall design requirements.
10. Seismic Design of Reinforced Concrete Structures ρn = reinforcement ratio corresponding to plane perpendicular to plane of Acv The maximum spacing of reinforcement is 18 in. At least two curtains of reinforcement, each having bars running in the longitudinal and transverse directions, are to be provided if the inplane factored shear force assigned to the wall exceeds 2Acv
f c′ . If the
(factored) design shear force does not exceed Acv
f c′ , the shear reinforcement may be
proportioned in accordance with the minimum reinforcement provisions of ACI Chapter 14. (b) Boundary elements: Boundary elements are to be provided, both along the vertical boundaries of walls and around the edges of openings, if any, when the maximum extremefiber stress in the wall due to factored forces including earthquake effects exceeds 0.2
f c′ . The boundary members
may be discontinued when the calculated compressive stress becomes less than 0.15
f c′ . Boundary elements need not be
provided if the entire wall is reinforced in accordance with the provisions governing transverse reinforcement for members subjected to axial load and bending, as given by Equations 106 and 107. Boundary elements of structural walls are to be designed to carry all the factored vertical loads on the wall, including selfweight and gravity loads tributary to the wall, as well as the vertical forces required to resist the overturning moment due to factored earthquake loads. Such boundary elements are to be provided with confinement reinforcement in accordance with Equations 106 and 107. Welded splices and mechanical connections of longitudinal reinforcement of boundary elements are allowed provided that: 1) they are used only on alternate longitudinal bars at a section;
521
2) the distance between splices along the longitudinal axis of the reinforcement is ≥ 24 in. The requirements for boundary elements in UBC97 and IBC2000 provisions which are essentially similar are much more elaborate and detailed in comparison with ACI95. In these two provisions , the determination of boundary zones may be based on the level of axial, shear, and flexural wall capacity as well as wall geometry. Alternatively, if such conditions are not met, it may be based on the limitations on wall curvature ductility determined based on inelastic displacement at the top of the wall. Using such a procedure, the analysis should be based on cracked shear area and moment of inertia properties and considering the response modification effects of possible nonlinear behavior of building. The requirements of boundary elements using these provisions are discussed in detail under item (f) below. (c) Shear strength of walls (and diaphragms): For walls with a heighttowidth ratio hw/lw ≥ 2.0, the shear strength is to be determined using the expression:
(
φVn = φAcv 2 f c ' + ρ n f y
)
where φ = 0.60, unless the nominal shear strength provided exceeds the shear corresponding to development of nominal flexural capacity of the wall A cv= net area as defined earlier hw = height of entire wall or of segment of wall considered lw= width of wall (or segment of wall) in direction of shear force For walls with hw/lw < 2.0, the shear may be determined from
(
φ Vn = φ Acv α c
f c ' + ρn f y
)
where the coefficient αc varies linearly from a value of 3.0 for hw/lw = 1.5 to 2.0 for hw/lw
522
Chapter 10
= 2.0. Where the ratio hw/lw <2.0 , ρv can not be less than ρn. Where a wall is divided into several segments by openings, the value of the ratio hw/lw to be used in calculating Vn for any segment is not to be less than the corresponding ratio for the entire wall. The nominal shear strength Vn of all wall segments or piers resisting a common lateral force is not to exceed 8Acv
f c′ where Acv is
the total crosssectional area of the walls. The nominal shear strength of any individual segment of wall or pier is not to exceed 10Acp
f c′ where Acp is the crosssectional
area of the pier considered. (d) Development length and splices: All continuous reinforcement is to be anchored or spliced in accordance with provisions governing reinforcement in tension, as discussed for flexural members. Where boundary elements are present, the transverse reinforcement in walls is to be anchored within the confined core of the boundary element to develop the yield stress in tension of the transverse reinforcement. For shear walls without boundary elements, the transverse reinforcement terminating at the edges of the walls are to be provided with standard hooks engaging the edge (vertical) reinforcement. Otherwise the edge reinforcement is to be enclosed in Ustirrups having the same size and spacing as, and spliced to, the transverse reinforcement. An exception to this requirement is when Vu in the plane of the wall is less than Acv
f c′ .
beams should be limited to 10φ
f c′
,
reinforcement in the form of two intersecting groups of symmetrical diagonal bars to be
f c′ where φ =
0.85. (f) Provisions of IBC2000 and UBC97 related to structural walls: These provisions treat shear walls as regular members subjected to combined flexure and axial load. Since the proportions of such walls are generally such that they function as regular vertical cantilever beams, the strains across the depth of such members (in the plane of the wall) are to be assumed to vary linearly, just as in regular flexural members, i.e., the nonlinear strain distribution associated with deep beams does not apply. The effective flange width to be assumed in designing I, L, C or Tshaped shear wall sections, i.e., sections formed by intersecting connected walls, measured from the face of the web, shall not be greater than (a) onehalf the distance to the adjacent shear wall web, or (b) 15 percent of the total wall height for the flange in compression or 30 percent of the total wall height for the flange in tension, not to exceed the total projection of the flange. Walls or portions of walls subject to an axial load Pu> 0.35 P0 shall not be considered as contributing to the earthquake resistance of a structure. This follows from the significantly reduced rotational ductility of sections subjected to high compressive loads (see Fig. 1011(b)). When the shear Vu in the plane of the wall exceeds Acv
(e) Coupling beams: UBC97 and IBC2000 provide similar guidelines for coupling beams in coupled wall structures. For coupling beams with ln/d≥ 4, where ln = clear length of coupling beam and d = effective depth of the beam, conventional reinforcement in the form of top and bottom reinforcement can be used. However, for coupling beams with ln/d< 4 , and factored shear stress exceeding 4
provided. The design shear stress in coupling
f c′ , the need to develop
the yield strength in tension of the transverse reinforcement is expressed in the requirement to have horizontal reinforcement terminating at the edges of shear walls, with or without boundary elements, anchored using standard hooks engaging the (vertical) edge reinforcement or alternatively, having the vertical edge reinforcement enclosed in “U” stirrups of the same size and spacing as, and spliced to, the horizontal reinforcement.
10. Seismic Design of Reinforced Concrete Structures Shear Wall Boundary Zones  The detailing requirements for boundary zones, to be described subsequently, need not be satisfied in walls or portions of walls where 0.10 Ag f c' Pu ≤ ' 0.05 Ag f c and either Mu ≤ 1.0 Vu lu
or
for geometrically symmetrical wall sections otherwise
Vu ≤ 3lw hw f c '
where lw is the length of the entire wall in the direction of the shear force, and hw is the height of the wall. Shear walls or portions of shear walls not meeting the above conditions and having Pu < 0.35 Po (so that they can be considered as contributing to the earthquake resistance of the structure) are to be provided with boundary zones at each end having a length varying linearly from 0.25lw for Pu = 0.35Po to 0.15lw for Pu = 0.15Po, with a minimum of 0.15lw and are to be detailed as will be described. Alternatively, the requirements of boundary zones not meeting the above conditions may be based on the determination of the compressive strain levels at wall edges using cracked section properties. Boundary zone detailing, however, is to be provided over the portions of the wall where compressive strains exceed 0.003. It is important to note that compressive strains are not allowed to exceed 0.015. For shear walls in which the flexural limit state response is governed by yielding at the base of the wall, the total curvature demand (φ t) can be obtained from:
φt =
∆i +φy ( h w − l p / 2) l p
where ∆i = inelastic deflection at the top of the wall
523
= (∆t  ∆y) ∆t = total deflection at the top of the wall equal ∆M, using cracked section properties, or may be taken as 2∆M , using gross section properties. ∆y = displacement at the top of wall corresponding to yielding of the tension reinforcement at critical section, or may be taken as (M′n/ME) ∆E , where ME equals unfactored moment at critical section when top of wall is displaced ∆E . M’n is nominal flexural strength of critical section at P′u. hw = height of the wall lp = height of the plastic hinge above critical section and which shall be established on the basis of substantiated test data or may be alternatively taken at 0.5lw φy = yield curvature which may be estimated at 0.003/lw If φt is less than or equal to 0.003/c′u, boundary zone details as defined below are not required. c′u is the neutral axis depth at P′u and M′n. If φt exceeds 0.003/c′u , the compressive strains may be assumed to vary linearly over the depth c′u , and have maximum value equal to the product of c′u and φt . The use of the above procedure is further discussed with the aid of the design example at the end of this Chapter. Shear wall boundary zone detailing requirements. When required as discussed above, the boundary zones in shear walls are to be detailed in accordance with the following requirements: (1) Dimensional requirements: (a) The minimum section dimension of the boundary zone shall be lw/16. (b) Boundary zones shall extend above the elevation where they are required a distance equal to the development length of the largest vertical bar in the boundary zone. Extensions of the boundary zone lateral reinforcement below its base shall conform to the same requirements as for columns terminating
524
Chapter 10
on a mat or footing. However, the transverse boundary zone reinforcement need not extend above the base of the boundary zone a distance greater than the larger of lw or Mu/4Vu. (c) Boundary zones shall have a minimum length of 18 inches (measured along the length) at each end of the wall or portion of wall. (d) In I, L, C or Tsection walls, the boundary zone at each end shall include the effective flange width and shall extend at least 12 in. into the web. (2) Confinement Reinforcement: (a) All vertical reinforcement within the boundary zone shall be confined by hoops or crossties having a steel crosssectional area Ash> 0.09 h fc′ / fyh (b) Hoops and crossties shall have a vertical spacing,
(b)Horizontal reinforcement shall not be lap spliced within the boundary zone. (4) Vertical reinforcement: (a) Vertical reinforcement shall be provided to satisfy all tension and compression requirements indicated by analysis. (Note again that, in contrast to earlier editions of the code, there is no longer the stipulation of rather arbitrary forces that “boundary elements”, and hence the vertical steel reinforcement in these, are to be designed for.) (b) Area of vertical reinforcement,
0.005 × (area of boundary zone) Av > Two No. 5 bars at each edge of the boundary zone (c)
6 in. S max < 6 × (diameter of largest vertical bar within boundary zone) (c) The lengthtowidth ratio of the hoops shall not exceed 3; and all adjacent hoops shall be overlapping. (d) Crossties or legs of overlapping hoops shall not be spaced farther apart than 12 in. along the wall. (e) Alternate vertical bars shall be confined by the corner of a hoop or crosstie. (3) Horizontal reinforcement: (a)All horizontal reinforcement terminating within a boundary zone shall be anchored as described earlier, i.e., when Vu > Acv
f c′ , horizontal
reinforcement are to be provided with standard hooks or be enclosed in Ustirrups having the same size and spacing as, and spliced to, the horizontal bars.
Lap splices of vertical reinforcement within the boundary zone shall be confined by hoops and crossties. The spacing of hoops and crossties confining lapspliced vertical reinforcement shall not exceed 4 in.
Discussion: (a) The use of two curtains of reinforcement in walls subjected to significant shear (i.e., > 2Acv fc′) serves to reduce fragmentation and premature deterioration of the concrete under load reversals into the inelastic range. Distributing the reinforcement uniformly across the height and width of the wall helps control the width of inclined cracks. (b) ACI Chapter 21 allows calculation of the shear strength of structural walls using a coefficient αc = 2.0. However, advantage can be taken of the greater observed shear strength of walls with low heighttowidth ratios hw/lw by using an αc value of up to 3.0 for walls with hw/lw = 1.5 or less.
10. Seismic Design of Reinforced Concrete Structures The upper bound on the average nominal shear stress that may be developed in any individual segment of wall (10
f c′ ) is
intended to limit the degree of shear redistribution among several connected wall segments. A wall segment refers to a part of a wall bounded by openings or by an opening and an edge. It is important to note that ACI Chapter 21 requires the use of a strengthreduction factor φ for shear of 0.6 for all members (except joints) where the nominal shear strength is less than the shear corresponding to the development of the nominal flexural strength of the member. In the case of beams, the design shears are obtained by assuming plastic end moments corresponding to a tensile steel stress of 1.25fy (see Figure 1016). Similarly, for a column the design shears are determined not by applying load factors to shears obtained from a lateral load analysis, but from consideration of the maximum probable moment strengths at the column ends consistent with the axial force on the column. This approach to shear design is intended to insure that even when flexural hinging occurs at member ends due to earthquakeinduced deformations, no shear failure would develop. Under the above conditions, ACI Chapter 21 allows the use of the normal strengthreduction factor for shear of 0.85. When design shears are not based on the condition of flexural strength being developed at member ends, the code requires the use of a lower shear strengthreduction factor to achieve the same result, that is, prevention of premature shear failure. As pointed out earlier, in the case of multistory structural walls, a condition similar to that used for the shear design of beams and columns is not so readily established. This is so primarily because the magnitude of the shear at the base of a (vertical cantilever) wall, or at any level above, is influenced significantly by the forces and deformations beyond the particular level considered. Unlike the
525 flexural behavior of beams and columns in a frame, which can be considered as closecoupled systems (i.e., with the forces in the members determined by the forces and displacements within and at the ends of the member), the state of flexural deformation at any section of a structural wall (a farcoupled system) is influenced significantly by the displacements of points far removed from the section considered. Results of dynamic inelastic analyses of isolated structural walls under earthquake excitation(103) also indicate that the base shear in such walls is strongly influenced by the higher modes of response. A distribution of static lateral forces along the height of the wall essentially corresponding to the fundamental mode response, such as is assumed by most codes,(101) will produce flexural yielding at the base if the section at the base is designed for such a set of forces. Other distributions of lateral forces, with a resultant acting closer to the base of the wall, can produce yielding at the base only if the magnitude of the resultant horizontal force, and hence the base shear, is increased. Results of the study of isolated walls referred to above,(103) which would also apply to frame—shearwall systems in which the frame is flexible relative to the wall, in fact indicate that for a wide range of wall properties and input motion characteristics, the resultant of the dynamic horizontal forces producing yielding at the base of the wall generally occurs well below the twothirdsoftotalheight level associated with the fundamentalmode response (see Figure 1024). This would imply significantly larger base shears than those due to lateral forces distributed according to the fundamental mode response. The study of isolated walls mentioned above indicates ratios of maximum dynamic shears to “fundamentalmode shears” (i.e., shears associated with horizontal forces distributed according to the fundamentalmode response, as used in codes) ranging from 1.3 to 4.0, the value of the ratio increases with
526 increasing fundamental period (see Figure 1023). (c) Since multistory structural walls behave essentially as vertical cantilever beams, the horizontal transverse reinforcement is called upon to act as web reinforcement. As such, these bars have to be fully anchored in the boundary elements, using standard 90° hooks whenever possible. (d) ACI Chapter 21 uses an extremefiber compressive stress of 0.2fc′, calculated using a linearly elastic model based on gross sections of structural members and factored forces, as indicative of significant compression. Structural walls subjected to compressive stresses exceeding this value are generally required to have boundary elements. Figure 1045 illustrates the condition assumed as basis for requiring that boundary elements of walls be designed for all the gravity loads (W) as well as the vertical forces associated with overturning of the wall due to earthquake forces (H). This requirement assumes that the boundary element alone may have to carry all the vertical (compressive) forces at the critical wall section when the maximum horizontal earthquake force acts on the wall. Under load reversals, such a loading condition imposes severe demands on the concrete in the boundary elements; hence the requirement for confinement reinforcement similar to those for frame members subjected to axial load and bending. Diaphragms of reinforced concrete, such as floor slabs, that are called upon to transmit horizontal forces through bending and shear in their own plane, are treated in much the same manner as structural walls. 6. Frame members not forming part of lateralforceresisting system. Frame members that are not relied on to resist earthquakeinduced forces need not satisfy the stringent requirements governing lateralloadresisting elements. These relate particularly to the transverse reinforcement requirements for confinement and shear. Nonlateralloadresisting elements,
Chapter 10 whose primary function is the transmission of vertical loads to the foundation, need comply only with the reinforcement requirements of ACI Chapter 21, in addition to those found in the main body of the code.
Figure 1045. Loading condition assumed for design of boundary elements of structural walls.
The 1994 Northridge earthquake caused the collapse or partial collapse of at least two parking structures that could be attributed primarily to the failure of interior columns designed to gravity loads only. Following the experience, the requirements for frame members not proportioned to resist forces induced by earthquake motions have been extensively rewritten for the ACI 95 code. A flow chart is provided in Figure 1046 for ease in understanding the new provisions. The requirements are as follows: A special requirement for nonlateralloadresisting elements is that they be checked for adequacy with respect to a lateral displacement representing the expected actual displacement of the structure under the design earthquake. For the purpose of this check, ACI Chapter 21 uses a value of twice the displacement calculated under the factored lateral loads, or 2×1.7 = 3.4 times the displacement due to the codespecified loads. This effect is combined with the effects of dead or dead and live load whichever is critical. If Mu and Vu for an element of gravity system are less than the
10. Seismic Design of Reinforced Concrete Structures corresponding nominal values, that element is going to remain elastic under the design earthquake displacements. If such an element is a beam (Pu≤ Ag fc′/10), it must conform to section 2 described earlier for minimum longitudinal reinforcement requirements. In addition, stirrups spaced at no more than d/2 must be provided throughout the length of the member. If such an element is a column, it must conform to some of the requirements listed under sections 2 and 3 for longitudinal and shear reinforcement. In addition, similar requirements for crossties under section 3(f), discussion, must be met. Also ties at a maximum spacing of so must not exceed six times the smallest longitudinal bar diameter, nor 6 in. Further, if Pu> 0.35 Po, the amount of transverse reinforcement provided must be no less than onehalf that required by 3(f). If Mu and Vu for an element of gravity system exceeds the corresponding nominal values, then it is likely to become inelastic under the design earthquake displacements. Also if deformation compatibility is not checked, this condition will be assumed to be the case. In that case, the structural material must satisfy the requirements described in section 1 and splices of reinforcement must satisfy section 2(e). If such an element is a beam (Pu ≤ Ag fc′ /10), it must conform to sections 2(b), and 2(g) (5) and (6). In addition, the stirrups at no more than d/2 must be provided throughout the length of the member. If it is a column, it must be provided with full ductile detailing in accordance with section 3(f), 3(g), and 4(a) as well as sections 2(g)(5) and (6). 7. Frames in regions of moderate seismic risk. Although ACI Chapter 21 does not define “moderate seismic risk” in terms of a commonly accepted quantitative measure, it assumes that the probable groundmotion intensity in such regions would be a fraction of that expected in a highseismicrisk zone, to which the major part of Chapter 21 is addressed. By the above description, an area of moderate seismic risk would correspond roughly to zone 2 as defined in UBC97(101) and
527
ASCE 795.(1072) For regions of moderate seismic risk, the provisions for the design of structural walls given in the main body of the ACI Code are considered sufficient to provide the necessary ductility. The requirements in ACI Chapter 21 for structures in moderaterisk areas relate mainly to frames and are contained in the last section, section 21.8. The same axial compressive force (Ag fc′ /l0) used to distinguish flexural members from columns in highseismicrisk areas also applies in regions of moderate seismicity. (a) Shear design of beams, columns, or twoway slabs resisting earthquake effects: The magnitude of the design shear is not to be less than either of the following: (1) The sum of the shear associated with the development of the nominal moment strength at each restrained end and that due to factored gravity loads, if any, acting on the member. This is similar to the corresponding requirement for highrisk zones and illustrated in Figure 1016, except that the stress in the flexural tensile reinforcement is taken as fy rather than 1.25fy. (2) The maximum factored shear corresponding to the design gravity and earthquake forces, but with the earthquake forces taken as twice the value normally specified by codes. Thus, if the critical load combination consists of dead load (D) + live load (L) + earthquake effects (E), then the design shear is to be computed from U = 0.75[1.4D + 1.7L + 2(1.87E)] (b) Detailing requirements for beams: The positive moment strength at the face of a joint must be at least onethird the negative moment capacity at the same section. (This compares with onehalf for highseismicrisk areas.) The moment strength—positive or negative—at any section is to be no less than onefifth the maximum moment strength at either end of a member. Stirrup spacing requirements are identical to those for beams in highseismicrisk areas.
528 However, closed hoops are not required within regions of potential plastic hinging. It should be noted that lateral reinforcement for flexural framing members subjected to stress reversals at supports to consist of closed ties, closed stirrups, or spirals extending around the flexural reinforcement as required according to chapter 7 of ACI 31895. (c) Detailing requirements for columns: The same region of potential plastic hinging (lo) as at the ends of columns in a region of high seismicity is defined at each end of a column. The spacing of ties within the region of potential plastic hinging must not exceed the smallest of 8 times the diameter of the smallest longitudinal bar enclosed; 24 times the diameter of the tie bar; or Onehalf the smallest crosssectional dimension of the column, and 12 in. Outside the region of potential plastic hinging, the spacing must not exceed twice the above value. The first tie must be located at no more than half the above spacing from the joint face. (e) Detailing requirements for twoway slabs without beams: As mentioned earlier, requirements for flat plates in ACI Chapter 21 appear only in the section relating to areas of moderate seismic risk. This suggests that ACI Chapter 21 considers the use of flat plates as acceptable components of the lateralloadresisting system only for areas of moderate seismicity. Specific requirements relating to flatplate and flatslab reinforcement for frames in moderaterisk zones are given in ACI Chapter 21 and illustrated in the corresponding Commentary.
Chapter 10
10.5
DESIGN EXAMPLES — REPRESENTATIVE ELEMENTS OF A 12STORY FRAME  SHEAR WALL BUILDING
10.5.1
Preliminaries
A significant part of the damage observed in engineered buildings during earthquakes has resulted from the effects of major structural discontinuities that were inadequately provided for. The message here is clear. Unless proper provision is made for the effects of major discontinuities in geometry, mass, stiffness, or strength, it would be prudent on the part of the engineer to avoid such conditions, which are associated with force concentrations and large ductility demands in localized areas of the structure. Where such discontinuities are unavoidable or desirable from the architectural standpoint, an analysis to obtain estimates of the forces associated with the discontinuity is provides recommended. IBC2000(1061) guidelines for estimating design forces in structures with various types of vertical and plan irregularities. In addition to discontinuities, major asymmetry, with particular regard to the disposition in plan of the lateralloadresisting elements, should be avoided whenever possible. Such asymmetry, which can result in a significant eccentricity between the center of stiffness and the center of mass (and hence of the resultant inertial force), can produce appreciable torsional forces in the structure. Torsional effects can be critical for corner columns or end walls, i.e., elements located far from the center of stiffness. Another important point to consider in the preliminary design of a structure relates to the effectiveness of the various lateralloadresisting components, particularly where these differ significantly in deformation capacity. Efficient use of structural components would suggest that the useful range of deformation of
10. Seismic Design of Reinforced Concrete Structures
529
Relevant Subsection of section 10.4.3 in this Chapter
Moments and shears due to 2 Not Computed
times displacements resulting from
1. Limitation on material strength 2(b). Limitation on flexural reinforcement ratio 2(e). Welded splices and mechanically connected reinforcement 2g(5),(6). Shear reinforcement 3(b). Limitations on longitudinal reinforcement 3(f). Transverse reinforcement for confinement and shear 3(g). Columns supporting discontinued walls 4(a). Transverse reinforcement for confinement
factored lateral forces
Computed and combined with effects of 1.05D + 1.28L or 0.9D, whichever is critical, resulting in Mu,Vu
Mu > φMn orVu> φVn
No
No
Pu >
Ag f c'
1
Yes
2(e)
Yes
10 No
Pu >
2(b) No
Stirrups @ d/2 or less throughout
Ag f c'
Yes
10
Yes
Pu > 0.35Po
the length of the member
3(f) ,3(g) 2(g)(5),(6)
3(b)
2(b)
2(g)(5),(6)
2(g)(5),(6)
Req’t for crossties 3(f), discussion S
≤
So for full height
So ≤ 6db (smallest long. Bar) ≤ 6 in.
Stirrups @ d/2 or less throughout Amount of transverse reinforcement
the length of the member
≥ 1/2 that required
by 3(f)
Figure 1046. Requirements for frame members not proportioned to resist forces induced by earthquake motions.
4(a)
530
Chapter 10
Figure 1047. Relative deformation capacity in lateralloadresisting elements in structure
the principal lateralloadresisting elements in a structure be of about the same magnitude whenever practicable. This is illustrated in Figure 1047a, which shows load—deformation curves of representative elements (1) and (2) in a structure. Such a design allows all the resisting elements to participate in carrying the induced forces over the entire range of deformation. In Figure 1047b, the resisting elements (1) and (2) not only possess different initial stiffnesses but, more importantly, exhibit different ductilities (not ductility ratios) or deformation capacities. In such a case, which is typical of a frame—shearwall structure, the design should be aimed at insuring that the maximum probable deformation or lateral displacement under dynamic conditions does not exceed the deformation capacity ∆2 of element (2); or, if the maximum expected deformation could exceed ∆2 , then element (1) should be so designed that it can support the additional load that may come upon it when element (2) loses a considerable part of its loadcarrying capacity. It is worth noting that, generally, the lateral displacements associated with full mobilization of the ductility of rigid (open) frames are such that significant nonstructural damage can be expected. For this
reason, the building codes limit the amount of deformation that can be tolerated in the structure. The need to tie together all the elements making up a structure or a portion of it that is intended to act as a unit cannot be overemphasized. This applies to the superstructure as well as foundation elements. Where a structure is divided into different parts by expansion joints, as when the various parts differ considerably in height, plan size, shape, or orientation, a sufficient gap should be provided between adjacent parts to prevent their pounding against each other. To avoid pounding between adjacent buildings or parts of the same building when vibrating out of phase with each other, a gap equal to the square root of the sum of the squares (SRSS) of the maximum lateral deflections (considering the deflection amplification factors specified in building codes) of the two structures under the design (codespecified) lateral forces, or the SRSS of the maximum deflections of the two structures as indicated by a dynamic analysis, would be desirable. A good basis for the preliminary design of an earthquakeresistant building is a structure proportioned to satisfy the requirements for
10. Seismic Design of Reinforced Concrete Structures gravity and wind loads. The planning and layout of the structure, however, must be undertaken with due consideration of the special requirements for earthquakeresistant design. Thus, modifications in both configuration and proportions to anticipate earthquakerelated requirements should be incorporated at the outset into the basic design for gravity and wind. Essential to the finished design is particular attention to details that can often mean the difference between a severely damaged structure and one with only minor, repairable damage. 10.5.2
Example Designs of Elements of a 12Story FrameShear Wall Building
The application of the earthquakeresistant design provisions of IBC2000 with respect to design loads and those of ACI 31895(1010) relating to proportioning and detailing of members will be illustrated for representative elements of a 12story frame—shear wall building located in seismic zone 4. The use of the seismic design load provisions in IBC2000, is based on the fact that it represents the more advanced version, in the sense of incorporating the latest revisions reflecting current thinking in the earthquake engineering profession. The typical framing plan and section of the structure considered are shown in Figure 1048ac and b, respectively. The columns and structural walls have constant crosssections throughout the height of the building. The floor beams and slabs also have the same dimensions at all floor levels. Although the dimensions of the structural elements in this example are within the practical range, the structure itself is hypothetical and has been chosen mainly for illustrative purposes. Other pertinent design data are as follows: Service loads — vertical: • Live load: c
Reproduced, with modifications, from Reference 1078, with permission from Van Nostrand Reinhold Company.
531 Basic, 50 lb/ft2. Additional average uniform load to allow for heavier basic load on corridors, 25 lb/ft2. Total average live load, 75 lb/ft2. Roof live load = 20 lb/ft2 • Superimposed dead load: Average for partitions 20 lb/ft2. Ceiling and mechanical 10 lb/ft2. Total average superimposed dead load, 30 lb/ft2. Material properties: • Concrete: fc′ = 4000 lb/in.2 wc = 145 lb/ft3. • Reinforcement: fy = 60 ksi.
Determination of design lateral forces On the basis of the given data and the dimensions shown in Figure 1048, the weights that may be considered lumped at a floor level (including that of all elements located between two imaginary parallel planes passing through midheight of the columns above and below the floor considered) and the roof were estimated and are listed in Tables 101 and 102. The calculation of base shear V, as explained in Chapter 5, for the transverse and longitudinal direction is shown at the bottom of Tables 101 and 102. For this example, it is assumed that the building is located in Southern California with values of Ss and S1 of 1.5 and 0.6 respectively. The site is assumed to be class B (Rock) and the corresponding values of Fa and Fb are 1.0. On this basis, the design spectral response acceleration parameters SDS and SMI are 1.0 and 0.4 respectively. At this level of design parameters, the building is classified as Seismic Group D according to IBC2000. The building consist of moment resisting frame in the longitudinal direction, and dual system consisting of wall and moment resisting frame in the transverse direction. Accordingly, the response modification factor, R, to be used is 8.0 in both directions.
532 Calculation of the undamped (elastic) natural periods of vibration of the structure in the transverse direction (NS) As shown in Figure 1049 using the story weights listed in Table 101 and member stiffnesses based on gross concrete sections, yielded a value for the fundamental period of 1.17 seconds. The mode shapes and the corresponding periods of the first five modes of vibration of the structure in the transverse direction are shown in Figure 1049. The fundamental period in the longitudinal (EW) direction was 1.73 seconds. The mode shapes were calculated using the Computer Program ETABS(1066), based on three dimensional analysis. In the computer model, the floors were assumed to be rigid. Rigid end offsets were assumed at the end of the members to reflect the actual behavior of the structure. The portions of the slab on each side of the beams were considered in the analysis based on the ACI 31895 provisions. The structure was assumed to be fixed at the base. The two interior walls were modeled as panel elements with end piers (26x26 in.). The corresponding values of the fundamental period determined based on the approximate formula given in IBC2000 were 0.85 and 1.27 seconds in the NS and the EW directions respectively. However, these values can be increased by 20% provided that they do not exceed those determined from analysis. On this basis, the value of T used to calculate the base shears were 1.02 and 1.52 seconds in the NS and the EW directions respectively. The lateral seismic design forces acting at the floor levels, resulting from the distribution of the base shear in each principal direction are also listed in Tables 101 and 102. For comparison, the wind forces and story shears corresponding to a basic wind speed of 85 mi/h and Exposure B ( urban and suburban areas), computed as prescribed in ASCE 795, are shown for each direction in Tables 101 and 102. Lateral load analysis of the structure along each principal direction, under the respective seismic and wind loads, based on three
Chapter 10 dimensional analysis were carried out assuming no torsional effects.
Figure 1048. Structure considered in design example. (a) Typical floor framing plan. (b) Longitudinal section
Figure 1049. Undamped natural modes and periods of vibration of structure in transverse direction
10. Seismic Design of Reinforced Concrete Structures
533
Table 101. Design Lateral Forces in Transverse (Short) Direction (Corresponding to Entire Structure).
Floor Level
Height, hx, ft
hxk k=1.26
story weight, wx, kips
148
543
2100
1140
0.162
136
488
2200
1073
124
434
2200
112
382
100
Roof 11 10 9 8 7 6 5 4 3 2 1
wx hxk ftkips x103
Seismic forces Cvx Lateral force,F xkips
Wind forces lateral Story shear force ΣHx, kips Hx, kips
Story shear ΣFx, kips
wind pressure lbs/ft2
208.8
208.8
21.1
23.0
23.0
0.152
196.0
404.8
20.9
45.6
68.9
955
0.135
174.0
578.8
20.5
44.8
113.4
2200
840
0.120
154.7
733.5
20.2
44.1
157.5
331
2200
728
0.103
132.8
866.3
19.8
43.2
200.7
88
282
2200
620
0.088
113.4
979.7
19.4
42.4
243.1
76
234
2200
515
0.073
94.1
1073.8
18.9
41.3
284.4
64
189
2200
415
0.059
76.1
1149.9
18.4
40.2
324.6
52
145
2200
320
0.045
58.0
1207.9
17.8
38.9
363.5
40
104
2200
230
0.033
42.5
1250.4
17.1
37.3
400.8
28
67
2200
147
0.021
27.1
1277.5
16.2
35.4
436.2
16
33
2200
72
0.010
12.9
1290.4
14.9
38.0
474.2

26,300
7055

1290.4


474.2

Total
Calculation of Design Base Shear in Transverse (Short) Direction Base shear, V= CS W where 0.1 SD1 I < CS =
S DS S D1 < R/I T (R / I )
SDS = 2/3 SMS, where SMS = Fa SS = 1.0 × 1.5 = 1.5 and SD1 = 2/3 SMI where SMI = Fv S1 = 1.0 × 0.6 = 0.6; SDS = 1.0, SD1 = 0.4; R=8; I=1.0;T=CT hn3/4 = 0.02 × (148)3/4 =0.849 sec; T can be increased by a factor of 1.2 but should be less than the calculated value (i.e. 1.17 sec). ∴ T = 0.849 × 1.2 =1.018<1.17 0.1 × 0.4 < CS =
1.0 0.4 < 8 / 1 1.018(8 / 1)
0.04 < CS = 0.125 < 0.0491 ∴ use CS = 0.0491 V = 0.0491 x 26,300 = 1290.4 kips
534
Chapter 10
Table 102. Design Lateral Forces in Longitudinal Direction (Corresponding to Entire Structure). Seismic forces Floor Leve l
Height, hx, ft
hxk k=1.51
story weight, wx, kips
wx hxk ftkips x103
Cvx
Lateral force, Fx, kips
Wind forces Story shear
wind pressure lbs/ft2
lateral force Hx, kips
Story shear ΣHx, kips
ΣFx, kips Roof
148
1893
2100
3975
0.178
154.5
154.5
17.2
6.8
6.8
11
136
1666
2200
3665
0.164
142.4
296.9
17.0
13.5
20.3
10
124
1449
2200
3188
0.142
123.3
420.2
16.6
13.1
33.4
9
112
1243
2200
2734
0.122
105.9
526.1
16.3
12.9
46.3
8
100
1047
2200
2304
0.103
89.4
615.5
15.9
12.6
58.9
7
88
863
2200
1899
0.085
73.8
689.3
15.5
12.3
71.2
6
76
692
2200
1522
0.068
59.0
748.3
15.0
12.0
83.2
5
64
534
2200
1174
0.052
45.1
793.4
14.5
11.5
94.7
4
52
390
2200
858
0.038
33.0
826.4
13.9
11.0
105.7
3
40
263
2200
578
0.026
22.6
849.0
13.2
10.5
116.2
2
28
153
2200
337
0.015
13.0
862.0
12.3
9.7
125.9
1
16
66
2200
145
0.006
5.2
867.2
11.0
10.2
136.1

26,300
22,379

867.2


136.1

Total
In longitudinal direction, Ct (for reinforced concrete moment resisting frames) = 0.03; T = Ct (hn)3/4 = (0.03) (148) = 1.27; T can be increased by a factor of 1.2, ∴ T = 1.2 × 1.27 = 1.524 < 1.73 0.1 × 0.4 < CS =
1.0 0 .4 < 8 / 1 1.524(8 / 1)
0.04 < CS = 0.125 < 0.0329∴ use CS = 0.0329 V = 0.033 × 26,300 = 867.2 kips
10. Seismic Design of Reinforced Concrete Structures (a) Lateral displacements due to seismic and wind effects: The lateral displacements due to both seismic and wind forces listed in Tables 101 and 102 are shown in Figure 1050 . Although the seismic forces used to obtain the curves of Figure 1050 are approximate, the results shown still serve to draw the distinction between wind and seismic forces, that is, the fact that the former are external forces the magnitudes of which are proportional to the exposed surface, while the latter represent inertial forces depending primarily on the mass and stiffness properties of the structure. Thus, while the ratio of the total wind force in the transverse direction to that in the longitudinal direction (see Tables 101 and 102) is about 3.5, the corresponding ratio
535 for the seismic forces is only 1.5. As a result of this and the smaller lateral stiffness of the structure in the longitudinal direction, the displacement due to seismic forces in the longitudinal direction is significantly greater than that in the transverse direction. By comparison, the displacements due to wind are about the same for both directions. The typical deflected shapes associated with predominantly cantilever or flexure structures (as in the transverse direction) and shear (openframe) buildings (as in the longitudinal direction) are evident in Figure 1050. The average deflection indices, that is, the ratios of the lateral displacement at the top to the total height of the structure, are 1/5220 for wind and 1/730 for seismic
Figure 1050. Lateral displacements under seismic and wind loads.
536 loads in the transverse direction. The corresponding values in the longitudinal direction are 1/9350 for wind and 1/590 for seismic loads. It should be noted that the analysis for wind was based on uncracked sections whereas that for seismic was based on cracked sections. The use of cracked section moment of inertia is a requirement by IBC2000 for calculation of drift due to earthquake loading. However, under wind loading, the stresses within the structure in this particular example are within the elastic range as can also be observed from the amount of lateral deflections. As a result, the amount of cracking within the members is expected to be insignificant. However, for the case of seismic loading, the members are expected to deform well into inelastic range of response under the design base shear. To consider the effects of cracked sections due to seismic loads, the moments of inertia of beams, columns and walls were assumed to be 0.5, 0.7 and 0.5 of the gross concrete sections respectively. (b) Drift requirements: IBC2000 requires that the design story drift shall not exceed the allowable limits. In calculating the drift limits, the effect of accidental torsion was considered in the analysis. On this basis, the mass at each floor level was assumed to displace from the calculated center of mass a distance equal to 5% of the building dimension in each direction. Table 103 shows the calculated displacements and the corresponding story drifts in both EW and NS directions. To determine the actual story drift, the calculated drifts were amplified using the Cd factor of 6.5 according to IBC2000. These increased drifts account for the total anticipated drifts including the inelastic effects. The allowable drift limit based on IBC2000 is 0.025 times the story height which corresponds to 3.6 in. and 4.8 in. at a typical floor and first floor respectively. The calculated values of drift are less than these limiting values. It is to be noted that using IBC2000 provisions, it is permissible
Chapter 10 to use the computed fundamental period of the structure without the upper bound limitation when determining the story drifts limits. However, the drift values shown are based on the calculated values of the fundamental period based on the code limits. Since the calculated drifts are less than the allowable values, further analysis based on the adjusted value of period was not necessary. In addition, the P∆ effect need not to be considered in the analysis when the stability coefficient as defined by IBC2000 is less than a limiting value. For the 12story structure, the effect of P∆ was found to be insignificant. (c) Load Combinations: For design and detailing of structural components, IBC2000 requires that the effect of seismic loads to be combined with dead and live loads. The loading combinations to be used are those prescribed in ASCE95 as illustrated in Equation (102) except that the effect of seismic loads are according to IBC2000 as defined in Equation (103). To consider the extent of structural redundancy inherent in the lateralforceresisting system, the reliability factor, ρ, is defined as follows for structures in seismic design category D as defined by IBC2000: ρ = 2−
20 rmax Ax
where rmax = the ratio of the design story shear resisted by the single element carrying the most shear force in the story to the total story shear, for a given direction of loading. For shear walls, rmax is defined as the shear in the most heavily loaded wall multiplied by 10/lw , divided by the story shear (lw is the wall length) Ax = the floor area in square feet of the diaphragm level immediately above the story
10. Seismic Design of Reinforced Concrete Structures
537
Table 103. Lateral displacements and Inerstory drifts Due to Seismic Loads (in.). EW Direction drift
NS Direction drift
drift ×
drift × Cd*
displacement
0.07
0.45
2.43
0.19
Cd* 1.24
2.96
0.12
0.78
2.24
0.20
1.30
10
2.84
0.16
1.04
2.04
0.21
1.37
9
2.68
0.20
1.30
1.83
0.23
1.50
8
2.48
0.24
1.56
1.60
0.24
1.56
7
2.24
0.27
1.76
1.36
0.24
1.56
6
1.97
0.28
1.82
1.12
0.23
1.50
5
1.69
0.31
2.02
0.89
0.23
1.50
4
1.38
0.32
2.08
0.66
0.22
1.43
3
1.06
0.33
2.15
0.44
0.18
1.17
2
0.73
0.34
2.21
0.26
0.15
0.98
1
0.39
0.39
2.54
0.11
0.11
0.72
Story Level Roof
displacement 3.03
11
* Cd = 6.5
When calculating the reliability factor for dual systems such as the frame wall structure in the NS direction, it can be reduced to 80 percent of the calculated value determined as above. However, this value can not be less that 1.0. In the NS direction, the most heavily single element for shear is the shear wall. Table 104 shows the calculated values for r over the 2/3 height of the structure. The maximum value of r occurs at the base of the structure where the shear walls carry most of the shear in the NS direction. On this basis, the maximum value of ρ determined was 1.0. The load combinations used for the design based on ρ= 1.0 and SDS=1.0 by combining
Table 104.Element story shear ratios for redundancy factor in NS direction. Story Level
Vi = shear force in wall
Vi x 10/Lw
story shear
ri
8 7 6 5 4 3 2 1
189 234 275 317 359 408 448 570
78 97 114 131 149 169 185 236
886 980 1074 1150 1208 1250 1278 1290
0.09 0.10 0.11 0.11 0.12 0.14 0.15 0.18
ρ = 2−
20 rmax
ρ =2−
Ax 20
0.18 × 66 × 182
= 0.99
but
ρ min = 1.0
538
Chapter 10
equations (102) and (103) are as follows: 1.2 D + 1.6 L + 0.5 L r U = 1.4 D ± 1.0 Q E + 0.5 L 0.7 D ± 1.0 Q E
(108)
The 3D structure was analyzed using the above load combinations. The dead and live loads were applied to the beams based on tributary areas as shown in Figure 1051. The effect of accidental torsion was also considered in the analysis. To protect the building against collapse, IBC2000 requires that in dual systems, the moment resisting frames be capable to resist at least 25% of prescribed seismic forces. For this reason, the building in the NS direction was also subjected to 25% of the lateral forces described above without including the shear
walls. An idea of the distribution of lateral loads among the different frames making up the structure in the transverse direction may be obtained from Table 105, which lists the portion of the total story shear at each level resisted by each of the three groups of frames. The four interior frames along lines 3, 4,5, and 6 are referred to as Frame T1, while the Frame T2 represents the two exterior frames along lines 1 and 8. The third frame, T3 represents the two identical frameshear wall systems along lines 2 and 7. Note that at the top (12th floor level), the lumped frame T1 takes 126% of the total story shear. This reflects the fact that in frameshearwall systems of average proportions, interaction between frame and wall under lateral loads results in the frame “supporting” the wall at the top, while at the base most of the horizontal shear is resisted by
W
45°
T ran sv erse B eam s
22'
26'
W In terio r B eam s
W
D L
= 3 .5 2 k /ft
W
= 1 .6 4 k /ft
W r = 0 .4 4 k /ft
W
D
= 1 .7 6 k /ft
E x terio r W = 0 .8 2 k /ft L B eam s W r = 0 .2 2 k /ft
L o ng itu d in al B eam s Figure 1051. Tributary area for beam loading.
10. Seismic Design of Reinforced Concrete Structures
539
Table 105. Distribution of Horizontal Seismic Story Shears among the Three Transverse Frames.
Story Level
Frame T1 (4 interior frames)
Frame T2 (2 exterior frames)
Frame T3 (2 interior frames with shear walls)
Total story shear,
Story shear
% of total
Story shear
% of total
Story shear
% of Total
Roof
263.6
126
102.1
49
156.9
75
kips 208.8
11
228.5
56
90.3
22
86.0
21
404.8
10
259.9
45
101.9
18
216.8
37
578.8
9
282.5
39
110.4
15
340.6
46
733.5
8
303.6
35
117.3
14
445.4
51
866.3
7
317.3
32
123.6
13
538.8
55
979.7
6
324.0
30
125.6
12
624.2
58
1073.8
5
320.0
28
124.0
11
705.9
61
1149.9
4
303.2
25
117.9
10
786.8
65
1207.9
3
269.6
22
104.4
8
876.4
70
1250.4
2
225.1
18
86.4
7
966.0
75
1277.5
1
96.0
7
34.8
3
1159.6
90
1290.4
the wall. Table 105 indicates that for the structure considered, the two frames with walls take 90% of the shear at the base in the transverse direction. To illustrate the design of two typical beams on the sixth floor of an interior frame, the results of the analysis in the transverse direction under seismic loads have been combined, using Equation 108, with results from a gravityload analysis . The results are listed in Table 106. Similar values for typical exterior and interior columns on the second floor of the same interior frame are shown in Table 107. Corresponding design values for the structural wall section at the first floor of frame on line 3 (see Figure 1048) are listed in Table 108. The
last column in Table 108 lists the axial load on the boundary elements (the 26 × 26in, columns forming the flanges of the structural walls) calculated according to the ACI requirement that these be designed to carry all factored loads on the walls, including selfweight, gravity loads, and vertical forces due to earthquakeinduced overturning moments. The loading condition associated with this requirement is illustrated in Figure 1045. In both Tables 107 and 108, the additional forces due to the effects of horizontal torsional moments corresponding to the minimum IBC2000 prescribed eccentricity of 5% of the building dimension perpendicular to the direction of the applied forces have been included.
540
Chapter 10
Table 106. Summary of design moments for typical beams on sixth floor of interior transverse frames along lines 3 through 6 (Figure 1048a).
1.2 D + 1.6 L + 0.5Lr U = 1.4 D + 0.5L ± 1.0 Q E 0.7 D ± 1.0 Q E BEAM AB 98 a Sides way to right 98 b Sides way to left 98 c
A 76 +91
(9 − 8a ) (9 − 8b) (9 − 8c )
Design moment, ftkips Midspan of AB B +100 202 +83 326
213
+85
19
Sides way to right
+127
+35
229
Sides way to left
177
+37
+79
BEAM BC 98 a Sides way to right 98 b Sides way to left Sides way to right 98 c Sides way to left
B 144 41
Design moment, ftkips Midspan of BC C +92 144 +77 282
282 +110 213
+77
41
+33
213
+33
+110
It is pointed out that for buildings located in seismic zones 3 and 4 (i.e., highseismicrisk areas), the detailing requirements for ductility prescribed in ACI Chapter 21 have to be met even when the design of a member is governed by wind loading rather than seismic loads. 2.Design of flexural member AB. The aim is to determine the flexural and shear reinforcement for the beam AB on the sixth floor of a typical interior transverse frame. The critical design (factored) moments are shown circled in Table 106. The beam has dimensions b = 20 in. and d = 21.5 in. The slab is 8 in. thick, f c′ = 4000 lb/in.2 and f y = 60,000 lb/in.2 In the following solution, the boxedin section numbers at the righthand margin correspond to those in ACI 31895 . (a) Check satisfaction of limitations on section dimensions: width 20 = depth 21.5
= 0.93 > 0.3 O.K 21.3.1.3 21.3.1.4 O.K. ≥ 10 in. ≤ (width of suuporting column width = 20 in. + 1.5 × depth of beam = 26 + 1.5(21.5) = 58.25 in. O.K.
Table 107. Summary of design moments and axial loads for typical columns on second floor of interior transverse frames along lines 3 through 6 (Figure 1048a).
1.2 D + 1.6 L + 0.5 L r U = 1.4 D + 0.5 L ± 1.0 Q E 0.7 D
98 b
98 c
(9 − 8a ) ( 9 − 8b )
± 1.0 Q E
98 a Sides way to right Sides way to left Sides way to right Sides way to left
( 9 − 8c )
Axial load, kips 1076
Exterior Column A Moment, ftkips Top Kips 84 +94
Interior Column B Axial load, Moment, ftkips kips Top Bottom 1907 +6 12
806
33
+25
1630
+73
108
1070
110
+134
1693
94
+119
280
+8
20
698
+79
111
544
69
+88
760
88
+116
10. Seismic Design of Reinforced Concrete Structures
541
Table 108. Summary of design loads on structural wall section at first floor level of transverse frame along line 2 (or 7) (Figure 1048a).
1.2 D + 1.6 L + 0.5 Lr U = 1.4 D + 0.5 L ± 1.0 Q E 0.7 D ± 1.0 Q E
( 9 − 8a ) ( 9 − 8b ) ( 9 − 8c ) Axial load# on boundary element, kips
Design forces acting on entire structural wall Axial Load, kips
Bending Horizontal (overturning) shear, Moment, ftkips kips 98 a 5767 Nominal Nominal 98 b 5157 30469 651 98 c 2293 30469 651 # Based on loading condition illustrated in Figure 1045 @ bending moment at base of wall
(b) Determine reinforcement:
required
flexural
(1) Negative moment reinforcement at support B: Since the negative flexural reinforcement for both beams AB and BC at joint B will be provided by the same continuous bars, the larger negative moment at joint B will be used. In the following calculations, the effect of any compressive reinforcement will be neglected. From C = 0.85fc′ba = T =Asfy,
As 60 As a= = = 0.882 As ' 0.85 f c b (0.85)(4)(20) M u ≤ φM n = φAs f y (d − a / 2 )
− (326)(12) = (0.90)(60) As × [21.5 − (0.5)(0.882As )] As 2 − 48.76 As + 164.3 = 0 or As = 3.64 in. 2
Alternatively, convenient use may be made of design charts for singly reinforced flexural members with rectangular crosssections, given in
2884 3963 2531
standard references. (1079) Use five No. 8 bars, As=3.95 in.2 This gives a negative moment capacity at support B of φMn = 351 ftkips. Check satisfaction of limitations on reinforcement ratio:
As 3.95 = bd ( 20)( 21.5) = 0.0092 200 > ρ min = = 0.0033 fy
ρ=
> ρ min =
3 fc ' fy
=
21.3.2.1
3 4000 = 0.0032 60,000
and <ρmax = 0.025
O.K.
(2) Negative moment reinforcement at support A: Mu = 213 ftkips As at support B, a = 0.882As. Substitution into M u = φAs f y (d − a / 2)
yields As = 2.31 in.2. Use three No. 8 bars, As = 2.37 in.2 This gives a negative moment capacity at support A of φMn = 218 ftkips.
542
Chapter 10 (3)Positive moment reinforcement at supports: A positive moment capacity at the supports equal to at least 50% of the corresponding negative moment capacity is required, i.e., 21.3.2.2 218 min M u (at support A) = = 109 ft − kips 2 which is less than M+max = 127 ftkips at A (see Table 106), but greater than the required Mu+ near midspan of AB (=100 ftkips).
min Mu+ (at support B for both spans AB and BC) =
351 = 176 ft − kips 2
Note that the above required capacity is greater than the design positive moments near the midspans of both beams AB and BC. Minimum positive/negative moment capacity at any section along beam AB or BC = 351/4 =87.8 ftkips. (4) Positive moment reinforcement at midspan of beam AB to be made continuous to supports: (with an effective Tbeam section flange width = 52 in.) a=
As f y 0.85 f c' b
=
60 As = 0.339 As (0.85)(4 )(52 )
> ρ min =
3 f c′ 3 4000 = fy 60,000
(c) Calculate required length of anchorage of flexural reinforcement in exterior column: Development length l dh
f d / 65 f ' c y b ≥ 8d b 6 in.
21.5.4.1
(plus standard 90º hook located in confined region of column). For the No. 8 (top) bars (bend radius, measured on inside of bar, ≥ 3d b = 3.0 in.),
l dh
(60,000)(1.0) 65 4000 = 15 in. ≥ (8)(1.0 ) = 8.0 in 6 in.
For the No. 7 bottom bars (bend radius ≥ 3d b = 2.7 in.), ldh = 13 in.
Figure 1052 shows the detail of flexural reinforcement anchorage in the exterior column. Note that the development length ldh is measured from the near face of the column to the far edge of the vertical 12bardiameter extension (see Figure 1035).
Substituting into a M u = (127 )(12 ) = φAs f y d − 2
yields As (required) = 1.35 in.2. Similarly, corresponding to the required capacity at support B, M u+ = 163 ftkips, we have As (required) =1.74 in.2. Use three No. 7 bars continuous through both spans. As = 1.80 in.2 This provides a positive moment capacity of 172 ftkips. Check: ρ=
1.8 = 0.0042 ( 20)( 21.5)
> ρ min =
200 = 0.0033 fy
O.K.
10.5.1
Figure 1052. Detail of anchorage of flexural reinforcement in exterior column
10. Seismic Design of Reinforced Concrete Structures
543
(d) Determine shearreinforcement requirements: Design for shears corresponding to end moments obtained by assuming the stress in the tensile flexural reinforcement equal to 1.25fy and a strength reduction factor φ = 1.0, plus factored gravity loads (see Figure 1016). Table 109 shows values of design end shears corresponding to the two loading cases to be considered. In the table,
Vb =
230 + 477 = 35.4 kips 20
’k 0 3 2
u W
A
which is approximately 50% of the total design shear, Vu = 69.6 kips. Therefore, the contribution of concrete to shear resistance can be considered in determining shear reinforcement requirements. At right end B, Vu = 69.6 kips. Using Vc = 2 f
WU = 1.2 WD + 1.6 WL = 1.2 × 3.52 + 1.6 × 1.64 = 6.85 kips/ft
'
c bw d
=
2 4000 ( 20)( 21.5) = 54.4kips 1000
we have
ACI Chapter 21 requires that the contribution of concrete to shear resistance, Vc, be neglected if the earthquakeinduced shear force (corresponding to the probable flexural strengths at beam ends calculated using 1.25fy instead of fy and φ = 1.0) is greater than onehalf the total design shear and the axial compressive force including earthquake effects is less than Ag f′c /20. 21.3.4.2 For sidesway to the right, the shear at end B due to the plastic end moments in the beam (see Table 109) is
φVs = Vu − φV c= 69.6 − 0.85 × 54.4 = 23.4 kips Vs = 27.5 kips
Required spacing of No. 3 closed stirrups (hoops), since Av (2 legs) = 0.22 in.2: s=
Av f y d
=
Vs
(0.22)(60)(21.5)
11.5.6.2
27.5
= 10.3 in. Maximum allowable hoop spacing within distance 2d = 2(21.5) = 43 in. from faces of supports:
Table 109. Determination of Design Shears for Beam AB.
Vu =
Loading 2 3 0 ’k
M prA + M prB l
±
wu l , (kips) 2
A
B
1.1
69.6
60.7
7.8
Wu
A
B 4 7 7 ’k
u W ’k 0 3 2 A
Wu 2 3 0 ’k A
B
2 9 9 ’k
Shear Diagram 6 0 .7
1 .1 A
11.1.1
B
7 .8
6 9 .6
544
Chapter 10
Figure 1053. Spacing of hoops and stirrups in right half of beam AB
s max
d / 4 = 21.5 / 4 = 5.4 in. 8 × (dia. of smallest long. bar) = = 8(0.875) = 7 in. 24 × (dia. of hoop bars) = 24(0.375) = 9 in. 12 in.
21.3.3.2 Beyond distance 2d from the supports, maximum spacing of stirrups: s max = d / 2 = 10.75 in.
21.3.3.4
Use No. 3 hoops/stirrups spaced as shown in Figure 1053. The same spacing, turned around, may be used for the left half of beam AB. Where the loading is such that inelastic deformation may occur at intermediate points within the span (e.g., due to concentrated loads at or near midspan), the spacing of hoops will have to be determined in a manner similar to that used above for regions near supports. In the present example, the maximum positive moment near midspan (i.e., 100 ftkips, see Table
106) is much less than the positive moment capacity provided by the three No. 7 continuous bars (172 ftkips). 21.3.3.1 (e) Negativereinforcement cutoff points: For the purpose of determining cutoff points for the negative reinforcement, a moment diagram corresponding to plastic end moments and 0.9 times the dead load will be used. The cutoff point for two of the five No. 8 bars at the top, near support B of beam AB, will be determined. With the negative moment capacity of a section with three No. 8 top bars equal to 218 ftkips (calculated using fs = fy = 60 ksi and φ = 0.9), the distance from the face of the right support B to where the moment under the loading considered equals 218 ftkips is readily obtained by summing moments about section a—a in Figure 1054 and equating these to 218 ftkips. Thus, 51.8 x − 477 − 3.2
x3 = −218 60
10. Seismic Design of Reinforced Concrete Structures Solution of the above equation gives x = 5.1 ft. Hence, two of the five No. 8 bars near support B may be cut off (noting that d = 21.5 in.> l2db = 12 × 1.0=12 in.) at 12.10.3 21.5 x + d = 5.1 + = 6.9 ft say 7.0 ft 12
from the face of the right support B. With ldh (see figure 1035) for a No. 8 top bar equal to 14.6 in., the required development length for such a bar with respect to the tensile force associated with the negative moment at support B is ld = 3.5 ldh = 3.5 × 14.6/12 = 4.3 ft < 7.0 ft. Thus, the two No. 8 bars may be cut off 7.0 ft from the face of the interior support B. 21.5.4.2 At end A, one of the three No. 8 bars may also be cut off at a similarly computed distance of 4.5 ft from the (inner) face of the exterior support A. Two bars are required to run continuously along the top of the beam. 21.3.2.3
545 splices have to be confined by hoops or spirals with a maximum spacing or pitch of d/4, or 4 in., over the length of the lap. 21.3.2.3 (1) Bottom bars, No. 7: The bottom bars along most of the length of the beam may be subjected to maximum stress. Steel area required to resist the maximum positive moment near midspan of 100 ftkips (see Table 106), As = 1.05 in.2 Area provided by the three No. 7 bars = 3 (0.60) = 1.80 in.2, so that As ( provided ) As ( required )
=
1.80 = 1.71 < 2.0 1.05
Since all of the bottom bars will be spliced near midspan, use a class B splice. 12.15.2 Required length of splice = 1.3 ld ≥ 12 in. where ld =
3 d b f y αβγλ 40 f ' c c + k tr db
12.2.3
where α = 1.0 (reinforcement location factor) β = 1.0 (coating factor) γ = 1.0 (reinforcement size factor) λ = 1.0 (normal weight concrete) c = 1.5 + 0.375 +
0.875 = 2.31 2
(governs)
(side cover, bottom bars) or c= Figure 1054. Moment diagram for beam AB
(f)Flexural reinforcement splices: Lap splices of flexural reinforcement should not be placed within a joint, within a distance 2d from faces of supports, or at locations of potential plastic hinging. Note that all lap
1 20 − 2(1.5 + 0.375) − 0.875 = 3.84 in. 2 2
(half the center to center spacing of bars) k tr =
Atr f yt 1500sn
where
546
Chapter 10
Atr = total area of hoops within the spacing s and which crosses the potential plane of splitting through the reinforcement being developed (ie. for 3#3 bars) fyt = specified yield strength of hoops = 60,000 psi s = maximum spacing of hoops = 4 in. n = number of bars being developed along the plane of splitting = 3
k tr =
(3 × 0.11)60,000
c + k tr db
∴ ld =
1500 × 4.0 × 3 =
2.31 + 1.1 = 3.90 > 2.5 , use 2.5 0.875
40
4000
2.5
ld =
3 d b f y αβγλ 40 f ' c c + k tr db
where α = 1.3 (top bars), β = 1.0, γ = 1.0, and λ = 1.0 c = 1.5 + 0.375 +
= 1 .1
3 0.875 × 60,000 1
bending moment (see Table 106), splices in the top bars should be located at or near midspan. Required length of class A splice = 1.0 ld. For No. 8 bars,
c=
1 .0 = 2.375 in. (governs) 2
1 20 − 2(1.5 + 0.375) − 1.0 = 3.81 in. 2 2
ktr= 1.1 = 24.9 in.
Required length of class B splice = 1.3 × 24.9 = 32.0 in. (2) Top bars, No. 8: Since the midspan portion of the beam is always subject to a positive
c + k tr 2.375 + 1.1 = = 3.5 >2.5 use 2.5 1 .0 db 3 1.0 x 60000 1.3 ∴ ld = = 37.0 in. 40 4000 2.5
Required length of splice = 1.0 ld = 37.0 in. (g) Detail of beam. See Figure 1055.
Figure 1055. Detail of reinforcement for beam AB.
10. Seismic Design of Reinforced Concrete Structures 3. Design of frame column A. The aim here is to design the transverse reinforcement for the exterior tied column on the second floor of a typical transverse interior frame, that is, one of the frames in frame T1 of Figure 1048. The column dimension has been established as 22 in. square and, on the basis of the different combinations of axial load and bending moment corresponding to the three loading conditions listed in Table 107, eight No. 9 bars arranged in a symmetrical pattern have been found adequate.(1080,1081) Assume the same beam section framing into the column as considered in the preceding section. f c' = 4000 lb / in. 2 and f y = 60,000 lb / in. 2
From Table 107, Pu(max) = 1076 kips: Pu (max ) = 1076 kips >
Ag f c' 10
=
(22)2 (4) = 194 kips 10
Thus, ACI Chapter 21 provisions governing members subjected to bending and axial load apply. 21.4.1 (a)Check satisfaction of vertical reinforcement limitations and moment capacity requirements: (1) Reinforcement ratio: 0.01 ≤ ρ ≤ 0.06
ρ=
Ast 8(1.0 ) = = 0.0165 Ag (22 )(22 )
547 M e (columns ) ≥
6 M g (beams ) 5
From Section 10.5.2, item 2, φM n− of the beam at A is 218 ftkips, which may be mobilized during a sidesway to the left of the frame. From Table 107, the maximum axial load on column A at the second floor level for sidesway to the left is Pu = 1070 kips. Using the PM interaction charts given in ACI SP17A,(1081) the moment capacity of the column section corresponding to Pu = φPn = 1070kips, fc′ = 4 ksi, fy = 60 ksi, γ = 0.75 (γ = ratio of distance between centroids of outer rows of bars to dimension of crosssection in the direction of bending, and ρ = 0.0165 is obtained as φMn = Me = 260 ftkips). With the same size column above and below the beam, total moment capacity of columns = 2(260) = 520 ftkips. Thus,
∑M
e
= 520 >
6 (6)(218) Mg = 5 5
= 262 ftkips O.K.
21.4.3.1 (2) Moment strength of columns relative to that of framing beam in transverse direction (see Figure 1056)
Figure 1056. Relative flexural strength of beam and columns at exterior joint transverse direction.
21.4.2.2
O.K.
(3) Moment strength of columns relative to that of framing beams in longitudinal direction (see Figure 1057): Since the columns considered here are located in the center portion of the exterior longitudinal frames, the axial forces due to seismic loads in the longitudinal direction are negligible. (Analysis of the longitudinal frames under seismic loads indicated practically zero axial forces in the exterior columns of the four transverse frames represented by frame on line 1 in Figure 1048) Under an axial load of 1.2 D + 1.6 L + 0.5 Lr = 1076 kips, the moment capacity of the column section with eight No. 9 bars is obtained as φMn= Me = 258 ftkips. If we assume a ratio for the negative moment reinforcement of about 0.0075 in the beams of the exterior longitudinal frames (bw = 20 in., d = 21.5 in.), then As = ρbw d ≈ (0.0075)(20)(21.5)
= 3.23 in.2
548
Chapter 10
Assume four No. 8 bars, As = 3.16 in. Negative moment capacity of beam: a=
(1) Confinement reinforcement (see Figure 1038). Transverse reinforcement for confinement is required over a distance l0 from column ends, where
As f y (3.16)(60) = = 2.79 in ' (0.85)(4 )(20) 0.85 f c bw
a φM n− = M g− = φAs f y d − 2 = (0.90)(3.16)(60)(21.5 − 1.39) / 12
= 286 ftkips
l0 ≥
depth of member = 22 in. ( governs ) 1 10 × 12 = 20 in. 21.4.4.4 (clear height ) = 6 6 18 in.
Maximum allowable spacing of rectangular hoops:
smax
1 4 (smallest dimension of column) 22 = = = 5.5 in. 4 4 in. (governs )
21.4.4.2 Figure 1057. Relative flexural strength of beam and columns at exterior joint longitudinal direction.
Assume a positive moment capacity of the beam on the opposite side of the column equal to onehalf the negative moment capacity calculated above, or 143 ftkips. Total moment capacity of beams framing into joint in longitudinal direction, for sidesway in either direction:
∑M ∑M >
g
= 286+ 143 = 429 ft − kips
e
= 2(258) = 516 ft − kips
6 5
∑M
g
=
6 (429 ) = 515 ft − kips 5
O.K. 21.4.2.2 (b) Orthogonal effects: According to IBC2000, the design seismic forces are permitted to be applied separately in each of the two orthogonal directions and the orthogonal effects can be neglected. (c) Determine transverse reinforcement requirements:
Required crosssectional area of confinement reinforcement in the form of hoops:
f c' 0 . 09 sh c f yh Ash ≥ ' 0.3sh Ag − 1 f c c f Ach yh 21.4.4.1 where the terms are as defined for Equation 106 and 107. For a hoop spacing of 4 in., fyh = 60,000 lb/in.2, and tentatively assuming No. 4 bar hoops (for the purpose of estimating hc and Ach)’ the required crosssectional area is (0.09 )(4 )(18.5)(4000) 60,000 2 = 0.44 in Ash ≥ (0.3)(4 )(18.5) 484 − 1 4000 361 60,000 2 = 0.50 in (governs)
21.4.4.3
No. 4 hoops with one crosstie, as shown in Figure 1058, provide Ash = 3(0.20) = 0.60 in.2
10. Seismic Design of Reinforced Concrete Structures
549 determine the design shear force on the column. Thus (see Figure 1042), Vu = 2 Mu/l = 2(293)/10 = 59 kips using, for convenience, Vc = 2 f c' bd =
Figure 1058. Detail of column transverse reinforcement.
(2)
Transverse reinforcement for shear: As in the design of shear reinforcement for beams, the design shear in columns is based not on the factored shear forces obtained from a lateralload analysis, but rather on the maximum probable flexural strength, Mpr (with φ = 1.0 and fs = 1.25 fy), of the member associated with the range of factored axial loads on the member. However, the member shears need not exceed those associated with the probable moment strengths of the beams framing into the column. If we assume that an axial force close to P = 740 kips (φ = 1.0 and tensile reinforcement stress of 1.25 fy, corresponding to the “balanced point’ on the PM interaction diagram for the column section considered – which would yield close to if not the largest moment strength), then the corresponding Mb = 601 ftkips. By comparison, the moment induced in the column by the beam framing into it in the transverse direction, with Mpr = 299 ftkips, is 299/2 = 150 ftkips. In the longitudinal direction, with beams framing on opposite sides of the column, we have (using the same steel areas assumed earlier), Mpr (beams) = Mpr (beam on one side) + M+pr (beam on the other side) = 390 + 195 = 585 ftkips, with the moment induced at each end of the column = 585/2 =293 ftkips. This is less than Mb = 601 ftkips and will be used to
2 4000 (22 )(19.5) = 54 kips 1000
Required spacing of No. 4 hoops with Av = 2(0.20) = 0.40 in.2 (neglecting crossties) and Vs = (Vu − φVc )/ φ = 14.8 kips :
s=
Av f y d Vs
=
(2)(2.0)(60)(19.5) = 31.6 in. 14.8
11.5.6.2 Thus, the transverse reinforcement spacing over the distance l0 = 22 in. near the column ends is governed by the requirement for confinement rather than shear. Maximum allowable spacing of shear reinforcement: d/2 = 9.7 in. 11.5.4.1 Use No. 4 hoops and crossties spaced at 4 in. within a distance of 24 in. from the columns ends and No. 4 hoops spaced at 6 in. or less over the remainder of the column. (d) Minimum length of lap splices for column vertical bars: ACI Chapter 21 limits the location of lap splices in column bars within the middle portion of the member length, the splices to be designed as tension splices. 21.4.3.2 As in flexural members, transverse reinforcement in the form of hoops spaced at 4 in. (
550
Chapter 10
ld =
3 d b f y αβγλ 40 f ' c c + k tr db
where α = 1.0, β = 1.0, γ = 1.0, and λ = 1.0 1.128 = 2.6 in. (governs) 2 1 22 − 2(1.5 + 0.5) − 1.128 or c = = 4.2 in. 2 2 Atr f yt (3 × 0.2) × 60,000 ktr= = = 2 .0 1500sn 1500 × 4 × 3 c + k tr 2.6 + 2.0 = = 4.1 >2.5 use 2.5 1.128 db 3 1.128 x 60,000 1.0 ∴ ld = = 32.1 in. 40 2.5 4000 c = 1.5 + 0.5 +
Thus, required splice length = 1.3(32.1) = 42 in. Use 44in, lap splices. (e) Detail of column. See Figure 1059.
(a) Transverse reinforcement for confinement: ACI Chapter 21 requires the same amount of confinement reinforcement within the joint as for the length l0 at column ends, unless the joint is confined by beams framing into all vertical faces of the column. In the latter case, only onehalf the transverse reinforcement required for unconfined joints need be provided. In addition, the maximum spacing of transverse reinforcement is (minimum dimension of column)/4 or 6 in. (instead of 4 in.). 21.5.2.1 21.5.2.2 In the case of the beamcolumn joint considered here, beams frame into only three sides of the column, so that the joint is considered unconfined. In item 4 above, confinement requirements at column ends were satisfied by No. 4 hoops with crossties, spaced at 4 in. (b) Check shear strength of joint: The shear across section xx (see Figure 1060) of the joint is obtained as the difference between the tensile force at the top flexural reinforcement of the framing beam (stressed to 1.25fy) and the horizontal shear from the column above. The tensile force from the beam (three No. 8 bars, As = 2.37 in.2) is (2.37)(1.25)(60) = 178 kips
Figure 1059. Column reinforcement details.
4.Design of exterior beam—column connection. The aim is to determine the transverse confinement and shearreinforcement requirements for the exterior beamcolumn connection between the beam considered in item 2 above and the column in item 3. Assume the joint to be located at the sixth floor level.
Figure 1060. Horizontal shear in exterior beamcolumn joint.
An estimate of the horizontal shear from the column, Vh can be obtained by assuming that
10. Seismic Design of Reinforced Concrete Structures the beams in the adjoining floors are also deformed so that plastic hinges form at their junctions with the column, with Mp(beam) = 299 ftkips (see Table 109, for sidesway to left). By further assuming that the plastic moments in the beams are resisted equally by the columns above and below the joint, one obtains for the horizontal shear at the column ends Vh =
M p (beam ) story height
=
551
section 21.4.4.3 of ACI Chapter 21 relating to a maximum spacing of 14 in. between crossties or legs of overlapping hoops (see Figure 1041) will not be satisfied. However, it is believed that this will not be a serious shortcoming in this case, since the joint is restrained by beams on three sides.
299 = 25 kips 12
Thus, the net shear at section xx of joint is 178 25 = 153 kips. ACI Chapter 21 gives the nominal shear strength of a joint as a function only of the gross area of the joint crosssection, Aj, and the degree of confinement provided by framing beams. For the joint considered here (with beams framing on three sides), φVc = φ 15 f c' A j
(0.85)(15)(
)
4000 (22 )2 1000 = 390 kips >Vu = 153 kips O.K. =
21.5.3.1 9.3.4.1 Note that if the shear strength of the concrete in the joint as calculated above were inadequate, any adjustment would have to take the form (since transverse reinforcement above the minimum required for confinement is considered not to have a significant effect on shear strength) of either an increase in the column crosssection (and hence Aj) or an increase in the beam depth (to reduce the amount of flexural reinforcement required and hence the tensile force T). (c) Detail of joint. See Figure 1061. (The design should be checked for adequacy in the longitudinal direction.) Note: The use of crossties within the joint may cause some placement difficulties. To relieve the congestion, No. 6 hoops spaced at 4 in. but without crossties may be considered as an alternative. Although the crosssectional area of confinement reinforcement provided by No. 6 hoops at 4 in. (Ash = 0.88 in.2) exceeds the required amount (0.59 in.2), the requirement of
Figure 1061. Detail of exterior beamcolumn connection.
5. Design of interior beamcolumn connection. The objective is to determine the transverse confinement and shear reinforcement requirements for the interior beamcolumn connection at the sixth floor of the interior transverse frame considered in previous examples. The column is 26 in. square and is reinforced with eight No. 11 bars. The beams have dimensions b = 20 in. and d = 21.5 in. and are reinforced as noted in Section item 2 above (see Figure 1055).
552
Chapter 10
(a) Transverse reinforcement requirements (for confinement): Maximum allowable spacing of