# Period Formulas for Concrete Shear Wall Buildings(18)

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## Applied And Interdisciplinary Physics

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PERIOD FORMULAS FOR CONCRETE SHEAR W ALL BUILDINGS By Rakesh K. Goel l and Anil K. Chopra 2 ABSTRACT Most seismic codes specify empirical fonnulas to estimate the fundamental vibration period of buildings. Evaluated first in this paper are the fonnulas specified in present U.S. codes using the available data on the fundamental period of buildings measured from their motions recorded during eight California - quakes, starting with the 1971 San Fernando earthquake and ending w ith the 1994 Northridge e rth~u~e t S shown that current code fonnulas for estimating the fundamental penod of concrete shear wall bUi1dmgs are grossly inadequate. Subsequently, an improved fonnula is developed by calibr~ting   a the.oretical fonnula, deriv~   using Dunkerley's method, against the measured period data through regress10n analys1s A]so recommended IS a factor to limit the period calculated by a rational analysis. such as Rayleigh's method. INTRODUCTION The fundamental vibration period of a building appears in the equation specified in building codes to calculate the design base shear and lateral forces. Because this building property cannot be computed for a structure that is yet to be designed, building codes provide empirical formulas that depend on the building material [steel, reinforced concrete (Re), etc.), building type (frame. shear wall. etc.). and overall dimensions. The empirical period formulas for concrete shear-wall (SW) buildings in the 1997 UBC ( Uniform 1997) and the 1996 SEAOC blue book ( Recommended 1996) were derived by modifying the ATC3-06 formulas ( Tentative 1978) during development of the 1988 SEAOC blue book to more accurately reflect the configuration and material properties of these systems ( Recommended 1988, Appendix 1E2b(1)-T). The period formulas in ATC3-06 ( Tentative ]978) are based largely on motions of buildings recorded during the 1971 San Fernando earthquake. However, motions of many more buildings recorded during recent earthquakes. including the 1989 Loma Prieta and 1994 Northridge earthquakes, are now available. These recorded motions provide an opportunity to expand greatly the existing database on the fundamenta.1 vi~ra- tion periods of buildings. To this end. the natural vlbr~tlon   periods of 21 buildings have been measured by system Identification methods applied to the motions of buildings recorded during the 1994 Northridge earthquake (Goel and Chopra 1997a). These data have been combined with similar data from the motions of buildings recorded during the 1971 San Fernando, 1984 Morgan Hill, 1986 Mt. Lewis and Palm prin~ 1987 Whittier, 1989 Loma Prieta, 1990 Upland, and 1991 S1- erra Madre earthquakes. The objective of this paper is to develop improved empirical formulas to estimate the fundamental vibration period of concrete SW buildings for use in equivalent lateral force analysis specified in building codes. Presented first is the expanded database for measured values of fundamental periods of SW buildings, against which the code formulas in present U.S. codes are evaluated; similar work on limited data sets has appeared previously (e.g., Arias and Husid 1962; Housner and Brady 1963; Cole et al. 1992; Li and Mau 1997). t is shown that current code formulas for estimating the fundamental period of concrete SW buildings are grossly inadequate. Subse-   Asst Prof., Dept of Civ. and Envir. Engrg., California Polytechnic Stale Univ., San Luis Obispo, CA 93407. 2Johnson Prof., Dept. of Civ. Engrg., Univ. of California, Berkeley, CA 94720. quently. an improved formula is developed by calibrating a theoretical formula, derived using Dunkerley's method, against the measured period data through regression analysis. Finally. a factor to limit the period calculated by a rational analysis. such as Rayleigh's method, is recommended. PERIOD D T B SE The data that are most useful. but hard to come by. are from structures shaken strongly but not deformed into the inelastic range. Such data are slow to accumulate because relatively few structures are installed with permanent accelerographs, and earthquakes causing strong motions of these instrumented buildings are infrequent. Thus, it is very important to investigate comprehensively the recorded motions when they do become available, such as during the 1994 Northridge earthquake. Unfortunately, this obviously important goal is not always accomplished, as indicated by the fact that the vibration properties of only a few of the buildings whose motions were recorded during post-1971 earthquakes have been determined. Available data on the fundamental vibration period of buildings measured from their motions recorded during several California earthquakes have been collected (Goel and Chopra 1997a). This database contains data for 106 buildings, including 21 buildings that experienced peak ground acceleration. o 0 15g during the 1994 Northridge earthquake. The remaining data come from motions of buildings recorded during the 1971 San Fernando earthquake and subsequent earthquakes (Hart et al. 1975; Hart and Vasudevan 1975; MacVerry 1979; Cole et al. 1992; Werner 1992; Gates et al. 1994; Marshall et al. 1994; Gael and Chopra 1996, 1997a). Shown in Table 1 is the subset of this database pertaining to 16 concrete SW buildings (27 data points); buildings subjected to peak ground acceleration. 0 15g are identified o with an asterisk (*). C and N denote buildings instrumented by the California Strong Motion Instrumentation Program (CSMIP) and National Oceanic and Atmospheric Administration (NOAA) and ATC denotes one of the buildings included in the ATC3-06 report ( Tentative 1978) for which the height and base dimensions were available from other sources, but these dimensions for other buildings could not be discerned from the plot presented in the ATC3-06 report. The number of data points exceeds the number of buildings, because the period of some buildings was determined from their motions recorded during more than one earthquake or was reported by more than one investigator for the same earthquake. CODE FORMUL S The empirical formula for the fundamental vibration ~er~od   of concrete SW buildings specified in current U.S. bUlldmg codes NEHRP-94 ( NEHRP 1994) SEAOC-96 ( Recom-  T BLE 1 Period Data for Concrete SW Buildings Peak Ground Acceleration Period T Number g) sec) Building ID of Height Longi-Trans-Longi-Trans-Width Length num er Location number stories ft) Earthquake tudinal verse tudinal verse ft) ft) 1 ) (2) 3) (4) 5) (6) 7) 8) 9) (10) (11 ) (12) 1 Belmont C58262 2 28.0 Lorna Prieta 0.10 0.11 0.13 0.20 NA NA 2* Burbank C24385 10 88.0 Northridge 0.26 0.30 0.60 0.56 75.0 215.0 3* Burbank C24385 10 88.0 Whittier 0.22 0.26 0.57 0.51 75.0 215.0 4 Hayward C58488 4 50.0 Lorna Prieta 0.05 0.04 0.15 0.22 NA NA 5 Long Beach C14311 5 71.0 Whittier 0.10 0.10 0.17 0.34 81.0 205.0 6 Los Angeles ATC 3 12 159.0 San Fernando NA NA 1.15 MRF 60.0 161.0 7* Los Angeles C24468 8 127.0 Northridge 0.16 0.11 1.54 1.62 63.0 154.0 8* Los Angeles C24601 17 149.7 Northridge 0.26 0.19 1.18 1.05 80.0 227.0 9 Los Angeles C24601 17 149.7 Sierra Madre 0.07 0.06 1.00 1.00 80.0 227.0 10* Los Angeles N253-5 12 161.5 San Fernando 0.26 0.19 1.19 1.14 76.0 156.0 11* Los Angeles N253-5 12 161.5 San Fernando 0.26 0.19 1.07 1.13 76.0 156.0 12 Palm Desert C12284 4 50.2 Palm Spring 0.07 0.12 0.50 0.60 60.0 180.0 13 Pasadena N264-5 10 142.0 Lytle Creek 0.02 0.02 0.71 0.52 69.0 75.0 14* Pasadena N264-5 10 142.0 San Fernando 0.18 0.22 0.98 0.62 69.0 75.0 15* Pasadena N264-5 10 142.0 San Fernando 0.18 0.22 0.97 0.62 69.0 75.0 16 Piedmont C58334 3 36.0 Lorna Prieta 0.08 0.07 0.18 0.18 NA NA 17 Pleasant Hill C58348 3 40.6 Lorna Prieta 0.08 0.13 0.38 0.46 77.0 131.0 18 San Bruno C58394 9 104.0 Lorna Prieta 0.11 0.13 1.20 1.30 84.0 192.0 19 San Bruno C58394 9 104.0 Lorna Prieta 0.11 0.13 1.00 1.45 84.0 192.0 20 San Jose C57355 10 124.0 Lorna Prieta 0.09 0.11 MRF 0.75 82.0 190.0 21 San Jose C57355 10 124.0 Morgan Hill 0.06 0.06 MRF 0.61 82.0 190.0 22 San Jose C57355 10 124.0 Mount Lewis 0.03 0.03 MRF 0.61 82.0 190.0 23 San Jose C57356 10 96.0 Lorna Prieta 0.10 0.13 0.73 0.43 64.0 210.0 24 San Jose C57356 10 96.0 Lorna Prieta 0.10 0.13 0.70 0.42 64.0 210.0 25 San Jose C57356 10 96.0 Morgan Hill 0.06 0.06 0.65 0.43 64.0 210.0 26 San Jose C57356 10 96.0 Mount Lewis 0.04 0.04 0.63 0.41 64.0 210.0 27* Watsonville C47459 4 66.3 Lorna Prieta 0.39 0.28 0.24 0.35 71.0 75.0 Note: *Denotes building with go 0.15g; NA indicates data not available; MRF implies moment-resisting frames form the lateral load resisting system; number followed by C or N indicates the station number and by ATC indicates the building number in ATC3-06 report ( Tentative 1978). mended 1996), and UBC-97 ( Uniform 1997) is of the form T= C H 3/4 (1) where H = the height of the building in feet above the base; and the numerical coefficient C, = 0.02. UBC-97 and SEAOC96 permit an alternative value for C, to be calculated from C, = 0.l/vA:: 2) where A c, the combined effective area (in square feet) of the shear walls, is defined as NW Ac = 2: Ai [0.2 + D,/H)2]; Di/H:S 0.9 (3) i I in which Ai = the horizontal cross-sectional area (in square feet); { = the dimension in the direction under consideration (in feet) of the ith SW in the first story of the structure; and NW = the total number of shear walls. The value of DJH in (3) should not exceed 0.9. ATC3-06 ( Tentative 1978) and earlier versions of other U.S. codes specify a different formula T= 0.05H (4) Vi where D = the dimension, in feet, of the building at its base in the direction under consideration. UBC-97 and SEAOC-96 codes specify that the design base shear should be calculated from v cw 5) in which W = the total seismic dead load; and C = the seismic coefficient defined as C v I 2.5C a C =- -; O.llCal:S C:S I T R 0.8ZNJ and for seismic zone 4 ~ - R (6) in which coefficients C v and C a depend on the near-source factors, N v and N a, respectively, along with the soil profile and the seismic zone factor Z; I = the importance factor; and R = the numerical coefficient representative of the inherent overstrength and global ductility capacity of the lateral-load resisting system. The upper limit of 2.5 Cal R on C applies to very short period buildings, whereas the lower limit of O.llCal (or 0.8ZNJ R for seismic zone 4) applies to very long period buildings. These limits imply that C becomes independent of the period for very short or very tall buildings. The upper limit existed, although in slightly different form, in previous versions of UBC and SEAOC blue book; however, the lower limit appeared only recently in UBC-97 and SEAOC-96. The fundamental period T calculated using the empirical formula (1) or (4), should be smaller than the true period to obtain a conservative estimate for the base shear. Therefore, code formulas are intentionally calibrated to underestimate the period by about 10-20% at first yield of the building ( Tentative 1978; Recommended 1988). The codes permit calculation of the period by established methods of mechanics (referred to as rational analyses in this paper), such as Rayleigh's method or computer-based eigenvalue analysis, but specify that the resulting value should not  be longer than that estimated from the empirical formula (I) or (4) by a certain factor. The factors specified in various U.S. codes are 1.2 in ATC3-06, 1.3 for high seismic region (Zone 4) and 1.4 for other regions (Zones 3, 2, and I) in UBC-97 and SEAOC-96, and a range of values with 1.2 for regions of high seismicity to 1.7 for regions of very low seismicity in NEHRP-94. The restriction in SEAOC-88 that the base shear calculated using the rational period shall not be less than 80% of the value obtained by using the empirical period corresponds to a factor of 104 (Cole et al. 1992). These restrictions are imposed to safeguard against unreasonable assumptions in the rational analysis, which may lead to unreasonably long periods and hence unconservative values of base shear. EVALUATION OF CODE FORMULAS For buildings listed in Table I, the fundamental period identified from their motions recorded during earthquakes (subsequently denoted as measured period) is compared with the values given by the code empirical formulas [Figs. I(a), 2(a), and 3(a)]. Also compared are the two values of the seismic coefficient for each building calculated according to (6), with = 1 for standard occupancy structures; R = 5.5 for concrete shear walls; and C v = 0.64 and Co = 0.44 for seismic zone 4 with Z = 004 soil profile type Sv, i.e., stiff soil profile with Concrete SW Buildings 75 100 125 150 175 200 Height H, ft (a) o_~ .150 oOgo < C .159 I . ,. ~. .~   ......  ,. .,. ..... . ., .... . ....... :}.-. 1 2~   ' ;.. -   ' .... .,..,. ,.,. . .... ; ::: 31   ... ~.- ~T   ~   \...  T= O.02H~4   ; ~ .. VI 0 '1 .25 0.5 2 lil1.25 1Il r :g ~0.75   1.75 1.5 25 50 0.3 ~alues   I C Iro ~   >--Cod ~   Perio( UO.25 ã MeE sured F rlod 0 10 ~   0.1 9 'E <Il o<O.Hg   MeE Isured F rlod 0 '0 E 02 <3 . 1 -. ·~0.15   ......... 1Il 'Q) l--. (/') r--- i -' ~   0.1 :1 ã D ::l 0.05 25 50 75 100 125 150 175 200 Height H, ft (b) FIG. 1. Comparison of: (a) Measured and Code Periods; (b) UBC-97 Seismic Coefficients from Measured and Code Periods Concrete SW Buildings 2 1.75 1.5 lil1.25 1Il r :g ~0.75   0.5 0.25 ã Q~   0.150 o Ogo ( 0.159 ã 0 0 . 0 .... >. .-:: ......   0 . .... ..... .... .... ã<111 ... ... ~   ............. --   ã 8 0 . ., ~   ...... ....  ::::   J ' :- ~T   = 0.1H I4 A~/~   :\~.   ..... 0 2 6 78 0.3 UO.25 'E <Il '0 ~   0.2 U ·~0.15   1Il jJ G; 0.1 ~   ::l 0.05 Blues ( C Iron I-- COl a Parlo arlod, (' arlod, ( ã Me o Ma sured sured go ~   0.1 go< 0.1 pg 9 ã 0 8 0 o 0 0 -. r ....... 8 I 0 ã 0 2 6 7 8 FIG. 2. Comparison of: (a) Measured and Code Periods; (b) UBC·97 Seismic Coefficients from Measured and Code Periods: Code Periods Are Calculated from Alternate Formula average shear wave velocity between 180 and 360 mis, and Nv = No = I [Figs. I(b), 2(b), 3(b)]. Code Formula: (1) with C l = 0.02 For all buildings in Table I, the periods and seismic coefficients are plotted against the building height in Fig. I. The measured periods in two orthogonal directions are shown by circles (solid for u go ~   0.15g, open for Ugo < 0.15g) connected by a vertical line, whereas the code period is shown by a solid curve because the code formula gives the same period in the two directions if the lateral-force resisting systems are of the same type. Also included are the curves for 1 2T and IAT representing the limits imposed by codes on a rational value of the period for use in high seismic regions like California. The seismic coefficients (6) corresponding to the measured periods in the two orthogonal directions are also shown by circles connected by a vertical line, whereas the value based on the code period is shown by a solid curve. Fig. 1 leads to the following observations. For a majority of buildings, the code formula gives a period longer than the measured value. In contrast, for concrete and steel momentresisting frame buildings, the code formula almost always gives a period shorter than measured value (Goel and Chopra 1996, 1997b). The longer period from the code formula leads to seismic coefficients smaller than the value based on the  Concrete SW Buildings 2 ãC  ~0.15g 1.75 o C go < 0.15g ã ã 1.5 0 1.21/ 0 ~   1.25 0 ã . .   l .. - 0,   · · 8 '1:: 75   . 0 . o.  ~ ., ã 0.5 0 -T=O 05H D112 o .25 0 5 1015 20 25 H D112 (a) 0.3 Valu s of C from 0.25 ) - . Code Perlcd Measured erlod, ago ~   0.15g 1:: 0 Measured erlod, ago < 0 15g 0.2 ã 0 ã ~   0 0 0.15 0 'E l CD n 0 1 0 v -   ~   ::::> ~   0 ã 0.05 5 1015 20 25 H D112 (b) FIG. 3. Comparison of: a) Measured and Code Periods; b) UBC-97 Seismic Coefficients from Measured and Code Periods: Code Periods Are Calculated from ATC3-06 Formula measured period if the period falls outside the flat portion of the seismic coefficient spectrum; otherwise, the two periods lead to the same seismic coefficient. For most of the remaining buildings, the code formula gives a period much shorter than the measured value and a seismic coefficient much larger than the value based on the measured period. Because the code period for many buildings is longer than the measured period, the limits of 1 2T or I 4T for the period calculated from a rational analysis are obviously inappropriate. The building height alone is not sufficient to estimate accurately the fundamental period of SW buildings because measured periods of buildings with similar heights can be very different, whereas they can be similar for buildings with very different heights. For example, in Table 1 the measured longitudinal periods of Buildings 4 and 12 of nearly equal heights differ by a factor of more than three; the heights of these buildings are 50 ft and 50.2 ft, whereas the periods are 0.15 sec and 0.50 sec, respectively. On the other hand, measured longitudinal periods of Buildings 13 and 23 are close even though Building 13 is 50% taller than Building 23; periods of these buildings are 0.71 sec and 0.73 sec, whereas the heights are 142 ft and 96 ft, respectively. The poor correlation between the building height and the measured period is also apparent from the significant scatter of the measured period data [Fig. l(a)]. Alternate Code Formula: 1) with C from 2) and 3) Table 2 lists a subset of nine buildings (17 data points) with their c values calculated from (3) using SW dimensions obtained from structural drawings; for details, see Appendix G of Goel and Chopra (1997a). These dimensions were not available for the remaining seven buildings in Table 1. In Fig. 2 the alternate code formula for estimating the fundamental period is compared with the measured periods of the nine buildings. The code period is determined from (1)-(3) using the calculated value of c and plotted against H 3I4 VIC. This comparison shows that the alternate code formula almost always gives a value for the period that is much shorter than the measured periods and a value for the seismic coefficient that is much higher than from the measured periods. The measured periods of most buildings are longer than the code imposed limits of 1 2T and 1 4T on the period computed from a rational analysis. Although the code period formula gives a conservative value for the seismic coefficient, the degree of conservatism seems excessive for most buildings considered in this investigation. ATC3-Q6 Formula In Fig. 3, the ATC3-06 formula for estimating the fundamental period is compared with the measured periods of all buildings listed in Table I. The code period is determined from (4) using the Hand D dimensions of the building (Table 1) and plotted against H + YD. This comparison demonstrates that (4) significantly underestimates the period and considerably overestimates the seismic coefficient for many buildings and the ATC3-06 imposed limit of 1 2T is too restrictive. The ratio H + Vl is not sufficient to estimate accurately the fundamental period of concrete SW buildings because measured periods of buildings with similar values of this ratio can be very different, whereas they can be similar for buildings with very different values of H + YD. For example, in Table 1 the measured transverse period of Building 18 and measured longitudinal period of building 27-two buildings with similar values of H + YD-differ by nearly a factor of five; H   YD = 7.51 and 7.87, and measured periods = 1.30 sec and 0.24 sec, respectively. On the other hand, the measured longitudinal and transverse periods of Building 9 are the same, equal to 1 sec, even though the values of H + YD in the two directions are 16.7 and 9.93. The poor correlation between the ratio H + YD and the measured periods is also apparent from a large scatter of the measured period data [Fig. 3(a)]. THEORETICAL FORMULAS The observations in the preceding section clearly indicate that the current code formulas for estimating the fundamental period of concrete SW buildings are grossly inadequate. For this purpose, equations for the fundamental period are derived using established analytical procedures. Based on Dunkerley's method (Jacobsen and Ayre 1958, pages 119-120 and 502 505; Veletsos and Yang 1977; Inman 1996, pages 442-449), the fundamental period of a cantilever, considering flexural and shear deformations, is (7) in which T F and T s are the fundamental periods of pure-flexural and pure-shear cantilevers, respectively. For uniform cantilevers T F and T s are given by (Jacobsen and Ayre 1958, pages 471-496; Timoshenko et al. 1974, pages 424-431; Chopra 1995, page 592) T ~   F - 3.516 YEi H 2 (8)
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