evaluation of building period formulas

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2010; 39:1569–1583Published online 23 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.998

Evaluation of building period formulas for seismic design

Oh-Sung Kwon∗,† and Eung Soo Kim

Department of Civil, Architectural and Environmental Engineering, Missouri Universityof Science and Technology, Rolla, MO 65409, U.S.A.

SUMMARY

Building period formulas in seismic design code are evaluated with over 800 apparent building periodsfrom 191 building stations and 67 earthquake events. The evaluation is carried out with the formulas inASCE 7-05 for steel and RC moment-resisting frames, shear wall buildings, braced frames, and otherstructural types. Qualitative comparison of measured periods and periods calculated from the code formulasshows that the formula for steel moment-resisting frames generally predicts well the lower bound of themeasured periods for all building heights. But the differences between the periods from code formula andmeasured periods of low- to-medium rise buildings are relatively high. In addition, the periods of essentialbuildings designed with the importance factor are about 40% shorter than the periods of non-essentialbuildings. The code formula for RC moment-resisting frames describes well the lower bound of measuredperiods. The formula for braced frames accurately predicts the lower bound periods of low-to-mediumrise buildings. The formula for shear wall buildings overestimates periods for all building heights. Forbuildings that are classified as other structural types, the measured building periods can be much shorterthan the periods calculated with the code formula. Based on these observations, it is suggested to useCr factor of 0.015 for shear walls and other structural types. Copyright q 2010 John Wiley & Sons, Ltd.

Received 27 August 2009; Revised 7 January 2010; Accepted 8 January 2010

KEY WORDS: building period formula; seismic design; system identification

1. INTRODUCTION

The fundamental period of a building is a key parameter for the seismic design of a buildingstructure using the equivalent lateral force procedure. As the building period cannot be analyticallycalculated before the building is designed, periods from the empirical period formulas recommendedin seismic design codes or from finite element analysis with assumed mass and stiffness are usedduring the preliminary design stage. In most building design projects, empirical building period

∗Correspondence to: Oh-Sung Kwon, Department of Civil, Architectural and Environmental Engineering, MissouriUniversity of Science and Technology, Rolla, MO 65409, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: Department of Civil, Architectural, and Environmental Engineering at the Missouri Universityof Science and Technology

Copyright q 2010 John Wiley & Sons, Ltd.

1570 O. KWON AND E. KIM

formulas are used to initiate the design process. The period from the empirical period formulaalso serves as a basis to limit the period from a finite element model by applying the upper boundfactor, Cu , suggested in the 2003 NEHRP Recommended Provisions for Seismic Regulations forNew Buildings ([1], referred to as NEHRP 2003 hereafter) and subsequently in ASCE 7-05 [2].

In the 1970s design codes, such as UBC-70 [3] and BOCA-75 [4], two formulas were used toestimate building periods: one for moment-resisting frames (MRFs hereafter) and the other onefor all other structural types as summarized in Table I. These formulas remained in the code untilUBC-82 [5]. From the ATC 3-06 project [6], the period formulas for reinforced concrete and steelmoment-resisting frames (RC MRFs and steel MRFs hereafter) were calibrated based on identifiedbuilding periods from the 1971 San Fernando Earthquake. Seventeen steel MRFs and 14 RC MRFswere used for this calibration. The form of the formulas for the RC and steel MRFs in ATC 3-06[6] were developed based on the assumption that lateral forces are distributed linearly over theheight of a building and that the deflections of the building are controlled by drift limitation.The calibrated building formulas in ATC 3-06 [6] were reflected in BOCA-87 [7] and UBC-88 [8] with minor refinement. The same form of the formula is also applied to other structuraltypes in UBC-88 [8]. More recently, Goel and Chopra [9–11] calibrated the formula for MRFsin the code and developed a new formula for shear wall buildings with measured (or apparent)building periods from several earthquake events. In their study, 42 steel MRFs, 27 RC MRFs, and9 shear wall buildings were used. In the study of shear walls [10], it was found that the buildingperiod formula for shear walls should be a function of equivalent shear area and building heightrather than a function of only building height. Hence, rather than calibrating parameters in theexisting code formula, a new formula was suggested. The suggested period formula for shear wallsand calibrated parameters for MRFs in Goel and Chopra [9–11] were reflected in NEHRP 2000[12] and 2003 which is the basis of the current minimum design load for buildings and otherstructures [2]. Table I summarizes the revision history of approximate building period formulasin design specifications (UBC, BOCA, NEHRP, ASCE 7, Eurocode, and ATC 3-06 [6]) since1970s.

While the building period formulas have been calibrated and revised over the past 30 years,the number of data points that were used for the previous calibrations was limited to take accountthe wide variability of structural periods over building heights. Housner and Brady [24] noteda wide spread in the available data set of periods for shear wall structures, mainly due to largevariations in building stiffness and to a lesser extent effective mass. In addition, well documentedcalibration or evaluation results are not reported for several structural types. For example, the currentapproximate period formula for eccentrically braced frames (EBF), shear walls, and other structuraltypes first appeared in UBC-88 [8]. But these formulas have not been calibrated. Furthermore, allinstrumented buildings are in strong seismic regions. To account for the effects of the design baseshear to structural periods, the upper limit coefficient, Cu , is proposed, which varies with designspectral acceleration level, purely based on engineering judgment. Hence, the empirical buildingperiod formulas derived from periods of buildings in strong seismic regions may or may not beapplicable to the buildings in a region with lower seismic activity.

The California Strong Motion Instrumentation Program (CSMIP) developed by the CaliforniaGeological Survey (CGS) has instrumented over 170 buildings in California since its establishmentin 1972. Among the instrumented buildings, around 40 buildings were instrumented after the1994 Northridge Earthquake. These instrumented buildings have recorded many minor to moderateseismic events in the US for the past several decades. The objective of this study is to evaluatethe current building period formulas in seismic design codes with measured (or apparent) building

Copyright q 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2010; 39:1569–1583DOI: 10.1002/eqe

EVALUATION OF BUILDING PERIOD FORMULAS FOR SEISMIC DESIGN 1571

Table I. Approximate fundamental period formulas.

RC MRF Steel MRF EBF RC/Masonry shear wall Other

UBC-70, 82∗ [13, 14]BOCA-75 [4] Ta =0.10N Ta =0.05hn/

√D

ATC 3-06 [6] Ta =Cth3/4n Ta =0.05hn/

√D

Ct =0.025 Ct =0.035

BOCA 87∗ [7] Ta =Cth3/4n Ta =0.05hn/

√D†

See note∗

Ct =0.030 Ct =0.035

UBC-88, 94, 97∗ Ta =Cth3/4n

Eurocode 8 [15]‖Ct =0.02 or,

Ct =0.030 Ct =0.035 Ct =0.030 Ct =0.1/√AC¶ Ct =0.020

ASCE 7-98 [16] Ta =Cth3/4n

BOCA-96 [17, 18] Ct =0.030 Ct =0.035 Ct =0.030∗∗ Ct =0.020 Ct =0.020NEHRP 94, 97 [19, 20]

or, Ta =0.10N‡ — — —

NEHRP 00 [12], 03ASCE 7-02,05 [2, 21] Ta =Crhxn

Cr =0.016 Cr =0.028 Cr =0.030 Cr =0.020 Cr =0.020x=0.9 x=0.8 x=0.75 x=0.75 x=0.75

or, Ta =0.10N‡ — or, Ta =0.0019hn/√Cw§ —

Note: ∗Rayleigh’s method is also suggested as a period formula for all structural types in BOCA-87 [7],UBC-82-97 [14, 22]. As the equation is not a function of geometry and needs a structural model a priori, theequation is not included in this table.†For shear walls or exterior concrete frames utilizing deep beams or wide piers, or both, D is the dimensionof the building in ft in a direction parallel to the applied force. For isolated shear walls not interconnectedby frames or for braced frames, D is the dimension of the shear wall or braced frame in a direction parallelto the applied force.

‡Applicable to structures not exceeding 12 stories in height and having a minimum story height of not lessthan 10 ft.§Refer to NEHRP 2003 for the definition of Cw .¶Refer to UBC-94 [23] for the definition of Ac.‖Eurocode 8 also suggests using Ta =2

√(d), where d is the lateral elastic displacement of top of the building

in m due to the gravity loads applied in the horizontal direction. Since the equation is not a function ofgeometry and needs a structural model a priori, the equation is not included in this table.∗∗BOCA-96 [17, 18] allows the use of Ct =0.03 for both EBF systems and dual systems using EBF.



periods from the instrumented buildings. To achieve this objective, 141 buildings are selectedfrom the CGS stations and apparent periods of the buildings are identified utilizing the transferfunction method. In addition, periods of 50 buildings from previous studies [6, 9, 10] are compiledto build a database of total 191 buildings. A total of 411 earthquake events recorded from thesebuildings are used to identify more than 800 building periods including periods in the transverse andlongitudinal directions. These measured periods are used to evaluate the building period formulasin [2]. The effects of Occupancy Category, building height, number of stories, and lateral load-resisting systems to the fundamental periods of the buildings are evaluated. From the evaluation,several recommendations for future studies are proposed.

In the following section, selected building stations and adopted system identification methodare presented. In Section 3, the identified building periods are used to evaluate building periodformula in the current seismic design code. This is followed by the conclusions and suggestionsfor future study in Section 4.

2. SELECTED BUILDINGS AND IDENTIFICATION OF PERIODS

To evaluate the approximate period formulas in the current seismic design code, a database of191 building periods is developed. There are about 170 building stations instrumented throughCalifornia Strong Motion Instrumentation Program. After reviewing the plans of the buildings,buildings with large irregularity, base isolation system, or energy dissipation system are excludedfrom the selection. Total 141 CGS stations are selected out of the 170 stations. The 50 non-CGSstations are from ATC 3-06 [6] (26 buildings), National Oceanic and Atmospheric Administration(NOAA, 14 buildings), United States Geological Survey (USGS, 4 buildings), and unknown sources(6 buildings). All buildings investigated in this study are located in California. The data from theselected buildings consist of a total of 67 earthquake events beginning with the 1970 Lytle CreekEarthquake (M 5.2) to the 2008 Yucaipa Earthquake (M 4.1).

The selected buildings have various lateral load-resisting systems including 125 steel MRFs, 58RC MRFs, 56 RC shear walls, 34 concentrically braced frames (CBFs), 8 EBFs, 23 reinforcedor unreinforced masonry shear walls (RM or URM shear walls), 12 precast concrete tilt-up shearwalls (PC shear walls), and 66 other structural types which cannot be classified as one of theabove lateral load-resisting systems. Some of the unclassified buildings consist of dual or multiplesystems. Since transverse and longitudinal lateral load-resisting systems are classified as sepa-rate systems, the total number of lateral load-resisting systems evaluated in this study is 382.The number of data points for RC MRFs (58) is not notably larger than the number of data points(50) that were used in the previous calibration by Goel and Chopra [9]. But other structural typesprovide a considerably large number of new data points that were not available in the previousstudies for the systematic evaluation of period formulas.

The selected buildings are classified into two occupancy categories, namely essential facilities(Occupancy Category IV) and non-essential facilities. Among the 141 CGS stations, 34 buildingsbelong to Occupancy Category III, such as hospitals and buildings for post-earthquake emergencyresponses and communications. Since buildings in Occupancy Category IV are designed withhigher design base shear than the buildings in other occupancy categories, the investigation on theperiods of these buildings can provide insight into the effects of seismic design level on structuralperiods, from which the applicability of the current empirical formula to buildings in low seismicareas can be indirectly evaluated.



0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

Tran

sfer

fu

nct

ion

Period, sec

Roof5th4th3rd2nd

N

S

Plan

Elevation

Ch. Avg = 0.51 sec

Figure 1. Transfer function in NS direction (C13214 station, 2008 Chinohills Earthquake).

The fundamental period in this study refers to the apparent first mode period identified fromrecorded accelerations. It is termed ‘apparent’ as the true fundamental period of a building is hardto be identified due to several factors including the variation of the periods from inelastic responseof soil and structure, soil–structure-interaction, inadequate distribution or insufficient number ofaccelerometers, and noise in the measurement system [25–27]. To minimize the effects of nonlinearresponses on the apparent periods, the periods from low-intensity seismic events are used in thisstudy. As there is no clear transition between linear elastic response and nonlinear response inmost structures, the cut-off PGA of 0.15g is selected, which was used for the calibration of theperiod formulas by Goel and Chopra [9]. By using the cut-off period consistent with the previousstudies, the periods identified from this study can be systematically compared with those in theprevious studies and with the periods calculated from approximate period formulas in the seismicdesign code.

Transfer functions of each building for each direction (transverse and longitudinal direction)are identified with one input channel (typically base acceleration) and several output channelsalong the height of the building. As the base acceleration is used as an input channel, the effectof soil properties or soil–structure-interaction on the apparent building periods is not consideredfor evaluation. Input and output channels of all 141 CGS stations are carefully selected afterinspection of the sensor locations. Output channels at a specific location of the buildings, whichmay independently vibrate or which may not capture the global vibration of the building, suchas at a penthouse or at one of the peculiar wings of the building, are not included in the systemidentification. The average of periods from multiple channels is defined as the apparent period ofthe building. For instance, the transfer functions of CGS station C13214 in Figure 1 show slightlydifferent periods depending on output channels. The average of these periods is used as the periodof the building in the considered direction.

3. EVALUATION OF BUILDING PERIOD FORMULAS

The following sections compare identified building periods with the approximate code formulas.To minimize the period variations due to inelastic deformation of structures, periods from earth-quake events with peak input channel acceleration less than 0.15g are selected for comparisons



except for CBFs. For the CBFs, a threshold of 0.20g is used due to limited number of data points.The effects of importance factors are investigated only for steel MRFs.

3.1. Steel MRFs

There are two approximate period formulas for steel MRFs in the current seismic design provisions.The first period formula, Equation (1), is based on the study by Goel and Chopra [9] where periodsfrom 42 steel buildings (a total of 81 lateral and transverse steel MRF systems) were used forthe calibration. The second formula, Equation (2), has been in the code since the 1970s and isapplicable to structures not exceeding 12 stories and with a minimum story height greater than 10ft.

Ta =Crhxn (1)

Ta = 0.10N (2)

where Cr and x are parameters whose values vary depending on the lateral load-resisting systemof the building, hn is height in ft, and N is number of stories. For the steel MRFs, Cr is 0.028and x is 0.8. The database in this study includes 65 steel MRF buildings (total 125 lateral andtransverse steel MRF systems). The newly added building periods in this study include periodsfrom several low-to-medium rise buildings lower than 100 ft in height. Figure 2 compares theperiods of steel MRFs with the approximate period formula in ASCE 7-05 [2]. The lower boundperiod is calculated using Equation (1). The upper bound period is calculated for a site withSD1�0.4g. Figure 2(a) shows that the current code formula conservatively predicts the lower boundof structural periods for all building heights. However, Figure 2(b) indicates that the differencebetween the periods from code formula and apparent periods is relatively large, especially for thebuildings with heights less than 100 ft, which corresponds to 6–8 story buildings. Based on thestudy by the US Department of Energy with randomly chosen commercial buildings in the UnitedStates, 98% of commercial buildings, including office buildings, have less than 100000ft2 of floorspace which is equivalent to 3–8 story office buildings [28]. Considering that the majority ofbuildings are low-to-medium rise buildings and that Figure 2(b) shows large uncertainty in the codeformula for these buildings, the code formula may need further refinement depending on buildingheights.

Figure 2(c) compares apparent building periods in terms of the number of stories with periodsusing Equation (2) which is applicable to buildings with a minimum story height not less than 10 ftand with the number of stories not exceeding 12. For the purpose of evaluation of the formula, allbuildings are plotted in Figure 2(c) including buildings with minimum story heights of less than10 ft or with more than 12 stories. The database of the selected buildings shows that the buildingsthat have a minimum story height less than 10 ft are mostly residential buildings and hotels. FromFigure 2(c), it can be seen that the formula that has been used for over 30 years predicts the lowerbound of building periods, especially for buildings with less than 5 stories. For buildings withmore than 12 stories, the code formula largely underestimates structural periods of all buildingsincluding buildings with minimum story height less than 10 ft. As the code formula providesconservative period (short periods), Equation (2) may not need to be restricted to buildings withnumber of stories less than 12 and story heights less than 10 ft unless the economic design is aconcern.

Seismic design codes mandate the use of higher design base shear for essential facilities whichare required for post-earthquake recovery or which contain substantial quantities of hazardous



0

1

2

3

4

5

6

7

8

0

Per

iod

, sec

Height, ft

New data pointsFrom literature

Ta=0.028h0.8, Cu=1.4

Ta=0.028h0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

Per

iod

, sec

Height, ft


Ta=0.028h0.8, Cu=1.4

Ta=0.028h0.8

0.0

1.0

2.0

3.0

4.0

0

Per

iod

, sec

Number of Stories

MinStoryHeight <10ftMinStoryHeight >=10ft

Ta=0.1N, Cu=1.4

Ta =0.1N

Code limit

200 400 600 800

50 100 150

5 10 15 20 25

(a)

(b)

(c)

Figure 2. Comparison of periods of steel MRFs from records: (a) comparison of Equation (1) with periodsfrom all buildings; (b) comparison of Equation (1) with periods from low-to-medium rise buildings; and

(c) comparison of Equation (2) with periods from steel MRFs.

substances. For the design of the essential facilities, which belong to Occupancy Category IV inthe ASCE 7-05 [2], higher occupancy importance factors are applied. The occupancy importancefactors were not specified in the design codes in the 1970s. The importance factors, I, from1.25 to 1.5 for essential facilities have been used since 1980s; UBC-82 [14] (I =1.5), BOCA-87[7] (I =1.5), UBC-88, 94 [8, 23] (I =1.25), and ASCE 7-02, 05 [21, 2] (I =1.5). Hence, it isexpected that essential facilities designed and constructed after 1980 were designed with 25–50%



0.0

0.4

0.8

1.2

1.6

2.0

0

Per

iod

, sec

Height, ft

Essential facilitiesNon-essential facilities

T = 0.034h0.8

T = 0.02h0.8

20 40 60 80 100

Figure 3. Effects of occupancy categories to steel MRFs.

higher design base shear than buildings in other Occupancy Category assuming that other designparameters are similar.

To investigate the effects of higher design load on the fundamental periods of steel MRFs,Figure 2 is re-plotted with two categories of periods for essential facilities, such as hospitals andemergency response agencies, and non-essential facilities. There are total 8 steel MRF buildingsthat belong to Occupancy Category IV which were designed after 1980. The heights of thesebuildings are less than 100 ft. The periods of the essential facilities and non-essential facilities inthis height range are compared in Figure 3 and regression analyses are conducted with the twosets of data points for qualitative comparison. For the purpose of comparison, Equation (1) withx=0.8 is used for the regression analysis. Figure 3 shows that the periods of the low-to-mediumrise essential facilities are about 40% shorter than that of the non-essential buildings in the similarheight range. It needs more data to generalize this observation to high-rise buildings. Based on theobservation though, it is suggested to use different building period formulas for low-to-mediumrise essential buildings and non-essential buildings in the same height range. As an alternative,a factor of 0.6 can be applied to the existing code formula for low-to-medium rise steel MRFessential buildings.

The decrease in period with the use of the importance factor is consistent with the findings ofTremblay [29] where it was found that CBF structures may reduce periods by as much as 42%due to application of the importance factor. Nakashima et al. [30] reported similar variations ofbuilding period with seismic hazard levels for high-rise steel frames built in Japan between 1968and 1988. These findings have two implications: (1) Use of the importance factor tends to shortenthe period which in general leads to a larger seismic demand depending on the period. Hence,buildings designed with importance factors may not provide the margin of safety required to ensurethe operational performance of facilities after an earthquake event. (2) The building period dependson the level of design base shear. Buildings designed for lower seismic regions are expected to havelonger periods than buildings in built in higher seismic regions. In the current code, the variationof building period is indirectly considered by allowing higher Cu factor (up to Cu =1.7) in regionswith low seismic base shear demand. With a lack of period data in low seismic regions in theUS, however, further data collection is required to calibrate empirical building period formulas forbuildings in low seismic regions.



Figure 4. Periods of concentrically braced frames.

3.2. Braced steel frames

In the current seismic design provisions, empirical period formula for EBF follows the format ofEquation (1) with Cr =0.030 and x=0.75. This equation has not been calibrated or evaluated sincelate 1980s when the equation was first recommended for EBFs in UBC-88 [8]. The period formulafor CBFs has the same format except Cr =0.020. Hence the EBFs could be designed for longerperiods than CBFs in order to account for the flexibility in the link beams. Recently, Tremblay[29] evaluated the period formula for braced steel frames by reviewing periods of 220 bracedframe buildings. Among the buildings, however, only two data points from a two-story buildingwere measured periods while all others are periods from analytical models. Through theoreticaland parametric studies, Tremblay [29] proposed an equation for CBFs as Ta =0.0076hn , wherehn is building height in feet.

Among the selected CGS stations, there are 34 CBFs and 8 EBFs excluding dual systems inwhich both braced frames and steel MRFs are used as lateral load-resisting systems. And amongthe 34 CBFs, 17 frames were installed with base isolation or seismic dampers. It is speculatedthat the CBFs were used to concentrate deformation demand to the isolation bearings in the baseisolated buildings. The buildings with isolators are not included in the period comparison. Hence,the measured periods of 17 CBFs and 8 EBFs are compared with the formulas in ASCE 7-05 [2]and by Tremblay [29].

Figure 4 compares the measured periods of CBFs with the periods from ASCE 7-05 [2] (Ta =0.02h0.75n ) and Tremblay (2005) (Ta =0.0076hn). Station C24643 has moment frames in the longi-tudinal direction and CBFs in the transverse direction. Since the main lateral load-resisting systemin the transverse direction is CBFs, the period in this direction is included in the comparison. It canbe clearly seen from Figure 4 that the equation for CBF in ASCE 7-05 [2] greatly underestimatesthe period of a high rise building (station C24643), which may lead to the uneconomical design.For low-to-medium rise buildings, the formula follows the lower bound of the measured periods.For all building heights, the formula developed by Tremblay [29] follows the apparent buildingperiods better than the code formula.

Figure 5 compares the measured periods of EBFs with period from ASCE 7-05 [2] (Ta =0.03H0.75

n ). Owing to the limited number of data points, it is difficult to properly evaluate thecurrent code formula. But it can be noted that for the given number of measured data points, theequation in ASCE 7-05 [2] describes the relationship between building height and the apparent



0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

Per

iod

, sec

Height, ft

T = 0.03h0.75

50 100 150 200 250 300

Figure 5. Periods of eccentrically braced frames.

building period. More data points are required to properly evaluate the formula for high-risebuildings.

3.3. RM, URM, and PC shear walls

In ASCE 7-05 [2], two equations, Equations (3) and (4), can be used to calculate the approximateperiods of shear wall structures. The Equation (3) has the same form as Equation (1) except thatCr is 0.02 and x is 0.75. In fact, Equation (3) is applicable to all structural systems as it is the mostconservative equation in the current design code. The equation has been in the code since UBC-88[8] and has not been recalibrated for shear walls. Goel and Chopra [10] evaluated Equation (3)with 16 buildings and found that for majority of buildings the code formula gives periods longerthan the measured values, which are non-conservative. The second formula in ASCE 7-05 [2],Equation (4), proposed by Goel and Chopra [10], is based on measured periods of nine shear wallbuildings. The format of Equation (4) was theoretically derived and calibrated with the measuredbuilding periods. In order to apply Equation (4) in a design process, however, a designer shouldhave the dimensions of the shear walls a priori.

Ta =Crhxn where Cr =0.02, x=0.75 (3)

Ta = 0.0019hn/√Cw (4)

where,

Cw = 100

AB

n∑i=1

(hnhi

)2 Ai[1+0.83

(hiDi

)2]

AB = base area of the structure in ft2

Ai = the area of shear wall i in ft2

Di = the length of shear wall i in fthi = the height of shear wall i in ftn = the number of shear walls in the building effective in resisting lateral forces

in the direction under consideration



0

1

2

3

0

Per

iod

, sec

Height, ft

RC SWPC SWRM & URM SW

C24541C58483

T = 0.02h0.75

T = 0.02h0.75, Cu =1.4

T = 0.015h0.75

50 100 150 200 250 300

Figure 6. Periods of shear wall structures.

Owing to a lack of detailed design information for the buildings in the database, Equation(4) could not be properly evaluated. Therefore, only Equation (3) is evaluated in this study.The measured periods of 56 RC shear walls, 23 RM and URM shear walls, and 12 PC shear wallsare compared with the code formula as shown in Figure 6. The figure shows that the data pointsof periods of shear walls form a trend except the two outlier data points, denoted as C24541 orC58483, which have longer periods than other structures. It can be seen from Figure 6 that fora large number of RC shear wall buildings, the code formula overestimates the lower bound ofstructural period which may lead to unconservative seismic design. This observation is consistentwith the findings of Goel and Chopra [10]. If the simple format of Equation (3) is to be usedfor the design, it is suggested to use Cr of 0.015 which predicts the lower bound of the apparentbuilding periods better than Equation (3) as shown in Figure 6. Another observation that can bemade is that the periods of PC, RM, and URM shear wall buildings are similar to the periods ofthe reinforced concrete shear wall. Hence the suggested Cr factor of 0.015 can be applied to othershear wall building types. As only two shear wall buildings are in Occupancy Category IV, theeffects of the importance factor on the fundamental periods are not investigated for the shear wallbuildings.

3.4. RC MRF

The period formula for RC MRFs has the same format as Equation (1) except Cr =0.016 andx=0.9. Equation (2) can be also applied to calculate the period of RC MRFs. The data pointsin Figure 7(a) show that the current period formula follows the lower bound of the measuredperiods. Figure 7(b) compares the measured periods with Equation (2), which is applicable to astructure with a minimum story height of not less than 10 ft and a maximum number of storiesnot exceeding 12. Equation (2) also follows the lower bound of periods of buildings even up to25 story buildings and even for buildings with a minimum story height less than 10 ft. However,the measured periods are more dispersed when the period is a function of the number of storiesthan when the period is a function of building height.



0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

Per

iod

, sec

Height, ft


Ta=0.016h0.9, Cu=1.4

Ta=0.016h0.9

0.0

0.5

1.0

1.5

2.0

2.5

0

Per

iod

, sec

Number of Stories

Min Story Height < 10ftMinStoryHeight >= 10ft

Ta=0.1N, Cu=1.4

Ta=0.1N

Code limit

50 100 150 200 250 300

5 10 15 20 25

(a)

(b)

Figure 7. Periods of RC MRFs with input ground motions less than 0.15g: (a) periods as a function ofbuilding height and (b) periods as a function of number of stories.

3.5. Other structural types

Equation (1) can also be applied to other structural configurations. Among the 142 selected CGSstations, 65 stations could not be classified as one of the structural systems evaluated in theprevious sections. These stations are in the ‘Other Structures’ category and have lateral load-resisting systems such as plywood shear walls, dual systems (steel brace and MRFs, RC shear walland MRFs), and frames with composite elements. The measured periods of these buildings arecompared with Equation (1) with Cr =0.02 and x=0.75 in Figure 8 where it can be clearly seenthat the approximate period in the code largely overestimates the periods of these buildings. Onereason for the overestimation is the fact that a large number of buildings having dual or multiplelateral load-resisting systems exhibit higher stiffness and strength than buildings with a singlelateral load-resisting system. Some of the outlier data points are denoted in Figure 8 with theirbuilding code. The buildings with shorter measured periods than the periods predicted with thecode formula include C58718 (distributed concrete shear walls and braced steel frames), C24514(concrete shear walls and perimeter steel shear walls), C03743 (concrete walls, concrete momentframes, and steel braces), and C58257 (composite steel plate shear walls and steel MRFs). Thebuildings with longer measured than predicted periods include C24581 (composite steel-concreteframe with infill URM walls on the perimeter), C14654 (chevron-type braced frames and steelMRF), and C57318 (steel MRF and RC shear wall). As there are not many data points for thesebuilding types, the structural periods of buildings in the ‘Other Structures’ category may need to



0

1

2

3

0

Per

iod

, sec

Height, ft

T = 0.02h0.75T = 0.02h0.75, Cu=1.4

←C03743

←C58718←C24514

←C58257

←C24581

←C57318←C14654

T = 0.015h0.75

50 100 150 200 250 300

Figure 8. Periods of other types of structures.

be estimated with a sound mechanical basis such as using the finite element method. To initiatedesign process, however, it is suggested to use Cr factor of 0.015 which predicts the lower boundbuilding period better than the current code formula as shown in Figure 8.

4. CONCLUSIONS AND SUGGESTED FUTURE STUDIES

This study evaluates the period formulas in seismic design provisions through comparison ofapparent periods of instrumented buildings. Periods from 141 building stations in the CGS andperiods from 50 buildings available in the literature are used for this evaluation. The measuredperiods of steel MRFs, braced frames, shear wall buildings, RC MRFs, and other types of structuresare compared with the code formulas. For steel MRFs, the effects of importance factors are alsoinvestigated. The following is a summary of the findings from this study:

• For steel MRFs, the current code formula, Equation (1), describes the general trend of thelower bound of the measured periods. For low-to-medium rise buildings, the uncertainty in thecode formula is high. The code formula may need further refinement depending on buildingheights.

• The essential steel MRF buildings designed with an importance factor tend to have periodsabout 40% shorter than the buildings designed without an importance factor. Hence it issuggested to apply a factor of 0.6 to the approximate building period for the essential steelMRFs. The intention of the importance factor in essential buildings is to increase the safetymargin (or to decrease probability of failure). Since the decreased period may increase seismicdemand depending on periods, the effect of importance factors to the seismic fragility ofessential buildings needs careful reevaluation.

• As was shown in the comparisons involving essential buildings, the level of design base shearaffects the structural period. Since the building period formulas have been derived primarilywith data obtained from buildings in strong seismic regions with large design base shear, theformulas can be conservatively applied to buildings in low seismic activity areas. But oncedata are available, the actual periods of buildings in low seismic region need to be evaluatedto further calibrate code formulas and to better estimate seismic fragility of those buildings.



• For CBFs and EBFs, there are not many available data points due to the limited number ofinstrumented buildings and due to the fact that large portion of the instrumented CBFs havebase isolation systems. But based on the available data points, the code formula, in the formof Equation (1), tends to underestimate the lower bound of the periods for buildings tallerthan 200 ft. For low-to-medium rise buildings, the code formula predicts well the lower boundperiods.

• For RM, URM, and RC shear walls, the code formula in the form of Equation (1) overestimatesthe lower bound of measured periods, which can be unconservative. This finding is consistentwith the research by Goel and Chopra [10]. RM, URM, and RC shear walls have similartrends so a single equation may be applicable to all shear wall types. If Equation (1) is usedfor design, a revision of Cr factor from 0.02 to 0.015 is suggested.

• For RC MRFs, the code formula predicts the lower bound of apparent periods.• For other structural types, a revision of Cr factor from 0.02 to 0.015 is suggested.

The approximate period formula in the current seismic provision ASCE 7-05 [2], has evolved forover three decades. With the new information from the instrumented buildings, the current codeformulas are evaluated in this study. Code formulas for clearly defined lateral load-resisting systemstend to predict the lower bound of the apparent periods. For other structural types, the equationsneed more refinement and calibration. In addition, there are about 30 or more building stationsin low-to-medium seismic region in the National Strong-Motion Network (NSMN) operated byUSGS. Recorded data from these stations may provide a good basis to extrapolate building periodformulas developed and calibrated with buildings in strong seismic region to the buildings in lowseismic region.

ACKNOWLEDGEMENTS

This investigation was carried out with the support from the Department of Civil, Architectural, andEnvironmental Engineering at the Missouri University of Science and Technology. The authors are gratefulfor this support. The authors also appreciate valuable comments by Mr. Lim in ABS Consulting, Inc. onseismic design practice in California.

REFERENCES

1. Building Seismic Safety Council (BSSC). NEHRP recommended provisions for the development of seismicregulations for new buildings. FEMA 450, Washington, DC, 2003.

2. American Society of Civil Engineers. Minimum design loads for buildings and other structures. ASCE 7-05,Reston, Virginia, 2005.

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