three-dimensional modal pushover analysis of buildings subjected to

8/12/2019 Three-dimensional Modal Pushover Analysis of Buildings Subjected To

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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn.2011; 40:789806Published online 30 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1060

Three-dimensional modal pushover analysis of buildings subjectedto two components of ground motion, including its evaluation for

tall buildings

Juan C. Reyes1 and Anil K. Chopra2,,

1Civil and Environmental Engineering, Universidad de los Andes, Bogota, Colombia2Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, U.S.A.

SUMMARY

The modal pushover analysis (MPA) procedure, presently restricted to one horizontal component of ground

motion, is extended to three-dimensional analysis of buildingssymmetric or unsymmetric in plansubjected to two horizontal components of ground motion, simultaneously. Also presented is a variantof this method, called the practical modal pushover analysis (PMPA) procedure, which estimates seismicdemands directly from the earthquake response (or design) spectrum. Its accuracy in estimating seismicdemands for very tall buildings is evaluated, demonstrating that for nonlinear systems this procedure isalmost as accurate as the response spectrum analysis procedure is for linear systems. Thus, for practicalapplications, the PMPA procedure offers an attractive alternative whereby seismic demands can be estimateddirectly from the (elastic) design spectrum, thus avoiding the complications of selecting and scaling groundmotions for nonlinear response history analysis. Copyright 2010 John Wiley & Sons, Ltd.

Received 10 December 2009; Revised 22 July 2010; Accepted 23 July 2010

KEY WORDS: multi-component; pushover; tall buildings; two components

1. INTRODUCTION

The earthquake engineering profession has been moving away from traditional code procedures

to performance-based procedures for evaluating existing buildings and proposed designs of new

buildings. Both nonlinear response history analysis (RHA) and nonlinear static procedure (NSP)

are now used for estimating engineering demand parameters (EDPs)floor displacements, story

drifts, internal forces, hinge rotations, etc.in performance-based engineering of buildings.

Nonlinear RHA of the structure is usually implemented for a few (say, seven or less) ground

motion records, which are selected and scaled appropriately to obtain results close to the median

demand due to a large ensemble of ground motions. Selection and scaling of ground motions

is fraught with several unresolved issues. In contrast, NSP (or pushover) procedures estimate

seismic demands directly from the earthquake design spectrum for the building site, avoiding the

complications associated with selecting and scaling records.To overcome the now well-known limitations of the NSP specified in most building evaluation

guidelines, researchers developed improved procedures [16]. Rooted in structural dynamics theory,

the modal pushover analysis (MPA) was developed to include the contributions of all modes

of vibration that contribute significantly to seismic demands [7]. The MPA procedure achieves

superior estimates of EDPs for steel and concrete moment-resisting-frame (MRF) buildings [8, 9],

Correspondence to: Anil K. Chopra, Civil and Environmental Engineering, University of California, Berkeley, CA94720-1710, U.S.A.

E-mail: [email protected]

Copyright 2010 John Wiley & Sons, Ltd.


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790 J. C. REYES AND A. K. CHOPRA

while retaining the conceptual simplicity and computational attractiveness of standard nonlinear

static procedures. To date, the MPA procedure has been restricted to determine seismic demands

due to a single component of ground motion, although at least the two horizontal components of

ground motion should be considered in computing seismic demands by three-dimensional analysis

of multistory buildings. To address this issue, the first part of the paper extends the MPA procedure

[10] to three-dimensional analysis of buildingssymmetric or unsymmetric in plansubjected to

two horizontal components of ground motion, simultaneously.The second part of the paper is in response to the design and construction of several tall buildings

on the west coast of the U.S., a region known for its high seismicity. Because many of these

buildings use high-performance materials or exceed the height limits specified in building codes,

they are designed using an alternative performance-based procedure [11]. This approach entails

nonlinear RHA of a computer model of the building for an ensemble of multi-component ground

motions to guarantee serviceability and safety equivalent to that of the prescriptive code provisions

for ordinary buildings. Based on a comprehensive investigation [12], this paper explores the

alternative of using the MPA procedure for estimating seismic demands for very tall buildings.

2. MPA FOR TWO COMPONENTS OF GROUND MOTION

2.1. Overall concept

The equations governing the displacements u of the Nfloor diaphragms (assumed to be rigid

in their own plane) of a building subjected to ground motion along two horizontal components

applied simultaneously are

Mu+cu+fS(u)=Mixugx(t)Miy ugy(t) (1)

in which M, a diagonal mass matrix of order 3N, includes three submatrices m, m, and Io; m

is associated with the x-lateral and y-lateral degrees-of-freedom (DOFs), and Io with torsional

DOFs. cis the damping matrix and fS is the vector of resisting forces. The influence vectors ixand iy associated with xand yground motions are

ix=

1

0

0

, iy =0

1

0

(2)where1 and 0 are vectors of dimension Nwith all elements equal to unity and zero, respectively.

Thus, the effective earthquake forces are

peff(t)=sxugx(t)sy ugy(t)=

m1

0

0

ugx(t)

0

m1

0

ugy(t) (3)

The spatial distribution of these forces over the building is defined by the vectors sx or sy ,

and the time variation by ug(t)= u

gx(t) or u

gy(t). These force distributions can be expanded as a

summation of modal inertia force distributions sn[13]:

sx=3N

n=1

snx=3N

n=1

nxMn, sy =3N

n=1

sny=3N

n=1

nyMn (4)

wherenis the n th natural vibration mode of the system vibrating at small amplitudes (within itslinearly elastic range), consisting of three subvectors xn, yn, and n,

n = L n

Mn, Mn =

TnMn, Ln =

TxnMix for ugx(t)

TynMiy for ugy(t)(5)

Copyright 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn.2011; 40:789806

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THREE-DIMENSIONAL MPA OF BUILDINGS 791

Thus,

peff,n(t)=snxugx(t)snyugy(t) (6)

is the n th-mode component of the effective earthquake forces.

In the MPA procedure, the peak response of the building to peff,n(t)=snxugx(t)or the peak

modal demand rnxto one component of ground motion (say, the x-component) is determinedby a nonlinear static analysis of the structure subjected to lateral forces and torques defined by the

modal force distribution

sn =sign(n)

mxn

myn

IOn

(7)

at the peak roof displacement ur nxassociated with the nth-mode inelastic SDF system; the

subscript xor yindicating the ground motion component is omitted from snand n. Because the

signs of the peak modal demands rnxare crucial to the accuracy of the CQC rule, Equation (7) must

include the sign ofnto obtain the correct algebraic sign forrnx. The peak modal demands rnxare

then combined by an appropriate modal combination rule, e.g. the CQC rule, to estimate the totaldemandrx. The MPA procedure applied to linearly elastic systems subjected to one component of

ground motion is equivalent to the standard response spectrum analysis (RSA) procedure.

Such analyses are implemented for each component of the ground motion (x and y), inde-

pendently, and the responses due to individual components are combined using one of the

available multi-component combination rules [1416]. This investigation selected the SRSS

multi-component combination rule assuming that the response quantitiesrxand ryare statistically

independent. This rule is accurate ifugx(t) and ugy(t) are along the principal axes of the ground

motion, or they are of equal intensity [17]. The first condition is usually not satisfied because

the recorded ground motion components were applied along the longitudinal and transverse axes

of the structure. Because the median response spectra for the two horizontal components of the

ensemble of ground motions considered in this investigation are of comparable intensity, the SRSS

rule is appropriate.

2.2. Step-by-step summary

The seismic demandsfloors displacements and story drifts at the C.M.for a symmetric- or

unsymmetric-plan building subjected to two horizontal components of ground motion can be

estimated by the MPA procedure, which is summarized in a step-by-step form (adapted from

Reference [12]):

1. Compute the natural frequencies, n, and modes, n, for linearly elastic vibration of thebuilding.

2. For thenth mode, develop the base shearreference displacement,Vnxur nx, pushover curve

by nonlinear static analysis of the building and applying the force distributionsnx[Equation

(7)]. The reference point is located at the C.M. of the roof, but other floors may be chosen; the

component chosen is in the direction of the dominant motion in the mode being considered.

Gravity loads are applied before the lateral forces, causing roof displacement urg.

3. Idealize the Vnxur nxpushover curve as a bilinear or trilinear curve, as appropriate, and

convert it into the forcedeformation, Fsn/LnDn , relation for then th-mode inelastic SDF

system by utilizing the following equations: Fsn/Ln =Vn/Mn and Dn = (urnurg)/nrn,

where Mn is the effective modal mass of the nth-mode. Starting with this initial loading

curve, define the unloading and reloading branches appropriate for the structural system

and material being considered.

4. Calculate the reference displacement ur n,x=urg+nxrnDnxfor the x-component of theground motion. The peak deformation Dnxof the nth mode inelastic SDF system, defined


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by the forcedeformation relation developed in Step 3 and estimated damping ratio n, isdetermined by nonlinear RHA.

5. From the pushover database (step 2), extract the response valuesrn+g,xdue to the combined

effects of gravity and lateral forces at reference displacements u r n,xdetermined in Step 4.

6. Compute the dynamic response due to the nth mode: rnx=rn+g,xrg, where rn+g,x are

the response quantities obtained in Step 5, and rg are the contributions of gravity loads

alone.7. Repeat Steps 26 for as many modes as required for sufficient accuracy.

8. Determine the total dynamic responses for the x-component of the ground motion by

combining the peak modal responses using the CQC rule, i.e. rx= (

i

n inrixrnx)

1/2.

9. Repeat Steps 28 to determine the total dynamic response ry due to the y-component of

ground motion.

10. Combine the responses rx and ry by the SRSS multi-component combination rule (i.e.

r=

r2x+r2

y ) to determine the dynamic response r, and, then, compute the total responses

rT=rgr; where rgare the responses due to gravity loads.

Steps 110 lead to an estimate of floor displacements and story drifts at the C.M., but not for

member forces or end rotations.

The MPA procedure for estimating seismic demands for buildings subjected to one component

of ground motion has been extended to estimate member forces [18], but it does not lend itself to

automation in a large computer code. To overcome this disadvantage, this procedure is implemented

by first applying gravity loads to the structure, and then imposing a set of displacements that are

compatible with the MPA estimate of story drifts at the C.M. of the building model. This nonlinear

static analysis provides internal forces if both ends of an element develop plastic hinges. Otherwise,

internal forces are determined by implementing Steps 110 as described above.

The preceding procedure, extended to two components of ground motion, is implemented in the

next two steps:

11. Estimate other deformation quantitiessuch as story driftsat locations other than the C.M.

and plastic hinge rotations from the story drifts. First apply gravity loads and then impose

at the C.M. a set of displacements uxand uythat are compatible with the drifts calculated

in Step 10, i.e. u

jx=j

i=1

Tjxand u

jy=

ji=1

Tjy. Four combinations of these displacements

should be imposed: ux+uy , u

xu

y , u

x+u

y , u

xu

y . The largest of the resulting

four values of a response quantity is taken as the MPA estimate. Required for this purpose

is a computer programsuch as SAP2000that allows displacements to be imposed on the

structure instead of loads. In computer programs that do not allow imposing displacements

such as PERFORM-3Da model combining lateral forces with bidirectional gap elements

at each floor could be implemented.

12. Using the plastic hinge rotations obtained in Step 11, internal forces are determined as follows.

If both ends of an element deform into the inelastic range, internal forces are those obtained

in Step 11. Otherwise, determine internal forces by implementing Steps 210. These results

are valid if the calculated bending moments, M, do not exceed the yield capacities, My , of

the structural members. If M>My , scale the internal forces by a factor equal to My/M; this

situation may arise when the rotation calculated in Step 11 is just below the yield rotation.The preceding computational steps were implemented for the two-component ground motions

and the mediandefined as the geometric meanof the resulting data set values for a response

quantity provides an estimate of the seismic demand.

2.3. Simplified MPA for practical applications

The MPA procedure summarized in Section 2.2 can be simplified into two ways. The first simpli-

fication comes in treating the building as linearly elastic in computing the response contributions

of modes higher than the first three modes, an approximation that is supported by Figure 9 that


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follows and by other work [12, 13, 19]; this leads to the modified MPA (MMPA) procedure [19].

Additionally, the contribution of torsional modes is negligible in the response of symmetric-plan

buildings to lateral ground motions; therefore, those modes can be ignored.

The second simplification comes in determining the median value of the peak deformation Dnof

thenth-mode inelastic SDF system that is needed to obtain the reference displacementu rn. Instead

of using nonlinear RHA for each excitation, Dncan be estimated by multiplying the median peak

deformation Dno of the corresponding linear system, known from the design spectrum, by theinelastic deformation ratio CRn. Several empirical equations for CRn are available [2024]; the

equation selected here is taken from Reference [21]. Using this approach to estimate the median

Dnleads to the PMPA procedure, which permits estimation of seismic demands directly from the

design spectrum without having to perform any nonlinear RHA of the modal SDF systems for

each ground motion.

3. STRUCTURAL SYSTEMS AND MODELING

The structural systems considered are 48- and 62-story buildings taken from the Tall Building Initia-

tive project at the Pacific Earthquake Engineering Research Center (PEER) [http://peer.berkeley.

edu/yang/]. The buildings are identified by the letters CW (concrete wall) followed by the numberof stories. The lateral resisting system of the buildings is a ductile concrete core wall consisting of

two channel-shaped walls connected to rectangular walls by coupling beams, as shown in Figures 1

and 2. The channel-shaped walls of the CW48 building have openings that create L-shaped walls

at some stories (Figures 1(a) and 2(a)). The typical floors are 8-in-thick post-tensioned slabs span-

ning between the core and perimeter concrete columns. The total height of the CW48 and CW62

buildings is 471 and 630 ft, respectively. The height-to-width aspect ratio of the core of the 62-story

building is 12:1 in the long direction (x-direction) and 18:1 in the short direction (y-direction), as

shown in Figure 2. To increase the aspect ratio and stiffness of the system in the short direction,

concrete outrigger columns are included. The outrigger columns are connected to the core with

buckling restrained braces (BRBs) at the 28th and 51st floors. Additionally, this building has a

tuned liquid mass damper at the roof to help reduce sway to acceptable comfort levels during

strong wind [25].These buildings were designed according to the 2001 San Francisco Building Code (SFBC) for

soil class Sd. Because these buildings exceed the height limit of 160 ft imposed in the SFBC for

concrete bearing wall systems, the buildings were designed by an alternative performance-based

procedure, allowed in Section 1629.10.1, to meet the equivalent criteria of Section 104.2.8 of

SFBC. The seismic forces were determined for site-specific design spectrum corresponding to a

concrete

core wall

gravity

columns

coupling

beams

CW62

outrigger

columns

CW48

concrete

core wall

coupling

beams

gravity

columns

x

y x

y

cmx

cmx

(a)

(b)

Figure 1. Schematic plans: (a) CW48 building and (b) CW62 building.


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concrete

shear wall

coupling

beams

x

z

y

z

basement

concrete

shear wall

BRB

outriggers

x

z

y

z

(a) (b)

Figure 2. Elevations: (a) CW48 building and (b) CW62 building.

design basis earthquake (DBE) and for a maximum considered earthquake (MCE). The earthquake

forces for preliminary design were determined by linear RSA of the building with the design

spectrum reduced by a response modification factor of 4.5. Subsequently, the preliminary design

was refined based on the results of nonlinear RHA of the buildings for seven ground motions.

The core walls were detailed according to Chapter 21 of the ACI 318-99 code. To ensure a

ductile response of the system, the design included several features: (1) capacity design of the

core walls to avoid shear failure and guarantee a predominantly flexural behavior at plastic hinge

zones; (2) increased moment strength was assigned above the base of the CW48 building to ensure

that the plastic hinge forms at the base of the wall, and (3) capacity design to determine the joint

shear strength, in order to avoid brittle shear failure at the slabcolumn and slabcore joints.

The computer models used in this research were provided by a private company to the TallBuilding Initiative project of the PEER. Because these computer models reflect the current state

of professional practice in California, we decided to adopt them. Both buildings were modeled in

the PERFORM-3D computer program [26] using the following nonlinear models for the various

structural components. The shear wall element in PERFORM-3D is an area finite element with

inelastic fibers, including P-delta effects for both in-plane and out-plane deformations. With four

nodes and 24 DOF, this element has multiple layers that act independently, but are connected at

the element nodes, as shown in Figure 3. The behavior of the element is defined by the combined

behavior of four layers [26]: (1) Axial-bending layer for the vertical axis (Figures 3(a)) with its

cross-section composed of concrete and steel fibers with linear or nonlinear stressstrain relations;

(2) axial-bending layer for the horizontal axis (Figures 3(b)) with its cross-section assumed to be

linearly elastic, because this effect is secondary in a slender wall; (3) conventional shear layer

(Figures 3(c)), assuming constant shear stress and a uniform wall thickness; and (4) out-of-plane

bending (Figure 3(d)), modeled as elastic because this effect is secondary in concrete walls. These

layers have the following additional characteristics. First, in-plane deformations, such as axial strain,

shear strain, and curvature, are assumed constant along the element length. Second, the hysteretic

behavior of the fibers in the axial-bending layer and the material of the shear layer are represented

by a tri-linear model with in-cycle strength deterioration and cyclic stiffness degradation as shown

schematically in Figure 4; this figure does not show the specific model used for the concrete or

steel fibers.

The core walls of the buildings were modeled using this shear wall element considering inelastic

fibers with in-cycle strength deterioration, but without cyclic stiffness degradation or buckling of

the steel bars. Figure 5 shows the finite element mesh for the top stories of the CW48 building.


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(a) Vertical

axial-bending

+

(b) Horizontal

axial-bending

(c) Conventionalshear

concrete

wall

(d) Out-of-plane

bending

+ +

Figure 3. Parallel layers of the shear wall element in Perform-3D; adapted from Reference [26].

excluding stiffness

degradation

including

stiffness

degradation

Force

Deformation

excluding in-cycle

strength deterioration

including in-cycle

strength deterioration

Force

Deformation

(a) (b)

Figure 4. Tri-linear forcedeformation relationships considering: (a) in-cycle strengthdeterioration and (b) cyclic stiffness degradation.

core wall

coupling

beams

Figure 5. Finite element mesh for the top stories of the CW48 building.

Coupling beams, slabs, and girders were modeled by a linear one-dimensional element with

tri-linear plastic hinges at the ends, with ductility capacities specified according to FEMA 356 [20].

The plastic hinges of the coupling beams include in-cycle strength deterioration (Figures 4(a)) and

cyclic stiffness degradation associated with the unloading and reloading stiffnesses (Figures 4(b))

adjusted to reduce the area of the hysteresis loops by 40%.


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Figure 6. Natural periods and modes of vibration of CW48 (top row) and CW62 (bottom row) buildings;shown in the three boxes are x-translational, y-translational, and torsional components of displacements.

The geometric nonlinear effects were considered by a standard P-Delta formulation for the

overall building using an equivalent leaning column to represent the gravity frames, and also locally

at the finite element level in the core wall.

The natural vibration periods and modes of the CW48 and CW62 buildings are shown in Figures 6

and 7, where xcmis the distance from the C.M. to a corner of the building (Figure 1). Figure 6

shows the height-wise variation of lateral displacements and torsional rotations, and Figure 7 shows

the motion of the roof in plan. Note: (1) lateral displacements dominate motion in the first mode of

lateral vibration in the xand ydirections of both buildings, whereas torsional rotations dominatemotion in the third mode; (2) the fundamental vibration periods of the CW48 and CW62 buildings

are 4.08 and 4.50s, respectively; the period of the dominantly torsional mode is much shorter

than that of the dominantly lateral modes; and (3) in modes 1 and 2, the CW48 building moves

simultaneously in the two lateral directions without torsion.

The damping of these buildings was modeled by the Rayleigh dampinga linear combination

of the mass and initial stiffness matrixwith its two constants selected to give 5% damping ratio

at the fundamental period of vibration T1and a period of 0.1T1. Damping ratios range from 2.9

to 7.2%, and from 2.9 to 7.6% for the first 12 vibration modes of the CW48 and CW62 buildings,

respectively.

4. GROUND MOTIONS

A total of 30 ground acceleration records from nine different earthquakes with magnitudes ranging

from 7.3 to 7.9 were selected according to the following criteria: (1) closest distance to the fault

200m/s; and (3) longest

usable period >6.4s. This is the period below which the response spectrum for the high-pass

filtered ground motion is unaffected by filtering of the data. This requirement ensures that selected

ground motions are appropriate for the analysis of these buildings with fundamental periods of 4.0

and 4.5 s. These selected records and their relevant data are listed in [12].

All the 30 records were scaled to represent the same seismic hazard defined by A(T1), the

pseudo-acceleration at the fundamental vibration period T1of the structure. Both components of


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T1=4.08 sec T2=3.77 sec T3=2.65 sec

T1=4.50 sec T2=3.91 sec T3=2.85 sec

CW48

CW62

Figure 7. First triplet of periods and modes of vibration of CW48 and CW62 buildings(only the motion at the roof is shown).

0 1 2 3 4 50

0.5

1

1.5

2

Natural vibration period Tn

, sec

Pseudo-accelerationA,g

CW48

T1

=4.08sec

Hazard spectrum

x-dir

y-dir

0 1 2 3 4 5Natural vibration period T

n

, sec

CW62

T1

=4.50sec

Hazard spectrum

x-dir

y-dir

Median spectrum

Figure 8. Seismic hazard spectrum for building site corresponding to 2% probability of exceedance in50 years (solid line), and the median response spectra of 30 scaled ground motions in the x and y

directions (dashed lines): CW48 and CW62 buildings.

a record were scaled by the same factor selected to match their geometric mean to the seismic

hazard. The geometric mean is defined as A(T1)=

Ax(T1)Ay(T1), where Ax(T1) and Ay(T1)

are the A(T1) values for the two horizontal components of the record. The selected seismic hazard

spectrum was determined as the average of three uniform hazard spectra for 2% probability of

exceedance in 50 years (return period of 2475 years), corresponding to three different attenuation

relationships [2729] as a part of the Next Generation of Ground-Motion Attenuation Models

(NGA) project.

The values of A(T1)2%/50selected to define ground motion ensembles are 0.148gand 0.137g

for the CW48 and CW62 buildings, respectively. Figure 8 shows the median response spectra for

the ensemble of 30 ground motions along the xand ydirections scaled to match A(T1)2%/50for

the two buildings, and the seismic hazard spectrum corresponding to 2% probability of exceedance

in 50 years. As imposed by the scaling criterion, the median pseudo-acceleration of the ground

motion ensemble at the fundamental vibration period of the building is matched to the seismic

hazard spectrum; because T1differs for each structure, the scaling factors for ground motions and

hence their median spectra vary with the building.

After evaluating the MPA procedure for the aforementioned ground motions, which already

represent a low-exceedance-probability hazard for a highly seismic region, these excitations are


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bn

V

rnu

1u

Idealized Actual

2u 3u

Slope 1

Slope 2

Slope 3

Figure 10. Pushover curve idealized as a tri-linear curve.

At displacements between u 1and u 2, sections of the wall around the 28th and 33rd stories yield.

From u2 to u3, the building strength degrades progressively until it abruptly looses its strength

at u3 due to damage in some coupling beams (Figure 9(f)). These coupling beams continue to

deteriorate at displacements larger than u 3until they reach their maximum deformation capacity.

Figure 9 also identifies reference displacements due to each of the 30 scaled ground motions and

their median value; these reference displacements were determined in Step 4 of the MPA procedure

(Section 2.2). All ground motions drive both buildings beyond the first yield displacement u1inthe first and second modes; the median displacement of the CW48 building exceeds its yield

displacement u1 by factors of 3.8 and 3.5 in the xand y directions, respectively; for the CW62

building, these factors are 2.4 and 1.9. More than half of the excitations drive the CW62 building

beyond its first yield displacement u1 in the second mode of lateral vibration in the x and y

directions, but the median displacement is only slightly larger than u1. Only a few ground motions

drive the CW48 building beyond its first yield displacement u 1in its fourth and fifth modes, and

the median displacement is smaller than u1. These results show that the median displacement in

modes higher than the first pair of modes is either close to or exceeds the first yield displacement

u1 only by a modest amount. This is consistent with the results of past research on steel MRF

buildings [8].

5.2. Higher mode contributions in seismic demandsFigure 11 shows the median (over the ground motion ensemble) values of floor displacements

and story drifts in the xand ydirections at the C.M., including a variable number of modes in

MPA, superimposed with the exact result from nonlinear RHA, for each building subjected to

both components of ground motion, simultaneously. Except for the rapid fluctuations near the top

of the CW62 building, which are due to reduction of core wall stiffness and presence of a tuned

liquid mass damper, electromechanical equipment and architectural setbacks, the variation of story

drifts over building height is typical of tall buildings. The observations presented in this section

and in Section 6 will exclude the upper three stories of the CW62 building and basements of both

buildings.

The first pair of modes alone is adequate in estimating floor displacements; including higher

modes does not significantly improve this estimate (Figure 11(a)). Although the first pair of

modes alone is inadequate in estimating story drifts, with the second pair of modes (fourth-and fifth-mode) included, story drifts estimated by the MPA procedure improve significantly, and

resemble nonlinear RHA results (Figure 11(b)). Despite this improved accuracy, notable discrep-

ancies between MPA and nonlinear RHA results remain for both buildings; these discrepancies

are discussed next.

5.3. Evaluation of MPA

The MPA procedure underestimates the x and y components of floor displacements for both

buildings, and the roof displacement is underestimated by about 16 and 12% for the CW48 and

CW62 buildings, respectively. The height-wise average underestimation of story drift is about 15%


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0 0.5 1G

12

24

36

48CW48

x-component

Floor

NL-RHA

MPA

1 pair

2 pairs

3 pairs

0 0.5 1

y-component

0 1 2G

12

24

36

48CW48

x-component

Story

0 1 2

y-component

0 0.5 1G

20

40

60 CW62

Floor Displacement / Building Height, %

Floor

NL-RHA

MPA

1 pair

2 pairs

3 pairs

0 0.5 1 0 1 2G

20

40

60 CW62

Story Drift Ratio, %

Story

0 1 2

)b()a(

Figure 11. Median (a) floor displacements and (b) story drifts at the C.M. of the CW48 (row 1) andCW62 (row 2) buildings determined by nonlinear RHA and MPA, with a variable number of modes.

for the CW48 building and about 18% for the CW62 building. Notable discrepancies remain for

y-direction drifts in the upper part of the CW62 building where the underestimation is around

30%; recall that in this direction, the core wall of the building interacts with BRBs and outrigger

columns, developing plastic hinges at various locations over building height (Section 5.1).

5.4. Evaluation of MMPA

The results of Figure 9 and their interpretation suggested that the buildings could be treated as

linear in estimating contributions of modes higher than the first triplet of modes (or first pair

of modes for a symmetric-plan building) to seismic demandsMMPAas used in Section 2.3to estimate seismic demands for the CW48 and CW62 buildings; torsional modes were ignored

because their effective modal mass is negligible for the x- and y-direction excitations. Figure 12

shows the median values of x and y components of floor displacements and story drifts at the

C.M., determined by nonlinear RHA, MPA, and MMPA for both buildings. The seismic demands

estimated by MMPA and MPA are very closeMMPA values are slightly largerimplying that it

is valid to treat the buildings as linearly elastic in estimating higher-mode contributions to seismic

demands.

6. EVALUATION OF PMPA PROCEDURE

6.1. Floor displacements and story drifts at the C.M.

In implementing the PMPA procedure (Section 2.3), the median value of the peak deformation Dnof the n th-mode inelastic SDF system was estimated by multiplying the median peak deformationDnoof the corresponding linear system by the inelastic deformation ratioCRn; Dnowas determined

from the median response spectrum for the ensemble of 30 ground motions corresponding to the

damping ratio appropriate for the particular mode. As expected, CRnwas essentially equal to 1.0

for all modes of these long-period buildings.

Figure 12 shows the median values of the xand ycomponents of floor displacements and story

drifts at the C.M., determined by nonlinear RHA, MPA, and PMPA procedures for both buildings.

In general, PMPA provides a larger estimate of seismic demands compared with MPA because


DOI: 10.1002/eqe


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0 0.5 1G

12

24

36

48CW48

x-component

Floor

NL-RHA

MPA

MMPA

PMPA

0 0.5 1

y-component

0 1 2G

12

24

36

48CW48

x-component

Story

0 1 2

y-component

0 0.5 1G

20

40

60 CW62

Floor Displacement / Building Height, %

Floor

NL-RHA

MPA

MMPA

PMPA

0 0.5 1 0 1 2G

20

40

60 CW62


Story

0 1 2

)b()a(

Figure 12. Median (a) floor displacements and (b) story drifts at the C.M. of the CW48 (row 1) andCW62 (row 2) buildings determined by nonlinear RHA, MPA, MMPA, and PMPA.

Table I. Ratio of Dndetermined by empirical equation for CRnand by nonlinear RHA of SDF systems.

Building First mode x-dir First mode y-dir

CW48 1.08 1.16CW62 1.13 1.27

the empirical value of Dn is larger than its value determined as the median of the Dn values

determined by nonlinear RHA of the modal SDF system for 30 ground motions (Step 4 of MPA);

the ratio of the two for the first pair of modes is shown in Table I. This is to be expected for

such long-period systems because the empirical equation does not permit values ofCRnbelow 1.0,

whereas the exact data do fall below 1.0 [21]. Despite this overestimation in one step of PMPA,

the method underestimates the seismic demands (Figure 12); however, this underestimation is

primarily due to approximations inherent in modal combination and multi-component combination

rules, as demonstrated next.

The PMPA procedure for concrete buildings subjected to two horizontal components of ground

motion, simultaneously, is based on six approximations, the first three of which are also inherent in

the MPA procedure: (1) neglecting the coupling of modal responses to peff,n(t) in Equation (6)

associated with x or y excitationwhich has been demonstrated to be weak for the structures

considered in this paper [12]; (2) using modal combination rules, originally developed for linear

systems, to determine the total response to one component of ground motion; (3) using multi-

component combination rules, originally derived for linear systems, to determine the response to

both components of ground motion, simultaneously; (4) estimating the peak deformation Dn of

the nth-mode inelastic SDF system from the elastic design spectrum using an empirical equation

for the inelastic deformation ratio; (5) treating the structure as linearly elastic in estimating higher-

mode contributions to seismic demand; and (6) ignoring cyclic stiffness degradation of structural

elements. Because approximations (2) and (3) are the only sources of error in the standard RSA

procedure for linear systems, the resulting error in the response of these systems serves as a

baseline for evaluating the additional errors in PMPA for nonlinear systems.


DOI: 10.1002/eqe


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gnidliub26WCgnidliub84WC

0 1 2G

12

24

36

48CW48

x-component

Story

0 1 2

y-component

RHA

RSA

0 0.5 1 1.5G

20

40

60 CW62

x-component

Story

0 0.5 1 1.5

y-component

RHA

RSA

(a)

0 1 2G

12

24

36

48CW48


Story

0 1 2

NL-RHA

PMPA

0 0.5 1 1.5G

20

40

60 CW62


Story

0 0.5 1 1.5

NL-RHA

PMPA

(b)

Figure 13. Median story drifts at the C.M. for: (a) linearly elastic systems determined byRSA and RHA procedures and (b) inelastic systems determined by PMPA and nonlinear RHA

procedures. Results are for CW48 and CW62 buildings.

Figure 13 compares the accuracy of PMPA in estimating the response of nonlinear systems

with that of RSA in estimating the response of linear systems. Observe that the RSA procedure

underestimates the median response for both buildings. This underestimation tends to be greater

in the upper stories of the buildings, consistent with the height-wise variation of contribution of

higher modes to response [13]. The height-wise average underestimation in story drifts is around

11% for both buildings. The errors introduced by the additional approximations in PMPA, which

are apparent by comparing parts (a) and (b) of Figure 13, are small. The height-wise average

underestimation in the story drifts determined by PMPA is 16 and 9% for the CW48 and the CW62buildings, respectively.

The PMPA is an attractive method to estimate seismic demands for tall buildings due to two

horizontal components of ground motion, simultaneously, not only because it calculates Dndirectly

from the design spectrum, but also because it leads to a slightly larger estimate of seismic demand,

thus reducing the unconservatism (relative to nonlinear RHA) of MPA results.

6.2. Other response quantities

The member forces and total end rotations corresponding to the median story drifts of Figure 12(b)

were estimated by implementing Steps 11 and 12 of the PMPA procedure (Section 2.2). Figure 14

presents such results for bending moments, shear forces, and plastic hinge rotations in the coupling

beams highlighted in Figure 1(a) of the CW48 building, together with the median values of these

responses determined by nonlinear RHA. The internal forces were estimated accurately, whereas

total rotations were underestimated just as the story drifts were underestimated.

The forces in the core wall can be estimated to a useful degree of accuracy by the PMPA

procedure. Figure 15 presents the height-wise variation of shear forces (Vx and Vy) in the core

wall of the CW48 buildings, including a variable number of modes in PMPA superimposed

with the results from nonlinear RHA. The first pair of modes alone is grossly inadequate in

estimating internal forces; however, with the second and third pair of modes included, internal

forces estimated by PMPA resemble nonlinear RHA results. By including the contributions of all

significant modes of vibration, PMPA is able to adequately capture the height-wise variation of

shear forces in the core shear wall. Thus, PMPA overcomes a well-known limitation of pushover


DOI: 10.1002/eqe


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0 1000 2000 3000 4000G

12

24

36

48

Bending moment, kip-ft

Floor

NL-RHA

PMPA

0 500 1000G

12

24

36

48

Shear force, kips

NL-RHA

PMPA

0 0.01 0.02G

12

24

36

48

Plastic hinge rotation, rad

Floor

NL-RHA

PMPA

(a) (b) (c)

Figure 14. (a) Bending moments; (b) shear forces; and (c) plastic rotations for the coupling beams of theCW48 building highlighted in Figure 1(a), determined by nonlinear RHA and PMPA.

0 1 2

x 104

G

12

24

36

48

Vx, kips

Sto

ry

0 1 2

x 104

G

12

24

36

48

Vy, kips

Sto

ry

NL-RHA

PMPA

1 "pair"

2 "pairs"

3 "pairs"

Figure 15. Shear forces Vxand Vyfor the core wall of the CW48 building, determined by nonlinear RHAand PMPA with variable number of modes.

procedures with a single invariant force distribution that are unable to consider the important higher

mode effects after formation of local mechanisms [30].

7. EVALUATION OF THE PMPA PROCEDURE FOR EXTREME SEISMIC HAZARD

Figure 16 shows pushover curves and their tri-linear idealization for each building associated with

their first and second modes in each lateral direction, wherein reference displacements due to

each of the 30 extreme ground motions (defined in Section 4) and their median value are identified.

These results lead to the following observations, with reference to the tri-linear idealization of

the pushover curves shown schematically in Figure 10. Almost all of the excitations drive the

CW48 building into the third slope of the pushover curve for the first- and second-mode, with

median displacements equal to 11.9 to 12.7 times u1the first yield displacementor 1.8 to 2.0

timesu2the second yield displacement; however, the median displacement in the fourth and fifth

modes is only 2.1 to 2.3 times u1. The ground motions drive the CW62 building into the third

and second slopes of the pushover curve for the first and second modes, respectively; the median

displacement is 3.7 to 5.6 times u 1or 0.7 to 1.6 times u 2. The median displacement in the fourth

and fifth modes is 2.7 to 3.1 times u 1.

Figure 17 compares the accuracy of PMPA in estimating the story drifts of nonlinear systems

with that of RSA in estimating the response of linear systems. As mentioned in Section 6.1, the

height-wise average underestimation in RSA is around 11% for both buildings. The additional

approximations in PMPA do not increase the errors fortuitously in this case; the height-wise average

underestimation of story drifts determined by PMPA is reduced to 5 and 7% for the CW48 and

the CW62 buildings, respectively. Figure 18 presents the height-wise variation of shear forces

(Vx and Vy) in the core wall of the CW48 building determined by PMPA and nonlinear RHA,


DOI: 10.1002/eqe


16/18


0 1 2 30

0.05

0.1

0.15

0.2

Baseshear/weight

CW48

"Mode" 1

Median=1.9

7

0 1 2 3

"Mode" 2

Median=1.9

7

0 0.2 0.40

0.1

0.2

0.3

0.4CW48

"Mode" 4

Median=0.1

6

0 0.2 0.4

"Mode" 5

Median=0.1

5

0 0.5 1 1.50

0.05

0.1

0.15

0.2

Baseshear/weight

CW62

Median=1.0

9

0 0.5 1 1.5

Median=0.8

9

0 0.5 10

0.1

0.2

CW62

Median=0.1

9

0 0.5 1

Median=0.1

8

Reference displacement/height,%

(e) (f) (g) (h)

(a) (b) (c) (d)

Figure 16. Modal pushover curves for the first four modes of lateral vibration of CW48 and CW62buildings; trilinear idealization is shown in dashed lines. The reference displacement due to 30 extreme

ground motions is identified and the median value is also noted.

gnidliub26WCgnidliub84WC

0 2 4 6G

12

24

36

48CW48

x-component

Story

0 2 4 6

y-component

RHA

RSA

0 2 4G

20

40

60 CW62

x-component

Story

0 2 4

y-component

RHA

RSA

(a)

0 2 4 6G

12

24

36

48CW48


Story

0 2 4 6

NL-RHA

PMPA

0 2 4G

20

40

60 CW62


Story

0 2 4

NL-RHA

PMPA

(b)

Figure 17. Median story drifts at the C.M. for: (a) linearly elastic systems determined by RSA and RHAprocedures and (b) inelastic systems determined by PMPA and nonlinear RHA procedures for an extreme

seismic intensity. Results are for CW48 and CW62 buildings.

where it is seen that the accuracy of PMPA is adequate. It is reassuring that, even for this extreme

seismic hazard, PMPA estimates the seismic demands to a useful degree of accuracy. As a side,

observe in Figure 17 the similarity between the story drifts for linearly elastic and inelastic systems.

This similarity seems to result for two reasons: (1) overall displacements of long-period systems

are known to be essentially independent of inelastic action and (2) yielding in the core wall is

essentially limited to the lower few stories above grade.


DOI: 10.1002/eqe


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0 2 4 6

x 104

G

12

24

36

48

Vx, kips

Story

0 2 4 6

x 104

G

12

24

36

48

Vy, kips

Story

NL-RHA

PMPA

Figure 18. Median shear forces Vxand Vyfor the core wall of the CW48 building, determined by nonlinearRHA and PMPA for an extreme seismic intensity.

8. CONCLUSIONS

The MPA procedure for estimating floor deformations and story drifts for buildings subjected to

one horizontal component of ground motion has been extended to estimate seismic demandsfloor

displacements, story drifts, internal forces, and plastic hinge rotationsfor buildings subjected totwo horizontal components of ground motion, simultaneously. The PMPA procedure, which is a

version of MPA that is especially suitable for practical application, has been developed to compute

seismic demands directly from the earthquake response (or design) spectrum. This procedure is

based on two principal simplifications: (1) the structure is treated as linearly elastic in estimating

higher-mode contributions to seismic demand; and (2) the median deformation of the nth-mode

inelastic SDF system is estimated directly from the design spectrum using an empirical equation

for the inelastic deformation ratio.

The median seismic demands for 48- and 62-story buildings with ductile concrete shear walls

designed according to the alternative provisions of the 2001 SFBCdue to an ensemble of 30

two-component ground motions were computed by MPA, PMPA, and nonlinear RHA procedures

and compared. The presented results led to the following conclusions:

1. Although, as expected, the first mode of lateral vibration in the x and y directions isinadequate in estimating story drifts, with the second mode of lateral vibration included,

the discrepancies (relative to nonlinear RHA results) reduce greatly and story drifts estimated

by MPA resemble nonlinear RHA results.

2. The MPA procedure may be simplified by treating the building as linearly elastic in estimating

higher-mode contributions to seismic demands. This approximation has been demonstrated

to be valid.

3. The modal combination and multi-component combination approximations inherent in RSA

of linearly elastic systemsa standard tool in structural engineering practicemay lead

to significant underestimation of story drift demands for tall buildings, especially in upper

stories.

4. For the selected tall buildings, the PMPA procedure for nonlinear systems is almost as accurate

as RSA for linear systems even for the extreme seismic hazard considered herein. Thus, PMPA

should be useful for practical application in estimating seismic demandsfloor displacements,

story drifts, plastic hinge rotations, and internal forcesfor tall buildings due to two horizontal

components of ground motion applied simultaneously. One attractive feature is that PMPA

estimates seismic demands directly from the (elastic) design spectrum, thus avoiding the

complications of selecting, scaling, and modifying ground motions for nonlinear RHA.

ACKNOWLEDGEMENTS

The first author acknowledge the fellowship from the Fulbright Commission, Colciencias, and the Univer-sidad de los Andes, Bogota, Colombia to pursue a PhD degree in Structural Engineering at the University


DOI: 10.1002/eqe


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of California, Berkeley. The authors are most grateful to Dr Tony Yang from the PEER for providing thestructural models and ground motion data that served as the basis for this study.

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