511_ftp.pdf

8/22/2019 511_ftp.pdf

1/16

ON THE APPLICABILITY OF PUSHOVER ANALYSIS FOR

SEISMIC EVALUATION OF MEDIUM- ANDHIGH-RISE BUILDINGS

KAI HUANG AND J. S. KUANG*Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology

SUMMARY

The assumption that the dynamic performance of structures is mainly determined from the corresponding single-degree-of-freedom system in pushover analysis is generally valid for low-rise structures, where the structuralbehaviour is dominated by the first vibration mode. However, higher modes of medium- and high-rise structureswill have significant effect on the dynamic characteristics. In this paper, the applicability of pushover analysis

for seismic evaluation of medium-to-high-rise shear-wall structures is investigated. The displacements and inter-nal forces of shear wall structures with different heights are determined by nonlinear response history analysis,where the shear walls are considered as multi-degree-of-freedom systems and modelled by fibre elements. Theresults of the analysis are compared with those from the pushover procedure. It is shown that pushover analysisgenerally underestimates inter-storey drifts and rotations, in particular those at upper storeys of buildings, andoverestimates the peak roof displacement at inelastic deformation stage. It is shown that neglecting higher modeeffects in the analysis will significantly underestimate the shear force and overturning moment. It is suggestedthat pushover analysis may not be suitable for analysing high-rise shear-wall or wall-frame structures. Newprocedures of seismic evaluation for shear-wall and wall-frame structures based on nonlinear response historyanalysis should be developed. Copyright 2009 John Wiley & Sons, Ltd.

1. INTRODUCTION

In the recent 79 Mw-magnitude earthquake in Sichuan, China, many public buildings, including

schools and hospitals, collapsed and were seriously damaged. How to evaluate and retrofit these build-

ings, in particular medium-rise and high-rise buildings, so that they can resist possible earthquake

attacks in the future, is an urgent need in the 512 Sichuan earthquake-stricken regions. Several inelas-

tic analysis procedures have been developed, where the most common one is pushover analysis, which

has been introduced to the framework of the performance-based seismic engineering and implemented

into both Applied Technology Council-40 (ATC, 1996) and Federal Emergency Management Agency-

356 (FEMA, 2000) in the USA for seismic evaluation of concrete buildings. One of the assumptions

in pushover analysis is that the performance of structures is mainly determined from the correspond-

ing single-degree-of-freedom (SDOF) system. This assumption is generally valid for low-rise struc-

tures, where the structural behaviour is mainly dominated by the first vibration mode. However, higher

vibration modes of a medium- or high-rise structure will have significant effect on the dynamiccharacteristics.

Copyright 2009 John Wiley & Sons, Ltd.

* Correspondence to:J. S. Kuang, Department of Civil Engineering, HKUST, Clear Water Bay, Kowloon, Hong Kong. E-mail:[email protected]

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build.19, 573588 (2010)Published online 2 April 2009 in Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/tal.511

8/22/2019 511_ftp.pdf

2/16

574 K. HUANG AND J. S. KUANG

Copyright 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 19, 573588 (2010)

DOI: 10.1002/tal

The applicability of pushover analysis for seismic evaluation of medium- to high-rise shear-wall

buildings is investigated. In this study, the displacements and internal forces of shear-wall structures

with different structural heights are first determined by nonlinear response history analysis (RHA),

where the shear walls are considered as multi-degree-of-freedom (MDOF) systems and modelled by

fibre elements. The results of RHA are then compared with those from the pushover procedure. The

comparisons include two parts. In the first part, the response quantities of the structures predicted by

the two methods are compared under the condition that the designate peak roof drift is equal to the

same predetermined drifts obtained from RHA and pushover analysis, while in the second part,

the peak roof drifts determined based on an MDOF system and the equivalent SDOF system are

compared.

2. STRUCTURES AND GROUND MOTIONS

Three shear-wall structures adopted in this investigation are 12-, 16- and 20-storey buildings, which

are all modified from the eight-storey shear-wall building used in the document FEMA-440/ATC-55

(FEMA, 2005). The original structure is modified in the following ways: (a) Node mass is changed

so that the fundamental periods of the modified structures are equal to those predicted by the empiri-cal formula in Structural Engineers Association of California-96 (SEAOC, 1996)

T A Hc= ( )0 1 1 2 3 4 (1)

whereHis the building height in feet;Acis the combined effective area of the shear walls,

A A D Hc i ii

NW

= + ( ) = 0 2 2

1

(2)

in which Ai is the horizontal cross-sectional area of the ith shear wall; Di is the dimension in the

direction under consideration of the ith shear wall at the first storey of the structure; and NW isthe total number of shear walls. (b) The gravity loads applied to the original structure are kept

the same to the new structures. Gravity loading induces compression in the concrete and steel

fibbers of the model, causing the wall to have an initial stiffness approximately equal to the gross

section stiffness.

The shear walls are modelled using fibre elements in the FEM software OpenSees (Mazzoni et al.,

2006). Figure 1 shows the OpenSees modelling of the 20-storey shear-wall structure, in which the

inelastic material properties of concrete and steel have been modelled. It is assumed that the walls

would have sufficient shear strength and that only elastic shear deformations are needed to be

represented.

From a preliminary pushover analysis of the three shear-wall structures, it is seen from their

capacity curves shown in Figure 2 that the yielding of all the structures will occur when the roof-drift

ratio, which is defined as a ratio of the top drift to the total height of the structure, top/H, reachesabout 05%. In the studies, three top drift levels for the structures with the roof-drift ratios of

02%, 1% and 2% are considered; thus, both elastic and inelastic performances of the structures

can be shown in the analysis. Whereas the roof-drift ratios equal to 1% and 2% can be considered

as the drift levels corresponding to the nominal life safety and collapse prevention performance

limits (FEMA, 2005).

In the analyses, 10 ground motions are selected from Pacific Earthquake Engineering Research

(PEER) Center strong motion database (PEER Center, 2000). The peak ground accelerations range

8/22/2019 511_ftp.pdf

3/16

APPLICABILITY OF PUSHOVER ANALYSIS FOR HIGH-RISE BUILDINGS 575

Copyright 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 19, 573588 (2010) DOI: 10.1002/tal

from 028 gto 041 g, and the peak ground displacements range from 10 cm to 145 cm. The detailed

information of the 10 ground motions is given in Table 1.

The selected ground motions are scaled so that the peak roof drifts are to be equal to the predeter-

mined target values. There are a total of nine sets of scaled factors for these three structures with

different drift levels. The scaled ground motions are used in the investigations on the MDOF effects

and the estimate of roof drift by the SDOF system.

2nd

Basement

1st

22

21

1

1

2

Concrete and Steel Fibers

Roof

20th

19th

21

20

19

19

18

20

1

1

Node Number

Element Number

Node

Element

Figure 1. OpenSees modelling of RC shear walls

8/22/2019 511_ftp.pdf

4/16



DOI: 10.1002/tal

0 20 40 60 800

50

100

150

200

250

300

350

Baseshear(kip)

Yield drift =5.25 inch

( Drift ratio = 0.401% )

Top storey drift (inch)

(a)

0 20 40 60 800

50

100

150

200

250

300

350

Baseshear(kip)



Top storey drift (inch)(b)

0 20 40 60 800

50

100

150

200

250

300

350

Baseshear(kip)



Top storey drift (inch)

(c)

Figure 2. Capacity curves of the shear-wall structures: (a) 12-storey shear-wall structure; (b) 16-storey shear-wall structure; (c) 20-storey shear-wall structure

8/22/2019 511_ftp.pdf

5/16



3. STRUCTURAL RESPONSES

The structural responses determined from the nonlinear RHA may be considered as exact responses

for comparison purposes. The exact responses have in fact reflected the contribution from MDOF

effects, while the pushover analysis is based on a SDOF system. Thus, when the roof displacements

obtained from RHA and pushover analysis are equal to the predetermined drift, the difference between

the response quantities obtained from the two methods is primarily attributable to the presence of

MDOF effects.

3.1 Storey displacement and internal forces

When conducting the pushover procedure that is presented in ATC-40, the recommended lateral force

pattern, which is proportional to the product of elastic first mode amplitude and floor mass, is appliedto the structures. The shear-wall structures are all pushed to a predetermined roof-drift level, and the

obtained storey displacements and internal forces are compared with the results from the nonlinear

time-history analysis. The exact responses of the structures under ground motion is determined by

nonlinear RHA using the computer program OpenSees, where a Rayleigh damping ratio of 2% is

applied to the first- and second-mode periods corresponding to the gross-section stiffness.

The response quantities determined by pushover analysis and RHA, which include the maximum,

minimum, mean and the mean plus and minus one standard deviation values of the dynamic response

quantities at each storey, are plotted in Figures 35. By comparing two sets of results from pushover

procedure and RHA, the findings can be summarized as follows.

(1) Pushover analysis provides reliable estimates of the maximum floor displacement and inter-storey

drift in the elastic range. However, the estimate becomes inaccurate when the structures haveinelastic performance. Pushover analysis underestimates the inter-storey drift, particularly at the

upper storeys of the buildings. This is mainly due to the yielding of some cross sections at

the upper storeys under the intensive ground motion, while this yielding behaviour cannot be

identified by pushover analysis as the higher mode contribution has been neglected.

(2) For the 12-storey shear-wall structure, pushover analysis can predict overturning moments well

in the lower part of the structure, and slightly underestimates those in the upper part. As the higher

model effect becomes significant with the increase in the height of a structure, it is shown that

Table 1. Ground motions

No. Earthquake Date Station location (number) PGA (g) PGV (cm/s) PGD (cm)

1 Northridge 1994/01/17 Canyon CountryW Lost Cany(90057)

041 43 1175

2 Northridge 1994/01/17 PardeeSCE 0406 436 1209 3 Chi-Chi, Taiwan 1999/09/20 TCU079 0393 488 1378 4 Westmorland 1981/04/26 Westmorland Fire Station (5169) 0368 487 1061 5 Imperial Valley 1979/10/15 Aeropuerto Mexicali (6616) 0327 428 101 6 Loma Prieta 1989/10/18 GilroyHistoric Bldg. (57476) 0284 42 111 7 Landers 1992/06/28 Joshua Tree (22170) 0284 432 1451 8 Chi-Chi, Taiwan 1999/09/20 CHY035 0252 456 1203 9 Imperial Valley 1979/10/15 Agrarias (6618) 0221 424 11710 Loma Prieta 1989/10/18 Alameda Naval Air Stn Hanger 23 0209 425 1407

PGA, peak ground acceleration; PGV, peak ground velocity; PGD, peak ground displacement.

8/22/2019 511_ftp.pdf

6/16



DOI: 10.1002/tal

Floor displacement (inch)

(a) (b) (c)

Storey

Floor displacement (inch)

Storey

Inter-storey drift (inch)

storey

MinSDSD

Mean

MaxPushover Analysis RHA

Figure 3. Comparison of floor displacements and inter-storey drifts of the 20-storey shear-wall structuredetermined by pushover analysis and RHA. (a) at 1% drift level; (b) at 2% drift level; (c) at 2% drift level

for the 16- and 20-storey buildings, pushover analysis underestimates the overturning moments

with either elastic or inelastic deformations.

(3) Pushover analysis is relatively poor for predicting shear forces at both elastic and inelastic per-

formance stages. The contribution of higher vibration modes has significant effect on the shear

forces. Neglecting the higher mode effect in the evaluation procedure may lead to significant

underestimation of shear forces of the structure.

3.2 Peak roof displacement

The preceding analysis focuses on the investigation of the accuracy of pushover analysis due to MDOF

effects when the structures are subjected to a predetermined drift level. The underlying assumption in

the previous analysis is that accurate estimate of the peak roof displacement can be obtained using a

model of an equivalent SDOF system. However, this assumption may not always be correct. It is

shown (Chopra et al., 2003) that the equivalent SDOF models is to potentially overestimate the peak

roof displacements of generic frame structures subjected to large ductility demand, but underestimate

for those with small ductility demand.

Based on the capacity curves shown in Figure 2, the structure can be simplified to an equivalent

SDOF system, and seismic performance can then be estimated. According to ATC-40, the spectral

displacement at yielding of the equivalent SDOF system is determined by

Sd yy roof

roof,

,= 1 1

(3)

8/22/2019 511_ftp.pdf

7/16



where y,roof is the roof displacement at yield, 1 is the first-mode participation factor and f1roof isamplitude of the first mode at the roof. The spectral acceleration at yielding of the equivalent SDOF

system is given by

SV

Wga y

y base

,

,=1

(4)

where

1

1

1

2

1

12

1

=( )

( )

=

= =

w g

w g w g

i i

i

N

i

i

N

i i

i

N (5)

where Vy,base is the base shear at yield, gis the gravity acceleration, W is the weight of the MDOF

system, a1is the modal mass coefficient, wi/gis mass assigned to level i, fi1is the amplitude of thefirst mode at level iandNis the uppermost level in the main portion of the structure. Based on the

spectral displacement and spectral acceleration at yield, an equivalent SDOF system representing

the shear-wall structure is developed using a bilinear hysteretic model.

Both the equivalent SDOF systems and the detailed MDOF systems of the structures are subjected

to the scaled ground motions. Nonlinear RHAs are then conducted. The ratio of the peak roof displace-

0 2 4

2

(a) (b) (c)

4

6

8

10

12

Storey

Moment (in-kip)

0 1 2 3

2

4

6

8

10

12

14

16

Storey

Moment (in-kip)0 2 4

2

4

6

8

10

12

14

16

18

20

Storey

Moment (in-kip)

Min

SDSD

Mean

MaxPushover Analysis RHA

Figure 4. Comparison of overturning moments at 2% drift level determined by pushover analysis and RHA.(a) 12-storey structure; (b) 16-storey structure; (c) 20-storey structure

8/22/2019 511_ftp.pdf

8/16



DOI: 10.1002/tal

ment estimated by the equivalent SDOF system to that obtained from the nonlinear RHA of the MDOF

system is calculated for each ground motion record. Statistical analysis results of this ratio are pre-

sented in Figure 6.

For a roof-drift level of 02%, the mean displacement ratios are between 080 and 085 for all three

structures. The underestimation of the roof drift in equivalent SDOF systems is due to the neglect of

higher vibration modes. However, the equivalent SDOF systems slightly overestimate the peak roof

displacement in the roof drift by 2%. By comparing with the results of the nine-storey steel-frame

structures given in FEMA-440 (FEMA. 2005), it can be seen that the equivalent SDOF system may

provide better estimate of the roof displacement of shear-wall structures than that of frame structures

with inelastic deformation. The reason is mainly due to the different deformation shapes of shear-wall

structures and frame structures.

4. CONTINUUM MODEL

4.1 Elastic continuum model

To understand the effect of higher vibration modes on the seismic behaviour of shear-wall structures,

the continuum model is used on the analysis. In general, the shear-wall structure can be simplified as

a flexural cantilever, where the shear deformation is neglected. The governing equation of flexural

cantilever with a fixed base subjected to horizontal ground excitation is given by

Shear force (kip)

Storey

Shear force (kip)

Storey

MinSDSD

Mean

MaxPushover Analysis

(a) (b)

RHA

Figure 5. Comparison of shear forces determined by pushover analysis and RHA for the 20-storey shear-wallstructure. (a) at 02% drift level; (b) at 2% drift level

8/22/2019 511_ftp.pdf

9/16



m xu x t

tc x

u x t

t H xEI x

u

xm( )

( )

+ ( ) ( )

+

( )

=

2

2 4

2

2

2

2

1, ,xx

u t

t

g( )

( )

2

2 (6)

where m(x) is the mass per unit length,u(x,t) is the relative displacement of cantilever at the height

ratiox, which is a ratio of the structural height zto the total height of the buildingH,z/H, at time t,

His the total height of the building, c(x) is the damping coefficient per unit length, ug(t) is the ground

displacement, andEI(x) is flexural rigidity along the structural height.

Elastic response of the structure can be computed from modal analysis. For a continuous flexural

cantilever, the displacement u(x,t) can be calculated as a linear combination of modal responses

u x t u x t ii

, ,( )= ( )=

1

(7)

where ui(x,t) is the contribution of the ith mode to the response. When classical damping is assumed

u x t x D t i i i i,( )= ( ) ( ) (8)

where iis the modal participation factor of the ith mode of vibration, fi(x) is the amplitude of theith mode shape of vibration and Di(t) is the deformation response of a SDOF system corresponding

to the ith model to the ground motion, whose response is computed with the following equation of

motion (Miranda and Taghavi, 2005)

d D t

dt

dD t

dtD t

d u t

dt

ii i

ii i

g2

2

22

22

( )+

( )+ ( )=

( ) (9)

For a flexural cantilever with uniformly distributed mass, the modal participation factor of the ith

mode of vibration is given by

Drift level

(a) (b) (c)

DESDOF

/DMDOF

Drift level Drift level

Min

SDSD

Mean

Max

Figure 6. Statistical distribution of roof-displacement ratios. (a) 12-storey structure; (b) 16-storey structure;(c) 20-storey structure

8/22/2019 511_ftp.pdf

10/16



DOI: 10.1002/tal

ii

i

x

x=

( )

( )

dx

dx

0

1

2

0

1 (10)

Since the response of the SDOF system can be obtained from spectrum analyses, the maximumcontribution of the ith mode can be computed by

u x x S T i i i d i( ) = ( ) ( ),max (11)

where Sd(Ti) is the value of the displacement response spectrum corresponding to the ith mode of

vibration. The overall displacement can then be computed by the square-root-of-sum-of-squares

u x u x ii

N

max ,max( ) ( )

=

21

1 2

(12)

Similarly, the seismic-equivalent static force associated with the ith mode is given by

F x m x S T i i i a i( )= ( ) ( ) (13)

where Sa(Ti) is the value of the displacement response spectrum corresponding to the ith mode of

vibration. Therefore, the shear force in the non-dimensional height ratio xcan be determined by

V x V x ii

N

( ) ( )

=

21

1 2

(14)

where

V x F x i ix

( )= ( ) dx1

(15)

The bending moment of the shear wall along the structural height is determined by

M x M xii

N

( ) ( )

=

21

1 2

(16)

M x H F x xi ix

( )= ( ) ( ) 11

dx (17)

4.2 Equivalent linearization technique

For structures with nonlinear behaviour in the intensive ground motion, the modal analysis method

for elastic structures is no longer valid. To understand the nonlinear behaviour of a shear wall where

a plastic hinge is formed at the bottom of the wall, the equivalent linearization techniques is used. As

shown in Figure 7(a), the basic assumption for equivalent linearization techniques is that the maximum

inelastic deformation of a nonlinear structure member can be approximated from the maximum defor-

mation of a linear elastic substitute member that has a stiffness given by (Shibata and Sozen, 1976)

8/22/2019 511_ftp.pdf

11/16



EIEI

eqa( ) =

( )

(18)

where (EI)eqis the equivalent flexural stiffness for the substitute member, (EI)ais the cracked-section

flexural stiffness and mis the damage ratio, which is comparable to but not exactly the same as duc-tility based on the ratio of maximum to yield rotation. Quantitatively, damage and ductility ratios are

identical only for elastoplastic response. It is assumed that the plastic zone is formed from the bottom

to the height of lHunder the horizontal ground motion, and the equivalent flexural stiffness is uniformin the plastic zone. Therefore, as shown in Figure 7(b), the flexural stiffness along the height of the

flexural cantilever is given by

EI x

EIx

EI x

a

a

( )=

<

1

(19)

where lis the relative plastic zone height. Moreover, the damping ratio for the equivalent linearizationelement is given by (Shibata and Sozen, 1976)

eff=

+ 0 2 11

0 021 2

(20)

Because the lateral stiffness along the height of the cantilever has two different values, a closed-

form solution for mode shape is difficult to be derived. Therefore, in order to study the influence of

Curvature

M

cy ctarget

My

( ) ( )

a

eq

EIEI

u=

: cracked sectionaEI

x

( )gu t:

H

H

(1-)H

EIe

q

EIa

Figure 7. Flexural cantilever model for shear-wall structures. (a) equivalent flexural stiffness for plastic zone;(b) stiffness distribution

8/22/2019 511_ftp.pdf

12/16



DOI: 10.1002/tal

plastic zone to the dynamic characteristic of the flexural cantilever, mode shapes, periods and modal

participation factors are calculated using finite element analysis. For this purpose, the model was

discretized into 100 equal-length elements. For the mass matrix, a uniformly distributed lumped-mass

approximation is used.

It is assumed that the length of plastic zone is 20% of the total structure height. The product of the

modal participation factor and model shape for the first three vibration modes are shown in Figure 8,

where the damage ratios are 2, 4 and 8. It is seen that the plastic zone existing in the bottom of the

shear wall has a negligible effect on the product of the modal participation factor and model shape.

Considering Equation (10), it can be thought that the difference between the ith modal deformation

contribution of shear wall with and without yielding is mainly determined by the value of spectral

displacement.

The periods of vibration modes will shift when the bottom of the shear wall yields. The period

ratios are defined as the ratio of vibration mode period of the structure with plastic zone to the cor-

responding period of elastic structure without plastic zone. The relation between period ratios and

damage ratios for the first three vibration modes are shown in Figure 9. It can be seen that the period

-1 0 1 -1 0 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

1f1

2f2

3f3

x

Elastic

=2=4

=8

=0.2

Figure 8. Effect of damage ratio on product of mode shape and modal participation factor of flexuralcantilever (=02)

0

1

2

1 3 5 7

Damage ratio

Period

ratio

T1T2

T3

Figure 9. Effect of damage ratio on period ratio of vibration mode (=02)

8/22/2019 511_ftp.pdf

13/16



ratio of the first mode is increasing, apparently with the increase in the damage ratio, whereas the

period ratios for the second and third modes do not increase significantly.

4.3 MDOF effect

To understand the MDOF effect on the nonlinear behaviour of shear-wall structures, the deformation

and internal force are calculated by only the first mode and by the first three modes, respectively,

according to the modal analysis method. The design spectrum of Uniform Building Code (International

Conference of Building Officials, 1997) shown in Figure 10, where the seismic zone is chosen to be

2A and the soil profile type is chosen to be Sc. The reduced acceleration spectrum with damping ratio

of 13% is also computed according to ATC-40, which is used for the response calculation of the

equivalent linearization system. For a flexural cantilever with a fundamental period of 23 s, the defor-

mation, shear force and overturning moment are computed by the first mode and the first three modes,

respectively, and the results are compared in Figures 1113.

Figure 11 shows the deformation shapes of the flexural cantilever when the cantilever remains elastic

and that the damage ratio mis equal to 8. It can be seen that the deformation shapes computed only

by the first mode agree well with that computed by the first three modes, showing that the highervibration modes have a negligible effect on the flexural cantilevers deformation. Therefore, although

pushover analysis is based on an equivalent SDOF system, it can generally predict the storey displace-

ment well, as shown in Figure 3(a).

By comparing the magnitudes of deformation, it is seen that the cantilever with the plastic zone at

the bottom has a much larger deformation than the cantilever that remains elastic. The main reason is

that the fundamental period of the cantilever becomes much longer with the formation of plastic zone,

as shown in Figure 9.

Considering Equation (10) and the little change in product of the participation factor and mode

shape, it can be thought that the deformation of the cantilever will increase with the increase in the

spectral displacement corresponding to the fundamental period. However, because the period for the

second and third vibration modes do not increase significantly as shown in Figure 9, their contribution

to the cantilevers deformation will be insignificant. Moreover, by comparing with the contributionincrement of the first model, it is seen that the MDOF effect on the deflection of the cantilever is

reduced when the bottom of the structure yields.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6Period (Sec)

Spectra

lAcceleration

(g)

=5%

=13%

Figure 10. UBC-97 acceleration design spectrum for zone 2A

8/22/2019 511_ftp.pdf

14/16



DOI: 10.1002/tal

0

a b

0.2

0.4

0.6

0.8

1

0 0.2 0.4

u(m)

x

elastic

3 modes

1st mode

0

0.2

0.4

0.6

0.8

1

0 0.4 0.8

u(m)

x

3 modes

1st mode

m=8

Figure 11. Deformation shape of flexural cantilever. (a) elastic deformation; (b) plastic deformation (m=8)

0

0.2

0.4

0.6

0.8

1

0 100 200

(a) (b)

F/m (m/s2)

x

elastic3 modes

1st mode

0

0.2

0.4

0.6

0.8

1

0 100 200

F/m (m/s 2)

x

3 modes

1st mode

=8

Figure 12. Shear force of flexural cantilever. (a) cantilever is elastic; (b) cantilever is plastic (m=8)

0

0.2

(a) (b)

0.4

0.6

0.8

1

0 30 60

M/(mH) (m/s 2)

x

elastic3 modes

1st mode

0

0.2

0.4

0.6

0.8

1

0 30 60

M/(mH) (m/s 2)

x

m=8

3 modes

1st mode

Figure 13. Normalized overturning moment of flexural cantilever. (a) cantilever is elastic; (b) cantilever isplastic (m=8)

8/22/2019 511_ftp.pdf

15/16



Figure 12 shows the normalized shear force of the flexural cantilever when the cantilever remains

elastic and the damage ratio m is equal to 8. It can be seen that the shear force computed only bythe first mode is greatly underestimated. The spectral acceleration corresponding to the second

and third modes is relatively large, comparing with that corresponding to the first mode. By consider-

ing Equation (13), it is shown that the contribution of the second and third vibration modes to the

seismic-equivalent static force is in the same magnitude order as that of the first mode. Therefore,

the MDOF effect cannot be neglected in the shear force calculation, as shown in Figure 5. It

is also seen from Figure 12, by comparing the shear force when the cantilever remains elastic with

that when the cantilever has a plastic zone, that the shear force decreases significantly when the

plastic zone is formed at the bottom of flexural cantilever. This is mainly due to the fact that the

spectral acceleration corresponding to the first mode is greatly reduced as the fundamental period

becomes longer.

Figure 13 shows the normalized overturning moment of the flexural cantilever when the cantilever

remains elastic and that the damage factor m is equal to 8. It can be seen that the overturningmoment computed only by the first mode agrees well with that computed by three modes at the lower

part of the cantilever, while the overturning moments are underestimated at the middle and upper

parts of the cantilever if it is computed only by the first mode, especially when the plastic zoneis formed at the bottom of the flexural cantilever. This may explain the reason why the

pushover analysis underestimates the overturning moment at the upper part of the structure, as shown

in Figure 4.

5. CONCLUSION

Based on the investigation of pushover analysis applied to seismic assessment of medium- and high-

rise shear-wall structures, the following conclusions can be drawn.

(1) Pushover analysis provides reliable estimates of the maximum floor displacement and inter-storey

drift in an elastic range, but underestimates the floor displacement and inter-storey drift in an

inelastic range, particularly at upper storeys of the buildings.(2) Pushover analysis can generally predict overturning moments well for low-rise shear-wall struc-

tures, but underestimate these moments for medium- and high-rise buildings with elastic or

inelastic deformations.

(3) Pushover analysis is poor for predicting shear forces.

(4) The equivalent SDOF model underestimates the peak roof displacement at the elastic stage

and may overestimate the peak roof displacement at the inelastic stage for shear-wall

structures.

(5) This investigation suggests that pushover analysis may not be suitable for the use of analysing

medium- and high-rise shear-wall structures, as the contributions from the higher vibration modes

to the structural responses cannot be ignored in seismic evaluation procedures. Since the param-

eters of vibration modes of a structure are varied with time in the nonlinear behaviour, the modal

analysis method for the elastic system cannot be applied. Methods based on nonlinear responsehistory analysis should be developed to facilitate the preliminary seismic evaluation of the

structures.

ACKNOWLEDGEMENT

The support of the Hong Kong Research Grant Council under grant No. 614308 is gratefully

acknowledged.

8/22/2019 511_ftp.pdf

16/16



DOI: 10.1002/tal

REFERENCES

ATC. 1996. Seismic Evaluation and Retrofit of Concrete Buildings, ATC-40. Applied Technology Council:Redwood City, CA.

Chopra AK, Goel RK, Chintanapakdee C. 2003. Statistics of single-degree-of-freedom estimate of displacementfor pushover analysis of buildings.Journal of Structural Engineering ASCE129(4): 459469.

FEMA. 2000. Prestandard and Commentary for the Seismic Rehabilitation of Buildings, FEMA 356. FederalEmergency Management Agency: Washington, DC.

FEMA. 2005.Improvement of Nonlinear Static Seismic Analysis Procedures, FEMA 440/ATC-55. Federal Emer-gency Management Agency: Washington, DC.

International Conference of Building Officials. 1997. Uniform Building Code, Vol. 2. International Code Council:Whittier, CA.

Mazzoni S, McKenna F, Scott MH, Fenves GL, et al. 2006. OpenSees Comand Language manual, http://opensees.berkeley.edu/OpenSees/manuals/usermanual [27/02/09].

Miranda E, Taghavi S. 2005. Approximate floor acceleration demands in multistory buildings. I: Formulation.Journal of Structural Engineering, ASCE131(2): 203211.

PEER Center. 2000. PEER strong motion database, http://peer.berkeley.edu/smcat/ [27/02/09].Shibata A, Sozen MA. 1976. Substitute-structure method for seismic design in RC.Journal of Structural Division,

ASCE102: 118.SEAOC. 1996.Recommended Lateral Force Requirement and CommentarySEAOC-96. Seismological Commit-

tee, SEAOC: San Francisco.

Documents

511_ftp.pdf