On the applicability of pushover analysis for seismic evaluation of medium- and high-rise buildings

ON THE APPLICABILITY OF PUSHOVER ANALYSIS FOR SEISMIC EVALUATION OF MEDIUM- AND

HIGH-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 structural behaviour is dominated by the fi rst vibration mode. However, higher modes of medium- and high-rise structures will have signifi cant 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 fi bre elements. The results of the analysis are compared with those from the pushover procedure. It is shown that pushover analysis generally underestimates inter-storey drifts and rotations, in particular those at upper storeys of buildings, and overestimates the peak roof displacement at inelastic deformation stage. It is shown that neglecting higher mode effects in the analysis will signifi cantly underestimate the shear force and overturning moment. It is suggested that pushover analysis may not be suitable for analysing high-rise shear-wall or wall-frame structures. New procedures of seismic evaluation for shear-wall and wall-frame structures based on nonlinear response history analysis should be developed. Copyright © 2009 John Wiley & Sons, Ltd.

1. INTRODUCTION

In the recent 7·9 Mw-magnitude earthquake in Sichuan, China, many public buildings, including schools and hospitals, collapsed and were seriously damaged. How to evaluate and retrofi t 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 5·12 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 fi rst vibration mode. However, higher vibration modes of a medium- or high-rise structure will have signifi cant effect on the dynamic characteristics.

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, 573–588 (2010)Published online 2 April 2009 in Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/tal.511

574 K. HUANG AND J. S. KUANG

Copyright © 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 19, 573–588 (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 fi rst determined by nonlinear response history analysis (RHA), where the shear walls are considered as multi-degree-of-freedom (MDOF) systems and modelled by fi bre elements. The results of RHA are then compared with those from the pushover procedure. The comparisons include two parts. In the fi rst 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 modifi ed from the eight-storey shear-wall building used in the document FEMA-440/ATC-55 (FEMA, 2005). The original structure is modifi ed in the following ways: (a) Node mass is changed so that the fundamental periods of the modifi ed 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)

where H is the building height in feet; Ac is 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 fi rst storey of the structure; and NW is the 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 fi bbers of the model, causing the wall to have an initial stiffness approximately equal to the gross section stiffness.

The shear walls are modelled using fi bre 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 suffi cient 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 defi ned as a ratio of the top drift to the total height of the structure, Δtop/H, reaches about 0·5%. In the studies, three top drift levels for the structures with the roof-drift ratios of 0·2%, 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 Pacifi c Earthquake Engineering Research (PEER) Center strong motion database (PEER Center, 2000). The peak ground accelerations range

APPLICABILITY OF PUSHOVER ANALYSIS FOR HIGH-RISE BUILDINGS 575


from 0·28 g to 0·41 g, and the peak ground displacements range from 10 cm to 14·5 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

211

1

2

Concrete and Steel Fibers

Roof

20th

19th

21

20

19

1918

20

1

1

Node Number

Element NumberNode

Element

Figure 1. OpenSees modelling of RC shear walls



0 20 40 60 800

50

100

150

200

250

300

350

Bas

e sh

ear

(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

Bas

e sh

ear

(kip

)


Top storey drift (inch)(b)

0 20 40 60 800

50

100

150

200

250

300

350

Bas

e sh

ear

(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



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 refl ected 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 fi rst mode amplitude and fl oor mass, is applied to 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 fi rst- 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 3–5. By comparing two sets of results from pushover procedure and RHA, the fi ndings can be summarized as follows.

(1) Pushover analysis provides reliable estimates of the maximum fl oor displacement and inter-storey drift in the elastic range. However, the estimate becomes inaccurate when the structures have inelastic 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 identifi ed 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 signifi cant 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 Country—W Lost Cany (90057)

0·41 43 11·75

2 Northridge 1994/01/17 Pardee—SCE 0·406 43·6 12·09 3 Chi-Chi, Taiwan 1999/09/20 TCU079 0·393 48·8 13·78 4 Westmorland 1981/04/26 Westmorland Fire Station (5169) 0·368 48·7 10·61 5 Imperial Valley 1979/10/15 Aeropuerto Mexicali (6616) 0·327 42·8 10·1 6 Loma Prieta 1989/10/18 Gilroy—Historic Bldg. (57476) 0·284 42 11·1 7 Landers 1992/06/28 Joshua Tree (22170) 0·284 43·2 14·51 8 Chi-Chi, Taiwan 1999/09/20 CHY035 0·252 45·6 12·03 9 Imperial Valley 1979/10/15 Agrarias (6618) 0·221 42·4 11·710 Loma Prieta 1989/10/18 Alameda Naval Air Stn Hanger 23 0·209 42·5 14·07

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



Floor displacement (inch)

(a) (b) (c)

Sto

rey

Floor displacement (inch)

Sto

rey

Inter-storey drift (inch)

sto

rey

MinSDSD

MeanMaxPushover Analysis RHA

Figure 3. Comparison of fl oor displacements and inter-storey drifts of the 20-storey shear-wall structure determined 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 signifi cant effect on the shear forces. Neglecting the higher mode effect in the evaluation procedure may lead to signifi cant 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 simplifi ed 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,,=

ΔΓ1 1θ

(3)



where Δy,roof is the roof displacement at yield, Γ1 is the fi rst-mode participation factor and f1roof is

amplitude of the fi rst mode at the roof. The spectral acceleration at yielding of the equivalent SDOF system is given by

SV

Wga y

y base,

,=α1

(4)

where

αφ

φ1

11

2

112

1

=( )⎡

⎣⎢⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )⎡⎣⎢

⎤⎦⎥

=

= =

∑

∑ ∑

w g

w g w g

i ii

N

ii

N

i ii

N (5)

where Vy,base is the base shear at yield, g is the gravity acceleration, W is the weight of the MDOF system, a1 is the modal mass coeffi cient, wi /g is mass assigned to level i, fi1 is the amplitude of the fi rst mode at level i and N is 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

Sto

rey

Moment (in-kip)0 1 2 3

2

4

6

8

10

12

14

16

Sto

rey

Moment (in-kip)0 2 4

2

4

6

8

10

12

14

16

18

20

Sto

rey

Moment (in-kip)

MinSDSD

MeanMaxPushover 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



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 0·2%, the mean displacement ratios are between 0·80 and 0·85 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 simplifi ed as a fl exural cantilever, where the shear deformation is neglected. The governing equation of fl exural cantilever with a fi xed base subjected to horizontal ground excitation is given by

Shear force (kip)

Sto

rey

Shear force (kip)

Sto

rey

MinSDSD

MeanMaxPushover Analysis

(a) (b)

RHA

Figure 5. Comparison of shear forces determined by pushover analysis and RHA for the 20-storey shear-wall structure. (a) at 0·2% drift level; (b) at 2% drift level



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

tg( )

∂ ( )∂

2

2 (6)

where m(x) is the mass per unit length, u(x,t) is the relative displacement of cantilever at the height ratio x, which is a ratio of the structural height z to the total height of the building H, z/H, at time t, H is the total height of the building, c(x) is the damping coeffi cient per unit length, ug(t) is the ground displacement, and EI(x) is fl exural rigidity along the structural height.

Elastic response of the structure can be computed from modal analysis. For a continuous fl exural cantilever, the displacement u(x,t) can be calculated as a linear combination of modal responses

u x t u x tii

, ,( ) = ( )=

∞

∑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 ti i i i,( ) = ( ) ( )Γ φ (8)

where Γi is the modal participation factor of the ith mode of vibration, fi(x) is the amplitude of the ith 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

dti

i ii

i ig

2

22

2

22

( )+

( )+ ( ) = −

( )ξω ω (9)

For a fl exural cantilever with uniformly distributed mass, the modal participation factor of the ith mode of vibration is given by

Drift level(a) (b) (c)

DE

SD

OF/D

MD

OF

Drift level Drift level

MinSDSD

MeanMax

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



Γi

i

i

x

x=

( )

( )

∫∫

θ

θ

dx

dx

0

1

2

0

1 (10)

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

u x x S Ti i i d i( ) = ( ) ( ),max Γ φ (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 xii

N

max ,max( ) ≈ ( )⎡⎣⎢

⎤⎦⎥=

∑ 2

1

1 2

(12)

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

F x m x S Ti i i a i( ) = ( ) ( )Γ φ (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 x can be determined by

V x V xii

N

( ) ≈ ( )⎡⎣⎢

⎤⎦⎥=

∑ 2

1

1 2

(14)

where

V x F xi ix

( ) = ( )∫ dx1

(15)

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

M x M xii

N

( ) ≈ ( )⎡⎣⎢

⎤⎦⎥=

∑ 2

1

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)



EIEI

eqa( ) =

( )μ

(18)

where (EI)eq is the equivalent fl exural stiffness for the substitute member, (EI)a is the cracked-section fl exural stiffness and m is 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 lH under the horizontal ground motion, and the equivalent fl exural stiffness is uniform in the plastic zone. Therefore, as shown in Figure 7(b), the fl exural stiffness along the height of the fl exural cantilever is given by

EI xEI

x

EI x

a

a

( ) = ≤

< ≤

⎧⎨⎪

⎩⎪μ

λ

λ 1 (19)

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

βμeff = ⋅ − ⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥+ ⋅0 2 1

10 02

1 2 (20)

Because the lateral stiffness along the height of the cantilever has two different values, a closed-form solution for mode shape is diffi cult to be derived. Therefore, in order to study the infl uence of

Curvatur

(a) (b)

e

M

cy ctarget

My

( ) ( )a

eq

EIEI

u=

: cracked sectionaEI

x

( )gu t

:

H

λH(1

-λ)H

EI e

qE

I a

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



plastic zone to the dynamic characteristic of the fl exural cantilever, mode shapes, periods and modal participation factors are calculated using fi nite 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 fi rst 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 defi ned 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 fi rst 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

Γ1f1 Γ2f2 Γ3f3

x

Elastic

μ=2

μ=4

μ=8

λ=0.2

Figure 8. Effect of damage ratio on product of mode shape and modal participation factor of fl exural cantilever (λ = 0·2)

0

1

2

1 3 5 7Damage ratio

Perio

d ra

tio

T1

T2

T3

Figure 9. Effect of damage ratio on period ratio of vibration mode (λ = 0·2)



ratio of the fi rst mode is increasing, apparently with the increase in the damage ratio, whereas the period ratios for the second and third modes do not increase signifi cantly.

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 fi rst mode and by the fi rst three modes, respectively, according to the modal analysis method. The design spectrum of Uniform Building Code (International Conference of Building Offi cials, 1997) shown in Figure 10, where the seismic zone is chosen to be 2A and the soil profi le 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 fl exural cantilever with a fundamental period of 2·3 s, the defor-mation, shear force and overturning moment are computed by the fi rst mode and the fi rst three modes, respectively, and the results are compared in Figures 11–13.

Figure 11 shows the deformation shapes of the fl exural cantilever when the cantilever remains elastic and that the damage ratio m is equal to 8. It can be seen that the deformation shapes computed only by the fi rst mode agree well with that computed by the fi rst three modes, showing that the higher vibration modes have a negligible effect on the fl exural cantilever’s 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 signifi cantly as shown in Figure 9, their contribution to the cantilever’s deformation will be insignifi cant. Moreover, by comparing with the contribution increment of the fi rst model, it is seen that the MDOF effect on the defl ection 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)

Spe

ctra

l Acc

eler

atio

n (g

)

ξ=5%

ξ=13%

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



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 fl exural 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/s 2)

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

μ=8

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



Figure 12 shows the normalized shear force of the fl exural 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 by the fi rst mode is greatly underestimated. The spectral acceleration corresponding to the second and third modes is relatively large, comparing with that corresponding to the fi rst 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 fi rst 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 signifi cantly when the plastic zone is formed at the bottom of fl exural cantilever. This is mainly due to the fact that the spectral acceleration corresponding to the fi rst mode is greatly reduced as the fundamental period becomes longer.

Figure 13 shows the normalized overturning moment of the fl exural cantilever when the cantilever remains elastic and that the damage factor m is equal to 8. It can be seen that the overturning moment computed only by the fi rst 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 fi rst mode, especially when the plastic zone is formed at the bottom of the fl exural 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 fl oor displacement and inter-storey drift in an elastic range, but underestimates the fl oor 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 response history 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.



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On the applicability of pushover analysis for seismic evaluation of medium- and high-rise buildings

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