análisis pushover

Análisis Pushover es un método estático, no lineal en el que se forma incremental la magnitud de la carga estructural incrementado de acuerdo con un cierto patrón predefinido. Con el aumento en la magnitud de la carga, los eslabones débiles y los modos de falla de la estructura se encuentran. La carga es monótona con los efectos del comportamiento cíclico y las transferencias de masas que se calcula mediante una deformación monotónica criterios modificados de fuerza y con amortiguación de análisis fácil de convencer approximations .Static es un intento por parte de la profesión de ingeniería estructural para evaluar la fuerza real de la estructura y promete ser una herramienta útil y eficaz para diseño prestacional. análisis Pushover es un espectáculo basado analysis.According al ATC de 40 años, hay dos elementos clave de un procedimiento de diseño basado en prestaciones - demanda y la capacidad. La demanda es la representación de los movimientos del suelo terremoto o temblor que el edificio es sometido. En los procedimientos de análisis estático no lineal, la demanda está representada por una estimación de los desplazamientos o deformaciones que la estructura se espera que undergo.Capacity es una representación de la estructura de la capacidad de resistir a la demanda sísmica. El rendimiento depende de la manera que la capacidad es capaz de manejar la demanda. En otras palabras, la estructura debe tener la capacidad de oponerse a las demandas del terremoto de tal manera que el rendimiento de la estructura es compatible con los objetivos del diseño. análisis Pushover se realiza por el método de coeficiente de desplazamiento o el método del espectro de capacidad .. El Método de la capacidad del espectro (CSM), una técnica de análisis de sísmica basada en el rendimiento, se puede utilizar para una variedad de propósitos tales como la evaluación rápida de un gran inventario de edificios, el diseño verificación para la nueva construcción de edificios individuales, la evaluación de una estructura existente para identificar estados daños, y la correlación de los estados daños de los edificios para diferentes amplitudes de movimiento del suelo. La correlación de los estados procedimiento de daños de los edificios para diferentes amplitudes de movimiento del suelo. El procedimiento se compara la capacidad de la estructura (en forma de una curva fácil de convencer) con las exigencias de la estructura .... Objetivo del método de coeficiente de desplazamiento es encontrar el desplazamiento objetivo que es el desplazamiento máximo que la estructura es susceptible de ser experimentada durante el terremoto de diseño. Proporciona un proceso numérico para estimar la demanda de desplazamientos en la estructura, utilizando una representación bilineal de la curva de capacidad y una serie de factores de modificación, o coeficientes, para calcular un desplazamiento de destino.

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 31:561–582 (DOI: 10.1002/eqe.144)

A modal pushover analysis procedure for estimating seismicdemands for buildings

Anil K. Chopra1;∗;† and Rakesh K. Goel2

1Department of Civil and Environmental Engineering; University of California at Berkeley; Berkeley;CA, 94720-1710; U.S.A.

2Department of Civil and Environmental Engineering; California Polytechnic State University; San Luis Obispo,CA; U.S.A.

SUMMARY

Developed herein is an improved pushover analysis procedure based on structural dynamics theory,which retains the conceptual simplicity and computational attractiveness of current procedures with in-variant force distribution. In this modal pushover analysis (MPA), the seismic demand due to individualterms in the modal expansion of the e<ective earthquake forces is determined by a pushover analysisusing the inertia force distribution for each mode. Combining these ‘modal’ demands due to the ?rst twoor three terms of the expansion provides an estimate of the total seismic demand on inelastic systems.When applied to elastic systems, the MPA procedure is shown to be equivalent to standard responsespectrum analysis (RSA). When the peak inelastic response of a 9-storey steel building determined bythe approximate MPA procedure is compared with rigorous non-linear response history analysis, it isdemonstrated that MPA estimates the response of buildings responding well into the inelastic range toa similar degree of accuracy as RSA in estimating peak response of elastic systems. Thus, the MPAprocedure is accurate enough for practical application in building evaluation and design. Copyright? 2001 John Wiley & Sons, Ltd.

KEY WORDS: building evaluation and retro?t; modal analysis; pushover; seismic demands

INTRODUCTION

Estimating seismic demands at low performance levels, such as life safety and collapse pre-vention, requires explicit consideration of inelastic behaviour of the structure. While non-linearresponse history analysis (RHA) is the most rigorous procedure to compute seismic demands,current civil engineering practice prefers to use the non-linear static procedure (NSP) orpushover analysis in FEMA-273 [1]. The seismic demands are computed by non-linear static

∗ Correspondence to: Anil K. Chopra, Department of Civil and Environmental Engineering, University of Californiaat Berkeley, Berkeley, CA 94720-1710, U.S.A.

† E-mail: [email protected]

Received 15 January 2001Revised 31 August 2001

Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 31 August 2001

562 A. K. CHOPRA AND R. K. GOEL

analysis of the structure subjected to monotonically increasing lateral forces with an invariantheight-wise distribution until a predetermined target displacement is reached. Both the forcedistribution and target displacement are based on the assumption that the response is controlledby the fundamental mode and that the mode shape remains unchanged after the structure yields.Obviously, after the structure yields, both assumptions are approximate, but investigations

[2–9] have led to good estimates of seismic demands. However, such satisfactory predictionsof seismic demands are mostly restricted to low- and medium-rise structures provided theinelastic action is distributed throughout the height of the structure [7; 10].None of the invariant force distributions can account for the contributions of higher modes

to response, or for a redistribution of inertia forces because of structural yielding and theassociated changes in the vibration properties of the structure. To overcome these limitations,several researchers have proposed adaptive force distributions that attempt to follow moreclosely the time-variant distributions of inertia forces [5; 11; 12]. While these adaptive forcedistributions may provide better estimates of seismic demands [12], they are conceptuallycomplicated and computationally demanding for routine application in structural engineeringpractice. Attempts have also been made to consider more than the fundamental vibration modein pushover analysis [12–16].The principal objective of this investigation is to develop an improved pushover analy-

sis procedure based on structural dynamics theory that retains the conceptual simplicity andcomputational attractiveness of the procedure with invariant force distribution—now commonin structural engineering practice. First, we develop a modal pushover analysis (MPA) pro-cedure for linearly elastic buildings and demonstrate that it is equivalent to the well-knownresponse spectrum analysis (RSA) procedure. The MPA procedure is then extended to inelas-tic buildings, the underlying assumptions and approximations are identi?ed, and the errors inthe procedure relative to a rigorous non-linear RHA are documented.

DYNAMIC AND PUSHOVER ANALYSIS PROCEDURES: ELASTIC BUILDINGS

Modal response history analysis

The di<erential equations governing the response of a multistorey building to horizontal earth-quake ground motion Oug(t) are as follows:

m Ou+ cu+ ku=−m� Oug(t) (1)

where u is the vector of N lateral Soor displacements relative to the ground, m; c; and k arethe mass, classical damping, and lateral sti<ness matrices of the systems; each element of theinSuence vector � is equal to unity.The right-hand side of Equation (1) can be interpreted as e<ective earthquake forces:

pe< (t)=−m� Oug(t) (2)

The spatial distribution of these e<ective forces over the height of the building is de?ned bythe vector s=m� and their time variation by Oug(t). This force distribution can be expanded

Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:561–582

ESTIMATING SEISMIC DEMANDS FOR BUILDINGS 563

as a summation of modal inertia force distribution sn [17, Section 13:2]:

m�=N∑n=1sn=

N∑n=1

TnmMn (3)

where Mn is the nth natural vibration mode of the structure, and

Tn=LnMn; Ln=MTnm�; Mn=MTnmMn (4)

The e<ective earthquake forces can then be expressed as

pe< (t)=N∑n=1pe< ; n(t)=

N∑n=1

− sn Oug(t) (5)

The contribution of the nth mode to s and to pe< (t) are:

sn =TnmMn (6a)

pe< ; n(t) =−sn Oug(t) (6b)

The response of the MDF system to pe< ; n(t) is entirely in the nth-mode, with no contribu-tions from other modes. Then the Soor displacements are

un(t)=Mnqn(t) (7)

where the modal co-ordinate qn(t) is governed by

Oqn + 2�n!nqn +!2nqn=−Tn Oug(t) (8)

in which !n is the natural vibration frequency and �n is the damping ratio for the nth mode.The solution qn of Equation (8) is given by

qn(t)=TnDn(t) (9)

where Dn(t) is governed by the equation of motion for the nth-mode linear SDF system, anSDF system with vibration properties—natural frequency !n and damping ratio �n—of thenth-mode of the MDF system, subjected to Oug(t):

ODn + 2�n!nDn +!2nDn=− Oug(t) (10)

Substituting Equation (9) into Equation (7) gives the Soor displacements

un(t)=TnMnDn(t) (11)

Any response quantity r(t)—storey drifts, internal element forces, etc.—can be expressed as

rn(t)= r stn An(t) (12)

where r stn denotes the modal static response, the static value of r due to external forces sn,and

An(t)=!2nDn(t) (13)



(a) Static Analysis ofStructure

(b) Dynamic Analysis ofSDF System

Forcessn

rnst

An(t )

ug(t )¨

ωn , ζn

Figure 1. Conceptual explanation of modal RHA of elastic MDF systems.

is the pseudo-acceleration response of the nth-mode SDF system [17, Section 13:1]. Thetwo analyses that lead to r stn and An(t) are shown schematically in Figure 1. Equations (11)and (12) represent the response of the MDF system to pe< ; n(t) [Equation (6b)]. Therefore,the response of the system to the total excitation pe< (t) is

u(t) =N∑n=1un(t)=

N∑n=1

TnMnDn(t) (14)

r(t) =N∑n=1rn(t)=

N∑n=1r stn An(t) (15)

This is the classical modal RHA procedure: Equation (8) is the standard modal equationgoverning qn(t), Equations (11) and (12) de?ne the contribution of the nth-mode to theresponse, and Equations (14) and (15) reSect combining the response contributions of allmodes. However, these standard equations have been derived in an unconventional way. Incontrast to the classical derivation found in textbooks (e.g. Reference [17, Sections 12:4and 13:1:3]), we have used the modal expansion of the spatial distribution of the e<ectiveearthquake forces. This concept will provide a rational basis for the MPA procedure developedlater.

Modal response spectrum analysis

The peak value rno of the nth-mode contribution rn(t) to response r(t) is determined from

rno= r stn An (16)

where An is the ordinate A(Tn; �n) of the pseudo-acceleration response (or design) spectrumfor the nth-mode SDF system, and Tn=2�=!n is the nth natural vibration period of the MDFsystem.The peak modal responses are combined according to the square-root-of-sum-of-squares

(SRSS) or the complete quadratic combination (CQC) rules. The SRSS rule, which is valid forstructures with well-separated natural frequencies such as multistorey buildings with symmetricplan, provides an estimate of the peak value of the total response:

ro≈(

N∑n=1r2no

)1=2

(17)



Modal pushover analysis

To develop a pushover analysis procedure consistent with RSA, we observe that static analysisof the structure subjected to lateral forces

fno=TnmMnAn (18)

will provide the same value of rno, the peak nth-mode response as in Equation (16) [17,Section 13:8:1]. Alternatively, this response value can be obtained by static analysis of thestructure subjected to lateral forces distributed over the building height according to

s∗n=mMn (19)

with the structure pushed to the roof displacement, urno, the peak value of the roof displacementdue to the nth-mode, which from Equation (11) is

urno=Tn�rnDn (20)

where Dn=An=!2n ; obviously Dn or An are readily available from the response (or design)

spectrum.The peak modal responses rno, each determined by one pushover analysis, can be combined

according to Equation (17) to obtain an estimate of the peak value ro of the total response.This MPA for linearly elastic systems is equivalent to the well-known RSA procedure.

DYNAMIC AND PUSHOVER ANALYSIS PROCEDURES: INELASTIC BUILDINGS

Response history analysis

For each structural element of a building, the initial loading curve can be idealized appropri-ately (e.g. bilinear with or without degradation) and the unloading and reloading curves di<erfrom the initial loading branch. Thus, the relations between lateral forces fs at the N Soorlevels and the lateral displacements u are not single-valued, but depend on the history of thedisplacements:

fs= fs(u; sign u) (21)

With this generalization for inelastic systems, Equation (1) becomes

m Ou+ cu+ fs(u; sign u)=−m� Oug(t) (22)

The standard approach is to directly solve these coupled equations, leading to the ‘exact’non-linear RHA.Although classical modal analysis is not valid for inelastic systems, it will be used next

to transform Equation (22) to the modal co-ordinates of the corresponding linear system.Each structural element of this elastic system is de?ned to have the same sti<ness as theinitial sti<ness of the structural element of the inelastic system. Both systems have the samemass and damping. Therefore, the natural vibration periods and modes of the correspondinglinear system are the same as the vibration properties of the inelastic system undergoing smalloscillations (within the linear range).



Expanding the displacements of the inelastic system in terms of the natural vibration modesof the corresponding linear system, we get

u(t)=N∑n=1Mnqn(t) (23)

Substituting Equation (23) into Equation (22), premultiplying by MTn , and using the mass- andclassical damping-orthogonality property of modes gives

Oqn + 2�n!nqn +FsnMn

=−Tn Oug(t); n=1; 2; : : : ; N (24)

where the only term that di<ers from Equation (8) involves

Fsn=Fsn(q; sign q)=MTnfs(u; sign u) (25)

This resisting force depends on all modal co-ordinates qn(t), implying coupling of modalco-ordinates because of yielding of the structure.Equation (24) represents N equations in the modal co-ordinates qn. Unlike Equation (8)

for linearly elastic systems, these equations are coupled for inelastic systems. Simultaneouslysolving these coupled equations and using Equation (23) will, in principle, give the sameresults for u(t) as obtained directly from Equation (22). However, Equation (24) is rarelyused because it o<ers no particular advantage over Equation (22).

Uncoupled modal response history analysis

Neglecting the coupling of the N equations in modal co-ordinates [Equation (24)] leads tothe uncoupled modal response history analysis (UMRHA) procedure. This approximate RHAprocedure was used as a basis for developing an MPA procedure for inelastic systems.The spatial distribution s of the e<ective earthquake forces is expanded into the modal

contributions sn according to Equation (3), where Mn are now the modes of the correspondinglinear system. The equations governing the response of the inelastic system to pe< ; n(t) givenby Equation (6b) are

m Ou+ cu+ fs(u; sign u)=−sn Oug(t) (26)

The solution of Equation (26) for inelastic systems will no longer be described by Equation (7)because ‘modes’ other than the nth-‘mode’ will also contribute to the solution. However,because for linear systems qr(t)=0 for all modes other than the nth-mode, it is reasonableto expect that the nth-‘mode’ should be dominant even for inelastic systems.This assertion is illustrated numerically in Figure 2 for a 9-storey SAC steel building

described in Appendix A. Equation (26) was solved by non-linear RHA, and the resulting roofdisplacement history was decomposed into its ‘modal’ components. The beams in all storeysexcept two yield when subjected to the strong excitation of 1:5×El Centro ground motion,and the modes other than the nth-mode contribute to the response. The second and thirdmodes start responding to excitation pe< ;1(t) the instant the structure ?rst yields at about 5:2s;however, their contributions to the roof displacement are only 7 and 1 per cent, respectively, ofthe ?rst mode response [Figure 2(a)]. The ?rst and third modes start responding to excitationpe< ;2(t) the instant the structure ?rst yields at about 4:2 s; however, their contributions to



80

0

80

(a) peff,1

= s1

× 1.5 × El Centro

u r1 (

cm)

Mode 1

48.24

0 5 10 15 20 25 30 80

0

80

u r2 (

cm)

Mode 2

3.37

0 5 10 15 20 25 30 80

0

80

u r3 (

cm)

Mode 3

Time (sec)

0.4931

20

0

20

(b) peff,2

= s2

× 1.5 × El Centro

u r1 (

cm)

Mode 1

1.437

0 5 10 15 20 25 30 20

0

20

u r2 (

cm)

Mode 2

11.62

0 5 10 15 20 25 30 20

0

20

u r3 (

cm)

Mode 3

Time (sec)

0.7783

_ _

_

_

_

_

_

_

Figure 2. Modal decomposition of the roof displacement for ?rst three modes due to 1:5×ElCentro ground motion: (a) pe< ; 1(t)= − s1 × 1:5×El Centro; (b) pe< ; 2(t)= − s2 × 1:5×El

Centro ground motion.

the roof displacement of the second mode response (Figure 2(b)) are 12 and 7 per cent,respectively, of the second mode response [Figure 2(b)].Approximating the response of the structure to excitation pe< ; n(t) by Equation (7), substi-

tuting Equation (7) into Equation (26), and premultiplying by MTn gives Equation (24) exceptfor the important approximation that Fsn now depends only on one modal co-ordinate, qn:

Fsn=Fsn(qn; sign qn)=MTnfs(qn; sign qn) (27)

With this approximation, the solution of Equation (24) can be expressed by Equation (9),where Dn(t) is governed by

ODn + 2�n!nDn +FsnLn

=− Oug(t) (28)

and

Fsn=Fsn(Dn; sign Dn)=MTnfs(Dn; sign Dn) (29)

is related to Fsn(qn; sign qn) because of Equation (9).Equation (28) may be interpreted as the governing equation for the nth-‘mode’ inelastic

SDF system, an SDF system with (1) small amplitude vibration properties—natural frequency!n and damping ratio �n—of the nth-mode of the corresponding linear MDF system; and (2)Fsn=Ln–Dn relation between resisting force Fsn=Ln and modal co-ordinate Dn de?ned by Equation(29). Although Equation (24) can be solved in its original form, Equation (28) can be solvedconveniently by standard software because it is of the same form as the standard equationfor an SDF system, and the peak value of Dn(t) can be estimated from the inelastic response(or design) spectrum [17, Sections 7:6 and 7:12:1]. Introducing the nth-‘mode’ inelastic SDFsystem also permitted extension of the well-established concepts for elastic systems to inelastic



(a) Static Analysis ofStructure

(b) Dynamic Analysis ofInelastic SDF System

Forcessn

rnst

An (t )

ug(t )¨

ωn, ζn, Fsn / Ln

Unit mass

Figure 3. Conceptual explanation of uncoupled modal RHA of inelastic MDF systems.

systems; compare Equations (24) and (28) to Equations (8) and (10), and note that Equation(9) applies to both systems.‡

Solution of the non-linear Equation (28) formulated in this manner provides Dn(t), whichwhen substituted into Equation (11) gives the Soor displacements of the structure associatedwith the nth-‘mode’ inelastic SDF system. Any Soor displacement, storey drift, or anotherdeformation response quantity rn(t) is given by Equations (12) and (13), where An(t) is nowthe pseudo-acceleration response of the nth-‘mode’ inelastic SDF system. The two analy-ses that lead to r stn and An(t) for the inelastic system are shown schematically in Figure 3.Equations (12) and (13) now represent the response of the inelastic MDF system to pe< ; n(t),the nth-mode contribution to pe< (t). Therefore, the response of the system to the total exci-tation pe< (t) is given by Equations (14) and (15). This is the UMRHA procedure.

Underlying assumptions and accuracy. Using the 3:0×El Centro ground motion for bothanalyses, the approximate solution of Equation (26) by UMRHA is compared with the ‘exact’solution by non-linear RHA. This intense excitation was chosen to ensure that the structureis excited well beyond its linear elastic limit. Such comparison for roof displacement andtop-storey drift is presented in Figures 4 and 5, respectively. The errors are slightly largerin drift than in displacement, but even for this very intense excitation, the errors in eitherresponse quantity are only a few per cent.These errors arise from the following assumptions and approximations: (i) the coupling

between modal co-ordinates qn(t) arising from yielding of the system [recall Equations (24)and (25)] is neglected; (ii) the superposition of responses to pe< ; n(t) (n=1; 2; : : : ; N ) accordingto Equation (15) is strictly valid only for linearly elastic systems; and (iii) the Fsn=Ln–Dnrelation is approximated by a bilinear curve to facilitate solution of Equation (28) in UMRHA.Although approximations are inherent in this UMRHA procedure, when specialized for linearlyelastic systems it is identical to the RHA procedure described earlier for such systems. Theoverall errors in the UMRHA procedure are documented in the examples presented in a latersection.

Properties of the nth-mode inelastic SDF system. How is the Fsn=Ln–Dn relation to bedetermined in Equation (28) before it can be solved? Because Equation (28) governing Dn(t)is based on Equation (7) for Soor displacements, the relationship between lateral forces fs

‡ Equivalent inelastic SDF systems have been de?ned di<erently by other researchers [18; 19].



150

0

150(a) Nonlinear RHA

u r1 (

cm)

n = 1

75.89

50

0

50

u r2 (

cm)

n = 214.9

0 5 10 15 20 25 30 10

0

10

u r3 (

cm)

n = 3

Time (sec)

5.575

150

0

150(b) UMRHA

u r1 (

cm)

n = 1

78.02

50

0

50

u r2 (

cm)

n = 214.51

0 5 10 15 20 25 30 10

0

10

Time (sec)

u r3 (

cm)

n = 3

5.112

_

_

_

_

_

_

Figure 4. Comparison of an approximate roof displacement from UMRHA with exactsolution by non-linear RHA for pe< ; n(t)= − sn Oug(t); n=1; 2 and 3, where Oug(t)=

3:0×El Centro ground motion.

20

0

20(a) Nonlinear RHA

r1 (

cm)

n = 1

6.33

20

0

20

r2 (

cm)

n = 27.008

0 5 10 15 20 25 30 10

0

10

r3 (

cm)

n = 3

Time (sec)

5.956

20

0

20(b) UMRHA

r1 (

cm)

n = 1

5.855

20

0

20

r2 (

cm)

n = 26.744

0 5 10 15 20 25 30 10

0

10

Time (sec)

r3 (

cm)

n = 3

5.817

∆∆

∆

∆∆

∆

_

_

_

_

_

_

Figure 5. Comparison of approximate storey drift from UMRHA with exact solution by non-linearRHA for pe< ; n(t)= − sn Oug(t), n=1; 2 and 3, where Oug(t)=3:0×El Centro motion.

and Dn in Equation (29) should be determined by non-linear static analysis of the structureas the structure undergoes displacements u=DnMn with increasing Dn. Although most com-mercially available software cannot implement such displacement-controlled analysis, it canconduct a force-controlled non-linear static analysis with an invariant distribution of lateralforces. Therefore, we impose this constraint in developing the UMRHA procedure in thissection and MPA in the next section.What is an appropriate invariant distribution of lateral forces to determine Fsn? For an

inelastic system no invariant distribution of forces can produce displacements proportional



(a) Idealized Pushover Curve

ur n

Vbn

ur n y

Vbny

Actual

Idealized

1kn

1αn kn

(b) Fsn / Ln Dn Relationship

Dn

Fsn / Ln

Dny

= ur n y

/ Γn

φr n

Vbny

/ M*n

1ω

n2

1αnωn

2

_

Figure 6. Properties of the nth-‘mode’ inelastic SDF system from the pushover curve.

to Mn at all displacements or force levels. However, before any part of the structure yields,the only force distribution that produces displacements proportional to Mn is given by Equa-tion (19). Therefore, this distribution seems to be a rational choice—even after the structureyields—to determine Fsn in Equation (29). When implemented by commercially available soft-ware, such non-linear static analysis provides the so-called pushover curve, which is di<erentthan the Fsn=Ln–Dn curve. The structure is pushed using the force distribution of Equation(19) to some predetermined roof displacement, and the base shear Vbn is plotted against roofdisplacement urn. A bilinear idealization of this pushover curve for the nth-‘mode’ is shownin Figure 6(a). At the yield point, the base shear is Vbny and roof displacement is urny.How to convert this Vbn–urn pushover curve to the Fsn=Ln–Dn relation? The two sets of

forces and displacements are related as follows:

Fsn=VbnTn; Dn=

urnTn�rn

(30)

Equation (30) enables conversion of the pushover curve to the desired Fsn=Ln–Dn relationshown in Figure 5(b), where the yield values of Fsn=Ln and Dn are

FsnyLn

=VbnyM ∗n; Dny =

urnyTn�rn

(31)

in which M ∗n =LnTn is the e<ective modal mass [17, Section 13:2:5]. The two are related

through

FsnyLn

=!2nDny (32)

implying that the initial slope of the bilinear curve in Figure 6(b) is !2n . Knowing Fsny=Ln

and Dny from Equation (31), the elastic vibration period Tn of the nth-‘mode’ inelastic SDFsystem is computed from

Tn=2�(LnDnyFsny

)1=2

(33)



This value of Tn, which may di<er from the period of the corresponding linear system, shouldbe used in Equation (28). In contrast, the initial slope of the pushover curve in Figure 6(a)is kn=!2

nLn, which is not a meaningful quantity.


Next a pushover analysis procedure is presented to estimate the peak response rno of theinelastic MDF system to e<ective earthquake forces pe< ; n(t). Consider a non-linear staticanalysis of the structure subjected to lateral forces distributed over the building height ac-cording to s∗n [Equation (19)] with structure pushed to the roof displacement urno. This valueof the roof displacement is given by Equation (20) where Dn, the peak value of Dn(t), is nowdetermined by solving Equation (28), as described earlier; alternatively, it can be determinedfrom the inelastic response (or design) spectrum [17, Sections 7:6 and 7:12]. At this roof dis-placement, the pushover analysis provides an estimate of the peak value rno of any responsern(t): Soor displacements, storey drifts, joint rotations, plastic hinge rotations, etc.

This pushover analysis, although somewhat intuitive for inelastic buildings, seems rationalfor two reasons. First, pushover analysis for each ‘mode’ provides the exact modal responsefor elastic buildings and the overall procedure, as demonstrated earlier, provides results thatare identical to the well-known RSA procedure. Second, the lateral force distribution usedappears to be the most rational choice among all invariant distribution of forces.The response value rno is an estimate of the peak value of the response of the inelastic

system to pe< ; n(t), governed by Equation (26). As shown earlier for elastic systems, rno alsorepresents the exact peak value of the nth-mode contribution rn(t) to response r(t). Thus, wewill refer to rno as the peak ‘modal’ response even in the case of inelastic systems.The peak ‘modal’ responses rno, each determined by one pushover analysis, is combined

using an appropriate modal combination rule, e.g. Equation (17), to obtain an estimate of thepeak value ro of the total response. This application of modal combination rules to inelasticsystems obviously lacks a theoretical basis. However, it provides results for elastic buildingsthat are identical to the well-known RSA procedure described earlier.

COMPARATIVE EVALUATION OF ANALYSIS PROCEDURES

The ‘exact’ response of the 9-storey SAC building described earlier is determined by thetwo approximate methods, UMRHA and MPA, and compared with the ‘exact’ results of arigorous non-linear RHA using the DRAIN-2DX computer program [20]. Gravity-load (andP-delta) e<ects are excluded from all analyses presented in this paper. However, these e<ectswere included in Chopra and Goel [21]. To ensure that this structure responds well into theinelastic range, the El Centro ground motion is scaled up a factor varying from 1.0 to 3.0.The ?rst three vibration modes and periods of the building for linearly elastic vibration areshown in Figure 7. The vibration periods for the ?rst three modes are 2.27, 0.85, and 0.49 s,respectively. The force distribution s∗n for the ?rst three modes are shown in Figure 8. Theseforce distributions will be used in the pushover analysis to be presented later.

Uncoupled modal response history analysis

The structural response to 1:5× the El Centro ground motion including the responsecontributions associated with three ‘modal’ inelastic SDF systems, determined by the UMRHA



1.5 1 0.5 0 0.5 1 1.5Mode Shape Component

Flo

orGround

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

T1 = 2.27 sec

T2 = 0.85 sec

T3 = 0.49 sec

_ _ _

Figure 7. First three natural-vibration periods and modes of the 9-storey building.

1 2 3s*

0.487

0.796

1.12

1.44

1.75

2.04

2.33

2.61

3.05

s*

1.1

1.67

2.03

2.1

1.8

1.13

0.0272

1.51

3.05

s*

2.31

2.94

2.37

0.728

1.38

2.93

2.72

0.39

3.05

_

_

_

_

_

_

_

_

_

_

Figure 8. Force distributions s∗n =mMn; n=1; 2 and 3.

procedure, is presented next. Figure 9 shows the individual ‘modal’ responses, the combinedresponse due to three ‘modes’, and the ‘exact’ values from non-linear RHA for the roof dis-placement and top-storey drift. The peak values of response are as noted; in particular, thepeak roof displacement due to each of the three ‘modes’, is ur10 = 48:3 cm, ur20 = 11:7 cmand ur30 = 2:53 cm. The peak values of Soor displacements and storey drifts including one,two, and three modes are compared with the ‘exact’ values in Figure 10 and the errors in theapproximate results are shown in Figure 11.Observe that errors tend to decrease as response contributions of more ‘modes’ are included,

although the trends are not as systematic as when the system remained elastic [22]. This isto be expected because in contrast to classical modal analysis, the UMRHA procedure lacksa rigorous theory. This de?ciency also implies that, with, say, three ‘modes’ included, theresponse is much less accurate if the system yields signi?cantly versus if the system remainswithin the elastic range [22]. However, for a ?xed number of ‘modes’ included, the errors instorey drifts are larger compared to Soor displacements, just as for elastic systems.Next we investigate how the errors in the UMRHA vary with the deformation demands

imposed by the ground motion, in particular, the degree to which the system deforms beyondits elastic limit. For this purpose the UMRHA and exact analyses were repeated for ground



80

0

80

•48.3

(a) Roof Displacement

u r1 (

cm)

80

0

80

•11.7

u r2 (

cm)

80

0

80

•2.53

u r3 (

cm)

80

0

80

•48.1

u r (cm

)

0 5 10 15 20 25 30 80

0

80

•44.6u r (cm

)

Time, sec

12

0

12

•3.62

(b) Top Story Drift

∆ r1 (

cm) "Mode" 1

12

0

12•5.44

∆ r2 (

cm) "Mode" 2

12

0

12

•2.88∆ r3 (

cm) "Mode" 3

12

0

12

•7.38∆ r (cm

) UMRHA 3 "Modes"

0 5 10 15 20 25 30 12

0

12•6.24

∆ r (cm

)

Time, sec

NL RHA

_

_

_

_

_

_

_

_

_

_

Figure 9. Response histories of roof displacement and top-storey drift due to 1:5×El Centroground motion: individual ‘modal’ responses and combined response from UMRHA, and total

response from non-linear RHA.

0 0.5 1 1.5 2Displacement/Height (%)

(a) Floor Displacements

Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

NL RHAUMRHA1 "Mode"2 "Modes"3 "Modes"

0 0.5 1 1.5 2 2.5Story Drift Ratio (%)

(b) Story Drift Ratios

Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

NL RHA

UMRHA1 "Mode"2 "Modes"3 "Modes"

_ _

Figure 10. Height-wise variation of Soor displacements and storey drift ratios from UMRHA andnon-linear RHA for 1:5×El Centro ground motion.

motions of varying intensity. These excitations were de?ned by the El Centro ground motionmultiplied by 0.25, 0.5, 0.75, 0.85, 1.0, 1.5, 2.0, and 3.0. For each excitation, the errors inresponses computed by UMRHA including three ‘modes’ relative to the ‘exact’ response weredetermined.



60 40 20 0 20 40 60Error (%)

(a) Floor DisplacementsF

loor

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


60 40 20 0 20 40 60Error (%)


Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


_ _ _ _ _ _

Figure 11. Height-wise variation of error in Soor displacements and storey drifts estimated by UMRHAincluding one, two or three ‘modes’ for 1:5×El Centro ground motion.

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

1

2

3

45

6

78

9

GM Multiplier


Err

or (

%)

Error EnvelopeError for Floor No. Noted

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

1

23

4

5

67

8

9

GM Multiplier

(b) Story Drifts

Err

or (

%)

Error Envelope

Error for Story No. Noted

Figure 12. Errors in UMRHA as a function of ground motion intensity:(a) Soor displacements; and (b) storey drifts.

Figure 12 summarizes the error in UMRHA as a function of ground motion intensity,indicated by a ground motion multiplier. Shown is the error in each Soor displacement [Figure12(a)], in each storey drift [Figure 12(b)], and the error envelope for each case. To interpretthese results, it will be useful to know the deformation of the system relative to its yielddeformation. For this purpose, pushover curves using force distributions s∗n [Equation (19)]for the ?rst three modes of the system are shown in Figure 13, with the peak displacementof each ‘modal’ SDF system noted for each ground motion multiplier. Two versions of thepushover curve are included: the actual curve and its idealized bilinear version. The locationof plastic hinges and their rotations, determined from ‘exact’ analyses, were noted but notshown here.Figure 12 permits the following observations regarding the accuracy of the UMRHA

procedure: the errors (i) are small (less than 5 per cent) for ground motion multipliers up



0 20 40 60 800

0.05

0.1

0.15

0.2

0.25

Roof Displacement (cm)

Bas

e S

hear

/Wei

ght

(a) "Mode" 1 Pushover Curve

uy = 36.3 cm; Vby /W = 0.1684; α = 0.19

Actual

Idealized0.25

0.5

0.75

0.85

11.5

23

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25


Bas

e S

hear

/Wei

ght

(b) "Mode" 2 Pushover Curve

uy = 9.9 cm; Vby /W = 0.1122; α = 0.13

Actual

Idealized

0.25

0.5

0.750.85

11.5 2 3

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25


Bas

e S

hear

/Wei

ght

(c) "Mode" 3 Pushover Curve

uy = 4.6 cm; Vby /W = 0.1181; α = 0.14

Actual

Idealized

0.25

0.5 0.75

0.851

1.5

2

3

Figure 13. ‘Modal’ pushover curves with peak roof displacements identi?ed for 0.25, 0.5, 0.75, 1.0,1.5, 2.0, and 3:0×El Centro ground motion.

to 0.75; (ii) increase rapidly as the ground motion multiplier increases to 1.0; (iii) maintainroughly similar values for more intense ground motions; and (iv) are larger in storey driftscompared to Soor displacements. Up to ground motion multiplier 0.75, the system remainselastic and the errors in truncating the higher mode contributions are negligible. Additionalerrors are introduced in UMRHA of systems responding beyond the linearly elastic limit forat least two reasons. First, as mentioned earlier, UMRHA lacks a rigorous theory and is basedon several approximations. Second, the pushover curve for each ‘mode’ is idealized by a bi-linear curve in solving Equation (28) for each ‘modal’ inelastic SDF system (Figures 6 and13). The idealized curve for the ?rst ‘mode’ deviates most from the actual curve near thepeak displacement corresponding to ground motion multiplier 1.0. This would explain whythe errors are large at this excitation intensity; although the system remains essentially elastic;the ductility factor for the ?rst mode system is only 1.01 [Figure 13(a)]. For more intenseexcitations, the ?rst reason mentioned above seems to be the primary source for the errors.


The MPA procedure, considering the response due to the ?rst three ‘modes’, was imple-mented for the selected building subjected to 1:5× the El Centro ground motion. The struc-



0 0.5 1 1.5 2

Displacement/Height (%)


loor

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

NL RHA

MPA1 "Mode"2 "Modes"3 "Modes"

0 0.5 1 1.5 2 2.5Story Drift Ratio (%)


Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

NL RHA


_ _

Figure 14. Height-wise variation of Soor displacements and storey drift ratios fromMPA and non-linear RHA for 1:5×El Centro ground motion; shading indicates errors

in MPA including three ‘modes’.

ture is pushed using the force distribution of Equation (19) with n=1; 2 and 3 (Figure 8)to roof displacements urno=48:3, 11.7 and 2:53 cm, respectively, the values determined byRHA of the nth-mode inelastic SDF system (Figure 9). Each of these three pushover analysesprovides the pushover curve (Figure 13) and the peak values of modal responses. Becausethis building is unusually strong—its yield base shear =16:8 per cent of the building weight[Figure 13(a)]—the displacement ductility demand imposed by three times the El Centroground motion is only slightly larger than 2. Figure 14 presents estimates of the combined re-sponse according to Equation (17), considering one, two, and three ‘modes’, respectively, andFigure 15 shows the errors in these estimates relative to the exact response from non-linearRHA. The errors in the modal pushover results for two or three modes included are generallysigni?cantly smaller than in UMRHA (compare Figures 15 and 11). Obviously, the additionalerrors due to the approximation inherent in modal combination rules tend to cancel out theerrors due to the various approximation contained in the UMRHA. The ?rst ‘mode’ alone isinadequate, especially in estimating the storey drifts (Figure 14). Signi?cant improvement isachieved by including response contributions due to the second ‘mode’, however, the third‘mode’ contributions do not seem especially important (Figure 14). As shown in Figure 15(a),MPA including three ‘modes’ underestimates the displacements of the lower Soors by up to8 per cent and overestimates the upper Soor displacements by up to 14 per cent. The driftsare underestimated by up to 13 per cent in the lower storeys, overestimated by up to 18 percent in the middle storeys, and are within a few per cent of the exact values for the upperstoreys [Figure 15(b)].The errors are especially large in the hinge plastic rotations estimated by the MPA pro-

cedures, even if three ‘modes’ are included [Figure 15(c)]; although the error is recorded as100 per cent if MPA estimates zero rotation when the non-linear RHA computes a non-zerovalue, this error is not especially signi?cant because the hinge plastic rotation is very small.Observe that the primary contributor to plastic rotations of hinges in the lower storeys is the?rst ‘mode’, in the upper storeys it is the second ‘mode’; the third ‘mode’ does not contributebecause this SDF system remains elastic [Figure 13(c)].



60 40 20 0 20 40 60Error (%)


loor

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


60 40 20 0 20 40 60Error (%)


Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


120 80 40 0 40 80 120Error (%)

(c) Hinge Plastic Rotations

Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


120 80 40 0 40 80 120Error (%)

(c) Hinge Plastic Rotations

Flo

or

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th


_ _ _ _ _ _

_ _ _

Figure 15. Errors in Soor displacements, storey drifts, and hinge plastic rotations estimated by MPAincluding one, two and three ‘modes’ for 1:5×El Centro ground motion.

The locations of plastic hinges shown in Figure 16 were determined by four analyses:MPA considering one ‘mode’, two ‘modes’, and three ‘modes’; and non-linear RHA. One‘mode’ pushover analysis is unable to identify the plastic hinges in the upper storeys wherehigher mode contributions to response are known to be more signi?cant. The second ‘mode’is necessary to identify hinges in the upper storeys, however, the results are not alwaysaccurate. For example, the hinges identi?ed in beams at the sixth Soor are at variance withthe ‘exact’ results. Furthermore, MPA failed to identify the plastic hinges at the column basesin Figure 16, but was successful when the excitation was more intense.Figure 17 summarizes the error in MPA considering three ‘modes’ as a function of ground

motion intensity, indicated by a ground motion multiplier. Shown is the error in each Soordisplacement [Figure 17(a)], each storey drift [Figure 17(b)], and the error envelope for eachcase. While various sources of errors in UMRHA also apply to MPA, the errors in MPAare fortuitously smaller than in UMRHA (compare Figures 17 and 12) for ground motionmultipliers larger than 1.0, implying excitations intense enough to cause signi?cant yieldingof the structure. However, errors in MPA are larger for ground motion multipliers less than



• •• •• •• ••• •• •• •• •• •• •• •• ••• •• •• •• •• • •• •• •

(a) MPA, 1 "Mode"

• •• •• •• ••• •• •• •• •• •• •• •• ••• •• •• •• •• • •• •• •

• •• •• •• •• •• •• •• •

(b) MPA, 2 "Modes"

• •• •• •• ••• •• •• •• •• •• •• •• ••• •• •• •• •• • •• •• •

• •• •• •• •• •• •• •• •

(c) 3 "Modes"

• •• •• •• ••• •• •• •• ••• •• •• •• ••• •• •• •• •• • •• • •

• •• •• •• •• •• •• •• ••

• • •

(d) Nonlinear RHA

_

_

_

• •

Figure 16. Locations of plastic hinges determined by MPA considering one, two and three ‘modes’ andby non-linear RHA for 1:5×El Centro ground motion.

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

12

3

4

56 7

8

9

GM Multiplier


Err

or (

%)

Error EnvelopeError for Floor No. Noted

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

12 3

45

6

7

8

9

GM Multiplier

(b) Story Drifts

Err

or (

%)

Error EnvelopeError for Story No. Noted

Figure 17. Errors in MPA as a function of ground motion intensity: (a) Soordisplacements; and (b) storey drifts.



0.75, implying excitations weak enough to limit the response in the elastic range of thestructure. Here, UMRHA is essentially exact, whereas MPA contains errors inherent in modalcombination rules.The errors are only weakly dependent on ground motion intensity (Figure 17), an observa-

tion with practical implications. As mentioned earlier, the MPA procedure for elastic systems(or weak ground motions) is equivalent to the RSA procedure—now standard in engineeringpractice—implying that the modal combination errors contained in these procedures are ac-ceptable. The fact that MPA is able to estimate the response of buildings responding well intothe inelastic range to a similar degree of accuracy indicates that this procedure is accurateenough for practical application in building retro?t and design.

CONCLUSIONS

This investigation aimed to develop an improved pushover analysis procedure based on struc-tural dynamics theory, which retains the conceptual simplicity and computational attractivenessof current procedures with invariant force distribution now common in structural engineeringpractice. It has led to the following conclusions:The standard response spectrum analysis for elastic multistorey buildings can be reformu-

lated as a modal pushover analysis (MPA). The peak response of the elastic structure due to itsnth vibration mode can be exactly determined by pushover analysis of the structure subjectedto lateral forces distributed over the height of the building according to s∗n=mMn, where mis the mass matrix and Mn its nth-mode, and the structure is pushed to the roof displacementdetermined from the peak deformation Dn of the nth-mode elastic SDF system; Dn is availablefrom the elastic response (or design) spectrum. Combining these peak modal responses by anappropriate modal combination rule (e.g. SRSS rule) leads to the MPA procedure.The MPA procedure developed to estimate the seismic demands on inelastic systems is

organized in two phases: First, a pushover analysis is used to determine the peak response rnoof the inelastic MDF system to individual terms, pe< ; n(t)=−sn Oug(t), in the modal expansion ofthe e<ective earthquakes forces, pe< (t)=−m� Oug(t). The base shear–roof displacement (Vbn −urn) curve is developed from a pushover analysis for the force distribution s∗n. This pushovercurve is idealized as a bilinear force–deformation relation for the nth-‘mode’ inelastic SDFsystem (with vibration properties in the linear range that are the same as those of the nth-mode elastic SDF system), and the peak deformation of this SDF system—determined by non-linear response history analysis (RHA) or from the inelastic response or design spectrum—isused to determine the target value of roof displacement at which the seismic response rno isdetermined by the pushover analysis. Second, the total demand ro is determined by combiningthe rno (n=1; 2; : : :) according to an appropriate modal combination rule (e.g. SRSS rule).

Comparing the peak inelastic response of a 9-storey SAC steel building determined bythe approximate MPA procedure—including only the ?rst two or three rno terms—with non-linear RHA demonstrated that the approximate procedure provided good estimates of Soordisplacements and storey drifts, and identi?ed locations of most plastic hinges; however,plastic hinge rotations were less accurate. Based on results presented for El Centro groundmotion scaled by factors varying from 0.25 to 3.0, MPA estimates the response of buildingsresponding well into the inelastic range to similar degree of accuracy as standard RSA iscapable of estimating peak response of elastic systems. Thus, the MPA procedure is accurate



enough for practical application in building evaluation and design. That said, however, allpushover analysis procedures considered do not seem to compute accurately local responsequantities, such as hinge plastic rotations.Thus the structural engineering profession should examine the present trend of comparing

computed hinge plastic rotations against rotation limits established in FEMA-273 to judgestructural performance. Perhaps structural performance evaluation should be based on storeydrifts that are known to be closely related to damage and can be estimated to a higher degreeof accuracy by pushover analyses. While pushover estimates for Soor displacements are evenmore accurate, they are not good indicators of damage.This paper has focused on developing an MPA procedure and its initial evaluation in

estimating the seismic demands on a building imposed by a selected ground motion, with theexcitation scaled to cover a wide range of ground motion intensities and building response.This new method for estimating seismic demands at low performance levels, such as lifesafety and collapse prevention, should obviously be evaluated for a wide range of buildingsand ground motion ensembles. Work along these lines is in progress.

APPENDIX A: SAC STEEL BUILDING

The 9-storey structure, shown in Figure A1, was designed by Brandow & Johnston Asso-ciates§ for the SAC¶ Phase II Steel Project. Although not actually constructed, this struc-ture meets seismic code requirements of the 1994 UBC and represents typical medium-risebuildings designed for the Los Angeles, California, region.A benchmark structure for the SAC project, this building is 45:73 m (150 ft) by 45:73 m

(150 ft) in plan, and 37:19 m (122 ft) in elevation. The bays are 9:15 m (30 ft) on centre, inboth directions, with ?ve bays each in the north–south (N–S) and east–west (E–W) directions.The building’s lateral force-resisting system is composed of steel perimeter moment-resistingframes (MRF). To avoid biaxial bending in corner columns, the exterior bay of the MRF hasonly one moment-resisting connection. The interior bays of the structure contain frames withsimple (shear) connections. The columns are 345MPa (50 ksi) steel wide-Sange sections. Thelevels of the 9-storey building are numbered with respect to the ground level (see Figure A1)with the ninth level being the roof. The building has a basement level, denoted B-1. TypicalSoor-to-Soor heights (for analysis purposes measured from centre-of-beam to centre-of-beam)are 3:96 m (13 ft). The Soor-to-Soor height of the basement level is 3:65 m (12 ft) and forthe ?rst Soor is 5:49 m (18 ft).The column lines employ two-tier construction, i.e. monolithic column pieces are connected

every two levels beginning with the ?rst level. Column splices, which are seismic (tension)splices to carry bending and uplift forces, are located on the ?rst, third, ?fth, and seventhlevels at 1.83 m (6 ft) above the centreline of the beam to column joint. The column bases aremodelled as pinned and secured to the ground (at the B-1 level). Concrete foundation wallsand surrounding soil are assumed to restrain the structure at the ground level from horizontaldisplacement.

§ Brandow & Johnston Associates, Consulting Structural Engineers, 1660 W. Third St., Los Angeles, CA 90017.¶ SAC is a joint venture of three non-pro?t organizations: The Structural Engineers Association of Califor-nia (SEAOC), the Applied Technology Council (ATC), and California Universities for Research in EarthquakeEngineering (CUREE). SAC Steel Project Technical OWce, 1301 S. 46th Street, Richmond, CA 94804-4698.



Figure A1. Nine-storey building (adapted from Reference [23]).

The Soor system is composed of 248 MPa (36 ksi) steel wide-Sange beams in actingcomposite action with the Soor slab. The seismic mass of the structure is due to vari-ous components of the structure, including the steel framing, Soor slabs, ceiling=Sooring,mechanical=electrical, partitions, roo?ng and a penthouse located on the roof. The seismicmass of the ground level is 9:65× 105 kg (66:0 kips-s2=ft), for the ?rst level is 1:01× 106 kg(69:0 kips-s2=ft), for the second through eighth levels is 9:89× 105 kg (67:7 kips-s2=ft), andfor the ninth level is 1:07× 106 kg (73:2 kips-s2=ft). The seismic mass of the above groundlevels of the entire structure is 9:00× 106 kg (616 kips-s2=ft).

The two-dimensional building model consists of the perimeter N–S MRF (Figure A1),representing half of the building in the N–S direction. The frame is assigned half of theseismic mass of the building at each Soor level. The model is implemented in DRAIN-2DX[20] using the M1 model developed by Gupta and Krawinkler [7]. The MI model is based oncentreline dimensions of the bare frame in which beams and columns extend from centrelineto centreline. The strength, dimension, and shear distortion of panel zones are neglected butlarge deformation (P–X) e<ects are included. The simple model adopted here is suWcientfor the objectives of this study; if desired, more complex models, such as those described inReference [7], can be used.



ACKNOWLEDGEMENT

This research investigation is funded by the National Science Foundation under Grant CMS-9812531, apart of the U.S.–Japan Co-operative Research in Urban Earthquake Disaster Mitigation. This ?nancialsupport is gratefully acknowledged.

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19. Han SW, Wen YK. Method of reliability-based seismic design. I: Equivalent non-linear system. Journal ofStructural Engineering, ASCE 1997; 123:256–265.

20. Allahabadi R, Powell GH. DRAIN-2DX user guide. Report No. UCB=EERC-88=06, Earthquake EngineeringResearch Center, University of California, Berkeley, CA, 1988.

21. Chopra AK, Goel R. Modal pushover analysis of SAC building, Proceedings SEAOC Convention, San Diego,California, 2001.

22. Chopra AK, Goel R. A modal pushover analysis procedure to estimate seismic demands for buildings: theory andpreliminary evaluation. Report No. PEER 2001=03, Paci?c Earthquake Engineering Research Center, Universityof California, Berkeley, CA, 2001.

23. Ohtori Y, Christenson RE, Spencer Jr BF, Dyke SJ. Benchmark Control Problems for Seismically ExcitedNonlinear Buildings, http:==www.nd.edu=∼quake=, Notre Dame University, Indiana, 2000.


13th World Conference on Earthquake Engineering Vancouver, B.C., Canada

August 1-6, 2004 Paper No. 2713

METHOD OF MODAL COMBINATIONS FOR PUSHOVER ANALYSIS OF BUILDINGS

Erol KALKAN1 and Sashi K. KUNNATH2

SUMMARY Nonlinear static procedures (NSP) are finding widespread use in performance based seismic design since it provides practitioners a relatively simple approach to estimate inelastic structural response measures. However, conventional NSPs using lateral load patterns recommended in FEMA-356 do not adequately represent the effects of varying dynamic characteristics during the inelastic response or the influence of higher modes. To overcome these drawbacks, some improved procedures have recently been proposed by several researchers. A method of modal combinations (MMC) that implicitly accounts for higher mode effects is investigated in this paper. MMC is based on invariant force distributions formed from the factored combination of independent modal contributions. The validity of the procedure is validated by comparing response quantities such as inter-story drift and member ductility demands using other pushover methods and also the results of nonlinear time history analyses. The validation studies are based on evaluation of three existing steel moment frame buildings: two of these structures were instrumented during the Northridge earthquake thereby providing realistic support motions for the time-history predictions. Findings from the investigation indicate that the method of modal combinations provides a basis for estimating the potential contributions of higher modes when determining inter-story drift demands and local component demands in multistory frame buildings subjected to seismic loads.

INTRODUCTION Current seismic design practice in the United States is still governed by force-based design principles. However, the emergence of performance-based seismic engineering has resulted in increasing use of nonlinear methods to estimate expected seismic demands in a building structure. A widely used and popular approach to establish these demands is a “pushover” analysis in which a mathematical model of the building is subjected to an inverted triangular distribution of lateral forces. While such a load distribution is based on the assumption that the response is primarily in its fundamental mode of vibration, it can lead to incorrect estimates for structures with significant higher mode contributions. This

1 PhD. Student, University of California, Davis, CA, USA. Email: [email protected] 2 Assoc. Prof., University of California, Davis, CA, USA. Email: [email protected]

accentuates the need for improved procedures that addresses current drawbacks in the lateral load patterns that are used in pushover analyses. Recently, several improved pushover procedures have been proposed [1,2]. These procedures have been shown to provide more accurate estimates of interstory drift values than conventional NSPs using inverted triangular, uniform or other lateral load patterns based on direct modal combination rules suggested in FEMA-356 [3]. The effort required to implement these procedures in routine analysis is significant and, therefore, not an attractive proposition in engineering practice. In order to investigate alternative simple schemes to represent realistic lateral force demands, a new lateral load configuration using factored modal combinations has been recently developed [4]. The accuracy of this approach, to be referred to in this paper as the Method of Modal Combinations (MMC), has been validated on two RC buildings [4]. Other pushover procedures, which are relevant to the work described in this paper, are the methods described by Paret et al. [5] and Sasaki et al. [6].

The principal objective of this paper is to extend the validation study to additional buildings and to compare predictions to other approaches such as the Modal Pushover Analysis (MPA) procedure proposed by Chopra and Goel [2] and to the results of detailed time-history analyses. Three existing buildings with varying story levels (4, 6 and 13) were selected for the evaluation. The six and thirteen story buildings were instrumented during the Northridge earthquake, therefore, recorded base motions were utilized in nonlinear time history (NTH) analyses. For the four-story building, which was not instrumented, three different ground motions were used to predict expected seismic demands.

NONLINEAR STATIC PROCEDURES

The MMC procedure is evaluated by comparing computed interstory drift demands to nonlinear time-history estimates and to other pushover procedures. One set of lateral load patterns was based on recommendations in FEMA-356 while the second methodology considered in the comparative study is the modal pushover analysis (MPA) of Chopra and Goel [2]. A brief overview is presented of the different NSP methodologies used in the study. Lateral Load Patterns Based on FEMA-356

In FEMA-356, several alternative invariant loading patterns are recommended for estimating equivalent seismic demands. In this study, two loading patterns are considered. These two patterns are permitted when more than 75 percent of the total mass participates in the fundamental vibration mode in the direction under consideration. The following notations are used in this paper to describe these patterns: NSP-1: The buildings are subjected to a lateral load distributed across the height of the building based on the following formula specified in FEMA-356:

V

hW

hWF

N

i

kii

kxx

x

∑

=

=1

(1)

where, Fx is the applied lateral force at level ‘x’, W is the story weight, h is the story height and V is the design base shear, and N is the number of stories. The summation in the denominator is carried through all story levels. This results in an inverted triangular distribution when k is set equal to unity. NSP-2: A uniform lateral load distribution consisting of forces that are proportional to the story masses at each story level.

Modal Pushover Analysis (MPA) Modal Pushover Analysis (MPA), developed by Chopra and Goel [2], is essentially the extension of single mode pushover analysis to multi-mode response, and use of the theory of response spectrum analysis to combine the modal contributions. The basic steps of the procedures are as follows:

1. Compute the natural frequencies, ωn and mode shapes using an elastic model of the system. 2. Run pushover analyses with the loading patterns (sn= m φn) based on each mode independently. 3. Idealize each pushover curve as bilinear curves considering negative post-yield stiffness if

necessary. 4. Convert the idealized pushover curves into a set of capacity spectrum curves of the corresponding

SDOF system using the ADRS conversion from MDOF to SDOF (Note that guidelines provided in ATC-40 [7] capacity spectrum procedure can be used for this purpose).

5. Compute the peak response corresponding to each SDOF system via a nonlinear response history analysis (NRHA) based on an input ground motion for each SDOF system or via inelastic design spectrum.

6. Convert the peak response of SDOF system to the target displacement of MDOF system for each mode separately.

7. From the pushover database (Step 2), extract the peak inelastic response quantities of interest, such as interstory drift and plastic hinge rotations independently for each mode.

8. By using SRSS, determine the combined peak response. At the first glance, MPA procedure is an adaptation of NRHA for inelastic static analysis. However, this process inherently requires considerable effort except if very few modes are considered in the evaluation. In its original form, MPA is not a static method since it requires repetitive runs of SDOF response history analyses for a given ground motion to identify the target displacement of each mode. Additionally, it requires the use of one or more ground motions unless an inelastic design spectrum is used. Running the pushover analyses independently in each mode and ignoring the contribution of other modes in the formation of plastic hinges is an issue of concern for MPA since it may result in inaccurate estimates of plastic hinge rotation, an important parameter for comparing acceptance criteria in performance-based evaluation.

Method of Modal Combinations (MMC) In this procedure (see Kunnath [4]) the spatial variation of applied forces is determined from:

(2) where nα is a modification factor that can assume positive or negative values; nΦ is the mode shape

vector corresponding to mode n; aS is the spectral acceleration at the period corresponding to mode n;

and

[ ] nT Mm /})]{[( ιΦ=Γ in which ]][[][ ΦΦ= mM T

n (3)

If only the first two modes are used in the combination process, then Equation 2 would have the following form:

j 1 1 1 a 1 1 2 2 2 a 2 2 S ( ,T ) S ( ,T )F α ζ α ζ= Γ Φ ± Γ Φm m (4)

j n n a n n S ( , T )nF α ζ= Σ Γ Φm

Therefore, the procedure requires multiple pushover analyses wherein several combinations of modal load patterns are applied. In order to arrive at estimates of deformation and force demands, it is necessary to consider peak demands at each story level and then establish an envelope of demand values for use in performance based-evaluation.

DESCRIPTION OF BUILDINGS Three special moment resisting steel frame buildings were selected as representative case studies to evaluate the MMC procedure. 4-Story Building The building was designed in according to 1988 UBC specifications. It is 16.15m in elevation and has a rectangular plan with plan dimensions of 33.27m x 19.2m. The structural system is composed of perimeter MRFs to resist lateral loads and interior gravity frames. The floor plan and elevation view of the building with beam and column sections are shown in Figure 1.

Figure 1. Structural configuration of 4-story building

1

2

3

4

BA

3@6.

1 m

DC

9.5 m 9.14 m 12.2 m 2.43 m

E

9.14 m

W14

x211

W14

x145

9.5 m

1st Floor

W14

x145

W14

x211

W24x104 W24x104

W14

x145

3rdFloor

2ndFloor W14

x145

A

Roof

W14

x211

W14

x211

W24x94

W24x76

W24x94

W24x76

W24x68

B

W24x68

W14

x211

W14

x145

3@3.

96 m

W14

x145

W24x104

W14

x211

W21x50

W14

x145

W14

x145

W14

x211

W24x76

W24x94W14

x211

W24x68

C

4.26 m

W21x50

W21x50

D E

W16x26

12.2 m

Moment resisting connection Simple hinge connection

(a) Plan view of perimeter frames

(b) Elevation

The columns of the MRFs are embedded into grade beams and anchored to the top of the pile cap, and the foundation system is composed of drilled concrete piers with pile caps, grade beams and tie beams. All columns are made of A-572 grade 50 steel. The girders and beams are made of A-36 steel. The floor system is composed of 15.9 cm thick slab (8.3 cm light weight concrete and 7.6 cm composite metal deck) at all floor levels and the roof. The outside walls are made of thin set brick veneer supported on a metal stud wall. This building suffered significant flange fracture damage in beam-flange to column-flange connections during 1994 Northridge earthquake [8]. All of the severely fractured beam-columns connections were concentrated in the NS direction in the moment frame on Line-D (Fig. 1). No fracture was observed in the NS direction moment frame on Line-A, and only one fracture was observed on Line-1 in the EW direction. This building was not instrumented. Further details of the building are given in Krawinkler et al. [8]. 6-Story Building This moment frame steel structure was designed in accordance with 1973 UBC requirements. The rectangular plan of the building measures 36.6m x 36.6m with a 8.2cm thick light weight concrete slab over 7.5cm metal decking. The primary lateral load resisting system is a moment frame around the perimeter of the building. Interior frames are designed to carry only gravity loads. The plan view and the elevation of a typical frame are shown in Figure 2. The building was instrumented with a total of 13 strong motion sensors at the ground, 2nd, 3rd and roof levels. The building performed well in the Northridge earthquake with no visible signs of damage. In constructing the building model, the columns were assumed to be fixed at the base level (all columns are supported by base plates anchored on foundation beams which in turn are supported on a pair of 9.75m - 0.75m diameter concrete piles). The specified minimum concrete compressive strength at 28 days was 20.7 MPa. Section properties were computed for A-36 steel with an assumed yield stress of 303 MPa. The total building weight (excluding live loads) was estimated to be approximately 34,644kN. Additional details including calibration of the building model is reported in Kunnath et al. [9].

Figure 2. Structural configuration of Burbank 6-story building


Moment resisting connection Simple hinge connection (b) Elevation

4

7

6

5

3

2

1

BA

6@6.

1m

DC E GF

[email protected] 6@ 6.1m

5@4m

5.3m

3rdFloor

2ndFloor

1st Floor

4th Floor

5th Floor

Roof

W14

x176

W14

x90

W14

x132

W24x68

W24x84

W24x68

W24x68

W27x102

W30x116

A C E F GDB

13-Story Building This building is located in South San Fernando Valley about 5 km southwest of the Northridge epicenter and is composed of one basement floor and 13 floors above ground. Built in accordance with the 1973 UBC code, this structure has been the subject of a previous investigation [9,10]. As shown in Figure 3, it has a 48.8m square plan and 57.5m in elevation. The lateral load resisting system is composed of four identical perimeter frames. The floor plan increases at the second floor to form a plaza level that terminates on three sides into the hillside thereby making this level almost fixed against translation. Recorded response of the building during the Northridge earthquake indicates a peak horizontal acceleration of 0.44g, 0.32g and 0.33g at the ground, 6th floor and roof levels. Weld fracture damage was observed primarily in the NS direction.

Figure 3. Structural configuration of the 13-story building

SUMMARY OF EVALUATION

The evaluation process consisted of comparing the computed demands using MMC with time-history analysis and with other pushover methods. For the two instrumented buildings, the recorded base motions served as the input accelerations for the time history analyses. Since the actual ground motions did not produce significant inelastic behavior, the records were scaled so as to induce a peak interstory drift of approximately 2 percent at any level. The target displacements for the pushover procedures were then based on the peak time-history induced story drifts. This approach provides a rational basis for comparing the demands obtained with different methods. Since instrumented information was not available for the four-story structure, three ground motion records were selected from the recommended set in ATC-40 [7]. Comparison of interstory drift demands comprised the primary basis for the evaluation. Typical member ductility demands (based on plastic rotations) were also evaluated for the MMC and FEMA procedures.

(b) Elevation


5

E

F

G

C

D

B

4 86 7 948

.8 m

48.8 m

Moment resisting connection Simple hinge connection

W33x118

W27x84

W33x141

W33x130

W33x130

W33x152

W33x152

W33x152

W33x141

W33x118

W36x230

W33x152

W33x152

W33x194

W14

x314

W14

x426

W14

x500

W14

x398

W14

x246

W14

x287

W14

x167

6th Floor

5th Floor

1st Floor

2nd Floor

3rd Floor

Plaza Level

4th Floor

12th Floor

Roof

9th Floor

10th Floor

11th Floor

7th Floor

8th Floor

12@

4.01

3 =

48.2

m

4.88

4.42

[email protected] = 48.8 m

The MPA procedure does not lend itself to a direct evaluation of component ductility. Details of the ground motions used for the time-history analyses of the three buildings are presented in Table 1.

Table 1. Details of ground motions used in nonlinear time history analyses

Year Earthquake Recording Station PGA (g) 4-Story 6-Story 13-Story

1989 Loma Prieta Hollister, South & Pine 0.370 1.0 - -1989 Loma Prieta Gilroy #2 0.350 2.1 - -1994 Northridge Moorpark 0.290 4.0 - -1994 Northridge Burbank 0.299 - 2.3 -1994 Northridge Woodland Hills 0.318 - - 1.9

* Scale factor used to achieve the target interstory drift ratio of two percent

EQ. Scale Factor *

Analytical Modeling The nonlinear evaluations were carried out using the open source finite element framework, OpenSees [11]. A nonlinear beam-column element that utilizes a layered ‘fiber’ section is used to model all components in the frame models since the interaction of axial force and flexure is automatically incorporated. The element is based on a force formulation that considers the spread of plasticity. Since the objective of the evaluation is to evaluate various pushover procedures rather than simulate local connection fracture, the modeling of the members and connections was based on the assumption of stable hysteresis derived from a bilinear stress-strain model. Since the buildings are symmetric in plan, only two-dimensional models of a single frame were developed for each building. In the case of the four-story building, the exterior frame along EW direction (Line-1) was modeled. Similarly, frame models for the six and thirteen story buildings were developed for the exterior frames in EW direction. The elastic models were validated using available recorded data and typical simulations are displayed in Figure 4.

-10

0

10

0 20 40 60Time (sec)

Dis

p. (

cm)

RecordedSimulated

6th Floor

-40

0

40

0 10 20 30Time (sec)

Dis

p. (

cm)

RecordedSimulated

12th Floor

Figure 4. Validation of analytical models: comparison of recorded and computed response (a) EW response at roof level of 6-story building; (b) EW response at roof level of 13-story building

(a)

(b)

Evaluation of Method of Modal Combination (MMC) The validation exercise presented in Figure 4 was obtained using elastic models indicating minimal or no inelastic behavior. Hence, it was necessary to scale the recorded ground motions so that interstory drifts reached a magnitude to cause yielding in elements and provide a more reasonable basis to evaluate the adequacy of the pushover methods. The scale factors used to produce 2 percent peak story drifts are given in Table 1 for each ground motion. The 5-percent damped elastic acceleration and displacement response spectra computed from the amplified motions are presented in Figure 5. Also marked on these figures are the first two modal periods for the four- and six-story models and three modal periods for the thirteen-story model.

0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0Period (sec)

Sa

(g)

Mean

Max Sa at Tn

T1

T2 4-Story Building

0

10

20

30

40

50

0.0 1.0 2.0 3.0Period (sec)

Sd

(cm

)Mean

Max Sd at Tn

T1

T2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0 3.0Period (sec)

Sa

(g)

T1

T2

6-Story Building

0

5

10

15

20

0.0 1.0 2.0 3.0Period (sec)

Sd

(cm

)

T1

T2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0 3.0 4.0Period (sec)

Sa (

g)

T1

T2

T3

13-Story Building

0

25

50

75

100

0.0 1.0 2.0 3.0 4.0Period (sec)

Sd (

cm)

T1

T2T3

Figure 5. 5-percent damped pseudo acceleration spectra and displacement spectra for (a) 4-story building (based on three ground motions); (b) 6-story building; (c) 13-story building

(c)

(b)

(a)

Evaluation of Interstory Drift Demands As indicated previously, three earthquake records were used for the nonlinear time history (NTH) analysis of the four-story building. The MMC response is based on the envelope of demands resulting from Mode 1 ± Mode 2 (using peak Sa of the three ground motions). The resulting lateral forces using such a modal combination is shown in Figure 6. Plots of the displacement and drift profile for both NTH runs and the various pushover methods are shown in Figure 7. The peak displacement profiles are generally similar for all methods. The variation of interstory drift indicates that both MPA and MMC capture the demands with reasonable accuracy though the demand at the first story is slightly over-estimated. Of the two FEMA methods, NSP-1 (first mode distribution) is a better indicator of seismic demands though the drifts at the uppermost level are under-estimated. Note that NTH gives the highest demand at third story that implies some contribution of the second mode in the response.

Figure 6. Spatial distribution of lateral forces for 4-story building, Sn = Γn m φn , for n = 1 and 2

0

1

2

3

4

0.00 0.01 0.02Roof Drift Ratio

Sto

ry L

evel

NTH MeanNSP-1NSP-2MMCMPA

0

1

2

3

4

0.00 0.01 0.02 0.03Interstory Drift Ratio

Sto

ry L

evel

NTH MeanNSP-1NSP-2MMCMPA

Figure 7. Comparison of roof drift and interstory drift ratio using various methods for 4-story building

0.00

-0.24

-0.34

-0.09

0.31

0

1

2

3

4 0.31

0.00

0.23

0.56

0.84

0.93

0

1

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4

± =

S1 (Mode 1)

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4

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0

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3

4

S2 (Mode 2) S1 +S2 S1 - S2

As in the case of the four-story building, the MMC procedure was developed from lateral forces using a modal combination based on Mode 1 ± Mode 2. This combination led to the lateral force distribution as indicated in Figure 8. As is evident from these distributions, the two combinations place increased demands on either the upper or middle stories and is a function of both the mode shape and the spectral demands at these modal periods. The resulting story demands are plotted in Figure 9 along with demand estimates from the other methods. Significant higher mode effects are apparent in Figure 9b. MMC captures the highest story drift as well as the other story drifts reasonably, however the drift demand is overestimated at the first story. The MPA procedure generally under-predicts the demands at most levels. For this particular building none of the FEMA procedures show good correlation with the time history results in terms of the story drifts, they overestimate the demand at the lower levels and underestimate it at upper levels.

Figure 8. Spatial distribution of lateral forces for 6-story building, Sn = Γn m φn , for n = 1 and 2

0

1

2

3

4

5

6


Sto

ry L

evel

NTHNSP-1NSP-2MMCMPA

0

1

2

3

4

5

6

0.00 0.01 0.02 0.03

Interstory Drift Ratio

Sto

ry L

evel

NTHNSP-1NSP-2MMCMPA

Figure 9. Comparison of roof and interstory drift ratios using various methods for 6-story building

0.00

-0.62

-0.71

-0.61

-0.23

0.38

0.64

0

1

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0.64

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S1 (Mode 1) S2 (Mode 2)

± =

S1 +S2 S1 - S2

For the thirteen-story building, the MMC procedure was developed from lateral forces using a modal combination based on Mode 1 ± Mode 2 + Mode 3. This combination led to the lateral force distribution as indicated in Figure 10. Inclusion of third mode indicates a significantly altered load distribution. For this building, the combinations based on the first two modes, as used for the other buildings, was also evaluated in addition to the two combinations shown in Figure 10, and analyses were conducted for two configurations separately. The resultant demands are given in Figure 11 for comparison. The predicted demands from the other methods are also presented. In general demands predicted by MMC are in agreement with the computed demand from NTH analysis. Incorporation of the third mode improved the capability of MMC to capture the time-history demands. Except for the lower story levels, neither FEMA procedures nor MPA show good correlation with the computed demand from NTH analysis.

Figure 10. Spatial distribution of lateral forces for 13-story building, Sn = Γn m φn , for n = 1, 2 and 3

0

1

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3

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ry L

evel

NTHNSP-1NSP-2MMCMPA

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8

9

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13

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ry L

evel

NTHNSP-1NSP-2MMCMPA

Figure 11. Comparison of roof and interstory drift ratios using various methods for 13-story building

0.00

0.83

2.05

2.60

3.18

3.48

3.50

3.29

2.97

2.66

2.52

2.58

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1.69

0

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0

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0

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14

S1 (Mode 1) S2 (Mode 2)

± + =

S3 (Mode 3) S1 +S2+S3 S1 -S2+S3 0.00

-0.29

-0.72

-0.93

-1.17

-1.30

-1.31

-1.18

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5

6

7

8

9

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11

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1.43

0.83

0.99

The displacement profile of the six and thirteen story buildings during the time history analysis is captured in Figures 12 and 13. Also shown in these figures are the corresponding story drift histories. Though not immediately evident from the figure, it was observed that the drift profile is initially representative of the modal contributions to the response based on spectral demands and that these demands change as the systems become inelastic and the modal periods shift along with the corresponding spectral demands.

Figure 12. Profile of response history for the six-story structure: (a) Roof drift ratio history; (b) Interstory drift ratio history

Figure 13. Profile of response history for the thirteen-story structure; (a) Roof drift ratio history; (b) Interstory drift ratio history

Evaluation of Component Ductility Demands Another important parameter in seismic response analysis is the estimate of ductility demands in individual components. FEMA-356, for example, uses component acceptance criteria using ductility demands as the fundamental basis of its performance-based evaluation methodology. In this section, the effectiveness of the MMC procedure to estimate component demands is investigated. Tables 2 and 3 show typical ductility demands for column elements at critical story levels in the six and thirteen story buildings experiencing the highest deformation demand. Also given are the global system ductility demands which are less than the observed local story and component ductility demands. Similar results were obtained in a more comprehensive study by the authors examining ductility demands of RC and steel buildings [12]. These results serve as evidence that designing a building to achieve a certain ductility demand may result in much larger demands at the local level. Comparisons of the ductility demand from pushover procedures with by nonlinear time-history analyses show that the predicted demands are remarkably similar to those estimated. A more visual comparison is provided in Figures 14 and 15 where the moment-rotation behavior of three typical column sections undergoing inelastic deformation is displayed. While cumulative effects cannot directly be incorporated into any static procedure, the ability of MMC to estimate component deformations is clearly demonstrated in these figures.

Table 2. Typical ductility demands in 6-story building

Location NSP-1 * NSP-2 * NTH MMC

Global - 1.53 - 1.92

5th Story - 0.0 0.0 - 2.02

5th Story Column Interior 0.0 0.0 2.81 2.73

* NSP-1:Inverted triangle; NSP-2: Mass proportional

Table 3. Typical ductility demands in 13-story building

Location NSP-1 * NSP-2 * NTH MMC

Global - 2.08 2.24 - 2.05

7th Story - 2.19 1.32 - 2.59


9th Story - 1.30 0.0 - 1.90


* NSP-1:Inverted triangle; NSP-2: Mass proportional

CONCLUSIONS The popularity of nonlinear static pushover analysis in engineering practice calls into question the validity of conventional lateral load patterns used to estimate inelastic demands. The aim of the present work is to develop alternative multi-mode pushover analysis procedures by indirectly accounting for higher mode contributions but yet retaining the simplicity of invariant distributions in a theoretically consistent manner. A new combination scheme is investigated in this paper and compared to both time-history procedures and other pushover methods. The evaluation is based on a series of analyses of existing steel moment frame buildings.

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5th Story InteriorColumn

Figure 14. Comparison of ductility demands in typical interior column from MMC and NTH for 6-story building (θ implies total rotation, θy implies yield rotation)

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Figure 15. Comparison of inelastic static demand using MMC and cyclic demand in typical interior column of the 13-story building

This study indicates that pushover methods utilizing lateral force distributions based on a single mode are not capable of predicting the story level at which the critical demands occur. On the other hand, the results of modal combination procedures based on ongoing research appears to be promising in terms of better estimating peak values of critical inelastic response quantities such as inter-story drifts and plastic hinge rotations. It is shown that considering sufficient number of modes, interstory drifts estimated by MMC is generally similar to trends noted from NTH analyses unless the building is deformed far into the inelastic range with significant strength and stiffness deterioration. Higher mode effects on seismic demand are dependent both on the frequency content of the ground motion and the characteristics of the structural system even for regular low-rise buildings (based on findings from the four-story building evaluation). In the present phase of the research, the force distributions are based on modal contributions in the elastic state of the system. The influence of higher modes in the inelastic phase of the response can be incorporated by introducing modification factors that

account for changes in spectral demands due to inelastic effects. Additional studies considering various structural systems and ground motions are ongoing to further validate the methodology.

ACKNOWLEDGEMNT Funding for this study provided by the National Science Foundation under Grant CMS-0296210, as part of the US-Japan Cooperative Program on Urban Earthquake Disaster Mitigation, is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

REFERENCES

1. Gupta, B. and Kunnath, S.K. (2000). “Adaptive spectra-based pushover procedure for seismic

evaluation of structures.”, Earthquake Spectra, 16(2), 367-392. 2. Chopra, A.K. and Goel, R.K. (2002). “A modal pushover analysis procedure for estimating seismic

demands for buildings.” Earthquake Engineering and Structural Dynamics, 31,561-582. 3. FEMA-356. (2000). Prestandard and Commentary for the seismic rehabilitation of buildings,

American Society of Civil Engineers (ASCE), Reston, VA. 4. Kunnath, S. K. (2004). “Identification of modal combinations for nonlinear static analysis of

building structures.” Computer-Aided Civil and Infrastructure Engineering.19: 282-295. 5. Paret T.F., Sasaki K.K., Eilbeck, D.H. and Freeman, S.A. (1996). “Approximate inelastic procedures

to identify failure mechanisms from higher mode effects.” Paper No. 966, Proc. of the 11th WCEE, Acapulco, Mexico.

6. Sasaki, K.K, Freeman, S.A. and Paret, T.F. (1998). “Multimode pushover procedure (MMP) – A method to identify the effects of higher modes in a pushover analysis.” Proc. of the U.S. National Conf. on Earthquake Engineering, Seattle, WA.

7. Applied Technology Council (ATC-40). (1996). Seismic evaluation and retrofit of concrete buildings, Redwood, CA.

8. Krawinkler, H. and Al-Ali, A. (1996). “Seismic demand evaluation for a 4-story steel frame structure damaged in the Northridge earthquake.” The Structural Design of Tall Buildings, 5(1), 1-27.

9. Kunnath, S.K., Nghiem, Q. and El-Tawil, S. (2004). “Modeling and response prediction in performance-based seismic evaluation: case studies of instrumented steel moment-frame buildings.” Earthquake Spectra. 20 (3).

10. Uang, C.M., Yu, Q.S., Sadre, A., Youseff, N. and Vinkler, J. (1997). “Seismic response of an instrumented 13-story steel frame building damaged in the 1994 Northridge Earthquake.” Earthquake Spectra, 13(1), 131-149.

11. OpenSees. (2004). Open system for earthquake engineering simulation, http://opensees.berkeley.edu.

12. Kunnath, S.K. and Kalkan, E. (2004). “Evaluation of seismic deformation demands using nonlinear procedures in multistory steel and concrete moment frames.” ISET Journal of Earthquake Technology. 41(1).

NONLINEAR STATIC (OR PUSHOVER) ANALYSIS

A pushover analysis is a nonlinear static procedure wherein monotonically increasing lateral loads are applied to the structure till a target displacement is achieved or the structure is unable to resist further loads. You can conduct a pushover analysis in SAP2000 as follows:

• Define the properties of the plastic hinges. A plastic hinge forms when a section reaches it moment capacity. While a “hinge” implies that the section cannot carry additional moments, it is possible to specify some amount of post-yield stiffness. It is advisable to specify some post-yield stiffness so that sudden system instability (when a mechanism is formed) is avoided.

• To specify hinge properties, select: DEFINE SECTION PROP HINGE PROP • Then select ADD NEW PROPERTY Select Steel/Concrete or User Defined

If you have calculated the yield moments yourself for beam sections, and generated P-M interaction curves for column sections, you can choose “User Defined” and input the hinge properties yourself.

To begin with, I suggest using the program generated properties. Hence, if you select STEEL

1. Enter a Hinge Property Name (to identify the hinge type), example “beam” 2. Choose M3 (this is a pure moment hinge) 3. This will bring up a new menu where you can specify the hinge property 4. Input values for C, D and E so that you get a bilinear curve with a constant slope; for example:

a. Moment/SF Rotation/SF 1. 0. 1.1 5 1.2 10

b. 1.5 25 c. Will generate a hinge with 2% post-yield slope

5. Also “check” the box “IS EXTRAPOLATED” 6. Click “OK” to complete the hinge definition Repeat the above steps for COLUMN hinges – but this time you will select a “P-M3” hinge and accept the default properties. You cannot specify moment rotation properties for columns, since they are dependent on the axial loads. NOTE: When you are back in the menu item titled “Define Frame Hinge Properties” you can check the box that states “Show Generated Props” This will enable you to come back to this menu (after assigning hinges to elements) to actually check the assigned properties. Next, you need to ASSIGN these hinge properties to the elements.

• Select all the beam elements • Select “ASSIGN” “FRAME” “HINGES” • From the drop-down menu select “beam” (defined in Step 1), choose “Relative Distance” as “0” and

“Add” then change relative distance to “1” and “Add” again. Now you should see the “beam” hinge property defined at 2 relative distances (0 and 1). This means you are assigning the beam hinge defined in Steps 1 – 6 to the 2 ends of the beam.

• Click “OK” – you should see the hinges assigned to the beam • Repeat the above process to assign the column hinges

A pushover analysis is associated with a load pattern. In this case, use the same later load pattern that was used for the seismic design. It is necessary to apply dead and live loads before doing a pushover analysis. Since the pushover procedure is nonlinear, it is necessary to setup a new load case for dead and live which is also nonlinear. Start by defining a new D+L case

• “DEFINE” “LOAD CASES” “ADD NEW LOAD CASE” • Specify a new load case (example D&L) • Select the correct Load Names and Scale Factors (example 1.2 D + 0.5 L) • Check the option for “Nonlinear” analysis • Click OK

The following steps outline the definition of a pushover analysis case:

• “DEFINE” “LOAD CASES” “ADD NEW LOAD CASE” • Specify a Load Case Name • Check box “Continue from State at End of Nonlinear Case” and pick “D&L” (defined in previous step) • Click the box “Nonlinear” • Choose the correct Load Pattern (in this case, you will select the Lateral Load Case) • Specify a small scale factor (<1.0), since we want the load to be applied in small steps • Under “Other Parameters”

o Click “Modify/Show” for “Load Application” and choose “Displacement Control” and “Conjugate Displacement” (Note that SAP will automatically select a node and displacement – this is usually a reasonable estimate, so you don’t need to change it)

o Click “Modify/Show” for “Results Saved” and select “Multiple States” o Click “Modify/Show” for “Nonlinear Parameters” and selected “Apply Local Redistribution”

Run the analysis Review results by first checking the pushover curve (DISPLAY Show Static Pushover Curve) If the pushover curve is linear, then none of the members have yielded under the specified lateral load. If you used the full lateral load as the load pattern for the pushover, this is unlikely – something else is wrong and you need to check your hinge specifications again. To make sure that the nonlinear analysis parameters are correctly specified, try running it again by scaling the lateral load with a factor greater than 1.0. If the pushover curve shows a sudden drop with limited inelastic behavior, you probably need to reduce the scale factor for the lateral load. Once you are satisfied with the pushover curve, you can examine the detailed results in the usual manner by viewing results step by step graphically or by outputting the results into a Table (all options are in the DISPLAY menu).

análisis pushover

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