10.1007@978-90-481-8746-120

Chapter 20Multi-Mode Pushover Analysis with GeneralizedForce Vectors

Halûk Sucuoglu and M. Selim Günay

20.1 Introduction

Considering the simplicity and conceptual appeal of conventional pushover analy-sis with a single mode, several researchers have attempted to develop multi-modepushover analysis procedures in order to replace nonlinear response history anal-ysis with an “inelastic” response spectrum analysis [3, 5]. Adaptive lateral forcedistribution schemes have further been proposed for overcoming the limitations ofconventional pushover analysis arising from an invariant lateral static load distribu-tion [1, 2, 4] which however require rigorous computations in the implementation.All multi mode pushover analysis procedures published in literature so far have twocommon features. First, they are adaptive except MPA [3] hence require an eigen-value analysis at each loading increment. Moreover, an adaptive algorithm cannot beimplemented with a conventional nonlinear structural analysis programming code.Second, all procedures combine modal responses statistically by SRSS, which is anapproximate rule developed for combining linear elastic modal responses. Internalforces should be checked at each load increment and be corrected if they exceed theassociated capacities.

A practical nonlinear static procedure is developed herein which accounts for thecontribution of all significant modes to inelastic seismic response. The procedureconsists of conducting a set of pushover analyses by employing different generalizedforce vectors. Each generalized force vector is derived as a different combinationof modal lateral forces in order to simulate the effective lateral force distributionwhen the interstory drift at a selected story attains its maximum value during seis-mic response. Hence the proposed procedure is called generalized pushover analysis(GPA). Target seismic demands for interstory drifts at the selected stories are calcu-lated from the associated generalized drift expressions where nonlinear responseis considered in the first mode only. Finally, the maximum value of a responseparameter is obtained from the envelope values produced by the set of generalized

H. Sucuoglu (B)Department of Civil Engineering, Middle East Technical University, 06531 Ankara, Turkeye-mail: [email protected]

213M.N. Fardis (ed.), Advances in Performance-Based Earthquake Engineering,Geotechnical, Geological, and Earthquake Engineering 13, DOI 10.1007/978-90-481-8746-1_20,C© Springer Science+Business Media B.V. 2010

214 H. Sucuoglu and M.S. Günay

pushover analysis conducted separately for each interstory drift. GPA can be imple-mented with any structural analysis software capable of performing displacementcontrolled nonlinear incremental static analysis. Seismic response of a twelve storyreinforced concrete frame structure under twelve ground motion records are esti-mated by GPA in this study, and compared with the results obtained from NRHA aswell as from the conventional pushover analysis.

20.2 Generalized Force Vectors

Different response parameters attain their maximum values at different times dur-ing seismic response. An effective force vector acts on the system instantaneouslyat the time when a specific response parameter reaches its maximum value. Thiseffective force vector is in fact a generalized force since it has contributions fromall modal forces at the time of maximum response for the specified response param-eter. Accordingly, if this force vector can be defined, then it can be applied eitherdirectly or incrementally to the investigated structural system in order to producethe maximum value of this response parameter.

The derivation of generalized effective force vectors is based on the dynamicresponse of linear elastic MDOF systems to earthquake ground excitation üg(t), byemploying the modal superposition procedure. The effective force vector f(tmax) attime tmax, when an arbitrarily selected response parameter reaches its maximumvalue, can be expressed as the superposition of modal forces fn (tmax):

f (tmax) =∑

nfn(tmax) (20.1)

The n’th mode effective force in Eq. (20.1) at time tmax is given by

fn(tmax) = ΓnmϕnAn(tmax) (20.2)

Here, Γ n = Ln /Mn; Ln = ϕTn ml; Mn = ϕT

n mϕn; ϕn is the n’th mode shape,m is the mass matrix, l is the influence vector, and An(tmax) is represented withEq. (20.3).

An(tmax) = ω2nDn(tmax) (20.3)

Here ω2n is the n’th mode vibration frequency and Dn(tmax) is the modal

displacement amplitude at tmax which satisfies

Dn(tmax) + 2ςnωnDn(tmax) + ω2nDn(tmax) = −ug(tmax) (20.4)

Dn(tmax) cannot be determined from Eq. (20.4) unless tmax is known. tmax is thetime when the selected response parameter becomes maximum, which depends onall modal responses. This response parameter is selected as the interstory drift�j atthe j’th story. Then,

20 Multi-Mode Pushover Analysis with Generalized Force Vectors 215

�j,max = �j (tmax) (20.5)

Its associated modal expansion is

�j (tmax) =∑

n�n Dn(tmax) (ϕn, j − ϕn, j−1) (20.6)

where ϕn, j is the j’th element of the mode shape vector ϕn. Equation (20.6) can benormalized by dividing both sides with �j(tmax), which yields

1 =∑

n

[�n

Dn(tmax)

�j (tmax)(ϕn, j − ϕn, j−1)

](20.7)

Each term on the right hand side of Eq. (20.7) under summation expresses thecontribution of n’th mode to the maximum interstory drift �j (tmax) at the j’th storyin a normalized form.

The maximum value of interstory drift at the j’th story in Eq. (20.5) can also beestimated by RSA through SRSS of the related spectral modal responses.

(� j, max)2 ≈∑

n

[�nDn(ϕn, j − ϕn, j−1)

]2 (20.8)

Dn in Eq. (20.8) is the spectral displacement of the n’th mode, which is directlyavailable from the displacement response spectrum of the ground excitation üg(t).Equation (20.8) can also be normalized similarly, by dividing both sides with(� j, max)2.

1 =∑

n

[�n

Dn

�j,max(ϕn,j − ϕn,j−1)

]2

(20.9)

Accordingly, the respective terms on the right hand sides of Eqs. (20.7) and (20.9)are the normalized contributions of the n’th mode to the maximum interstory driftat the j’th story. Equating them and assuming the equality �j,max = �j (tmax) fromEq. (20.5) leads to

Dn(tmax) = Dn

�j,max

[�n Dn(ϕn,j − ϕn,j−1)

](20.10)

Since the term in the parentheses in Eq. (20.10) is equal to�j,n from Eq. (20.8),

Dn(tmax) = �j,n

�j,maxDn (20.11)

It should be noted that �j,n in Eq. (20.11) is the n’th mode contribution to themaximum interstory drift at the j’th story determined from RSA, and �j,max inEq. (20.11) is the quadratic combination of the �j,n terms as given by Eq. (20.8).The generalized force vector is obtained by first calculating An(tmax) by substituting


Dn(tmax) from Eq. (20.11) into Eq. (20.3), then substituting An(tmax) from Eq. (20.3)into Eq. (20.2), and finally substituting fn(tmax) from Eq. (20.2) into Eq. (20.1).

fj(tmax) =∑

n

(�nmϕn An

�j,n

�j,max

)(20.12)

An is the pseudo spectral acceleration of the n’th mode in Eq. (20.12). Since theformulation that is developed in Eqs. (20.1)–(20.11) is employed for obtaining thegeneralized force vector which acts on the structural system when the interstory driftat the j’th story becomes maximum, the associated generalized force vector in Eq.(20.12) is identified with the subscript j.

20.3 Target Seismic Deformation Demand

In linear elastic response spectrum analysis the target drift demand at the j’th storyΔjt is calculated from the SRSS combination of modal drifts Δn,j expressed by

Δ2jt =

∑n

[Γn (ϕn, j − ϕn, j−1)Dn

]2 (20.13)

The displacement shape of a nonlinear system during seismic response can beexpanded in terms of the linear elastic mode shapes if the modal amplitudes (coordi-nates) can be calculated appropriately. It has been observed by Chopra and Goel [3]that the coupling between modal coordinates due to yielding of the system is neg-ligible. Therefore Eq. (20.13) may be employed for estimating the inelastic driftdemands, provided that linear elastic modal spectral displacement demands Dn inEq. (20.13) are replaced by the inelastic modal spectral displacement demands Dn

∗.Replacing only D1 in Eq. (20.13) with D1

∗ while retaining the linear elasticmodal spectral displacements for the second and higher modes improves the tar-get displacement demand significantly. D1

∗ can then be estimated from either theNRHA of the nonlinear SDOF system representing the first mode contribution,or from the associated R-μ-T relations. Then the target drift demand Δjt in GPAbecomes;

Δjt =([�1 (ϕ1, j − ϕ1, j−1)D∗

1

] 2 +N∑

n=2

[Γn (ϕn, j − ϕn, j−1)Dn

]2)1/2

(20.14)

The contribution of higher modes to a maximum interstory drift parameter ismore significant than their contribution to a maximum displacement parameter. GPAuses the interstory drift parameters as target demands. Accordingly, interstory drift isnot obtained from GPA, but from an independent response spectrum analysis. Whenthe associated generalized force vector pushes the system to this target drift, thesystem adopts itself in the inelastic deformation range while the further higher order


deformation parameters (rotations, curvatures) take their inelastic values as in modalpushover, but by receiving more appropriate contributions from the higher modes.

20.4 Generalized Pushover Algorithm

The GPA algorithm is composed of the five basic steps summarized below.

1. Eigenvalue analysis: Natural frequencies ωn (natural periods Tn), modal vec-tors ϕn and the modal participation factors �n are determined from eigenvalueanalysis.

2. Response spectrum analysis: Modal spectral amplitudes An, Dn are obtainedfrom the corresponding linear elastic spectra and modal interstory drift ratiosat the j’th story, �j,n are determined from RSA. The maximum interstory driftratio at the j’th story, �j,max is obtained by SRSS.

3. Generalized force vectors: Generalized force vectors fj which produce themaximum response �j are calculated from Eq. (20.12).

4. Target interstory drift demands: Maximum inelastic modal displacement demandD1

∗ for the first mode under an earthquake excitation is obtained from eitherNRHA or inelastic response spectrum of the inelastic SDOF system idealizedwith a bi-linear force–displacement relation. For the higher modes n = 2–N, Dn

values are obtained from the linear elastic response spectrum. Finally, D1∗ and

Dn (n = 2, N) are substituted into Eq. (20.14) for calculating the target interstorydrift demands Δjt.

5. Generalized pushover analysis: A total number of N generalized pushover anal-yses are conducted. In the j’th GPA (j = 1–N), the structure is pushed in thelateral direction incrementally with a force distribution proportional to fj. Atthe end of each loading increment i, the interstory drift �ji obtained at thej’th story is compared with the target interstory drift �jt calculated from Eq.(20.14). Displacement controlled incremental loading (i = 1, 2. . .) at the j’thGPA continues until �ji reaches �jt.

All member deformations and internal forces are directly obtained from the j’thGPA at the target interstory drift Δjt. Once all GPA is completed for j = 1–N, theenveloping values of member deformations and internal forces are registered as themaximum seismic response values.

20.5 Test Case: 12 Story RC Symmetrical Plan Building

The proposed GPA procedure is tested on a 12 story reinforced concrete buildingwith symmetrical plan, where the floor plan is shown in Fig. 20.1. The building isdesigned according to the regulations of 2007 Turkish Earthquake Code in accor-dance with the capacity design principles. An enhanced ductility level is assumedfor the building. The design spectrum is shown in Fig. 20.1. Concrete and steel char-acteristic strengths are 25 and 420 MPa, respectively. Slab thickness for all floors is


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3T (s)

A (g

)

5 m

5 m

Frame A

Frame BB5 m

6 m 4 m 6 m

a) b)

Fig. 20.1 (a) Typical plan of the 12 story building, (b) design spectrum

140 mm and live load is 3.5 kN/m2. Dimensions of the beams at the first four, thesecond four and the last four stories are 300 × 550, 300 × 500 and 300 × 450 mm2

respectively, whereas dimensions of the columns at the first four, the second fourand the last four stories are 500 × 500, 450 × 450 and 400 × 400 mm2 respec-tively. There is no basement; height of the ground story is 4 m while the height ofall other stories is 3.2 m.

Plane frame models consisting of Frames A and B are constructed for the analy-sis. They are analyzed by using the nonlinear analysis software Drain-2DX. Crackedsection stiffness is employed for the initial linear segment of the moment-curvaturerelations. Gross moments of inertia are multiplied with 0.6 and 0.4 for the columnsand beams, respectively, in order to represent cracking. Free vibration periods forthe first three modes of the frame are 2.38, 0.86 and 0.50 s.

20.5.1 Strong Ground Motions

The building is analyzed under twelve different strong motion components. Theseground motions are selected from a larger set in accordance with the FEMA 356criteria limiting the applicability of conventional nonlinear static procedure. Theyproduce significant higher mode effects on the investigated 12 story frame; henceFEMA 356 does not allow using standard NSP. The basic characteristics of thetwelve ground motion components are presented in Table 20.1. The first nine containa significant pulse whereas the last three are without a pulse. All ground motionswere downloaded from the PEER strong motion database.

20.5.2 Implementation of the GPA algorithm

The generalized pushover analysis algorithm is presented herein with an implemen-tation to the 12 story building, subjected to the ground motion 7 in Table 20.1. Theresulting force distributions fj along height, obtained from Eq. (20.12) are presentedin Fig. 20.2. It can be observed that the second mode contribution is significant onthe force distributions which produce maximum interstory drifts at all 12 stories.


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(g)

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(cm

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(7.1

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3/92

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4.4

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496

64.3

21.9

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0414

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407.

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0.48

537

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1090

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95(6

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Port

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278

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52.0

15.3

Ord

inar

y


0123456789

101112

–1000 –500 0 500 1000 1500

Lateral Force (kN)

Sto

ry #

f1f2f3f4f5f6f7f8f9f10f11f12

Fig. 20.2 Generalized force distributions fj for the 12 story building

20.5.3 Results

Beam plastic rotations and member internal forces are calculated respectively underthe 12 ground motions in Table 20.1. Beam plastic rotations at member ends cal-culated with NRHA, GPA and PO-1 (conventional pushover analysis) are shown inFig. 20.3. Beam plastic rotations are calculated for each story, as the average valueof all beam-end plastic rotations in that story. Plastic actions did not develop at thecolumns of the building under any of the twelve ground motion excitations exceptthe ground story column bases.

The performance of GPA in predicting beam plastic rotations from NRHA issatisfactory. PO-1 is not able to capture plastic deformations in any of the beams atthe upper five stories since first mode is not sufficient by itself for developing plasticrotations. GPA is quite successful in predicting the plastic beam deformations atboth upper and the lower stories. Simultaneous combination of all modes in GPAleads to realistic estimations of plastic deformations. There are some unsuccessfulcases at the lower stories however for GPA, such as GM2 and GM9, where highermode contributions at the lower stories do not develop apparently during the NRHAcontrary to the GPA prediction.

20.6 Conclusions

The generalized pushover analysis (GPA) procedure presented here has three basicadvantages over the other multi-mode pushover procedures available in literature.

• GPA (like MPA) is not adaptive. It can be implemented conveniently by employ-ing a general purpose nonlinear static analysis tool. There is no need fordeveloping a special programming code.


GM 10123456789

101112

0 0.005 0.01 0.015

Sto

ry #

GM 20123456789

101112

0 0.005 0.01 0.015 0.02

GM 30123456789

101112

0 0.005 0.01 0.015

Sto

ry #

GM 40123456789

101112

0 0.005 0.01 0.015

GM 50123456789

101112

0 0.005 0.01

Sto

ry #

GM 60123456789

101112

0 0.005 0.01 0.015

GM 70123456789

101112

0 0.005 0.01 0.015 0.02

Plastic Rotation (rad)

Sto

ry #

GM 80123456789

101112

0 0.005 0.01


Fig. 20.3 Beam plastic rotations


GM 90123456789

101112

0 0.005 0.01 0.015

Sto

ry #

GM 100123456789

101112

0 0.005 0.01

GM 110123456789

101112

0 0.002 0.004 0.006 0.008 0.01


Sto

ry #

GM 120123456789

101112

0 0.005 0.01 0.015


NRHA GPA PO-Mode1

Fig. 20.3 (continued)

• The computational effort required by GPA is much less compared to the adaptivepushover procedures.

• GPA does not suffer from the statistical combination of individually calculatedinelastic modal responses. It activates all modes of a multi degree of freedomsystem simultaneously. So, all response parameters are obtained directly from ageneralized pushover analysis at the associated target drift demand.

The number of pushover analysis required in GPA under a single ground excita-tion expressed by its response spectrum is equal to one plus the number of storiesin symmetrical plan buildings (1+N), where N is the number of lateral dynamicdegrees of freedom, or the number of stories. However 1+N pushover in GPA isonly repetitive.

References

1. Antoniou S, Pinho R (2004) Development and verification of a displacement-based adaptivepushover procedure. J Earthq Eng 8(5):643–661

2. Aydınoglu MN (2003) An incremental response spectrum analysis procedure based on inelas-tic spectral displacements for multi-mode seismic performance evaluation. Bull Earthq Eng1(1):3–36


3. Chopra AK, Goel RK (2002) A modal pushover analysis procedure for estimating seismicdemands for buildings. Earthq Eng Struct Dyn 31(3):561–582

4. Gupta B, Kunnath SK (2000) Adaptive spectra-based pushover procedure for seismic evaluationof structures. Earthq Spectra 16(2):367–391

5. Sasaki F, Freeman S, Paret T (1998) Multi-mode pushover procedure (MMP). Proceedings of6th U.S. NCEE, Seattle, WA, USA

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