An Equivalent Model for Seismic Analysis of Masonry and Reinforced Concrete Buildings

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Construction and Building Materials 23 (2009) 40–53

and Building

MATERIALS

Review

An equivalent frame model for seismic analysis of masonryand reinforced concrete buildings

Y. Belmouden *, P. Lestuzzi

Ecole Polytechnique Federale de Lausanne, ENAC-IS-IMAC, EPFL, Station 18, CH-1015 Lausanne, Switzerland

Received 13 August 2007; received in revised form 29 October 2007; accepted 29 October 2007Available online 19 December 2007

Abstract

In this paper a novel equivalent planar-frame model with openings is presented. The model deals with seismic analysis using the Push-over method for masonry and reinforced concrete buildings. Each wall with opening can be decomposed into parallel structural wallsmade of an assemblage of piers and a portion of spandrels. As formulated, the structural model undergoes inelastic flexural as wellas inelastic shear deformations. The mathematical model is based on the smeared cracks and distributed plasticity approach. Both zeromoment location shifting in piers and spandrels can be evaluated. The constitutive laws are modeled as bilinear curves in flexure and inshear. A biaxial interaction rule for both axial force–bending moment and axial force–shear force are considered. The model can supportany shape of failure criteria. An event-to-event strategy is used to solve the nonlinear problem. Two applications are used to show theability of the model to study both reinforced concrete and unreinforced masonry structures. Relevant findings are compared to analyticalresults from experimental, simplified models and finite element models such as Drain3DX and ETABS finite element package.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Seismic evaluation; Unreinforced masonry; Reinforced concrete; Structural wall; Equivalent frame

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412. A model for structural walls with openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

0950-0

doi:10.

* CoE-m

2.1. Description and hypotheses of the structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2. Formulation of a structural wall model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3. A nonlinear analysis of framed structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444. Pushover analysis of a RC building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1. Description of the structural model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2. Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5. Pushover analysis of an URM building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1. General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2. Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Appendix 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

618/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

1016/j.conbuildmat.2007.10.023

rresponding author. Tel.: +41 21 693 6382; fax: +41 21 693 47 48.ail addresses: [email protected] (Y. Belmouden), [email protected] (P. Lestuzzi).

mailto:[email protected]

mailto:[email protected]

F

Intermediate wall

Rigid zone

Deformablepart

Opening

F

ih

iL

spl

ph

Edge wallEdge wall

Fig. 1. A schematic representation of equivalent frame model for planarwalls with openings.

Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 40–53 41

1. Introduction

Earthquakes are considered to be the major cause ofstructural failure of buildings in Europe. Despite their rar-ity and moderate intensity, earthquakes in the interior ofnorthwest and central Europe have the potential to causeextensive damage and associated financial losses, due tothe vulnerability of the local building stock. The mitigationof earthquake hazard involves the collaboration of manyspecialists with different tasks. One of these topics is struc-tural engineering providing and advancing the knowledgefor earthquake resistant construction. However, a problemarises for existing buildings analysis. In this context, in thefew last decades, technical advances have been made inseismic engineering and particularly in the seismic vulnera-bility assessment of existing buildings. The vulnerabilityassessment focuses on the study of the extent of damagefor different earthquake scenarios.

In almost all countries, the majority of the buildingstock is classified as existing buildings. This is why exten-sive assessment of such structures is motivated since theyhave been generally designed to resist gravity loads. Never-theless, the seismic vulnerability of existing buildingsdesigned against wind loads, is found to be very low.

This paper makes a contribution to the seismic vulnera-bility assessment of existing buildings through the develop-ment of a simplified analytical model. The need for suchmodels is always motivated by first, the large amount ofstructures that should be analyzed in a very short timeand second, the search for optimal solutions for structuralretrofitting.

For vulnerability assessment purposes, the analysis of alarge number of existing buildings requires relatively simpleapproaches that are capable of representing their essentialcharacteristics. The models should be able to evaluate theultimate strength, maximum displacements and the failuremodes. Different models are developed based on analyticaland finite element approaches [1]. The analytical modelsare found to be very simple to use and require lesseramount of data. However they are very limited, particu-larly for large building analysis in terms of structuralbehavior (coupling effect, distribution of the nonlinearity,modes of failures prediction). The performed analysis showthat they are conservative and are not able to represent allfeatures of such buildings [2]. On the other side, finite ele-ment approach is a powerful tool for seismic analysis but itis time consuming and requires a large amount of data.Moreover, refined models based on either discrete or con-tinuum approaches suffer from the strong mesh-depen-dency and require numerous parameters that may not bedirectly extractable from structural analysis. Hence, thesemodels are very sensitive to the parameter calibration thataffects closely the reliability of the results and the analysisstability (lack of convergence, flip-flop occurrence, suddenload falling, and so on). With such methods it is not possi-ble to treat a stock of buildings. Thus, these methods arecumbersome due to the high analytical skills required for

their numerical implementation and they are restricted onlyto practitioners with a high level of knowledge.

A widely used model for structural analysis is the linear(beam-column element) finite element or the equivalentframe models. Despite of some limitations in the equivalentframe model, it is very attractive in comparison to complexfinite element models [1,3–5]. Moreover, they have shownsatisfactory results particularly for RC structures. In thiscontext, the proposed model is based on beam-column ele-ment and distributed of nonlinearity approaches. It isadapted to analytical methods without use of finite elementmethod.

In this paper, the developed model deals with the seismicvulnerability assessment of existing multistoried buildings.

2. A model for structural walls with openings

2.1. Description and hypotheses of the structural model

The mathematical model can represent solid walls,frame structural elements (made in beams and columns),coupled walls and perforated walls (or framed walls) [6].The model can represent different openings. However, thevertical axis should lie through all vertical piers elementsas well as for the horizontal axes that should lies throughall spandrels.

The structural model consists of an assemblage of verti-cal plane walls with openings that form a single perforatedwall. Each structural wall is made of pier elements with orwithout rigid offsets and a portion of spandrels such thatthere are two kinds of individual walls: exterior walls andinterior walls (Fig. 1). The length of these parts of span-drels is equal to the zero moment length, and can beupdated at each step depending on the bending momentsat the spandrel ends.

In the equivalent frame models that are based on finiteelement method, nonlinear flexural springs (lumped plastic-ity) are inserted into the model at the ends of the piers and/or spandrel elements. These elements are defined in terms

42 Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 40–53

of moment–rotation laws. Translational shear springs areadded at each pier and spandrel at mid-points. Thesesprings are expressed in terms of shear force–displacementlaws. However, the occurrence of yielding is unlikely alongspandrel spans and piers heights. For that reason, nonlin-earity should be distributed along the clear pier heightand clear spandrel length. Thus, the proposed model isbased on the spread nonlinearity approach. Each pierand spandrel can be discretized into a series of slices [7]while cross-sections are considered as homogeneous. Thestructural element behavior is monitored at the center ofthe slices [7] while bending moments are evaluated at sliceends.

The mechanical model undergoes flexural as well asshear deformation. In the current formulation, the modelonly considers a biaxial interaction between axial forces–bending moments (N–M) and axial forces–shear forces(N–V) only. The so-called shifting of the primary curvetechnique is used in a simple manner [8]. The axial forceis evaluated in a simple manner based on initial axial forcesplus vertical shear forces produced in spandrels at joints. Atriaxial interaction rule, (N–M–V), is not currently consid-ered. At present, only interaction curves that representsbending moment or shear force interaction with regardsto a compressive axial force are considered. The major fea-tures of this model are summarized as follows:

1. All previous attempts to use simplified models based onstatic equilibrium method, always consider a constantzero moment location [9–14], and others. The wall for-mulation herein permits the capture of the couplingeffect in elevation due to the nonlinearity distributionin both piers and spandrels. Thus, the zero momentlocation in both piers and spandrels can be mitigatedduring the nonlinear analysis.

2. In the current development, the variation of the axialvertical loads are considered for piers only and theyare based on an over-simplified approach. The axialloads on piers are updated based on the initial axialforces at each storey plus the shear forces developedon spandrel ends.

3. The nonlinearity is treated using a smeared plasticityapproach [7]. Thus, the piers and spandrels are discret-ized into finite homogenized slices [15]. Variable sectionscan be specified over either spandrels or piers. In pierelements the axial forces can increase or decrease. Inthat case, the pier slices can shift either from elastic-to-plastic or from plastic-to-elastic state depending onthe axial force distribution.

4. The model can take into account both flexural and shearbehavior in the inelastic range. The interaction effect canbe defined by using experimental and phenomenologicalmodels. These equations are considered as failure crite-ria that can be defined by points and linear segments.The non linear constitutive model for both flexuraland shear behavior is considered as a bilinear envelopcurve with a very small post-yield stiffness to avoid

numerical problems. The flexural behavior is modeledas a moment-curvature law that is based on an equilib-rium statement in a cross-section.

5. The present formulation deals with a Pushover analysis.It is based on the well-known event-to-event strategy. Asimplified algorithm for systems with interaction effect ispresented through an equilibrium correction at each stepof calculation. The analysis is performed by a force-con-trolled technique. The change of sign in a structural ele-ment is permitted only in the elastic range. In theinelastic range, this leads to stoppage of the analysis.

6. The structural wall is a planar structure (two-dimen-sional). However, the sum of all capacity curves, onthe basis of the equal top displacement assumption, per-mits to analyze an entire building and to develop capac-ity curves.

2.2. Formulation of a structural wall model

The structural walls, composed in an assemblage of piersand spandrels, are modeled as an equivalent frame struc-ture. Rigid offsets can be added at the top and/or the bot-tom of the piers, or the left side and/or right side ofspandrels. The storey rotations and lateral displacementsare calculated so that flexural and shear deformationscan be considered. It should be noted that only in-planedeformations and rotations of the entire walls are consid-ered. In the following a general method for structural anal-ysis of a multistory building is given. The wall base isconsidered as fixed. The storey moment–lateral force for-mulation of a structural wall element is expressed by:

fMbsg ¼ ½KFrame�fPg ð1ÞfPg ¼ ½LP–T�½LT–F�fF g ð2Þ

½KFrame��1 ¼

B1 C1 0 0 . . . 0 0 0

A2 B2 C2 0 . . . 0 0 0

0 A3 B3 C3 . . . 0 0 0

..

. ... ..

. ... . .

. ... ..

. ...

0 0 0 0 . . . AN�1 BN�1 CN�1

0 0 0 0 . . . 0 AN BN

26666666664

37777777775ð3Þ

P n ¼ DnT n�1 þ EnT n ð4Þ

{Mbs} represents the base storey bending moments vector,[KFrame] is the equivalent frame stiffness matrix, {P} is thereduced shear forces vector, [LP–T] is the reduced storeyshear forces–storey shear force transformation matrix,[LT–F] is the storey shear force–applied lateral force trans-formation matrix in absence of vertical distributed loads,{F} is the lateral load pattern vector. The equivalent framematrix and the reduced shear forces vector are defined bythe expressions An, Bn, Cn, Dn, and En (with n = 1, N; N isthe number of storeys) (Eqs. (3), and (4)) such as

An ¼ ðkspeq;n�1Þ

�1 ð5Þ


Bn ¼ �1

kspeq;n�1

þ 1

kspeq;n

þ 2

kpn

!ð6Þ

Cn ¼ ðkspeq;nÞ

�1 ð7Þ

Dn ¼�hn�1

kspeq;n�1

ð8Þ

En ¼ hn1

kpn

þ 1

kspeq;n

!ð9Þ

The equivalent stiffness of a spandrel for an individualstructural wall ‘i’ at level ‘n’ (Fig. 2) is:

For an interior wall:

kspeq;n ¼

1

cspandreli�1

þ 1

aspandreli

!ðnÞ

ð10Þ

For a left side exterior wall:

kspeq;n ¼ ða

spandreli Þ�1

ðnÞ ð11Þ

For a right side exterior wall:

kspeq;n ¼ ðc

spandreli�1 Þ�1

ðnÞ ð12Þ

The equivalent stiffness of a pier for an individual struc-tural wall element at level ‘n’ is:

kpn ¼ ðcpier

n þ bpiern Þ

�1 ð13ÞThe superscripts ‘p’ and ‘sp’ mean pier and spandrel ele-ments respectively. The superscripts ‘flx’ and ‘shr’ meanflexure and shear respectively. The nonlinearity redistribu-tion coefficients are defined as follows:

an ¼ aflxn � dshr

n ð14Þbn ¼ bflx

n þ dshrn ð15Þ

cn ¼ cflxn � dshr

n ð16Þ

aflxn ¼ hn

Xm¼Nslices

m¼1

Z gm

gm�1

ð1� nÞ2dnKvðnmÞ

ð17Þ

nh

1−nh

Wall i Wall i+1

1, −nnM1, +nnM

spniLM ,,

nT

1−nT

spniLM 1,, −

spniLM 2,, −

2,1 −− nnM

3,2 −− nnM

nnM ,1−

1,2 −− nnM

niLL ,,0

1,,0 −niLL

2,,0 −niLL2,,0 −niRL

1,,0 −niRL

niRL ,,0 niLL ,1,0 +

1,1,0 −+ niLL

2,1,0 −+ niLL

Level n-2

Level n-1

Level n

Span i

spniRM ,,

spniRM 1,, −

spniRM 2,, −

Fig. 2. Moment diagrams in an equivalent frame model and decompo-sition into simplified individual wall element. L0L,i,n and L0R,i,n are the leftside and right side zero moments lengths for the wall ‘i’ at level ‘n’.

bflxn ¼ hn

Xm¼Nslices

m¼1

Z gm

gm�1

ð1� nÞndnKvðnmÞ

ð18Þ

cflxn ¼ hn

Xm¼Nslices

m¼1

Z gm

gm�1

n2dnKvðnmÞ

ð19Þ

dshrn ¼

1

hn

Xm¼Nslices

m¼1

Z gm

gm�1

dnKdðnmÞ

ð20Þ

Herein, the expressions (17)–(20) are formulated for pierelements; hn is the storey height at level ‘n’, n is anormalized variable. gm�1 and gm are the left side and theright side coordinates of a slice respectively; Nslices is thenumber of slices in a wall element, nm = 0.5(gm�1 + gm)represents the slice centre at which the slice behavior ismonitored. The same expressions are also used for the cal-culation of the nonlinearity redistribution coefficients in thespandrels with respect to their zero moment lengths L0R

and L0L (Fig. 2). The redistribution coefficients are thendiscretized such that a structural element is decomposedinto finite slices [7]. The inter-storey rotations between level‘n’ and ‘n-1’, for a given single wall, omitting the index ‘i’for clarity, are given by:

hn � hn�1 ¼ �ðbpiern þ apier

n ÞMn�1;n þ ðbpiern þ cpier

n ÞMn;n�1

ð21Þ

hn � hn�1 ¼M sp

n;tot

kspeq;n

�M sp

n�1;tot

kspeq;n�1

ð22Þ

M spn;tot and M sp

n�1;tot represent the total bending momentdeveloped in spandrels at the nth and n � 1th level respec-tively (Fig. 2). ksp

eq;n and kspeq;n�1 represent the equivalent stiff-

ness of spandrels at the nth and n � 1th level respectively(Fig. 2) for an individual wall ‘i’.

The lateral story-displacements for each individual wall,in a given section yj, are given as follows:

vðyjÞ ¼ hj�1yj þZ yj

0

vðnÞðyj � nÞdnþZ yj

0

dðnÞdn ð23Þ

The term ‘level’ means the centre-line (or the neutral axis)of the spandrels between two adjacent storeys that form thepier-to-spandrel joints. The wall curvatures and sheardeformations are expressed as follows:

vðnÞ ¼ MðnÞ=KvðnÞ ð24ÞdðnÞ ¼ T ðnÞ=KdðnÞ ð25Þ

M(n) is the bending moment and T(n) is the shear force in aslice, Kv(n) and Kd(n) are the flexural and shear stiffness ofa slice, respectively. The curvatures are assumed linear overeach slice.

Moreover, the equilibrium equation at a rigid joint isformulated as follows:X

Mpiers ¼X

M spandrels ð26Þ

The pier-to-spandrel joint equilibrium for an individualwall ‘i’ (Fig. 2) is expressed as follows:

Slice Slice elasticelastic beforebefore loadingloading (state k(state k--1) 1)

Slice Slice yieldsyields afterafter loadingloading (state k)(state k)

1S

2S

( )11

−kS

( )kS1

( )12

−kS( ) =kS2

1−k

k

( ) ↓Δ kS2

( ) ↑Δ kS1

( )kyS −

2

( )12

−+ kyS

EquilibriumEquilibriumcorrectioncorrection

c

( ) ( )ky

k SS −= 22

( )kyS +

2( )1

2−− k

yS

CompressionCompression

TensionTension

Fig. 3. General representation of an interaction process between S1 and S2

parameters.


M spn;tot ¼ �ðMn;n�1 þMn;nþ1Þ ð27Þ

M spn;tot ¼ M sp

L;i;n þM spR;i�1;n ð28Þ

For a left side exterior wall ‘i’:

M spR;i�1;n ¼ ksp

eq;R;i�1;nhn ð29ÞT sp

R;i�1;n ¼ M spR;i�1;n=L0R;i�1;n ð30Þ

For a right side exterior wall ‘i’:

M spL;i;n ¼ ksp

eq;L;i;nhn ð31ÞT sp

L;i;n ¼ M spL;i;n=L0L;i;n ð32Þ

L0R,i,n and L0L,i,n define the zero moment location for theith spandrel element at the nth level, M sp

L;i;n and M spR;i�1;n

are their corresponding spandrel ends bending momentsin the span ‘i’ (Fig. 2). The initial values for zero momentlocations are calculated according to the relative stiffnessbetween spandrels and the adjacent piers. For an interiorwall, the expressions from Eqs. (29) to (32) are consideredsimultaneously.

3. A nonlinear analysis of framed structures

In the following, a simplified approach of the Pushovermethod based on the event-to-event strategy is presented(Table 1). In this method, the event factor for the entireframe subjected to the predefined lateral load pattern is cal-culated such as ak = Min(ak

i , i = 1, Nwall), Nwall is thenumber of parallel individual walls. A wall event factor,ak

i , is calculated for the nominal event plus the tolerance[7]. A wall event factor is extracted from the lowest slice

Table 1Flow chart of the proposed ‘Pushover’ analysis for the equivalent planar-fram

Step 1 Structural modeling: Frame geometry, lateral loads pattern {F}, initStep 2 Slices state initialization: Yielding bending moments and yielding sh

zero moment locations L0Land L0R.Step 3 For each individual structural wall i = 1, Nwall: (ak = 1)

3-1 Apply a load increment ak{DF}3-2 Having Lspan;k�1

0L , Lspan;k�10R ,Nk�1 and Mk�1

y , and T k�1y

3-3 Calculate an, bn and cn

3-4 Form [KFrame]�1, and {DP}

3-5 Solve Eq. (1)3-6 Calculate {DMbs} and {DT}3-7 Extract the bending moments at the top of all storeys3-8 Calculate the total bending in spandrels DM sp

n;tot, for n =3-9 Calculate Dhn � Dhn�1, for n = 1,N (Eqs. (21) and (22))3-10 Calculate DM sp

R;i�1;n, DM spL;i;n, DT sp

R;i�1;n and DT spL;i;n, for n =

3-11 Calculate the lateral displacementsDv(Eq. (23))3-12 Calculate the event factor for the entire frame (parallel w3-13 Adjust and update the applied lateral force, the storey rot

and their corresponding deformations, D, (slice curvaturDk = Dk�1 + akDDk

3-14 Update axial loads on piers and spandrels by force equi3-15 Equilibrium correction using Nk: determine corrected yie3-16 With respect to the current yielding limits (Fig. 3), updat

the updated slice stiffnesses3-17 New estimation of Lspan;k

0 using spandrel ends bending m3-18 Check for structural stability: Quit the analysis if the str

subjected to a tension axial load. Otherwise perform theStep 4 Plot the capacity curve for the entire structure

event factors. The occurrence of an event depends on theyield surfaces function and yielding limit states for all slicesin flexure and in shear (see Figs. 3 and 4).

4. Pushover analysis of a RC building

4.1. Description of the structural model

A three-dimensional multistoried building made in RCstructural walls is studied (Fig. 5). The structure was mod-eled on both Drain3DX [16] using a fiber beam element(type 15) and ETABS [17] using a point hinge beam ele-

e model

ial compressive axial loads on piers N, failure criterion (N–M) and (N–V).ear forces, elastic stiffnesses, event overshoot tolerances, initial values for

1,N (Eq. (26))

1,N (Eqs. (29)–(32))

alls) ak

ations, the lateral displacements, the bending moments and shear forces, S,es and slice distortions) in both piers and spandrels: Sk = Sk�1 + akDSk,

librium:Nk

lding states Mky and T k

y for the next iteration (Figs. 3 and 4)e the shear and/or flexural stiffness of the yielded slices (Fig. 4): Determine

omentsucture becomes unstable (significant loss of stiffness), or if a given pier isnext iteration, k + 1 (go to step 3-1)

EL

EL

EL

EL

EL

PL

PL

PL

PL

PL

1−> kyMM

1−> kyMM

tolky MMM δ+= −1

1−kstep

kstep

1−< kyMM

kyMM ≤

1−< kyMM

kyMM >

Curvature

Bending moment

1−≥ kyMM

kyMM >

EL

EL

EL

EL

EL

PL

PL

PL

kyMM >

Equilibriumcorrection

For : N ↑ and M ↓

EL

1

2

3

1−kyM

kyM

1−> kyMM

tolky MMM δ+= −1

PL

PL

PL

PL

kyMM ≥

kyMM >

kyMM >

Fig. 4. Bending moment redistribution and the equilibrium correction inthe case of a compressive axial force loading.

Fig. 5. Plan view of the RC structure.

Fig. 6. Plan view of a RC wall section with fiber discretization onDrain3DX model (dimensions in m).

Fig. 7. A view of the structural model developed on ETABS.


ment. The fiber beam is a nonlinear finite element model inflexure but linear elastic in shear. The behavior of fibers isdefined by a stress–strain relationship for both steel andconcrete materials. The use of fibers to model cross-sec-tions accounts rationally for axial force–biaxial bendingmoments (Fig. 6). A detailed description of the element isgiven in the reference including related capabilities,assumptions and limitations [16]. In this application, anelastic perfectly plastic model was adopted for steel fibers.However, a parabolic–rectangular stress block wasadopted for concrete material. The floor was modeled asa grid system using the elastic linear beam element type17 [16]. A validation of the finite element type 15 againstexperimental results can be found in the reference [2].

On the other side, ETABS provides a flexural pointhinge finite element model (PHFE) called P-M2-M3. Thismodel considers an interaction between two-way momentcurves and axial forces. Since the structural model behavesin the in-plane direction (Z-direction, Fig. 5), the pointhinge model performs with a biaxial interaction rule. Thehinges are located at the ends of all beam elements, at

top and bottom storeys. In the equivalent frame model(EFM), the nonlinear behavior for each slice is definedby a moment-curvature relationship in compression only.

The analysis performed on Drain3DX was force-con-trolled. On the other hand, the analysis performed onETABS was displacement controlled in the presence of agiven lateral load pattern. The adopted load pattern repre-sents the distribution of inertia forces corresponding to thefirst mode of vibration.

4.2. Application

The building studied herein is a six storey torsionallybalanced reinforced concrete structure with a total heightequal to 6.0 · 3.4 m and a total floor area equal to

yM

uM

Moment

Curvature'yφ yφ

y

uyy M

M'φφ =

Elastic plastic

Bilinear

Fig. 8. Bilinear idealization of the moment-curvature law.

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

0 1000 2000 3000 4000 5000 6000 7000 8000

Axi

al f

orc

e (k

N)

Tension

Compression

Bending moment (kNm)

Fig. 9. Axial force–bending moment interaction law for RC wall sectionsdefined by linear segments and extracted from ETABS according to theEurocode design code [17].


30.0 · 18.0 m2 (Fig. 7). The floor is of a grid-type with RCslab. In the case studied, the spandrels consist of beam ele-ments representing a grid floor with underbeams. For esti-mating the stiffness of the floors with underbeams, theeffective width of the floor slab is calculated according tothe rules suggested by Bachmann and Dazio [18]. The sec-ond moment of inertia of the floor section is equal to0.0262 m4 [11].

The floor load carried by each wall is equal to 188 kN,300 kN and 900 kN for walls ‘1’, ‘2’ and ‘3’, respectively.The resulting normal forces acting on each floor level aresummarized as follows: the axial forces acting on wallsA1, A2, and A3 at the 6th floor are equal to 211 kN,324 kN, and 924 kN, respectively, while for lower levels,these forces are equal to 235 kN, 347 kN, and 947 kN,respectively. Three cases are investigated to study the axialforce redistribution and the axial force–bending momentinteraction rule. They are: (1) rigid-floor-type structurewith 100% of the floor stiffness, IFloor, (2) semi rigid-floor-type with 50% IFloor and (3) flexible floor-type with10% IFloor. For each floor-type model, two cases were stud-ied for the EFM and four cases for the PHFE model onETABS. The case studies are defined as follows:

1. PHFE M1 and M2: Bilinear and elastic–plasticmoment–rotation law respectively, without (N–M)interaction,

2. PHFE M3 and M4: Bilinear and elastic–plasticmoment–rotation law respectively, including (N–M)interaction,

3. EFM 1 and EFM2: Without and with (N–M) interac-tion, respectively.

In this simulation, the spandrels represent the floorbeams that were elastic linear. Thus, the zero momentlengths were chosen to be at the middle of the span length,without rigid offsets, and kept constant during the analysis.The material properties were defined by the tensile strengthof concrete, the compressive strength of concrete, the steelmaximum strength, the concrete Young modulus and thesteel elastic modulus that are equal to 5 MPa, 45 MPa,500 MPa, 37,500 MPa, and 210,000 MPa, respectively.Additional details on the material properties are found in[11,19].

For the EFM, the walls at each storey are discretizedinto 20 slices over a storey height. The EFM necessitatesmoment-curvature laws that are considered idealized forboth elastic perfectly plastic and bilinear curves (Fig. 8).For the RC wall section, the mechanical properties(Fig. 8) are summarized as follows: The yielding moment,My, the ultimate moment, Mu, the first yield curvature,/0y, the nominal curvature for a bilinear idealization, /y,and the ultimate curvature, /u, are equal to 3034 kN m,4786 kN m, 1.648 m�1, 2.60 m�1, and 28.5 m�1, respec-tively. On ETABS, only yielding moments should be spec-ified for the flexural hinges since the yielding rotations arecalculated by the program. The axial deformations of the

beams are neglected on both ETABS and Drain3DXmodels.

With regards to the biaxial failure criteria used herein(Fig. 9), the hinge yielding in both PHFE model and slicesof the EFM depends closely on the axial load. For lowaxial loads, the yielding of the sections is delayed, whilefor high axial loads the yielding is anticipated. These mech-anisms are closely related to the axial load redistribution,the floor stiffness and wall coupling. Hence, the globalresponse of the structure is affected (elastic stiffness, struc-tural displacement, damage occurrence).

Figs. 10–12 display capacity curves for the three floor-type models analyzed by Drain3DX, ETABS and the pro-posed EFM. The (N–M) interaction effect increases withthe total base shear. The (N–M) interaction has small effectin the first stage of the analysis. As the floor stiffnessincreases, the force redistribution capacity of the structureincreases, the normal forces increase, and then the effect of(N–M) interaction becomes significant. When axial force isstill small, the (N–M) interaction is negligible. In other

0

5000

10000

15000

20000

25000

30000

0 0.05 0.1 0.15 0.20 0.25 0.3

Roof lateral displacement (m)

To

tal b

ase

shea

r (k

N)

Fiber model100%Ipl

Fiber model50%Ipl

Fiber model10%Ipl

PHFE model M1

PHFE model M2

PHFE model M3

PHFE model M4

EFM1

EFM2

10%IFloor

50%IFloor

Fig. 10. Capacity curves for rigid-floor-type model.

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15000

20000

25000

30000

0 0.05 0.1 0.15 0.20 0.25 0.3

To

tal b

ase

shea

r (k

N)

Fiber model100%Ipl

Fiber model50%Ipl

Fiber model10%Ipl

PHFE model M1

PHFE model M2

PHFE model M3

PHFE model M4

EFM1

EFM2

100%IFloor

10%IFloor


Fig. 11. Capacity curves for semi rigid-floor-type model.


words, the (N–M) interaction rule has no effect for flexiblefloor-type structures (Fig. 12). This application tends todemonstrate the ability of the EFM, in comparison toETABS’s results, to reproduce the interaction betweenthe floor stiffness, the structural wall coupling, the forceredistribution, and the failure criteria on the globalresponse of the building.

The order of occurrence of the plastic hinges can beobtained for all the steps of the displacement control froma pushover analysis. The collapse sequence at ultimate stateis presented in the Figs. 13 and 14 [20]. The comparisonshould be drawn in term of general behavior of the struc-ture, not in terms of exact location of the hinges. Note that,

the EFM is compared only to the elastic perfectly plasticPHFE model.

In the first steps of analysis, the diagram of moments(elastic diagram), correspond to the relative stiffnessbetween beam-floors and walls. However, as the nonlinear-ity grows, the wall element stiffness decreases while thebeam-floor stiffness remains elastic linear. This means thatthe shape of the moment diagram tends gradually to aframe-type moment diagram. When a plastic hinge formsat a given wall base in the EFM, the bending momentremains constant at this slice. Generally, as the loadincreases, the zero moment location shifts to the mid-storeyheight as explained above. The plastic hinge presented

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14000

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0 0.05 0.1 0.15 0.20 0.25 0.3


To

tal b

ase

shea

r (k

N)

Fiber model100%Ipl

Fiber model50% Ipl

Fiber model10% Ipl

PHFE model M1

PHFE model M2

PHFE model M3

PHFE model M4

EFM1

EFM2

100%IFloor

50%IFloor

Fig. 12. Capacity curves for flexible floor-type model.

Fig. 13. Plastic hinges sequence for rigid-type floor and with no (N–M)interaction. Right: PHFE model at Droof = 0.1348 m case M2. Left: EFM1case for Droof = 0.149 m.

Fig. 14. Plastic hinges sequence for rigid-type floor and with (N–M)interaction. Right: PHFE model at Droof = 0.2094 m case M4. Left: EFM2case for Droof = 0.209 m.


Fig. 15. Geometry of the model building.

145.2

99.2

835.3

07.5

7525.5

235.1 1088.2

365.1

69.1

5314.2

1P 2P 3P

398.3

5069.5

94.094.0

7625.2

5314.2

435.6

Fig. 16. Elevation view of the wall D and geometry (in m). Exterior wallslength and axial loads on the bottom and top levels are equal to 1.15 m,56 kN, 26.9 kN, respectively. Interior wall length and axial loads on thebottom and top level are equal to 1.82 m, 133 kN, and 64.5 kN,respectively.

0

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500

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700

800

900

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0 100 200 300 400 500 600

Axi

al f

orc

e (k

N)

Exterior wallsInterior walls

Bending moment (kN)


herein corresponds to the step for which the EFM achievesits ultimate state under force control.

For an elastic perfectly plastic moment rotation modelwithout (N–M) interaction, the EFM behaves as a pointhinge model with lumped plasticity. The use of force inter-action permits modeling the effect of the axial redistribu-tion on the yielding capacity of the structure. Figs. 13and 14 show the difference in the number of hinges andthe yielded slices when the (N–M) interaction was acti-vated. The results extracted from capacity curves and struc-tural damage assessment point out that the use of forceredistribution in simplified models is necessary in particularfor existing building analysis more than for structuraldesign.

5. Pushover analysis of an URM building

5.1. General assumptions

The proposed model can be used also for URM struc-tures modeling. The URM piers and spandrels are subdi-vided into a series of slices. The slices represent anhomogeneous bricks and mortar one-phase material. Asknown, the masonry material is a weak isotropic materialwith very limited ductility. Thus, the softening behavioris very burdensome for computation and causes failure ofconvergence particularly when the analysis is force-con-trolled. The post-peak behavior with softening is beyondthe scope of this model. The yield criteria considered areexpressed for flexure (Eq. (33)) and for shear behavior(Eq. (34)) according to the Magenes model [21–23] asfollows:

f ðN ;MÞ ¼ M þ aN þ bN 26 0 ð33Þ

f ðN ; V Þ ¼MIN V � a1N þ b1N 2

c1 þ N

�

V þ a2 þ b2N V � a3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b3N

p �6 0 ð34Þ

Two constants a and b are required for flexure failure cri-teria, while nine constants ai, bi and ci (for i = 1,3) are re-quired for shear failure criteria [21–23]. N is the axialcompressive load acting on a pier element. The same failurecriteria can be found in many other procedures for ma-sonry assessment [12–14, 24] and others). These equationsdeal with elastic perfectly plastic models in terms of mo-ment–rotation and shear force–displacement laws. In thisstudy, the behavior of the spandrel is assumed to be elasticlinear both in flexure and shear.

5.2. Application

A full-scale two-storey unreinforced masonry tested atthe Pavia University was chosen for model validation

Fig. 17. Flexural criteria for rocking mode of failure in piers.


[21–23]. This structure has been extensively studied in liter-ature. A remarkable feature of this structure, is that theaxial load in piers varies during the experimental test.The variation of the axial load in the considered structureis exploited to study the sensitivity of the model to the axialforce variation on piers. The structural model is subjectedto increasing lateral forces that are applied at the floor lev-els, keeping a 1:1 ratio between the force at the first and thesecond floor. In this application the door wall D (Figs. 15and 16) was chosen because of no flange effect isconsidered.

The elastic properties of the structure used in the modelare summarized as follows:

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350

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Axi

al f

orc

e (k

N)

Vshear mode1 Vshear mode2 Vshear mode3 Minimum Shear

Mode 1

Mode 2

Mode 3

Shear force (kN)

Fig. 18. Shear failure criteria for piers only. Mode 1: (cs) shear failurealong bed-joints at the end section cracked in flexure; Mode 2: (ws)diagonal cracking at the centre of the panel due to mortar joint failure;Mode 3: (b) diagonal cracking at the centre of the panel due to brickfailure.

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0 2 4 8

Roof lateral disp

To

tal b

ase

shea

r (k

N)

6

Fig. 19. Capacity curves of the

The maximum compressive strength of a masonry prismorthogonal to the mortar bed, fm, is equal to 7.9 MPa [23],the shear modulus (Geff = 90fm) is equal to the effectivevalue. For full data see Ref. [23].

In the current application, The (N–M) interaction isshown in Fig. 17. However, the (N–V) interaction wasnot activated. In fact, it was found that axial forces inthe second storey are confined to the first failure modedomain of validity (Fig. 18) since the variation in axialforces for this storey is very low. The second and thirdmodes of failures are not activated. However, despitethe variation in axial forces in the first storey, they areconfined to the second failure mode domain ofoccurrence during the analysis (Fig. 18). Hence, an elas-tic perfectly plastic model without (N–V) interaction isused.

The use of rigid offsets is a crucial issue in equivalentframe modeling. The dimensions of rigid offsets in piersare calculated based on an empirical approach proposedby Dolce [25]. In this study, full rigid offsets are considered.The capacity curves (total base shear versus top lateral dis-placement) are developed for different cases (Table 2,Appendix 1).

In the light of the obtained results, the following recom-mendations are made:

1. The effect of the axial force–bending moment, (N–M),interaction is showed by the cases ‘1’ and ‘2’. As dis-played in Fig. 19, as axial compressive load increases,flexural strength of the piers also increases with regardsto the failure criteria (Fig. 17).

2. The nonlinear effect of shear mechanism is illustratedby cases ‘3’ and ‘4’ in the absence of rigid offsets, andby cases ‘7’, ‘8’ and ‘9’ in the presence of rigid offsets(Fig. 20). As expected, the contribution of shear

10 12 14 16 18

lacement (mm)

Experiment

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Experiment

Case 3

Case 2Case 1, 4 and 6

Case 5

wall D with no rigid offsets.

S

S

SS

F F F

F

FF

F

SS: Shear

cracks

F: Flexuralcracks

Fig. 21. Crack patterns from the experimental test of the URM building(at failure state (top displacement equal to 24 mm)).

Fig. 22. Crack patterns from numerical results at 17 mm.

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0 6 10 12 14 16 18

Roof lateral displacement (mm)

To

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shea

r (k

N)

Experiment

Case 7

Case 8

Case 9

Case 10

Case 11

Case 12

Case 13

Experiment

Case 8

Case 12

2 4 8

Fig. 20. Capacity curves of the wall D with rigid offsets.


mechanism tends to decrease the capacity of the struc-ture due to the occurrence of shear damage. This featureis successfully captured by the simplified model.

3. As displayed in Figs. 19 and 20, the rigid offsets havea significant effect on the global response not only onstiffness, but also on strength capacity of the structure[6,15,22]. This is expected as the horizontal elementstiffness closely affects the contribution of the framemechanism to structural response (cases ‘10’, ‘11’and ‘12’). The capacity curves obtained from EFM(case ‘12’) versus PHFE model (case ‘13’) aresatisfactory.

4. In cases ‘12’ and ‘13’, the two capacity curves are closeto a certain extent in spite of the smeared approach inthe EFM. Both cases ‘5–6’, and ‘12–13’, show the com-

parison of the modeling performance, including sheareffect and (N–M) interaction rule and using either theEFM and the PHFE model with or without rigid zones.

Due to the coupling effect, the resulting crack patternsdisplayed by the numerical simulation are different on theleft and right sides (Figs. 21 and 22). On the other hand,the crack pattern predicted by the EFM is symmetrical(Fig. 23). The shear cracks on spandrels were not obtainedsince the spandrels were modeled as elastic linear. It is clearthat the numerical results should be more accurate, in com-parison to the EFM, since both the axial deformation andaxial force redistribution were not considered in the EFM.The axial force and deformation are crucial issues whenusing failure criteria for the plastic hinge formation and

c c

c

F F F

F

FF

S

FF

Fig. 23. Crack patterns from analytical results (on the EFM) F: flexuralcrack in one slice, S: shear crack over a pier.


damage occurrence. Moreover, in comparison to the testcrack pattern, the numerical and analytical results wereextracted at 17 mm while the tests represent a crack patternat 24 mm. Also, the shear failure was neglected in the lintelsin both models. These are the reasons for the lack of matchbetween the experimental, the analytical and the numericalcrack patterns.

6. Conclusions

This paper presents a simplified formulation of anequivalent frame model. The model permits to considermany relevant features of structural behavior such as struc-tural wall coupling, zero moment location shifting, axialforce–bending moment interaction, axial force–shear forceinteraction, and failure modes prediction. However, in thecase of URM buildings, it is well known that smeared

Table 2Case studies for both EFM and PHFE models

Case Model type Rigid zone in pier Rigid zone in spandrel (N–

1 PHFE – – –2 PHFE – – ·3 EFM – – ·4 PHFE – – –5 PHFE – – ·6 EFM – – ·7 EFM · (with 2Em) – ·8 EFM · (with 2Em) – ·9 EFM · (with 4Em) – ·

10 EFM · (with 10Em) – ·11 EFM · (with 2Em) · (with 2Em) ·12 EFM · (with 10Em) · (with 10Em) ·13 PHFE · (with 10Em) · (with 10Em) ·

Legend: PHFE: point hinge finite element model, EFM: equivalent frame mostrength ratio = analytical/experimental maximum strengths %, Em is the mas

crack approach suffers from a few limitations. The smearedcrack model is enable to represent effectively the rockingand bed joint sliding mode of failures.

For the development of capacity curves, the obtainedresults from the proposed model show good agreementwith experiment and numerical results (Figs. 19 and 20).The model has proven its capability to satisfactorily predictthe maximum strength. The calculated maximum strengths,in particular for the masonry structure (in the range of 9%),could be judged as good results since the model is based onsimplified approaches in comparison to finite element mod-els. However, the post-peak behavior with softening is notyet obtained since the model is force-controlled. Careshould be taken when modeling dual buildings as frame-wall structures in particular with respect to the initial zeromoment lengths assumption. In all cases, obtained resultsshould be considered from an engineering point of viewas is generally done for all simplified existing models.

It is evident that the failure mode identification is a chal-lenging task even if finite element models are used. Thisfeature is sensitive to various analysis parameters such asthe modeling of shear mechanisms, the lateral load patternand force redistribution capabilities. The proposed modelworks well for RC structure. However, it requires furtherimprovements for URM structural modeling (displacementcontrol, variation of axial force in spandrels, multilinearmodels with softening).

Finally, the proposed model is formulated in order toextract capacity curves with damage identification. It canbe implemented readily using any programming platform.The model can be used to assess URM structures, RCstructures as well as dual structures that are commonlyadopted in many countries.

Acknowledgements

This work is a part of a project dealing with seismicvulnerability assessment of unreinforced masonry existingbuildings in Switzerland. The project was supported by

M) failure criteria V shear effect Maximum strength ratio (*) (%)

– �20.1– �15.5– �10.7· �21.7· �22.4· �20.5· �9.9– +18· �9.9· �9.0· �8.5· �7.1· �9.3

del, (–) option considered, (·) option not considered, (*) the maximumonry Young Modulus.


the Federal Office for the Environment, FOEN (Depart-ment of the Environment, Transport, Energy andCommunications). This financial support is gratefullyacknowledged.

Appendix 1

See Table 2.

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[1] Tzamtzis AD, Asteris PG. FE analysis of complex discontinuous andjointed structural systems: part 1: presentation of the method – Astate-of-the-art review. Electron J Struct Eng 2004:1.

[2] Belmouden Y, Lestuzzi P. Seismic vulnerability assessment of existingbuildings in Switzerland. Applied Computing and Mechanics Labo-ratory. Structural Engineering Institute. Ecole Polytechnique Fede-rale de Lausanne. Research Report No 6, April 2006.

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[4] Salonikios T, Karakostas C, Lekidis V, Anthoine A. Comparativeinelastic pushover analysis of masonry frames. Eng Struct J2005;25:1515–23.

[5] Kappos A, Penelis G, Drakopoulos C. Evaluation of simplifiedmodels for lateral load analysis of unreinforced masonry buildings. JStruc Eng, ASCE 2002;128(7):890–7.

[6] ATC-40. Seismic evaluation and retrofit of concrete buildings, vol. 1.Redwood City, California: Applied Technology Council; 1996.

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[8] ElMandooh K, Ghobarah A. Flexural and shear hysteretic behaviorof reinforced concrete columns with variable axial load. Eng Struc2003;25:1353–67.

[9] Nuray Aydinoglu M. An incremental response spectrum analysisprocedure based on inelastic spectral displacements for multi-modeseismic performance evaluation. Bull Earthquake Eng 2003;1:3–36.

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[11] Lang K. Seismic vulnerability of existing buildings. PhD thesis.Institute of Structural Engineering, Zurich, Switzerland, SwissFederal Institute of Technology, February 2002.

[12] FEMA 356. Prestandard and commentary for the seismic rehabili-tation of buildings. Washington (DC): Federal Emergency Manage-ment Agency; 2000.

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[14] Paquette J, Bruneau M. Pseudo-dynamic testing of unreinforcedmasonry building with flexible diaphragm and comparison withexisting procedures. Construct Build Mater 2006;20:220–8.

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[17] ETABS. Integrated building design software, computers and struc-tures, Inc. Berkeley, California, USA, Version 9, November 2005.

[18] Bachmann H, Dazio A. A deformation-based seismic designprocedure for structural wall buildings. In: Proceedings of theinternational workshop on seismic design methodologies for the nextgeneration of codes, Bled/Slovenia, A.A. Balkema: Rotterdam; 24–27 June 1997.

[19] Dazio A. Entwurf und Bemessung von Tragwandgebauden unterErdbebeneinwirkung. Institut fur Baustatik und Konstruktion, ETHZurich, Bericht Nr. 254. Birkhauser Verlag: Basel; 2000 [in German].

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[22] Magenes G, Della Fontana A. Simplified non-linear seismic analysisof masonry buildings. In: Proceedings of the fifth internationalmasonry conference, London; 13–15th, 1998.

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An Equivalent Model for Seismic Analysis of Masonry and Reinforced Concrete Buildings