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 walls

2. A model for structural walls with openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

* Corresponding author. Tel.: +41 21 693 6382; fax: +41 21 693 47 48.E-mail addresses: youssef.belmouden@ep.ch (Y. Belmouden), pierino.lestuzzi@ep.ch (P. Lestuzzi).

Available online at www.sciencedirect.com Constructionand Building

MATERIALS2.1. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1. General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2. Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Appendix 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53made of an assemblage of piers and a portion of spandrels. As formulated, the structural model undergoes inelastic exural 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 exure and inshear. A biaxial interaction rule for both axial forcebending moment and axial forceshear 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 ndings are compared to analyticalresults from experimental, simplied models and nite element models such as Drain3DX and ETABS nite element package. 2007 Elsevier Ltd. All rights reserved.

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

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Abstract0950-0

doi:10.Received 13 August 2007; received in revised form 29 October 2007; accepted 29 October 2007Available online 19 December 2007Ecole Polytechnique Federale de Lausanne, ENAC-IS-IMAC, EPFL, Station 18, CH-1015 Lausanne, SwitzerlandY. Belmouden *, P. Lestuzziand reinforced concrete buildingsReview

An equivalent frame model for seismic analysis of masonrywww.elsevier.com/locate/conbuildmat

Construction and Building Materials 23 (2009) 4053618/$ - see front matter 2007 Elsevier Ltd. All rights reserved.1016/j.conbuildmat.2007.10.023

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) nite element or the equivalentframe models. Despite of some limitations in the equivalentframe model, it is very attractive in comparison to complexnite element models [1,35]. 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 nite elementmethod.

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

2. A model for structural walls with openings

n and Building Materials 23 (2009) 4053 411. 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 nancial losses, due tothe vulnerability of the local building stock. The mitigationof earthquake hazard involves the collaboration of manyspecialists with dierent 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 dierent earthquake scenarios.

In almost all countries, the majority of the buildingstock is classied 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 simplied analytical model. The need for suchmodels is always motivated by rst, the large amount ofstructures that should be analyzed in a very short timeand second, the search for optimal solutions for structuralretrotting.

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. Dierent models are developed based on analyticaland nite 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 eect, 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, nite ele-ment approach is a powerful tool for seismic analysis but itis time consuming and requires a large amount of data.Moreover, rened models based on either discrete or con-tinuum approaches suer 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 thataects closely the reliability of the results and the analysisstability (lack of convergence, ip-op occurrence, suddenload falling, and so on). With such methods it is not possi-

Y. Belmouden, P. Lestuzzi / Constructioble to treat a stock of buildings. Thus, these methods arecumbersome due to the high analytical skills required for2.1. 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 dierent 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 osets 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 niteelement method, nonlinear exural springs (lumped plastic-ity) are inserted into the model at the ends of the piers and/or spandrel elements. These elements are dened in terms

F

Intermediate wall

Rigid zone

Deformablepart

Opening

F

ih

iL

spl

ph

Edge wallEdge wallFig. 1. A schematic representation of equivalent frame model for planarwalls with openings.

n aof momentrotation laws. Translational shear springs areadded at each pier and spandrel at mid-points. Thesesprings are expressed in terms of shear forcedisplacementlaws. 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 exural as well asshear deformation. In the current formulation, the modelonly considers a biaxial interaction between axial forcesbending moments (NM) and axial forcesshear forces(NV) 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, (NMV), 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 simplied models based onstatic equilibrium method, always consider a constantzero moment location [914], and others. The wall for-mulation herein permits the capture of the couplingeect 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-simplied 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 nite homogenized slices [15]. Variable sectionscan be specied 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 exural and shearbehavior in the inelastic range. The interaction eect canbe dened by using experimental and phenomenologicalmodels. These equations are considered as failure crite-ria that can be dened by points and linear segments.The non linear constitutive model for both exural

42 Y. Belmouden, P. Lestuzzi / Constructioand shear behavior is considered as a bilinear envelopcurve with a very small post-yield stiness to avoidnumerical problems. The exural 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. Asimplied algorithm for systems with interaction eect 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 osets 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 exural 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 xed. The storey momentlateral force for-mulation of a structural wall element is expressed by:

fMbsg KFramefPg 1fPg LPTLTFfF g 2

KFrame1

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 . . . AN1 BN1 CN10 0 0 0 . . . 0 AN BN

26666666664

377777777753

Pn DnT n1 EnT n 4{Mbs} represents the base storey bending moments vector,[KFrame] is the equivalent frame stiness matrix, {P} is thereduced shear forces vector, [LPT] is the reduced storeyshear forcesstorey shear force transformation matrix,[LTF] is the storey shear forceapplied 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 dened bythe expressions An, Bn, Cn, Dn, and En (with n = 1, N; N isthe number of storeys) (Eqs. (3), and (4)) such as

nd Building Materials 23 (2009) 4053An kspeq;n11 5

Bn 1kspeq;n1 1kspeq;n

2kpn

!6

Cn kspeq;n1 7

Dn hn1kspeq;n18

En hn 1kpn 1kspeq;n

!9

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

For an interior wall:

kspeq;n 1

cspandreli1 1aspandreli

!n

10

Y. Belmouden, P. Lestuzzi / Construction aFor a left side exterior wall:

kspeq;n aspandreli 1n 11For a right side exterior wall:

kspeq;n cspandreli1 1n 12The equivalent stiness of a pier for an individual struc-tural wall element at level n is:

kpn cpiern bpiern 1 13The superscripts p and sp mean pier and spandrel ele-ments respectively. The superscripts x and shr meanexure and shear respectively. The nonlinearity redistribu-tion coecients are dened as follows:

an aflxn dshrn 14bn bflxn dshrn 15cn cflxn dshrn 16

aflxn hnXmNslicesm1

Z gmgm1

1 n2dnKvnm

17

nh

1nh

Wall i Wall i+1

1, nnM1, +nnM

spniLM ,,

nT

1nTsp

niLM 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 simplied 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 hnXmNslicesm1

Z gmgm1

1 nndnKvnm

18

cflxn hnXmNslicesm1

Z gmgm1

n2dnKvnm

19

dshrn 1

hn

XmNslicesm1

Z gmgm1

dnKdnm

20

Herein, the expressions (17)(20) are formulated for pierelements; hn is the storey height at level n, n is anormalized variable. gm1 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(gm1 + 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 coecients in thespandrels with respect to their zero moment lengths L0Rand L0L (Fig. 2). The redistribution coecients are thendiscretized such that a structural element is decomposedinto nite slices [7]. The inter-storey rotations between leveln and n-1, for a given single wall, omitting the index ifor clarity, are given by:

hn hn1 bpiern apiern Mn1;n bpiern cpiern Mn;n121

hn hn1 M spn;totkspeq;n

Mspn1;tot

kspeq;n122

M spn;tot and Mspn1;tot represent the total bending moment

developed in spandrels at the nth and n 1th level respec-tively (Fig. 2). kspeq;n and k

speq;n1 represent the equivalent sti-

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:

vyj hj1yj Z yj0

vnyj ndnZ yj0

dndn 23

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:

vn Mn=Kvn 24dn T n=Kdn 25M(n) is the bending moment and T(n) is the shear force in aslice, Kv(n) and Kd(n) are the exural and shear stiness ofa slice, respectively. The curvatures are assumed linear overeach slice.

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

Mpiers X

M spandrels 26

nd Building Materials 23 (2009) 4053 43The pier-to-spandrel joint equilibrium for an individualwall i (Fig. 2) is expressed as follows:

M spn;tot Mn;n1 Mn;n1 27M spn;tot M spL;i;n M spR;i1;n 28For a left side exterior wall i:

M spR;i1;n kspeq;R;i1;nhn 29T spR;i1;n M spR;i1;n=L0R;i1;n 30For a right side exterior wall i:

M spL;i;n kspeq;L;i;nhn 31T spL;i;n M spL;i;n=L0L;i;n 32L0R,i,n and L0L,i,n dene the zero moment location for theith spandrel element at the nth level, M spL;i;n and M

spR;i1;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 stinessbetween 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 simplied approach of the Pushover

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

4. Pushover analysis of a RC building

ram

initg sh

sn = 1,N (Eq. (26))2))n = 1,N (Eqs. (29)(32))

lel walls) ak

rotations, the lateral displacements, the bending moments and shear forces, S,tures and slice distortions) in both piers and spandrels: Sk = Sk1 + akDSk,

quilibrium:Nk

yielding states Mky and Tky for the next iteration (Figs. 3 and 4)

date the shear and/or exural stiness of the yielded slices (Fig. 4): Determine

g momentsstructure becomes unstable (signicant loss of stiness), or if a given pier is

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

1k

k

( ) kS2

( ) kS1

( )kyS

2( )1

2+ k

yS

EquilibriumEquilibriumcorrectioncorrection

c

( ) ( )ky

k SS = 22

( )kyS

+2

( )12

kyS

CompressionCompression

TensionTension

Fig. 3. General representation of an interaction process between S1 and S2parameters.

44 Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 4053method based on the event-to-event strategy is presented(Table 1). In this method, the event factor for the entireframe subjected to the predened lateral load pattern is cal-culated such as ak =Min(aki , i = 1, Nwall), Nwall is thenumber of parallel individual walls. A wall event factor,aki , 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-f

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

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;k10L , L

span;k10R ,N

k1 and Mk1y , and Tk1y

3-3 Calculate an, bn and cn3-4 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 storey3-8 Calculate the total bending in spandrels DM spn;tot, for3-9 Calculate Dhn Dhn1, for n = 1,N (Eqs. (21) and (23-10 Calculate DM spR;i1;n, DM

spL;i;n, DT

spR;i1;n and DT

spL;i;n, for

3-11 Calculate the lateral displacementsDv(Eq. (23))3-12 Calculate the event factor for the entire frame (paral3-13 Adjust and update the applied lateral force, the storey

and their corresponding deformations, D, (slice curvaDk = Dk1 + akDDk

3-14 Update axial loads on piers and spandrels by force e3-15 Equilibrium correction using Nk: determine corrected3-16 With respect to the current yielding limits (Fig. 3), up

the updated slice stinesses3-17 New estimation of Lspan;k0 using spandrel ends bendin3-18 Check for structural stability: Quit the analysis if thesubjected to a tension axial load. Otherwise perform theStep 4 Plot the capacity curve for the entire structure4.1. 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 ber beam element(type 15) and ETABS [17] using a point hinge beam ele-

e model

ial compressive axial loads on piers N, failure criterion (NM) and (NV).ear forces, elastic stinesses, event overshoot tolerances, initial values fornext iteration, k + 1 (go to step 3-1)

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

n and Building Materials 23 (2009) 4053 45EL

EL

EL

EL

EL

PL

PL

PL

PL

PL

1> kyMM

1> kyMM

tolky MMM += 1

1kstep

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

31k

yM

kyM

1> kyMM

tolky MMM += 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.

Y. Belmouden, P. Lestuzzi / Constructioment. The ber beam is a nonlinear nite element model inexure but linear elastic in shear. The behavior of bers isdened by a stressstrain relationship for both steel andconcrete materials. The use of bers to model cross-sec-tions accounts rationally for axial forcebiaxial 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 bers.However, a parabolicrectangular stress block wasadopted for concrete material. The oor was modeled asa grid system using the elastic linear beam element type17 [16]. A validation of the nite element type 15 againstexperimental results can be found in the reference [2].

On the other side, ETABS provides a exural pointhinge nite 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

Fig. 5. Plan view of the RC structure.top and bottom storeys. In the equivalent frame model(EFM), the nonlinear behavior for each slice is denedby 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 therst 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 oor area equal toFig. 7. A view of the structural model developed on ETABS.

increases, the normal forces increase, and then the eect of(NM) interaction becomes signicant. When axial force isstill small, the (NM) interaction is negligible. In other

yM

uM

Moment

' y

y

uyy M

M' =

Elastic plastic

Bilinear

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

0 1000 2000 3000 4000 5000 6000 7000 8000

Axi

al fo

rce

(kN)

Tension

Compression

Bending moment (kNm)

Fig. 9. Axial forcebending moment interaction law for RC wall sections

n and Building Materials 23 (2009) 405330.0 18.0 m2 (Fig. 7). The oor is of a grid-type with RCslab. In the case studied, the spandrels consist of beam ele-ments representing a grid oor with underbeams. For esti-mating the stiness of the oors with underbeams, theeective width of the oor slab is calculated according tothe rules suggested by Bachmann and Dazio [18]. The sec-ond moment of inertia of the oor section is equal to0.0262 m4 [11].

The oor 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 oor level aresummarized as follows: the axial forces acting on wallsA1, A2, and A3 at the 6th oor 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 forcebending momentinteraction rule. They are: (1) rigid-oor-type structurewith 100% of the oor stiness, IFloor, (2) semi rigid-oor-type with 50% IFloor and (3) exible oor-type with10% IFloor. For each oor-type model, two cases were stud-ied for the EFM and four cases for the PHFE model onETABS. The case studies are dened as follows:

1. PHFE M1 and M2: Bilinear and elasticplasticmomentrotation law respectively, without (NM)interaction,

2. PHFE M3 and M4: Bilinear and elasticplasticmomentrotation law respectively, including (NM)interaction,

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

In this simulation, the spandrels represent the oorbeams that were elastic linear. Thus, the zero momentlengths were chosen to be at the middle of the span length,without rigid osets, and kept constant during the analysis.The material properties were dened 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 rst 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 m1, 2.60 m1, and 28.5 m1, respec-tively. On ETABS, only yielding moments should be spec-

46 Y. Belmouden, P. Lestuzzi / Constructioied for the exural hinges since the yielding rotations arecalculated by the program. The axial deformations of thebeams 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 oor stiness and wall coupling. Hence, the globalresponse of the structure is aected (elastic stiness, struc-tural displacement, damage occurrence).

Figs. 1012 display capacity curves for the three oor-type models analyzed by Drain3DX, ETABS and the pro-posed EFM. The (NM) interaction eect increases withthe total base shear. The (NM) interaction has small eectin the rst stage of the analysis. As the oor stinessincreases, the force redistribution capacity of the structure

Curvaturey

Fig. 8. Bilinear idealization of the moment-curvature law.dened by linear segments and extracted from ETABS according to theEurocode design code [17].

n a15000

20000

25000

30000

tal b

ase

shea

r (kN

)

50%IFloor

Y. Belmouden, P. Lestuzzi / Constructiowords, the (NM) interaction rule has no eect for exibleoor-type structures (Fig. 12). This application tends todemonstrate the ability of the EFM, in comparison toETABSs results, to reproduce the interaction betweenthe oor stiness, 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,

0

5000

10000

0 0.05 0.1 0.15

Roof lateral displacem

To

Fig. 10. Capacity curves for

0

5000

10000

15000

20000

25000

30000

0 0.05 0.1 0.15

Tota

l bas

e sh

ear (

kN)

100%IFloor

Roof lateral displaceme

Fig. 11. Capacity curves for seFiber model100%Ipl

Fiber model50%Ipl

Fiber model10%Ipl

PHFE model M1

PHFE model M2

PHFE model M3

nd Building Materials 23 (2009) 4053 47the EFM is compared only to the elastic perfectly plasticPHFE model.

In the rst steps of analysis, the diagram of moments(elastic diagram), correspond to the relative stinessbetween beam-oors and walls. However, as the nonlinear-ity grows, the wall element stiness decreases while thebeam-oor stiness 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

0.20 0.25 0.3

ent (m)

PHFE model M4

EFM1

EFM2

10%IFloor

rigid-oor-type model.

0.20 0.25 0.3

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

nt (m)

mi rigid-oor-type model.

02000

4000

6000

8000

10000

12000

14000

16000

18000

0 0.05 0.1 0.15 0.20 0.25 0.3

Roof lateral displacement (m)

Tota

l bas

e sh

ear (

kN)

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 exible oor-type model.

Fig. 13. Plastic hinges sequence for rigid-type oor and with no (NM)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 oor and with (NM)interaction. Right: PHFE model at Droof = 0.2094 m case M4. Left: EFM2case for Droof = 0.209 m.

48 Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 4053

f N ; V MIN V a1N b1N2

c1 N

V a2 b2N V a31 b3N

p 6 0 34

Two constants a and b are required for exure failure cri-teria, while nine constants ai, bi and ci (for i = 1,3) are re-quired for shear failure criteria [2123]. N is the axialcompressive load acting on a pier element. The same failurecriteria can be found in many other procedures for ma-sonry assessment [1214, 24] and others). These equationsdeal with elastic perfectly plastic models in terms of mo-mentrotation and shear forcedisplacement laws. In thisstudy, the behavior of the spandrel is assumed to be elasticlinear both in exure and shear.

5.2. Application

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

145.2

99.2

835.3

07.5

7525.5

235.1 1088.2

365.1

69.1

5314.2

398.3

5069.5

7625.2

5314.2

435.6

n and Building Materials 23 (2009) 4053 49herein corresponds to the step for which the EFM achievesits ultimate state under force control.

For an elastic perfectly plastic moment rotation modelwithout (NM) interaction, the EFM behaves as a pointhinge model with lumped plasticity. The use of force inter-action permits modeling the eect of the axial redistribu-tion on the yielding capacity of the structure. Figs. 13and 14 show the dierence in the number of hinges andthe yielded slices when the (NM) interaction was acti-

Fig. 15. Geometry of the model building.

Y. Belmouden, P. Lestuzzi / Constructiovated. The results extracted from capacity curves and struc-tural damage assessment point out that the use of forceredistribution in simplied 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 exure (Eq. (33)) and for shear behavior(Eq. (34)) according to the Magenes model [2123] asfollows:

f N ;M M aN bN 2 6 0 33

94.094.01P 2P 3P

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

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600

Axi

al fo

rce

(kN)

Exterior wallsInterior wallsBending moment (kN)

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

[2123]. 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 oor lev-els, keeping a 1:1 ratio between the force at the rst and thesecond oor. In this application the door wall D (Figs. 15and 16) was chosen because of no ange eect isconsidered.

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

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

In the current application, The (NM) interaction isshown in Fig. 17. However, the (NV) interaction wasnot activated. In fact, it was found that axial forces inthe second storey are conned to the rst 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 rst storey, they areconned to the second failure mode domain ofoccurrence during the analysis (Fig. 18). Hence, an elas-tic perfectly plastic model without (NV) interaction isused.

The use of rigid osets is a crucial issue in equivalentframe modeling. The dimensions of rigid osets in piersare calculated based on an empirical approach proposed

0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200 250 300

Axi

al fo

rce

(kN) 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 exure; 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.

140

160

50 Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 40530

20

40

60

80

100

120

0 2 4 8

Tota

l bas

e sh

ear (

kN)

6Roof lateral disp

Fig. 19. Capacity curves of theby Dolce [25]. In this study, full rigid osets are considered.The capacity curves (total base shear versus top lateral dis-placement) are developed for dierent cases (Table 2,Appendix 1).

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

1. The eect of the axial forcebending moment, (NM),interaction is showed by the cases 1 and 2. As dis-played in Fig. 19, as axial compressive load increases,exural strength of the piers also increases with regardsto the failure criteria (Fig. 17).

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

10 12 14 16 18

Experiment

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Experiment

Case 3

Case 2Case 1, 4 and 6

Case 5lacement (mm)

wall D with no rigid osets.

n a40

60

80

100

120

140

160

180

Tota

l bas

e sh

ear (

kN)

Y. Belmouden, P. Lestuzzi / Constructiomechanism tends to decrease the capacity of the struc-ture due to the occurrence of shear damage. This featureis successfully captured by the simplied model.

3. As displayed in Figs. 19 and 20, the rigid osets havea signicant eect on the global response not only onstiness, but also on strength capacity of the structure[6,15,22]. This is expected as the horizontal elementstiness closely aects the contribution of the framemechanism to structural response (cases 10, 11and 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 56, and 1213, show the com-

SS

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)).

0

20

0 6

Roof lateral dis2 4 8

Fig. 20. Capacity curves of thExperimentCase 7

Case 8Case 9Case 10Case 11

Case 12Case 13

Experiment

Case 8

Case 12

nd Building Materials 23 (2009) 4053 51parison of the modeling performance, including sheareect and (NM) interaction rule and using either theEFM and the PHFE model with or without rigid zones.

Due to the coupling eect, the resulting crack patternsdisplayed by the numerical simulation are dierent 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

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

10 12 14 16 18

placement (mm)

e wall D with rigid osets.

crack approach suers from a few limitations. The smearedcrack model is enable to represent eectively 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 onsimplied approaches in comparison to nite 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 zero

c c

c

F

FF

FF

52 Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 4053damage 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 simplied formulation of an

F F F

S

Fig. 23. Crack patterns from analytical results (on the EFM) F: exuralcrack in one slice, S: shear crack over a pier.equivalent frame model. The model permits to considermany relevant features of structural behavior such as struc-tural wall coupling, zero moment location shifting, axialforcebending moment interaction, axial forceshear 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 nite element model, EFM: equivalent frame mostrength ratio = analytical/experimental maximum strengths %, Em is the masmoment lengths assumption. In all cases, obtained resultsshould be considered from an engineering point of viewas is generally done for all simplied existing models.

It is evident that the failure mode identication is a chal-lenging task even if nite 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 identication. 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 eect 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 9.3del, () option considered, () option not considered, (*) the maximumonry Young Modulus.

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

Appendix 1

See Table 2.

References

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[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.

[3] RocaP,MolinsC,MarAR.Strengthcapacity ofmasonrywall structures

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

[13] FEMA 306. Evaluation of earthquake damaged concrete andmasonry wall buildings, basic procedures manual. Washington(DC): The Partnership for response and recovery; 1999.

[14] Paquette J, Bruneau M. Pseudo-dynamic testing of unreinforcedmasonry building with exible diaphragm and comparison withexisting procedures. Construct Build Mater 2006;20:2208.

[15] Penelis GRG. An ecient approach for pushover analysis ofunreinforced masonry (URM) structures. J Earthquake Eng2006;10(3):35979.

[16] Prakash V, Powell GH, Campbell S. DRAIN3DX: Base programdescription and user guide, UCB/SEMM-1994/07. Berkeley: Depart-ment of Civil Engineering, University of California; August 1994.

[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

Y. Belmouden, P. Lestuzzi / Construction and Building Materials 23 (2009) 4053 53by the equivalent frame method. J Struct Eng 2005;131(10):160110.[4] Salonikios T, Karakostas C, Lekidis V, Anthoine A. Comparative

inelastic pushover analysis of masonry frames. Eng Struct J2005;25:151523.

[5] Kappos A, Penelis G, Drakopoulos C. Evaluation of simpliedmodels for lateral load analysis of unreinforced masonry buildings. JStruc Eng, ASCE 2002;128(7):8907.

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

[7] Belmouden Y, Lestuzzi P. Analytical model for predicting nonlinearreversed cyclic behaviour of reinforced concrete structural walls. EngStruc 2007;29(7):126376.

[8] ElMandooh K, Ghobarah A. Flexural and shear hysteretic behaviorof reinforced concrete columns with variable axial load. Eng Struc2003;25:135367.

[9] Nuray Aydinoglu M. An incremental response spectrum analysisprocedure based on inelastic spectral displacements for multi-modeseismic performance evaluation. Bull Earthquake Eng 2003;1:336.

[10] Kilar V, Fajfar P. Simple pushover analysis of asymmetric buildings.Earthquake Eng Struc Dyn 1997;26:23349.

[11] Lang K. Seismic vulnerability of existing buildings. PhD thesis.Institute of Structural Engineering, Zurich, Switzerland, SwissFederal Institute of Technology, February 2002.27 June 1997.[19] Dazio A. Entwurf und Bemessung von Tragwandgebauden unter

Erdbebeneinwirkung. Institut fur Baustatik und Konstruktion, ETHZurich, Bericht Nr. 254. Birkhauser Verlag: Basel; 2000 [in German].

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[23] Magenes G, Calvi GM. In-plane response of brick masonry walls.Earthquake Eng Struc Dyn 1997;26:1091112.

[24] Tomazevic M. Seismic resistance verication of buildings: followingthe new trends, seismic methodologies for the next generation ofcodes. In: Fajfar P, Krawinkler H, editors. Balkema: Rotterdam;1997.

[25] Dolce M. Schematizzazione e modellazione per azioni nel piano dellepareti (Models for in-plane loading of masonry walls). Corso sulconsolidamento degli edici in muratura in zona sismica, Ordine degliIngegneri, Potenza; 1989.