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Engineering Structures 33 (2011) 19551965

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Nonlinear response of masonry wall structures subjected to cyclic anddynamic loading

Jos Fernando Sima , Pere Roca, Climent MolinsDepartment of Construction Engineering, Technical University of Catalonia, Barcelona, Spain

a r t i c l e i n f o

Article history:

Received 3 December 2009Received in revised form28 January 2011Accepted 28 February 2011Available online 9 April 2011

Keywords:

Masonry wallNonlinear analysisCyclic loadingDynamic loadingGeneralized Matrix Formulation

a b s t r a c t

The assessment of the dynamic or seismic performance of complex structures often requires theintegration in the time domain of the structural equation of motion in the frame of a nonlinearanalysis. Although sophisticated methods have been developed for the nonlinear analysis of masonrywall structures, including the macro- and micro-modeling approaches, these require large computationaleffortstilllimitingthe extentand complexity of thestructures analyzed. This paper presents an alternativemethod based on the Generalized Matrix Formulation for masonry skeletal structures and load bearingwall systems, which has been proved as an efficient formulation for the analysis of the strength capacityof these kinds of structures (Roca et al. (2005) [17]). The basic formulation has been complementedwith a uniaxial cyclic constitutive model for masonry and a time integration scheme. The ability of theresulting approach to predict the nonlinear dynamic response of masonry structures is shown throughits application to the time domain analysis of an experimental scale masonry building with availableexperimental results on its dynamic response.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

During the past two decades, structural analysis of masonrystructures has experienced a significant progress thanks to thedevelopment of sophisticate computer methods for both staticand dynamic analysis. These methods are based on two mainapproaches, namely macro- and micro-modeling. The first doesnot take into consideration any distinction between masonry unitsand joints, masonry being regarded as an equivalent continuousand homogeneous material. The average material properties areusually obtained by means of homogenization techniques (Pegonand Anthoine [1], Luciano and Sacco [2], Milani and Loureno [3],

amongothers). The micro-modeling approach consists of modelingindividually the mortar joints and the masonry units [4]. Insome cases, simplifications on the micro-modeling have beenintroduced by using zero-thickness interfaces for the joints [5,6].Although the two approaches afford the simulation of manyaspects of the complex nonlinear behavior of masonry, they both(an especially micro-modeling) require still large computationaleffort preventing their application to the study of large andcomplex masonry structures. This limitation is even more evidentwhen the assessment of complex masonry structures by means oftime history analysis is considered. Alternative efficient methods,allowing the time history nonlinear analysis of masonry wall

Corresponding author. Tel.: +34 934017380; fax: +34 93 4054135.E-mail address: [email protected](J.F. Sima).

structures with a reasonable grade of accuracy and computationaleffort, are still necessary. Some of these alternative methods arebased on modeling wall masonry buildings as equivalent framesystems.

Due to its large advantage in terms of computational effort, thepossibility of using equivalent frames to model masonry wall sys-tems has been explored since long time ago. Among the first at-tempts, Karantoni and Fardis [7], utilized this approach to carrypushover analysis using a nonlinear static analysis and comparedthe results with more detailed FEM models. These authors identi-fied the main limitations and problems of the method, which werelater overcome thanks to further sophistications in the description

of the material and connections. Through their pioneering work,Magenes andCalvi [8] and Magenes and Della Fontana [9] provideda powerful tool (the SAM method) for the seismic pushover anal-ysis of masonry buildings using rigid links to improve the descrip-tion between piers and lintels and appropriate inelastic models totake into account the walls flexural and shear failure. The methodwas validated by comparison with full scale experiments and FEMnumerical results on the seismic response of multi-story masonrybuilding faades. The method was later extended to the analysis of3D systems [10]. Further applications were developed by Kapposet al. [11] and Salonikios et al. [12], among others.

Seismic analysis of wall systems by frame equivalent modelingis at present experiencing significant attention. Belmouden and

Lestuzzi [13,14] have formulated a linear finite element includingtwo multilayer connection hinges to model the deformations

0141-0296/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.02.033


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1956 J.F. Sima et al. / Engineering Structures 33 (2011) 19551965

Notation

f, m Parameters of the generalized- integration method Parameter of the Newmark integration methodC Damping matrix0 Strain at the elastic limit of masonry in compressionc Strain at peak of the stressstrain curve of masonry

in compressionct Tensile strain corresponding to the tensile strength

of masonryun Unloading strain on envelope curve for masonry in

compressionre Reloading strain on envelope curve after a complete

cycle of masonry in compressionpl Residual plastic strain after the unloading curve of

masonry in compression

s = (,)T =

x, y, z, x, y, z

TVector of sectional

strains and curvatures

0s Vector of initial strains of the section

XY(s) = (,)T Strain vector of the section with curvilin-ear coordinate s

d0 Vector of displacements in B dueto theinitial strainsand stresses

d Vector of displacements in B produced by thedeformation of the element in its basic isostaticconfiguration under the effect of distributed loads

Masonry damage in compression+ Masonry damage in tensionun Damage at the unloading strain on envelope curve

for masonry in compressionre Damage at the reloading strain on envelope curve

for masonry in compressionE0 Initial elastic modulus of masonry

Epl Stiffness at zero stress after unloading of masonryEre Reloading stiffness of masonry in compressionEtre Unloadingreloading stiffness of masonry in tension

f0 Stress at the elastic limit in compressionfc Compressive strength of masonryfct Tensile strength of masonryF Flexibility matrix of the element Parameter of the Newmark integration methodGf Fracture energy of the masonry in tension = (X, Y,Z) Reference vector of points belonging to the

axial curveKs Sectional stiffness matrixK Stiffness matrix of the elementl Characteristic length or crack band width(s) Vector of displacements reference axial curve at the

point with curvilinear coordinate s( ) = (X, Y, Z) Vector of sectional strains in the curvi-

linear coordinate point M Mass matrix of the elementN(s, sB) Interpolation matrix that describes the exact equi-

librium forces between the transverse sections Band s

(s) Vectorof rotations of thereference axial curve at thepoint with curvilinear coordinate s

( ) = (X, Y, Z) Vector of sectional curvatures in thecurvilinear coordinate point

p(t) Vector of applied loads[p,m]T Vector of distributed forces and momentsP Vector of forces at the extreme of the element

P0 Vector of reactions corresponding to the perfectclamping at the ends of the element due to initialstrains and stresses.

PA Vector of forces at the end APB Vector of forces at the end B

s = (R,M)T =

N, Vy, Vz, Mx, My, Mz

T

Vector of sectionalforces

0s Vector of forces due to initial stressesXY(s) Vector of sectional forces at a section of curvilinear

coordinate s

XY(s) Vector of forces produced by the distributed loads

on a cantilever isostatic configurationu Vector of displacements at the extremes of the

elementu(t) Vector of displacements at instant tui, ui+1 Vectors of displacements at time ti and ti+1ui, ui+1 Vectors of velocities at time ti and ti+1ui, ui+1 Vectors of accelerations at time ti and ti+1u(s) Vector of accelerations of a point on the axis with

curvilinear coordinate s.

occurring in beam-to-column, column-to-footing or wall-to-footing connections. A plastic fracturing model is used to describethe cyclic response of concrete. The model has been validatedby comparison with experimental results on the cyclic responseof reinforced concrete structural walls modeled as a single beamelement. Pasticier et al. [16] have used equivalent frame modelingfor the study of the seismic performance for an existing two-storymasonry building faade. In their approach, the inelastic behavioris lumped to plastic devices (flexural and shear plastic hinges andnonlinear links) available in SAP 2000 r v.10 package. Using thesearrangements, the authors have successfully carried out pushoverand time history analyses.

Recently, Milani et al. [15] have proposed a new strategy for thepushover analysis of in-plane loaded 2D wall systems based on theequivalent frame model. Their method involves two steps, the firstone utilized to determine the ultimate resistance of the masonrywalls using a heterogeneous limit analysis, and the second one forassembling and analyzing the equivalent frame. Flexural hinges areintroduced at both ends of the coupling beams when the capacityis exceeded according to the failure loads determined in the firststep.

The aim of this paper is to present an alternative method fornonlinear analysis of masonry wall structures in the time domainbased using equivalent frame systems. The basis of the model pro-posed herein has been presented by Roca et al. [17]. In this work,the authors proposed an efficient method on the framework of theGeneralized Matrix Formulation, which allows the modelization ofwall panels as a unique element with only two nodes with six de-

grees of freedom per node. Molins et al. [18] extended the basicformulation to linear dynamic analysis. These authors proposed aconsistent element mass matrixwhich takes into account theexactstiffness distribution throughout the element. The approach pre-sented in this paper involves the extension of the formulation pro-posed byRoca et al. [17] to nonlinear dynamic analysis of masonrystructures. The mainadvantageof the GeneralizedMatrix Formula-tion, compared with traditional FEM beam models, is that providesa virtually exact solution. It consists of a generalization of theflexibility-based conventional matrix methods, in which the framedeformation shape is a result of the exact integration of the equi-librium and compatibility equations of an element. The absence ofinterpolation errors allows using large elements without the ne-cessity of intermediate nodes. However, it is necessary to use a

great number of control sections for the integration along the ele-ment which may introduce precision problems and errors similarto those obtained with the finite element method when the struc-ture is discretized into a large number of elements.


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J.F. Sima et al. / Engineering Structures 33 (2011) 19551965 1957

In order to extend the method to nonlinear dynamic analysis, aphenomenological uniaxial cyclic constitutive model for masonryis proposed. Additionally, an efficient time integration procedure(the -generalized method) has been adapted and implementedinto the Generalized Matrix Formulation. In order to show the

capabilities of the proposed approach to predict the nonlineardynamic response of masonry wall structures, results obtainedwith this method are compared with experimental resultsobtained by Tomazevic and Weiss [19] on a three-story masonryexperimental model building subjected to a simulated groundmotion on a shaking table.

Compared to the previous developments based on frameequivalent systems, the proposed method shows some advantagesbecause of possibility of carrying out pushover analysis or eventime history analysis on complex 2D and 3D systems involvingnot only regular piers and lintels but also deformable floor slabs,arches or vaults. With respect to the previous proposals, thedescription of the walllintel connection is improved thanks tothe adoption of Kwans [20] approach for the modeling of the

rotational compatibility condition. As mentioned, the adoption ofGMF as frame formulation grants large accuracy in combinationwith numerical efficiency even in the case of large and complexsystems.

2. Basic formulation

As in [17], the Generalized Matrix Formulation (GMF), a fle-xibility-based formulation for 3D-framed structures composed ofcurved elements with variable cross section, has been adoptedas a framework to model masonry linear or wall members. Thisapproach, firstly derived by Mar [21] and later extended byMolins and Roca [22], consists of a hybrid formulation in which

the sectional internal forces across the element are expressedas an interpolation of the external end forces acting on it.This interpolation is not arbitrarily defined; actually, it is anexact expression directly derived from the consideration of theequilibrium between external and internal forces at any pointwithin an element, with no additional assumptions. Given thatthe basic formulation has been presented in detail in the referredworks, only the fundamentals are outlined herein.

The equilibrium equation of the element is derived by consid-ering a slice of element with infinitesimal length ds belonging to amember subjected to a distributed load p and a moment m alongits axis, as is shown in Fig. 1. Acting on both ends of the slice, theforces and moments R, MandR + dR,M+ dMat point O andO respectively, arein equilibrium with the applied loads.The equi-

librium condition may be expressed as follows:

dR + pds = 0

R ds + dM+ mds = 0.(1)

The equations of equilibrium of an element can be obtained byintegrating the ordinary system of first-order equations (1). Thesolution of the system is formed by a solution of the homogeneoussystem plus a particular solution of the complete system. As aparticular solution of the system, a cantilever configuration withfree end at point B has been selected. Under these assumptions,integration of(1) leads to the following equation:

RXRY

RZMXMYMZ

=

1 0 0 0 0 00 1 0 0 0 0

0 0 1 0 0 00 (Z ZB) (Y YB) 1 0 0

(Z ZB) 0 (X XB) 0 1 0(Y YB) (X XB) 0 0 0 1

Fig. 1. Equilibrium over a slice of beam with a differential thickness.

Fig. 2. Cantilever basic static determined configuration.

RXBRYBRZBMXBMYBMZB

+

RXRYRZMXMYMZ

. (2)

Eq. (2) may be rewritten in the following compact form:

XY(s) = N(s, sB)PB +

XY(s) (3)

where N(s, sB) is an interpolation matrix that describes the exactequilibrium forces between the transverse sections B and s, XY(s)is the vector of sectional forces at any section of curvilinearcoordinate s, and XY(s) is the vector of forces produced bythe distributed loads, p and m, on a cantilever basic staticallydetermined configuration (Fig. 2) configuration, calculated as

XY

(s) = Bs

N(s, )[p,m]Td . (4)

The kinematic compatibility is introduced by means of the NavierBresse equations [21]:

(s) = (s0) +

ss0

( )d (5)

(s) = (s0) + (s0) ((s0) (s))

+

ss0

(( ) + ( ) (( ) (s))) d (6)

where (s) and (s) are the vector of displacements and rotationsof the reference axial curve at the point with curvilinear coordinates;( ) = (X, Y, Z) and ( ) = (X, Y, Z) are the vectorsof sectional strains and sectional curvatures, in the curvilinear

coordinate point , and = (X, Y,Z) is the reference vector ofpoints belonging to the axial curve (Fig. 2).

The constitutive relationship between sectional forces and sec-tional strains in local coordinates may be expressed as follows:


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s = Kss

0s

+ 0s (7)

where s = (R,M)T =

N, Vy, Vz,Mx,My,Mz

Tare the sectional

forces, and s = (,)T =

x, y, z, x, y, z

T

are respec-tively the sectional axial strain, the shear strains in they andzaxis,

the twist and the bending curvatures, Ks is the sectional stiffnessmatrix, which depends on the shape of the section and on the elas-tic properties of the materials, 0s is the vector of forces due to ini-

tial stresses and 0s is the vector of initial strains of the section. Thefollowing matrix expression is obtained by operating and combin-ing expressions (1)(7):[

PAPB

]=

[N(sA, sB)F

1NT (sA, sB) N(sA, sB)F1

F1NT (sA, sB) F1

]

[uAuB

]

[N(sA, sB)F

1d + d0

PA

F1d + d0

] . (8)Refer to Molins et al. [18] for a detailed explanation of the deriva-tion of Eq. (8). This may be rewritten in a more compact form as

P = Ku + P0 (9)

where P is the vector of forces at the extreme points, K is thestiffness matrix of the element, u is the vector of displacementsat the extreme points and P0 is the vector of reactions corre-sponding to the perfect clamping at the ends of the element due toinitial strains and stresses. This last equation allows the construc-tion of the global equation of the structural problem following theconventional assemblage processes. The solution method for thenonlinear problem in the framework of the Generalized MatrixFormulation adopted hereinwas proposed byMolins andRoca[22].It consists of a combination of an incremental process, in whichsome predefined load increments are gradually provided to thestructure, together with an iterative method to solve the problem

at each increment (NewtonRaphson). A remarkable feature of thissolution strategy is that it combines a set of iterations at the globallevel with a secondary set for each individual element. Molins andRoca [22] observed that this procedure is significantly advanta-geous in terms of total computer cost.

A consistent mass matrix, based on the present formulation,has been derived by Molins et al. [18] by defining a sectional massmatrix and providing a description of a displacement field insidethe element. A remarkable feature of the resulting formulation isthat the element mass matrix incorporates information related tothe geometry, stiffness distribution and mass density within theelement.

Roca et al. [17] proposed an equivalent frame method, whereboth the walls and lintels beams are treated as discrete framemembers with the introduction of a set of special devices to

represent more realistically the shear deformation of the wallsbased on the technique proposed by Kwan [20]. A detailedexplanation of this approach for the treatment of wall systems canbe found in the referred work. Integration along the axis of theelement is carried outusinga multiple Simpsonsrule with variablenumber of points. In order to obtain the stiffness matrix of anyelement with curved 3D axis and variable cross section is usuallysufficient to consider 11 integration points. On the other hand, thecomputation of the mass matrix includes three integration levelsand needs more accurate rules, together with a set of additionalintegration points [18].

3. Cyclic constitutive model for masonry

A significant number of papers have been presented on thecharacterization of the behavior of masonry subjected to mono-tonic compression or shearcompression loading (Priestley andElder [23] Magenes [24], Ewing and Kowalski [25] among others).

However, only a few works have been presentedon the behavior ofmasonry under cyclic compressive loadings. This is the case of theworks by Naraine and Sinha [26], AlShebani and Sinha [27,28] andOliveira et al. [29]. Naraine and Sinha [26] investigated the defor-mationcharacteristics of fired clay brick masonry with lowlevels of

compressive strength under cyclic loading. This investigation waslater extended to the deformation characteristics of sand plast (aform of calcium silicate) brick masonry with higher levels of com-pressive strength subjected to uniaxial cyclic loading [27] and bi-axial cyclic loadings AlShebani and Sinha [28]. Oliveira et al. [29]researched on brittleness, energy dissipation and stiffness degra-dation of masonry prisms under cyclic loading. Other authors, asChen et al. [30] or Macchi [31], have also reported on the cyclicbehavior of brick masonry with focus on seismic design of build-ings. In all cases, the behavior shown by brick masonry subjectedto uniaxial cyclic loading presents significant similarity to that ofconcrete.The latterhas been investigatedsincelong time byKarsanand Jirsa [32], among others.

Based on these results, the basic procedure proposed by Simaetal.[33] for the derivation of a uniaxial cyclic constitutive loading

of concrete hasbeen adopted herein. This procedure consists of themodelization of experimental results available in the literature inorder to obtain, by means of linear regression, the necessary modelparameters for the material.

The present method considers the following equation, previ-ously used for concrete, to model the envelope curve of masonryin cyclic compression:

= E0 0 =

1

E0 > 0

(10)

where 0 is a strain value limiting the initial branch, E0 is deforma-tion modulus of the initial linear branch, and

= 1 0

(1 A) Ae

0

c

(11)

is the compression damage parameter which represents the mate-rial degradation in compression, varying from 0 (material withoutdeterioration) to 1 (completely damaged material), and

A =fc 0E0

E0

ce

0c

1 0

(12)wherefc is thecompressionstrength and

c isthe strain atthe peak

of the stressstrain curve.The cyclic behavior is characterized by a set of equations that

reproduce a complete loop of unloadingreloading.The proposed unloading curve is given by the equation:

= D1eD2

1

plun pl

E0

pl

(13)

where

D1 =r(1 un)

(r 1)D2 = Ln

[R (1 un) (r 1)

r

](14)

with r = pl un andR = Epl/E0. pl is the strain at zero stress, Epl isthe stiffnessat the end of the unloading curve and (Fig. 3), un is thecompressive damage at the unloading point which is the onlyparameter used here to define the complete unloadingreloadingpath.

The dependence of the other variables with this parameter hasbeen determined in a semi-empirical way. The cyclic compressionexperimental tests performed by Naraine and Sinha [26] over

brick masonry panels have been reproduced with this model andstatistical regression has been performed in order to obtain theproper dependencyof the variables with the level of damage whenthe unloading starts.


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Fig. 3. Complete unloadingreloading cycle in compression.

Fig. 4. Relationship between the reloading damage and the unloading damageobtained by means of statistical regression on experimental results by Naraine andSinha [26].

Significant differences are found in the shape of the reload-ing curves in compression obtained by different authors. Oliveiraet al. [29] obtained rather straight reloading branches, whileNaraine and Sinhas [26] experiments yielded strongly concavecurves. The character of the reloading curves may be very de-pendent upon the specific properties of the masonry components(brick and mortar). Unfortunately, a general description can behardly formulated based of the very limited available experi-mental results. For the sake of simplicity, the constitutive equa-tion adopted for the present research considers straight reloadingbranches. Such a simple approach has obvious advantages on re-quired input data and computer effort. It must be noted, however,that it may produce, in some cases, an inaccurate estimation of the

overall energy dissipation throughout the hysteretic cycles.It has been observed in test results [26] that the reloading curve

does not return to the envelope curve at the previous maximumunloading strain and further straining in needed to taking up againtheenvelopecurve. It hasbeen found that therelationship betweenthe reloading compressive damage re (defined as the compressivedamage at the reloading strain re) and the unloadingcompressive damage un shows a linear behavior (Fig. 4).

The difference between un and re represents the damageaccumulated in each cycle. Theeffectof cyclic stiffness degradationand its dependence with the unloading compressive damage maybe observed in Fig. 4. Figs. 5 and 6 give the obtained relationshipbetween the parameter r (unloading strainplastic strain ratio)and the unloading compressive damage, and the parameter R

(final unloading stiffnessinitial unloading stiffness ratio) andthe unloading compressive damage, respectively. Fig. 7 relatesthe reloading stiffness degradation with the level of damage incompression.

Fig. 5. Relationship between the reloading stiffness and the unloading damageobtainedby statisticalregressionon experimental resultsby Naraine andSinha [26].

Fig. 6. Relationship between the unloading strainplastic strain ratio and theunloading damage obtained by means of statistical regression on experimentalresults by Naraine and Sinha [26].

A linear variation between the unloading damage un and thereloading damage re has been considered for the compressivedamage during the unloading path. For the reloading path, thecompressive damage has been maintained as a constant, which isin agreement with experimental data (i.e. the test results suggestthat the reloading curve becomes highly nonlinear only beyondthe point of intersection with the unloading curves, often referredto as the common point). These relationships can be expressed asfollows:

= un +re un

pl un( un) (15)

for the unloading path, and

= re (16)

for the reloading path.Very good adjustments have been possible for the different

parameters of the model, with correlation coefficients varyingbetween 0.98 and 1 (Figs. 47). As a result, the proposed modelcan be applied to simulate the experimental results presented byNaraine andSinha[26] anda good comparison is obtained betweenthe cyclic experimental and numerical results (Fig. 8).

Due to the lack of experimental information regarding thegeneral case of partial unloadingreloading, as well as forthe cyclicbehavior in tension, the envelope curve in tension proposed for

concrete in [33] has been adopted herein. The basics features ofthe cyclic tension model are outlined in the following paragraphs.

The tensile envelope curve adopted for the present formulationconsists of a linear elastic relationship until reaching the tensile


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Fig. 7. Relationship between the final unloading stiffnessinitial unloading stiffness ratio and the unloading damage obtained by means of statistical regression on resultsby Naraine and Sinha [26].

Fig. 8. Application of the proposed model for masonry under cyclic compressiveloading to one of the tests presented by Naraine and Sinha [26].

strength, followed by an exponential curve to represent thesoftening branch. This curve may be expressed as follows:

= E0 ct

=

1 +

E0 > ct(17)

where

+ = 1 ct

e

1 ct

(18)

is the tensile damage parameter, which measures the materialdegradation in tension and varies from 0 (material withoutdeterioration) to 1 (completely damaged material), where ct is thetensile strainthat corresponds with thetensile strength (Fig.9)and is defined by the following expression:

=GfE0

lf2ct

1

21

0 (19)

where Gf is the fracture energy (considered as a material prop-erty),fct is thetensile strength of masonry and l

is a characteristic

Fig. 9. Proposed cyclic envelope and unloading/reloading path for masonry intension.

length or crack band width introduced to guarantee the objec-tivity of the results with respect to the size of the finite elementmesh [34].

The tensile cyclic behavior is modeled herein in a simplifiedway. A straight line is used forthe unloading branch in tension. Thesame curve is considered for the reloading branch when there isno incursion in compression during a cycle. The criterion proposedby Sima et al. [33] is considered, accounting for the stiffnessdeterioration due to cyclic loading:

Etre

E0=

ct

1.05(20)

where Etre is defined in Fig. 9.

4. Integration of the dynamic equation of motion

A great number of integration schemes have been proposedto date for the dynamic equation of motion. The selection of the


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proper integration algorithm becomes an important issue whenthe assessment of complex masonry structures is considered.The main characteristics of an appropriated method have beenaddressed by Hilber and Hughes [35]. These include unconditionalstability, accuracy and the introduction of algorithmic damping.

Unconditional stability may be achieved by considering implicitalgorithms. The introduction of the algorithmic damping allowscontrollable numerical dissipation over the higher-frequencymodes (to avoid spurious, non-physical oscillations caused bythe spatial discretization due to excitation of spatially unresolvedmodes). In addition,it has been observed thatusing high-frequencydissipation schemes may improve the convergence of equationsolvers in the case of highly nonlinear problems [36].

A basic difficulty in designing this kind of algorithms lays in ad-equately balancing high-frequency and low-frequency dissipation.High-frequency dissipation must be attained without introducingexcessive algorithmic damping over the important low-frequencymodes. The so called generalized- algorithm proposed by ChungandHulbert [37] hasbeen selected to be implemented in this work,

due to its optimal combination of high-frequency dissipation andlow-frequency dissipation. Moreover, its second-order accuracyand stability properties have been proved even in the nonlinearcase [36].

The generalized- algorithm is well known and detailed expla-nation of this scheme can be found in the referred works. However,the implementation of the method for the nonlinear case deservesto be commented herein. This has been carried out through a New-tonRaphson procedure which for the iteration k can be expressedas:

u(k+1)i+1

r

(k)i+1, u

(k)i+1, u

(k)i+1

=K

eff(K)i+1

1R

(k)i+1 (21)

where u(k+1)

i+1

is the vector of displacement increment, R(k)

i+1

is the

vector of residual forces and the matrix Keff(k)i+1 is the effective stiff-

ness matrix, which takes into account inertial andviscous damping

effects and the tangent stiffness matrix of the structureK(k)

i+1 as fol-lows:

Keff(K)

i+1 =1 m

1 f

t2M+

1 m1 f

t

C+ K(k)

i+1 (22)

whereMand Care the mass and damping matrices of the element.

The vector of residual forces of the iteration k is calculated as:

R(k)i+1 = Fi+1 +

f

1 fFi

1 m

1 fMu

(k)i+1

m

1 fMui

Cu(k)

i+1

f

1 f Cui r

(k)

i+1

f

1 f ri (23)

where f and m are the parameters of the generalized- inte-gration method, is the parameter of the Newmark integrationmethod, t is the time increment and ri is the vector of nonlinearinternal forces.

Once the incremental displacement vector is evaluated bymeans of(21) and the total displacements are actualized through

u(k+1)i+1 = u

(k)i+1 + u

(k+1)i+1 (24)

then, the velocities and accelerations can be actualized by meansof the well-known Newmark relationships.

The residual forces indicated in Eq. (23) are dissipated until the

desired level of accuracy is obtained, by means of convergencecriteria in both forces and displacements. The internal forces areevaluated during a global iteration with a second set of iterationsat the element level as is shown in [17].

Fig. 10. Test setup (reproduced from [19]).

Fig. 11. Input ground accelerogram.

5. Application: three-story masonry building under seismic

loading

The example presented concerns the numerical simulation ofa three-story 1:5 scale plain masonry building subjected to asimulated ground motion, experimentally tested at the Institutefor Testing and Research in Materials and Structures in Ljubljana,Slovenia by Tomazevic and Weiss [19].

Fig. 10 describes the main characteristics of the experimentalmodel and test. In order to meet the requirements of similitude inmass distribution and vertical stresses in the load bearing walls,concrete blocks were fixed to the floor slabs (300 kg mass at eachfloor level) and additional vertical stresses at the load bearingwalls were introduced by means of prestressed steel ropes at everycorner of the model, each providing a force of 12 kN, fixed to the

top slab and anchored into the foundation.The structure was subjected to a series of ground shakingsimulations corresponding to the northsouth component of theearthquake acceleration record of the Montenegro earthquake of1979, with a peak ground acceleration of 0 .43 g. The intensity ofthe shaking was controlled by adjusting the maximum amplitudeof the shaking table displacement. The latter was obtained bynumerical integration of the earthquake accelerogram scaledaccording to the laws of similitude.

The test identified as R47 has been modeled herein in orderto assess the capability of the proposed model. The shakingtable motion, in this case, presents a maximum acceleration of1.10 g and the duration of the test was 5.5 s. The input groundacceleration is shown in Fig. 11.

In order to apply the method presented, a structural model wasdeveloped consisting of an equivalent system of beam elements tomodel the entire system. To model the walls, a series of verticalelements divided longitudinally into 13 integration sections were


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a

b

c

Fig. 12. Building model (a) overview (b) line model (c) cross section of the walls.

used. At the sectional level, the number of trapezoidal sub-divisions used varied for each element, a higher number of sub-divisions being included where high stress concentration wasexpected (Fig. 12).

The mechanical properties of the masonry reported by Tomaze-vic and Weiss [19] are compressive strengthfc = 6.33 MPa, tensilestrength ft = 0.40 MPa and modulus of elasticity E = 6450 MPa.The concrete slab presented a medium compressive strength of

fc = 25 MPa. Details of reinforcement for the slab were notprovided in the original paper. The following additional materialproperties were assumed for the masonry walls: shear modulusG = 1400 MPa and fracture energy Gf = 120 J/m

2. The ma-sonry shear modulus considered herein is based on laboratory ob-servations by Tomazevic [38] showing that the ratio G/E (shear toYoungs moduli) obtained in experiments, is lower than that rec-ommended in Eurocode 6 [39] (6%25% against 40%).The value as-sumed herein for the fracture energy is in agreement with values

obtained experimentally by other authors [40]. The concrete slabswere also modeled by means of a series of linear elements and thematerial was assumed linear elastic with a modulus of elasticityE = 31 500 MPa.

Fig. 13. Time history horizontal displacement results.

A linear elastic equation is adopted to describe the responseof the material in shear. Interaction between normal and shearresponses is considered through the use of a MohrCoulombcriterion as axial strength envelope. In addition, theresulting shearbehavior is modified depending on the level of tensile damageaccumulated at each portion of the section. For tensile damageequal to 0, the portion collaborates fully to the shear response,while fortensile damageequalto 1, theportion does notcontributeat all. A linear interpolation is considered between these extremevalues.

As reported by Tomazevic and Weiss [19], the response of thebuilding is governed by the first mode of vibration. Therefore,an eigenvalue analysis was performed before the test simulationyielding a first natural frequency of 13.84 Hz, which is very similar

to the value obtained experimentally (13.81 Hz).The time history analysis has been performed considering a

time step of t = 0.006 s and parameters for the generalized- method m = 0.05, f = 0.1, = 0.33 and =0.65. The selection of these values has been made by takinginto consideration the relationships between the values of theparameters recommended by Erlicher et al. [36] to obtain anadequate combination of accuracy, stability and dissipation overhigher modes.

Fig. 13 shows the results obtained for the time history analysisin terms of horizontal displacements for each one of the floors. Agood overall agreement between the model predictions and thetest results was obtained, with the amplitudes and the frequenciesexhibiting acceptable deviations. The difference between the

maximum computed displacements on the 3rd floor and thoseobtained experimentally was 8%, while at the 2nd floor was 3%and at the first floor was a 25%. Figs. 14 and 15 show the damagedistribution at the masonry walls after the test. Even that the


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a

b

Fig. 14. Tensile damage distribution in the direction of the strong motion after thetest (a) center walls (b) lateral walls.

crack pattern for this test was not reported in the original work ofTomazevic and Weiss [19] (i.e. the authors only provided the crackpattern for the test that produced the collapse of the structure)the damage distribution follows the trend observed at the crackpattern at the ultimate state (Fig. 16).

Fig. 17 shows the displacement-response Fourier spectra forthe top floor. It can be observed that the higher contributionto the response is produced at a lower value in the computermodel than the experimental result (3.45 Hz against 4.99 Hz).Compared with the frequency measured for the intact structure(13.81 Hz in the experiment vs. 13.84 in the simulation), it turnsout that the measured decrease in the value of the frequencydue to cumulated damage is of 64% in the experiment and 75%in the numerical simulation. Given the large complexity of thephenomena analyzed, this numerical value can be regarded as anacceptable estimation of the experimental result. The differencebetween both values can be attributed to a slight overestimationof damage degree in the numerical simulation.

Interstory drift is an important parameter which is closely

related to damage sustained by the building during the seismicexcitation. Fig. 18 shows the interstory drift profile obtained bymeans of the time history analysis. The maximum drift anglenumerically obtained for the first level was 0.81% corresponding to

a

b

Fig. 15. Compressive damage distribution in the direction of the strong motionafter the test (a) center walls (b) lateral walls.

a maximum base shear coefficient of 23.5%, given as a percentageof the total gravity load, while the values obtained experimentallyby Tomazevic and Weiss [19] were 0.86% and 19% for the first floordrift and base shear coefficient respectively.

A remarkable feature of the proposed formulation is that the

computational cost as well as the time required for the analysisis very low due to the reduced number of elements necessary fortheanalysisand the simplicity of the nonlinearmaterialmodels. Asa matter of example, in this application the time required for thecomplete analysis was less than two hours in a standard personalcomputer (dual core 2 GHz, 4 GB RAM).

6. Conclusions

A novel method for the nonlinear dynamic analysis of masonrystructures in the time domain has been presented. The method isapplicable to skeletal structures or to load bearing wall systemsusing, in the later case, equivalent frame configurations to modelthe walls. The basic formulation for the description of linear

spatial systems, namely the Generalized Matrix Formulation, hasbeen complemented with a uniaxial cyclic constitutive model formasonry adjusted with available experimental data. A significantfeature of the proposed model is that the required input data can


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a b

Fig. 16. Crack pattern at the ultimate state (a) lateral walls (b) center walls [19].

Fig. 17. Displacement-response Fourier spectra for the top floor.

Fig. 18. Analytical drift profile obtained through the time history analysis.

be obtained through conventional monotonic compression andtension tests. Additionally, a time integration scheme has beenadapted and implemented into the frame formulation.

The ability of the method to accurately and efficiently sim-

ulate the response in the time domain of a complex masonryconstruction is shown through the application to a three-story ex-perimental scaled building subjected to groundmotion. The resultsobtained show the capacity of the model to adequately predict the

dynamic and cyclic response of masonry structures at very reason-able computer cost.

Acknowledgements

The studies presented here were developed within the researchproject BIA2006-04127 funded by DGE of the Spanish Ministry ofScience and Technology, whose assistance is gratefully acknowl-edged.The first authorexpresses hisgratitude to theEducation andScience Ministry of the Spanish Government for the financial sup-port by means of a Grant for the Formation of University Teachers.

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