Construction and Building Materials

Transcription

1 Construction and Building Materials 41 (213) Contents lists available at SciVerse ScienceDirect Construction and Building Materials journal homepage: An orthotropic damage model or the analysis o masonry structures Luca Pelà, Miguel Cervera, Pere Roca Technical University o Catalonia (UPC), Campus Norte, Jordi Girona 1-3, 834 Barcelona, Spain highlights " A Continuum Damage Mechanics model or FE analysis o masonry is presented. " The orthotropic behavior is modeled using a tensor mapping procedure. " The theoretical ormulation and the implementation into a FE code are detailed. " The proposed method presents a simple ormulation and provides computational advantages. " The model can simulate the behavior o dierent types o orthotropic masonry. article ino abstract Article history: Available online 23 August 212 Keywords: Continuum damage mechanics Orthotropy Mapping Transormation tensor Masonry FE analysis Tensile cracking This paper presents a numerical model or nonlinear analysis o masonry structural elements based on Continuum Damage Mechanics. The material is described at the macro-level, i.e. it is modeled as a homogeneous orthotropic continuum. The orthotropic behavior is simulated by means o an original methodology, resulting rom the concept o mapped tensors rom the anisotropic ield to an auxiliary workspace. The application o this idea to strain-based Continuum Damage Models is innovative and leads to several computational beneits. The suitability o the model or representing the behavior o dierent types o brickwork masonry is shown via the simulation o experimental tests. Ó 212 Elsevier Ltd. All rights reserved. 1. Introduction The assessment o the structural capacity o masonry constructions is still a challenging task. Numerical approaches oer interesting possibilities to deal with such a diicult problem. At present, several methods and computational tools are available or the assessment o the structural behavior [1] and the choice by the analyst depends on the searched inormation (serviceability, damage, collapse, ailure mechanisms, etc.), the required level o accuracy (local or global behavior o the structure), the necessary input data (detailed or rough inormation about material characteristics) and the computational cost (processing time and memory requirements or the analysis). Thereore, trying to individuate a unique model o general validity is not realistic. Simpliied modeling o masonry structures through the equivalent rame method [2] or two-dimensional macro-elements [3] ensures eicient computations, due to drastic reduction o the Corresponding author. Tel.: ; ax: addresses: luca.pela@upc.edu (L. Pelà), miguel.cervera@upc.edu (M. Cervera), pere.roca.abregat@upc.edu (P. Roca). structure degrees o reedom, but provides only an approximate description o the masonry element behavior. Contrariwise, micro-modeling [4,5] is considered the most accurate tool available to analyze masonry, since the discretization is carried down to the level o the constituents, viz. unit (brick, block, etc.), mortar and their mutual interaces. Such high level o reinement requires intensive computational eort, which limits today s micro-models applicability to the analysis o small elements (e.g. laboratory specimens) or to structural details. Macro-modeling is a valuable approach in practice-oriented analyses, where a compromise between accuracy and eiciency is needed. The material is regarded as a homogeneous orthotropic continuum and this implies considerable computational advantages due to reduced time and memory requirements as well as a user-riendly mesh generation. The mechanical behavior o the continuum can be described by plasticity or Continuum Damage Mechanics (CDM) constitutive laws. Macro-models have been extensively used with the aim o analyzing the seismic response o complex masonry structures, such as arch bridges [6], historical buildings [7] and cathedrals [8]. In the case o CDM inite element models, isotropic criteria are usually preerred because o their /$ - see ront matter Ó 212 Elsevier Ltd. All rights reserved.

2 958 L. Pelà et al. / Construction and Building Materials 41 (213) simplicity and the need or only ew material parameters. Isotropic material models or masonry can be combined with sophisticated algorithms able to account or cracking localization and to achieve proper structural ailure mechanisms [9]. The orthotropic macroscopic behavior o masonry arises rom the spatial organization o its constituents, their nature and the complex units-mortar interaction. Also, masonry exhibits geometrical irregularities in the orm o weak planes along the bed and head joints. The degree o anisotropy may increase due to the presence o horizontal or vertical openings in blocks or bricks. According to the macro-modeling strategy, an appropriate relationship is established between average strains and stresses. The continuum parameters can be assessed by means o tests on specimens o suiciently large size, under homogeneous states o stress, see or instance [1]. As an alternative to diicult laboratory tests, it is possible to assess experimentally the individual components or simple wallets and cores [11] and consider the obtained data as input parameters or numerical homogenization techniques [12]. Several ailure criteria have been proposed [13 18] as phenomenological ormulations based on the interpretations o comprehensive experimental tests. The diiculties in deining reliable and accurate suraces or the description o the shape o the admissible ield have been evident since the irst attempts [19]. In spite o the mentioned problems, single ailure suraces have been considered to reproduce approximately the material strength [2,21]. On the other hand, the conventional ormulations or isotropic quasi-brittle materials [22] have been extended [23] to describe the orthotropic behavior, with a material admissible ield bounded by a Hill-type yield criterion or compression and a Rankine-type yield criterion or tension, according to dierent ailure mechanisms, i.e. cracking and crushing. The inclusion o the orthotropic behavior in the non-linear range causes intrinsic complexities to the macro-model ormulation. In the ramework o Plasticity, the model proposed in Res. [1,23] considers the principal directions o damage ixed and aligned with the initial orthotropy axes. In tension an exponential sotening law or the stress strain diagrams is adopted, with dierent racture energies along each material axes. In compression, an isotropic parabolic hardening law is adopted, ollowed by a parabolic/exponential sotening law with dierent compressive racture energies along the material axes. In a similar way, but through a CDM model, the natural axes o the masonry (i.e. the bed joints and the head joints directions) are assumed coincident with the principal axes o the damage in Re. [21]. Consequently, or x and y directions, two independent damage parameters are assumed, one or compression and one or tension. Their evolution is described by unctions similar to those used or isotropic damage o concrete. Aiming at more accurate but still eicient macro-modeling approaches, this paper presents an implicit orthotropic model based on the classical CDM models. The orthotropic behavior is simulated by means o an original methodology, which establishes a conveniently deined mathematical relationship between the anisotropic real space and an auxiliary mapped one. In this way, it is possible to solve the problem in the mapped space and to return the results to the real ield, with considerable beneits in terms o simplicity and computational eiciency. The paper is organized as ollows: irstly, the mapping theory at the basis o the proposed orthotropic CDM model is described; secondly, the implementation o the algorithm into the ramework o standard nonlinear inite element programs is detailed; inally, the model perormance is demonstrated by means o the comparison between experimental and numerical results, with respect to orthotropic ailure domains and a shear-wall testing. Fig. 1. Relationship between the real anisotropic space and the mapped isotropic space [26]. 2. Orthotropic damage model This section presents the ormulation o a model based on CDM or the inite element analysis o masonry structures. The orthotropic behavior o the material is simulated using the concept o mapped stress tensor, irstly introduced in [24] and reined in [25,26] aterwards. The method consists in studying the behavior o a real anisotropic solid by solving the problem in an auxiliary space. The two spaces are related by means o a linear transormation, deined by a symmetric and rank-our transormation tensor, which allows a one-to-one mapping o an image o the stress (or strain) tensor deined in one space into the other and vice versa. In this way, the dierent behavior along each material axis can be reproduced by means o a very simple ormulation, taking advantage o the well-known isotropic damage models and criteria, while all the inormation concerning the orthotropy o the material is included in the transormation tensor Space transormation tensors The present methodology is based on assuming a real anisotropic space o stresses r and a conjugate space o strains e, such that each o these spaces has its respective image in a mapped space o stresses r and strains e, respectively (see Fig. 1). The relationship between these spaces is deined by Fig. 2. Global (x y) and material (1) and (2) coordinate systems.

3 L. Pelà et al. / Construction and Building Materials 41 (213) Fig. 3. Damage suraces assumed in the real space and in the mapped space. r ¼ A r : r ð1þ r þ ¼ A rþ : r þ ð8þ e ¼ A e : e where A r and A e are the transormation tensors, or stresses and strains, respectively, relating the mapped space and the real one. These rank our-tensors embody the natural anisotropic properties o the material. In order to account or dierent material behavior in tension and compression, a split o the stress tensor into tensile and compressive components is introduced, according to [27 33]: r þ ¼ X3 i¼1 r ¼ r r þ hr i ip i p i where r i denotes the ith principal stress value rom tensor r and p i represents the unit vector associated with its respective principal direction. The ramp unction indicated by the Macaulay brackets hi returns the value o the enclosed expression i positive, but sets a zero value i negative. The split shown by Eqs. (3) and (4) can be expressed in an alternative compact orm as ollows r þ ¼ P : r ð5þ r ¼ ði PÞ : r ð6þ where I is the rank-our identity tensor and P is a projection operator such that P ¼ X3 i¼1 ð2þ ð3þ ð4þ Hðr i Þp i p i p i p i ð7þ where H(r i ) denotes the Heaviside unction computed or the ith principal stress r i. The ollowing transormations o the tensile and compressive stress components rom the real to the mapped space are introduced, according to [34,35]: r ¼ A r : r where A r+ and A r are the stress transormation tensors, or positive and negative components r + and r, respectively, relating the mapped and real spaces. Such tensors are non-singular and positive-deinite. The assumption o two distinct stress transormation tensors permits to map the real stresses into the auxiliary space and solve the problem there, by adopting two dierent isotropic damage criteria or tension and compression. The stress transormations (8) and (9), making reerence to the (local) material coordinate system (denoted by axes 1 and 2, see Fig. 2), can be expressed in Voigt s notation as ollows: r r g ¼ A r g ; 8 >< >: r 11 r 22 s >= 11 >< >; ¼ >: r 11 r 22 s 12 9 >= >; ð9þ ð1þ Such mapping transormations are related to in-plane stress conditions, even i the approach can be easily extended to the three dimensional case [26]. Note that rom Eq. (1) on apex ( ) denotes tensors reerred to the material coordinate system. The parameters ij represent the intersections o the mapped ailure suraces with axes 1, 2 and 3. Since two distinct isotropic criteria are assumed in the mapped space, it results that þ 11 ¼ þ 22 ¼ þ and 11 ¼ 22 ¼. The choice o + and - is arbitrary. Parameters þ 12 and 12 derive rom the particular isotropic criteria adopted or tension and compression. The parameters ij represent the intersections with axes 1, 2 and 3 o the real orthotropic ailure suraces. Making r ij ¼ cosðx i ; x jþ, where x i and x i denote the global and local coordinates, the relationship between A r± and (A r± ) is deined as ollows

4 96 L. Pelà et al. / Construction and Building Materials 41 (213) Table 1 Algorithm used or the proposed model. START LOAD INCREMENTAL LOOP: n = 1, NINCR EQUILIBRIUM ITERATION LOOP: i = 1, NITER IF (n > 1 or i > 1) GOTO 2 (1) Deine strengths, constitutive tensors and rotation tensors þ 11 ; þ 22 ; þ 12 ; þ 11 ; þ 22 ; þ ; 22 ; 12 ; 11 ; 22 ; 12 C C (2) Calculate the transormation tensors: ða rþ Þ, ða r Þ, A rþ, A rþ (3) Compute tangent stiness: A r ijkl ¼ r pi r qj r rk r sl A r pqrs ð11þ It is possible to relate the positive and negative stress transormation tensors to the global stress transormation tensor. In act, ater the deinitions (8) and (9), the condition r ¼ r þ þ r must still apply. Thereore, the previous expression yields A r : r ¼ A rþ : r þ þ A r : r A r : r ¼ A rþ : P : r þ A r : ði PÞ : r and hence n ðk ðeþ Þ i 1 ¼ R V B : n ðc tan Þ i 1 : BdV n ðkþ i 1 ¼ A ne e¼1 n ðk ðeþ Þ i 1 (4) Compute displacement and strains: n ðduþ i ¼ n ðk 1 Þ i 1 n ðf resid Þ i 1 n ðduþ i ¼ n ðduþ i 1 þ n ðduþ i n ðeþ i ¼ B : n ðuþ i (5) Calculate real eective stresses and split: n ðrþ i ¼ C : n ðeþ i ðpþ i ¼ P 3 Hðr j¼1 jþp j p j p j p j n ðr þ Þ i ¼ðPÞ i : n ðrþ i n ðr Þ i ¼ n ðrþ i n ðr þ Þ i ¼½I ðpþ i Š : n ðrþ i (6) Transorm real eective stresses to the mapped space: n ðr þ Þ i ¼ A rþ : n ðr þ Þ i n ðr Þ i ¼ A r : n ðr Þ i (7) Compute damage indexes and total stresses in the mapped space: n ðr þ Þ i ¼ð1 d þ Þ n ðr þ Þ i n ðr Þ i ¼ð1 d Þ n ðr Þ i (8) Return to the real orthotropic stress space: n ðr þ Þ i ¼ðA rþ Þ 1 : n ðr þ Þ i n ðr Þ i ¼ðA r Þ 1 : n ðr Þ i n ðrþ i ¼ n ðr þ Þ i þ n ðr Þ i (9) Compute residual orces: n ðf ðeþ resid Þi ¼ R V BT : n ðrþ i dv ext n ðf resid Þ i ¼ A ne e¼1 n ðf ðeþ resid Þi IF kn ðf residþ i k > tol ) i ¼ i þ 1 GO BACK TO 3 ext else: END EQUILIBRIUM ITERATION LOOP Converged solution or the nth increment. Compute new incremental solution: n ¼ n þ 1 END LOAD INCREMENTAL LOOP Table 2 Material properties or uniaxial tension/compression test. Material properties E 1 = E 3 MPa þ 11 =+.35 MPa 1 = 7. MPa ð12þ ð13a; bþ E 2 2 MPa þ MPa 2 3. MPa v 12 = v.1 þ 12.2 MPa MPa v G þ ;1 =G+ 1 J/m 2 G ;1 = G 4, J/m2 G 12 9 MPa G þ ; J/m 2 G ;2 551 J/m2 A r ¼ A rþ : P þ A r : ði PÞ ð14þ The strain space transormation tensor A e results ater simple calculations: A e ¼ ðc Þ 1 : A r : C ð15þ where C and C are the (ourth-order) linear constitutive tensors in the real and mapped space, respectively. The ormer is expressed in the global reerence system as ollows: C ijkl ¼ r pi r qj r rk r sl C pqrs 2.2. Underlying damage model and damage criteria ð16þ The constitutive model considered in the mapped space is based on the concept o eective stress tensor, introduced in connection with the hypothesis o strain equivalence [36]. The eective stresses r can be computed in terms o the total strain tensor, as r ¼ C : e ð17þ We recall that apex ( ) is assigned to variables related to the mapped space. The tension compression Damage Model adopted in the mapped space is based on a split [27] o the eective stress tensor into tensile and compressive components, r þ and r. The constitutive equation is deined as. r ¼ 1 d þ r þ þ ð1 d Þr ð18þ where the damage indexes d + and d are internal variables, each related with the sign o the stress and thus with tension and compression. Individual criteria or tension and compression are considered in the mapped space, in order to describe dierent ailure mechanisms or masonry, i.e. cracking and crushing o the material. The two damage criteria U + and U are deined as ollows: U þ ðs þ ; r þ Þ ¼s þ r þ ð19þ U s ; r ð Þ ¼s r ð2þ Scalar norms s ± are postulated in order to identiy loading, unloading or reloading situations: s þ ¼hr 1 i p s ¼ iii 3 K r oct þ s oct ð21þ ð22þ The ormer expression represents a tensile Rankine criterion, being r 1 is the largest principal eective stress. The latter equation is the compressive criterion proposed in [31], which is directly inspired on the Drucker Prager criterion. Symbols r oct and s oct are the octahedral normal stress and the octahedral shear stress obtained rom r, while constant K controls the aperture o the inherent Drucker-Prager cone. Variables r + and r in Eqs. (19) and (2) are the internal stresslike variables representing the current damage thresholds in tension and compression. Their values control the size o each (monotonically) expanding damage surace. The expansion o the damage bounding suraces or loading, unloading and reloading conditions is related to the evolution law o the internal variable, explicitly deined in the ollowing way: r ¼ max r ; max s ð23þ where the initial values o the tensile and compressive damage thresholds are r þ ¼ þ ð24þ

5 L. Pelà et al. / Construction and Building Materials 41 (213) by axes r x ; r y ; s xy. Transormations o stresses (8) and (9) scale in distinct manners the two isotropic damage suraces assumed in the mapped space. By means o such a mapping operation, the desired real orthotropic criteria are reproduced in the coordinate system denoted by axes r x, r y, s xy. Owing to the choices o the Rankine and Faria isotropic criteria in the mapped space, the stress transormation tensors (1) take the diagonal orms in Voigt s notation 2 3 þ A þ 11 rþ ¼ þ 6 4 þ þ þ 12 ð26þ A r ¼ ii p p ð ii 7 2 K 5 Þ= 6 12 ð27þ The choice o + and is arbitrary. It is advisable to assume þ ¼ þ 11 and ¼ 11, in order to obtain ðarþ 11 Þ ¼ðA r 11 Þ ¼ 1. The transormation o space is easible only i the six parameters þ 11 ; 11 ; þ 22 ; 22 ; þ 12 ; 12, i.e. the strengths o the real orthotropic material, are known. Such parameters also represent the intersections o the real damage threshold suraces with axes 1, 2 and 3, see Fig. 3. The irst group o our strength parameters ( þ 11 ; 11 ; þ 22 ; 22 ) can be estimated by means o uniaxial experimental tests. I such tests are perormed under displacement control conditions, it is possible to obtain also the inelastic parameters that deine the model, viz. the our independent racture energies. The parameters þ 12 and 12 can be derived by the experimental tests proposed in [1], which weight the shear stress contribution to tensile and compressive ailure. Finally, a biaxial compressive test is required in order to assess the value o the K parameter termed in (25) Evolution laws or damage variables Fig. 4. Stress strain responses to uniaxial tension (a) and unixial compression (b) or dierent angles o orthotropy. Uniaxial response under cyclical displacement history (c). r ¼ piii 3 p K iii 2 3 ð25þ Note that Eq. (24) allows one to compute the current values or r ± in terms o the current values o s + and s, which depend explicitly on the current total strains, see Eqs. (17), (21), and (22) Damage suraces in the real orthotropic space Expressions (19) (22) lead to the equations o two threedimensional suraces deined in the coordinates system denoted The damage indexes d ± reported in (18) are monotonically increasing unctions such that 6 d ± (r ± )61. They are equal to zero when the material is undamaged and equal to one when it is completely damaged. In strict dependence to the deinitions given in Section 2.2 or the thresholds r ±, appropriate evolution laws are considered or the damage variables d ± to reproduce both the tensile sotening and the compressive hardening/sotening observable in masonry. In this work, we assume in the mapped space the detailed expressions given in [29,3] that will not be reiterated here. The post-peak behavior is deined by means o the racture energies G, normalized with respect to the inite element characteristic length, in order to ensure the FEM solution mesh-independency [37,38]. For urther details the reader is reerred to the cited reerences and to the validation example o Section Numerical implementation o the proposed model The steps or implementing the orthotropic damage model into the ramework o standard nonlinear inite element programs are given in Table 1. The proposed model adopts a strain-driven ormalism consistent with standard displacement-based inite element codes. This eature provides high algorithmic eiciency, which is o primary importance when practice-oriented analyses are carried out.

6 962 L. Pelà et al. / Construction and Building Materials 41 (213) Fig. 5. Calculated damage surace or solid clay brick masonry (h = ), according to experimental tests conducted by Page [16,17]. 4. Validation examples The irst example discusses the nonlinear behavior in tension and compression o the proposed model. Then, the experimental ailure domains ound in literature or dierent types o orthotropic masonry are reproduced numerically. Finally, the structural application to a shear-wall is presented. Calculations are perormed with an enhanced version o the inite element program COMET [39], developed at the International Center or Numerical Methods in Engineering (CIMNE, Barcelona). Pre- and post-processing is done with GiD [4], also developed at CIMNE Inelastic orthotropic behavior under tension and compression This example explores the capacity o the proposed model to model the inelastic orthotropic behavior o masonry. For this purpose, a masonry element subjected to uniaxial tension is considered. The material properties, reerred to the material axes 1 and 2, are listed in Table 2. The values chosen or the mate- Fig. 6. Comparisons between the proposed model, the model o Lourenço et al. [23] and the experimental results rom Page [17]: (a) h = ; (b) h = 22.5 and (c) h =45.

7 L. Pelà et al. / Construction and Building Materials 41 (213) Table 3 Comparison between the proposed model and the experimental results obtained by Ganz and Thürlimann [42]. Panel Experimental results Present model Ratio r x r y s xy r x r y s xy K K K K K K K K K K Fig. 7. Calculated damage surace or hollow clay brick masonry and experimental results obtained by Ganz and Thürlimann [42]. rial parameters illustrate the act that completely dierent behaviors along the two material axes can be reproduced. Fig. 4a shows the stress strain responses or angles o orthotropy equal to, 45 and 9. The present model considers an exponential sotening law, which is convenient or a quasi-brittle material such as masonry. Once the racture energy is exhausted, a no-tension material is recovered. As a second step, a masonry specimen subjected to uniaxial compression is considered. The same observations made or the tension test hold. The only exception concerns the compressive nonlinear behavior. A parabolic hardening ollowed by exponential sotening is considered or the stress strain diagrams, according to the assumed compressive racture energy, see Fig. 4b. The peak strength value is assumed to be reached simultaneously on both materials axes, i.e. isotropic hardening, ollowed by orthotropic sotening as determined by the dierent racture energies. The model allows one to set an ultimate value o the strain, rom which the material begins to soten. As a third step, the behavior o the proposed model under unloading/reloading conditions is studied. In compliance with the CDM classical theory, in case o unloading the damage does not rise and, consequently, unloading occurs until the origin according to a damaged Young modulus. The damage constitutive law diers rom the plasticity constitutive law in that no plastic irreversible deormation occurs: all the deormation is recovered during the unloading, so that the unloading paths are not parallel. In addition, the two-parameter damage model is able to capture the unilateral behavior exhibited by the material when passing rom tension to compression [27 33]. This is due to the assumption o the stress split to the deinition o two dierent variables to describe tensile and compressive damage, see Eq. (18). This peculiarity o the model is emphasized in Fig. 4c, which shows the numerical response o a masonry specimen subjected to tensile-compressive cycles. A cyclical displacement history is applied to the specimen with horizontal bed joints. As can be seen rom Fig. 4c, the unloading occurs until the origin o the stress strain diagram, according to a damaged stiness. A successive reloading ollows the same unloading branch, until the damage threshold is reached again. When reversing the sign o the external loading, the constitutive model is able to distinguish tension rom compression. In particular, the stiness recovery upon loading reversal is correctly represented. For instance, when passing rom tension to compression, the model accounts or the crack closure phenomenon in masonry. Concerning the representation o irreversible deormation upon unloading, which is not considered in the model at this stage, it is worth mentioning the CDM models o Res. [31,32] that include inelastic strains in problems with reversal loading Comparison with experimental data o masonry strength The capability o the proposed model to reproduce the strength o dierent masonry types is shown next. A comparison with dierent available experimental data is carried out. Firstly, the biaxial tests conducted by Page [16,17] on solid clay brick masonry are considered. The tests were conducted or ive dierent orientations,, 22.5, 45, 67.5 and 9, o the principal stress with respect to the direction o the mortar beds, in order to assess the directional strength characteristics o masonry panels subjected to in-plane monotonic loading.

8 964 L. Pelà et al. / Construction and Building Materials 41 (213) Fig. 8. Calculated damage surace or hollow concrete block masonry and experimental results obtained by Lurati et al. [43]. Table 4 Comparison between the proposed model and the experimental results obtained by Lurati et al. [43]. Panel Experimental results Present model Ratio r x r y The values assumed or real orthotropic strengths are þ 11 ¼ :43 MPa, þ 22 ¼ :32 MPa and þ 12 ¼ :33 MPa or tension and 11 ¼ 8:74 MPa, 22 ¼ 8:3 MPa and 12 ¼ 2:71 MPa or compression. The parameter K o Eq. (25) has been considered equal to.118. All the aorementioned values have been selected according to data given by Page [17] and parameters calibrated in Re. [1]. The composite damage criterion eatures a low degree o anisotropy (x þ= y þ ¼ 1:34 and x = y ¼ 1:9), as shown in Fig. 5. For all the tests, the material properties in the 1-axis have been selected or the mapped isotropic behavior. The comparisons between the experimental values and the model ones are given in Fig. 6a c, corresponding to orientations o the bed joints equal to, 22.5 and 45, respectively. Globally, good agreement is ound. The results obtained by the proposed model are also consistent with the simulations obtained Table 5 Material properties adopted in the numerical analysis o TU Eindhoven Shear Walls [44]. Material properties E 1 = E 752 MPa þ 11 =+.35 MPa 1 = 6.3 MPa E MPa þ MPa MPa v 12 = v.9 þ 12.3 MPa MPa v 21.5 G þ ;1 =G+ 5 J/m 2 G ;1 = G 2, J/m2 G MPa G þ ;2 s xy r x r y s xy ZSW ZSW ZSW ZSW ZSW ZSW ZSW ZSW J/m 2 G ;2 19,4 J/m2 with the plasticity model o Lourenço et al. [23]. The two-parameters damage model beneits rom more large eiciency, thanks to its intrinsic simplicity. Moreover, the avorable strain-driven ormat provides robustness and high algorithmic eiciency, avoiding the problem o possible ill-conditioning o the return-mapping algorithm in stress-driven orthotropic plasticity models [41]. Secondly, the biaxial tests conducted by Ganz and Thürlimann [42] on hollow clay brick masonry are considered. The values assumed or real orthotropic strengths are þ 11 ¼ :28 MPa, þ 22 ¼ :1 MPa and þ 12 ¼ :4 MPa or tension and 11 ¼ 1:83 MPa, 22 ¼ 7:63 MPa and 12 ¼ 3:41 MPa or compression. The parameter K o Eq. (25) has been considered equal to.72. All the aorementioned values have been selected according to data given by Ganz and Thürlimann [42] and parameters calibrated in Re. [1]. The composite damage criterion eatures a high degree o anisotropy (x þ= y þ ¼ 28 and y = x ¼ 4:17). These high ratios are due to the high peroration o the clay bricks. For all the tests, the material properties in the 1-axis have been selected or the mapped isotropic behavior. Fig. 7 shows the shape o the adopted composite damage criterion both with the points representing the set o strength experimental data. It appears that the tension regime represents the majority o the composite damage surace domain. The test results, the proposed model results and the ratio between experimental and predicted ailure are given in Table 3. Notice that this ratio is a measure o the norm o the stress vector in the (r x, r y, s xy )-space which equals ðr 2 x þ r2 y þ s2 xy Þ1=2. Panels K5 and K9 are not included because the boundary conditions aected the ailure mode o panel K5 and panel K9 included reinorcement. The model seems to be able to reproduce the strength behavior o this type o anisotropic masonry with good accuracy. The error is bounded by a maximum value o 5%, corresponding to test K8. The mean o the ratios is equal to.995. Finally, the biaxial tests conducted by Lurati et al. [43] on hollow concrete block masonry are considered. The values assumed or real strengths are þ 11 ¼ :1 MPa, þ 22 ¼ :1 MPa and þ 12 ¼ :1 MPa or tension and 11 ¼ 5:78 MPa, 22 ¼ 9:12 MPa and 12 ¼ 3:98 MPa or compression. This type o masonry is practically a no-tension material. The parameter K o Eq. (25) has been considered equal to.. All the aorementioned values have been selected according to data given by Lurati et al. [43] and parameters calibrated in Re. [1]. The composite damage criterion eatures a reasonable degree o anisotropy in compression, with y = x ¼ 1:58.

9 L. Pelà et al. / Construction and Building Materials 41 (213) Fig. 9. TU Eindhoven Walls [44]: (a) smeared damage contour; (b) localized damage contour [9]; (c) compressive damage contour and (d) deormed mesh (3). For all the tests, the material properties in the 1-axis have been selected or the mapped isotropic behavior. Fig. 8 shows the shape o the adopted composite damage criterion both with the points representing the set o strength experimental data. The comparison between experimental and numerical results is reported in Table 4. Panel ZSW3 is not considered because the head joints were not illed. The model has shown its ability to simulate the strength behavior o this type o anisotropic masonry with good accuracy. The error is bounded by a maximum value o 7%, corresponding to test ZSW7. The mean o the ratios is equal to TU Eindhoven shear-walls The shear walls J2G and J3G with a central opening tested at TU Eindhoven [44] are here considered. They have dimensions o 99 1 mm 2 and are constituted by 18 courses, o which 16 courses are active and 2 courses are clamped in steel beams. The walls are made o wire-cut solid clay bricks with dimensions mm 3 and 1 mm thick mortar, prepared with a volumetric cement:lime:sand ratio o 1:2:9. Vertical precompres- Fig. 1. Comparison between experimental and numerical load vs. displacement diagrams or walls J2G and J3G.

10 966 L. Pelà et al. / Construction and Building Materials 41 (213) Fig. 11. Comparison between numerical results: (a) micro-model [5] and (b) proposed macro-model. sion uniormly distributed orces p =.3 N/mm 2 are applied to the walls, beore a horizontal load is monotonically increased under top displacement control in a conined way, i.e. keeping the bottom and top boundaries horizontal and precluding any vertical movement. For the numerical analysis, the wall is represented by 5982 bidimensional plane-stress 3-noded linear triangular elements. The computational domain is discretized with an unstructured mesh with average mesh size o h e = 2 mm (3128 nodes). The discrete problem is solved incrementally, in a (pseudo) time step-by-step manner. The analysis is completed by means o 5 equal time steps. Within each step, a modiied Newton Raphson method (using the secant stiness matrix), together with a line search procedure, is used to solve the corresponding non-linear system o equations. Convergence at a particular time step is attained when the ratio between the norm o the iterative residual orces and the norm o the total external orces is lower than 1%. The values o the mechanical parameters used in the numerical analysis to describe the masonry behavior are summarized in Table 5. Some o them are the mechanical characteristics o masonry provided in [44], others are data obtained via a homogenization procedure [45]. Fig. 9a shows iso-contours or the tensile damage, which arises rom the opening and propagates towards the top and the bottom o the wall. In addition, tensile damage arises rom the vertical external sides o the wall, involving the top let pier next to the opening and the bottom right one. Such approximate representation o the tensile damage as a smeared phenomenon can be considerably improved resorting to the crack-tracking technique proposed in Re. [9], which orces the tensile damage to develop along a single row o inite elements. In this way, the tensile damage is represented in the orm o localized cracks, similar to the ones typically observed on masonry structures. Fig. 9b shows the discrete tensile cracks predicted by the proposed model combined with the aorementioned crack-tracking algorithm. Compared with the smeared approach, the localized one shows a better capacity to predict the real collapsing mechanism. Fig. 9c depicts the compressive smeared damage contour. As shown, the model predicts correctly the location o the areas aected by material compressive ailure. The ailure mechanism is properly represented, with the compressed struts located next to the opening which ail at both o their ends. Fig. 9d shows the computed deormed shape corresponding to an imposed horizontal displacement o 2 mm, with a displacement ampliication actor o 3. The comparison between the calculated and experimental load displacement diagrams is shown in Fig. 1. Although both walls J2G and J3G were tested under the same conditions, the latter one resisted a lower ultimate loading. The numerical results agree reasonably with wall J2G, as also ound in other studies [5,46]. Finally, Fig. 11a and b shows the comparison between the result obtained in [5] with a micro-model including the distinct representation o constituents and the visualization o the maximum principal strain vectors both with the compressive damage isocontours derived rom the proposed model. The concentration o the displacement gradients (strains) in the elements lying along the computed crack is evident. Thereore, the resolution o the cracks is optimal or the mesh used. 5. Conclusions In the present paper, an original method is proposed or the inite element analysis o masonry structures. This working strategy, based on CDM and on the concept o space mapping, allows the establishment o an implicit orthotropic damage criterion in the real anisotropic space by using the damage criterion ormulated in an auxiliary mapped space, with all the advantages implied by this. The model is able to capture the stiness, the strength and the inelastic dissipation in each material direction. The implementation o this theory in inite element codes is straightorward. The procedure can be applied to the analysis o masonry structures, such as horizontally and vertically in-plane loaded masonry walls. Acknowledgments The studies presented here have been developed within the research projects BIA and SEDUREC (CSD26-6), unded by DGE o the Spanish Ministry o Science and Technology, whose assistance is grateully acknowledged. The authors thank Pro. Sergio Oller or his helpul suggestions. Reerences [1] Roca P, Cervera M, Gariup G, Pelà L. Structural analysis o masonry historical constructions. Classical and advanced approaches. Arch Comput Methods Eng 21;17: [2] Roca P, Molins C, Marí AR. Strength capacity o masonry wall structures by the equivalent rame method. J Struct Eng 25;131(1): [3] Brencich A, Lagomarsino S. A macro-element dynamic model or masonry shear walls. In: Pande GN, Middleton J, editors. Computer methods in structural masonry 4, Proc o the int symp. London: E&FN Spon; [4] Loti HR, Shing PB. Interace model applied to racture o masonry structures. J Struct Eng 1994;12(1):63 8. [5] Lourenço PB, Rots JG. Multi-surace interace model or the analysis o masonry structures. J Eng Mech 1997;123(7):66 8. [6] Pelà L, Aprile A, Benedetti A. Seismic assessment o masonry arch bridges. Eng Struct 29;31(8):

11 L. Pelà et al. / Construction and Building Materials 41 (213) [7] Mallardo V, Malvezzi R, Milani E, Milani G. Seismic vulnerability o historical masonry buildings: a case study in Ferrara. Eng Struct 28;3: [8] Martínez G, Roca P, Caselles O, Clapés J. Characterization o the dynamic response or the structure o Mallorca Cathedral. In: Lourenço PB, Roca P, Modena C, Agrawal S, editors. Structural analysis o historical constructions. New Delhi: 26. [9] Cervera M, Pelà L, Clemente R, Roca P. A crack-tracking technique or localized damage in quasi-brittle materials. Eng Fract Mech 21;77(13): [1] Lourenço PB, Rots JG, Blaauwendraad J. Continuum model or masonry: parameter estimation and validation. J Struct Eng 1998;124(6): [11] Benedetti A, Pelà L, Aprile A. Masonry properties determination via splitting tests on cores with a rotated mortar layer. In: Sinha B, Tanaçan L, editors. Proceedings o 8th international seminar on structural masonry. Istanbul: 28. [12] Lourenço PB, Milani G, Tralli A, Zucchini A. Analysis o masonry structures: review and recent trends o homogenisation techniques. Can J Civ Eng 27;34: [13] Sinha B, Hendry AW. Racking tests on storey-height shear-wall structures with openings, subjected to pre-compression. Designing engineering & construction with masonry products. Houston: Gul Publishing Co.; p [14] Yokel FY, Fattal SG. Failure hypothesis or masonry shear walls. J Struct Div 1976;12(3): [15] Hamid AA, Drysdale RG. Proposed ailure criteria or concrete block masonry under biaxial stresses. J Struct Eng 1981;17(8): [16] Page AW. The biaxial compressive strength o brick masonry. Proc Inst Civil Eng 1981;71(2): [17] Page AW. The strength o brick masonry under biaxial tension compression. Int J Mas Constr 1983;3(1): [18] Mann W, Müller H. Failure o shear-stressed masonry-an enlarged theory, tests and application to shear walls. Proc Brit Ceramic Soc 1982;3: [19] Dhanasekar M, Page AW, Kleeman PW. The ailure o brick masonry under biaxial stresses. Proc Inst Civil Eng 1985;79(2): [2] Syrmakesis CA, Asteris PG. Masonry ailure criterion under biaxial stress state. J Mater Civil Eng 21;13: [21] Berto L, Saetta A, Scotta R, Vitaliani R. An orthotropic damage model or masonry structures. Int J Numer Methods Eng 22;55: [22] Feenstra PH, De Borst R. A composite plasticity model or concrete. Int J Solids Struct 1996;33(5):77 3. [23] Lourenço PB, De Borst R, Rots JG. Plane stress sotening plasticity model or orthotropic materials. Int J Numer Methods Eng 1997;4: [24] Betten J. Applications o tensor unctions to the ormulation o yield criteria or anisotropic materials. Int J Plast 1988;4: [25] Oller S, Botello S, Miquel J, Oñate E. An anisotropic elastoplastic model based on an isotropic ormulation. Eng Comput 1995;12(3): [26] Oller S, Car E, Lubliner J. Deinition o a general implicit orthotropic yield criterion. Comput Methods Appl Mech Eng 23;192: [27] Cervera M, Oliver J, Faria R. Seismic evaluation o concrete dams via continuum damage models. Earthq Eng Struct D 1995;24(9): [28] Cervera M, Oliver J, Manzoli O. A rate-dependent isotropic damage model or the seismic analysis o concrete dams. Earthq Eng Struct D 1996;25: [29] Cervera M, Oliver J, Prato T. Thermo-chemo-mechanical model or concrete. ii: Damage and creep. J Eng Mech 1999;125(9): [3] Cervera M. Viscoelasticity and rate-dependent continuum damage models. Monograph no Barcelona: CIMNE; 23. [31] Faria R, Oliver J, Cervera M. A strain-based plastic viscous-damage model or massive concrete structures. Int J Solids Struct 1998;35(14): [32] Faria R, Oliver J, Cervera M. On isotropic scalar damage models or the numerical analysis o concrete structures. Monograph PI198. Barcelona: CIMNE; 2. [33] Faria R, Oliver J, Cervera M. Modeling material ailure in concrete structures under cyclic actions. J Struct Eng 24;13(12): [34] Pelà L. Continuum damage model or nonlinear analysis o masonry structures. PhD thesis. Technical University o Catalonia, University o Ferrara; 29. [35] Pelà L, Cervera M, Roca P. Continuum damage model or orthotropic materials: application to masonry. Comput Methods Appl Mech Eng 211;2: [36] Lemaitre J, Chaboche JL. Aspects phénoménologiques de la rupture par endommagement. J Méc Appl 1978;2: [37] Bazant ZP, Oh BH. Crack band theory or racture o concrete. Mater Struct 1983;16: [38] Cervera M, Chiumenti M. Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Comput Methods Appl Mech Eng 26;196:34 2. [39] M Cervera, C Agelet de Saracibar, M Chiumenti. COMET: COupled MEchanical and thermal analysis Data input manual version 5.. Technical report IT- 38. Barcelona: CIMNE; 22. [4] Website o CIMNE. Barcelona. Technical University o Catalonia. 22. < gid.cimne.upc.es/> [41] Lourenço PB, Rots JG, Feenstra PH. A tensile Rankine-type orthotropic model or masonry. In: Pande GN, Middleton J, editors. Computer methods in structural masonry 3. Swansea: Books & Journals International; [42] Ganz HR, Thürlimann B. Tests on the biaxial strength o masonry. Report no ETH Zurich: Institute o Structural Engineering; 1982 [in German]. [43] Lurati F, Gra H, Thürlimann B. Experimental determination o the strength parameters o concrete masonry. Report no ETH Zurich: Institute o Structural Engineering; 199 [in German]. [44] Raijmakers TMJ, Vermeltoort ATh. Deormation controlled tests in masonry shear walls. Research report TNO-Bouw. Report B Delt; 1992 [in Dutch]. [45] Lourenço PB. Computational strategies or masonry structures. PhD thesis. Delt University Press; [46] Van Zijl GPAG. Modeling masonry shear-compression: role o dilatancy highlighted. J Eng Mech 24;13(11):