Damage assessment using modeling of large-scale confined masonry building

DAMAGE ASSESSMENT ON THE MODELING OF A LARGE SCALE1 CONFINED MASONRY BUILDING2 Cuauhtemoc Escudero1, Sergio Oller2, Xavier Martinez3, Alex Barbat4, and Cesar Dávalos5 3 1Departamento de Resistencia de Materiales y Estructuras en la Ingeniería, ETSECCPB,4 Technical University of Catalonia, Spain. Email: [email protected]5 2Departamento de Resistencia de Materiales y Estructuras en la Ingeniería, ETSECCPB,6 Technical University of Catalonia, Spain. Email: ser[email protected]7 3Departamento de Ciencia e Ingeniería Náutica, FBN, Technical University of Catalonia, Pla de8 Palau 18, 08003 Barcelona, Spain. Email: xmar[email protected]9 4Departamento de Resistencia de Materiales y Estructuras en la Ingeniería, ETSECCPB,10 Technical University of Catalonia, Spain. Email: ale[email protected]11 5I.I.T. Departamento de Ingeniería Civil y Ambiental, Universidad Autónoma de Ciudad Juárez12 Av. del Charro 450 norte, 32310, Ciudad Juárez, México13 ABSTRACT14 The seismic behavior of a representative medium-rise building of Mexico City has been eval-15 uated using the capacity spectrum method. This method is widely used nowadays in the seismic16 assessment of buildings, since it allows obtaining fragility curves which permit evaluating the17 ability of a building to resist earthquakes.18 A real full-height multi-story model is proposed to test the capabilities of the algorithm herein19 exhibited. The model is outlined through structural drawings; sized and structured following the20 building code regulations for masonry structures in Mexico City. Computational requirements21 for the analysis of large structures are indicated, in addition to the improvements to a non-linear22 computing code for a better performance in terms of memory management and execution times.23 1 Escudero, September 23, 2019 Finally, a comparison between obtained results and the building code regulation is carried out,24 highlighting differences in the obtained results.25 The need to handle meshes with high amount of finite elements pushed us to develop a new26 layered finite element (FE), that can reproduce the non-linear behavior of its constituent materials27 when there are out-of-plane stresses without having to introduce additional degrees of freedom. The28 proposed FE has been compared with standard FE, presenting different kinematics, and excellent29 results have been obtained.30 This work emerges as the need to combine and improve existing technologies in the field of31 finite element analysis. One of such technologies is the numerical simulation of the behaviour32 of composite materials. That is why it has also been necessary to develop a computing program33 capable of reading both finite element meshes and patterns of fibers represented with convex34 polygons, and as a result of areas intersections between polygons returns volumetric participation35 of fiber and matrix of constituents materials for each layer, in addition had to return the fiber36 orientation with respect to the local axis of the finite element.37 Keywords: Laminated element, Composite materials, large reinforced concrete (RC) structures,38 Masonry structures, Mexican building code39 INTRODUCTION40 Since the last half of 20th century, masonry building construction has been widely used in41 countries and regions of extremely high seismic activity – used for construction from one-story42 single-family housing to medium-rise apartment buildings up to six-story high. The persistence of43 masonry building construction, in these regions, has a context in two main areas: economics and44 its structural performance.45 From the economical point of view, a low-cost construction is the most significant aspect that46 makes masonry structures reliable. Hence, construction of housing buildings made out of reinforced47 masonry bearing walls is highly promoted by governmental entities. Regarding its structural48 performance, masonry structures, designed in compliance with current code requirements, present49 a high performance even in cases of significant seismic activity (Meli 1990;Marques and Lourenço50 2 Escudero, September 23, 2019 2014). Despite such wide range of application for this construction method, few resources have51 been put in structural masonry research while compared with steel or reinforced concrete (RC)52 construction.53 The numerical simulation in the field of civil engineering, while widely used in structural54 analysis, has not benefited from the full potential offered by the new technologies for the analysis of55 composite materials. Within the framework of the finite element method, this technology is fully56 adopted in industries such as automotive, aerospace and shipbuilding. The present work emerges57 from the need of improving the existing computational technologies in the field of structural58 behavior of reinforced concrete buildings with masonry in-fills using finite element analysis for59 composite materials.60 The need to handle meshes with high amount of finite elements pushed us to develop a new61 layered finite element (FE), that can reproduce the non-linear behavior of its constituent materials62 when there are out-of-plane stresses without having to introduce additional degrees of freedom63 (Escudero et al. 2017). The proposed FE has been compared with standard FE, presenting different64 kinematics, and excellent results have been obtained. The robustness and efficiency of the proposed65 methodology in the analysis of confined masonry (CM) buildings is conditioned by the ability of66 using different patterns of steel reinforcement, which are typically present in such structures.67 The objective of this work is to present a more exhaustive contribution on the numerical68 modeling of CM using a One-phase material model, (Lourenço 1996;Rots 1997), to reproduce69 the overall behavior of masonry structures, using both classical and enhanced mixing theories,70 (Martínez 2008;Rastellini et al. 2008), on laminated composite materials (Escudero et al. 2016)71 within the finite element method framework for real case buildings.72 A real full-height multi-story model is proposed to test the capabilities of the algorithm herein73 exhibited. The model is outlined through structural drawings – sized and structured following74 the building code regulations for masonry structures in Mexico City. Computational requirements75 for the analysis of large structures are indicated, in addition to the improvements to a non-linear76 computing code (CIMNE 2014) for a better performance in terms of memory management and77 3 Escudero, September 23, 2019 execution times. Finally, a comparison between obtained results and the building code regulation78 is carried out, highlighting differences in the obtained results.79 NUMERICAL FORMULATION80 The constructive process of CM involves the interaction of several materials; in this work, it81 is fundamental to establish the mechanical behavior and constitutive models, from a mathematical82 point of view, of three different simple materials – steel, concrete, and masonry – which are briefly83 summarized in this section.84 Steel85 The classical macro-mechanical theories of plasticity are based on the notions of a yield surface,86 a hardening rule, and on the stress-plastic strain relations of the given material (Simo and Hughes87 1998). Such notions are used to formulate a material model for the calculation of its response88 during plastic deformation.89 Let us consider Fig. 1.(a), where the idealized uniaxial Strain-Stress relationship of steel is90 depicted. Both Elastic (OA) and Plastic (AB) ranges can be clearly distinguished. Elastic range can91 be mathematically modeled by generalized Hooke’s Law for restoring forces, where the stress is92 linearly proportional to the strain. However, stresses larger than the elastic limit, or yield strength93 σy(point A) cause a permanent deformation known as plastic or plasticity. On the other hand, a94 typical representation, in the Westergard stress space, of the von Mises yield criterion is given in95 1.(b). Within this work, the mechanical behavior of steel will be reproduced following a plasticity96 model, combined with this yield function. The governing equations of classical rate-independent97 plasticity follows the formulation presented in Ref. (Simo and Hughes 1998).98 The general formulation follows the additive decomposition of the strain tensor εinto an elastic99 εeand plastic part εp. For more details see references (Simo and Hughes 1998;Escudero et al.100 2017).101 4 Escudero, September 23, 2019 Concrete102 Concrete is a composite material, since it is produced of a granular material (aggregate)103 embedded in a hard matrix of material (cement); however, it is a common practice to model it as104 a simple quasi-brittle geo-material, even though its high non-linear performance achieved due to105 the formation of micro-cracks and slipping among the aggregate particles (Oller 1998). Although106 Continuum Damage Mechanics (CDM) was settle first as a scheme to model quasi-brittle materials107 (Chaboche 1988a;Chaboche 1988b;Ju 1989;Oliver et al. 1990), it has been used for the last thirty108 years to model concrete materials (Mazars and Pijaudier-Cabot 1989;Lubliner et al. 1989;Oller109 1998;Jason et al. 2006;Tao and Phillips 2005).110 Continuum Damage Models are defined by a degradation, as an internal variable, of both111 resistance and elasticity modulus – this variable being either a scalar or a tensor; for an isotropic112 damage model, such internal variable is represented by a scalar (Simo and Ju 1989;Oliver et al.113 1990). For this case, the micro-cracking orientation is neglected, which leads to skip the effect of114 anisotropic damage, or the effect of material’s anisotropy in the direction of the micro-crack.115 An idealization of the concrete expected behavior, during a cycling loading process, is repre-116 sented in Fig. 2.(a). The loading process goes from traction to compression, starting in an elastic117 range OA, once it reaches the maximum traction resistance A, a degradation of stiffness is activated118 in AB, finally stress state Bis reached and it starts an unloading process BO. Now, let us consider119 the unloading process has reached point Oagain, which shows the advantage of damage models120 with two scalar damage variables. Whereas d+,0,d−=0(since they are independent from121 each other), and loading process along OC has an elastic behavior again, and until it reaches the122 maximum compression stress Ccompression degradation will not be activated. As expected, once123 loading process reaches stress Cdegradation of stiffness in compression is activated CD until it124 reaches Dand it starts an unloading process again.125 Fig. 2.(b) shows a Westergard stress space where two yield surfaces are superimposed. Several126 yield functions, appropriated for concrete, have been proposed. For instance, in Ref (Faria et al.127 1998) was used a yield surface based on energy combined with a modification of the Drucker-128 5 Escudero, September 23, 2019 Prager yield function for the only-compression case; finally, the obtained results by (Faria et al.129 1998) were compared with the experimental results reported in Ref. (Kupfer et al. 1969) showing130 good accuracy among the results.131 Damage models for concrete are based in the fracture energy which is assumed to be a material132 property (Oliver et al. 1990). This work follows a damage model given in Ref. (Paredes 2013)133 which is a modification of the damage model presented by Faria et al (Faria et al. 1998) where two134 scalar damage variables are used: a variable for traction d+and another for compression d−. Such135 modification raises for quadrants II and IV of a given yield surface with traction and compression136 combined, see Fig. 2.(b), consisting on:137 •Additive decomposition of the stress tensor σ. In the case that a tensional state is either in138 quadrant II or IV, the weight between traction σ+and compression stresses σ−is computed139 to obtain a dominant stress.140 •Stresses σ+and σ−will not be treated separately. The constitutive equation will be141 integrated using σtaking into account the dominant stress.142 •Constitutive Damage conditions detailed in (Paredes 2013) are applied.143 A more in-depth review about the topic, for the reeder, is given in Ref. (Faria et al. 1998).144 Masonry145 Masonry is a composite material resulting of individual units laid (brick or rock) bound together146 by mortar. From now on, it will be implied that whenever it is referred to masonry it means brick147 masonry.148 In order to settle a damage criteria to predict non-linear behavior of masonry, it is necessary149 to determine its basic failure mechanisms, under the most elemental loading conditions, and its150 mechanical properties through experimental tests. Large amount of test over masonry materials,151 mortar and units, have been performed to study their interaction once they are bond together as a152 masonry prism.153 Masonry is stated as an anisotropic material; depending on the accuracy and the simplicity154 6 Escudero, September 23, 2019 desired, masonry can be modeled using the following strategies (Lourenço 1996):155 •Detailed Micro-Modeling: Units and mortar in the joints are represented by continuum156 elements whereas the unit-mortar interface is represented by discontinuous elements.157 •Simplified Micro-Modeling: Expanded units are represented by continuum elements158 whereas the behavior of the mortar joints and unit-mortar interface is lumped in dis-159 continuous elements.160 •Macro-Modeling: Units, mortar, and unit-mortar interface are bonded in the continuum.161 •Homogenized Modeling: A midway strategy between micro and macro-modeling. This162 method deals with the global problem of composite material in a two-scale context. The163 macroscopic scale uses the composite materials to obtain the global response of the structure;164 composites are treated as homogeneous materials in this scale. The microscopic scale165 corresponds to an elemental characteristic volume in which the microscopic fields inside166 the composite are obtained; this scale deals with the component materials. Homogenization167 theory assumes a periodical configuration of the composite material to relate these two scales168 (Oller 2016).169 •Mixing Theory: Based in two main hypothesis: all composite constituents have the170 same strains and each constituent collaborates to the composite behavior according to its171 volumetric participation. The main problem of this approach is the iso-strain condition,172 which forces a parallel distribution of the constituents in the composite. Some improvements173 to the original formulation can be found in reference (Oller 2016).174 Advantages and disadvantages of each approach are discussed in (Lourenço et al. 2007;175 Lourenço 1996;Pelà 2009;Lopez et al. 1999;Quinteros et al. 2012). Herein, a Macro-Modeling176 approach is followed due to the reduced time and memory requirements as well as for the implied177 user-friendly mesh generation. This modeling scheme is the most valuable when a compromise178 between accuracy and efficiency is needed.179 Several approaches have been suggested for the determination of Young’s modulus. This180 7 Escudero, September 23, 2019 parameter is rather variable even for nominally identical specimens, and, as an approximation,181 several researches have derived expressions assuming linear elastic behavior for both materials and182 equating the compressive deformation of masonry to the sum of the deformation of the brick and183 mortar joints. For more details see references (Lopez et al. 1999;Quinteros et al. 2012).184 Extension to this work is given in (Lourenço 1996;Pelà 2009) where composite yield criteria,185 suitable for modeling anisotropic materials under plane stress conditions, are presented. In such186 cases, individual yield criterion for both tension and compression have been considered, according187 to different failure mechanisms. Although both cases miss to reproduce the uniaxial compressive188 strength parallel to the bed joints, globally, good accuracy can be notice from such yield criteria.189 In this work, it is adopted the scheme to model the masonry as an orthotropic material – which190 is an implicit definition of the orthotropic yield function based on the transformed-tensor method191 proposed by Oller et al. (Oller et al. 2003). The reader may abound in the subject consulting192 reference (Pelà 2009), where an extensive review of the state of the art has been carried out; also,193 the underlying principles and basis of the numerical implementation follows the same reference.194 The objective of the approach, proposed by Oller et al. (Oller et al. 2003), is to adjust an195 arbitrary isotropic yield criterion to the real behavior of an anisotropic material using a mapping196 method based in the existence of a real anisotropic space of stresses σi j and a conjugate space of197 strains εij, such that each of these spaces has its respective image in a fictitious isotropic space of198 stresses σij and strains εi j respectively, Fig. 3.199 NUMERICAL DESCRIPTION OF THE STUDIED MODEL200 In order to show the potential of the proposed method, the model of a real-life masonry building201 has been proposed, which corresponds to the geometry, plan distribution, and elevation of typical202 buildings of Mexico City. Such building will be referred, from now on, as the B-SSC model, and,203 fictitiously, located in the seismic zone ZIIIa (placed at the former lake area of Mexico City)204 according the classification depicted in reference (Gobierno del Distrito Federal 2004a).205 8 Escudero, September 23, 2019 Building description206 The proposed housing building is a six-story construction, with two apartments for each level –207 a total of twelve (12) apartments per building with an area of 58m2each. Architectural drawings,208 showing the distribution in plan and elevation of the structural members (Fig. 4 and Fig. 5),209 demonstrate the complexity of the structural system which is analyzed .210 The general aspect of the designing process, for structural masonry, is summarized as follows:211 1. Definition of the location of the building, in order to define seismic hazard.212 2. Bearing masonry walls, with structural purpose, have to be distinguished – discarding walls213 with only architectural purpose, like dividing walls or short walls.214 3. Loading conditions such as dead loads, additional dead loads, and live loads have to be215 defined using regional codes – for this case (Gobierno del Distrito Federal 2004d).216 4. The first performed design is through gravity loads, paying special attention to walls in the217 ground floor, since those are the ones subjected to the highest vertical stresses.218 5. An iterative design due to lateral loads is performed for each orthogonal direction. The219 starting point of such analysis is the assumption that all walls are made of confined masonry.220 6. Computation of in-plane bending moment of the masonry walls, where the confining221 columns forming the masonry wall have to be endow with the required amount of steel222 reinforcement.223 7. Structural documentation is generated using the information obtained throughout the de-224 signing process.225 The structural design based in the regional construction code was obtained, for comparison226 purpose, using the commercial software ANEMgcW, (G.C. Ingeniería y Diseño, S. C. 2007). The227 structural analysis program ANEMgcW follows the steps mentioned above and it has been used as228 an aid for analysis and design process, and consequently, to generate the structural drawings shown229 in figure 6. Hence, this structural layout become the basis of the upcoming comparisons among the230 designing process using a code regulation and the analysis scheme proposed in this work. Obtained231 9 Escudero, September 23, 2019 in Fortran developed to treat a large variety of composite materials through the use of Rules of386 Mixtures. The obtained results for model B-SSC applying the pushover analysis, described in387 previous section, are summarized and discussed in this section with the aid of figures described388 down below.389 The displacement-force response of model B-SSC is given in Figure 20, where a schematic390 representation of the structural stiffness, set by equations 2 and 3, is shown. In dashed lines, the391 stiffness of both models are shown, and as can be seen, a good approximation has been obtained392 between them since the difference is only 12.95%.393 Figure 20 also shows us separately the stiffness of walls placed perpendicular to the application394 of displacement (bending walls) and walls parallel to the application of displacement (membrane395 walls), which confirms reference (Gobierno del Distrito Federal 2004c) regarding the low stiffness396 exerted by the bending-acting walls, while their stiffness is compared with the membrane-acting397 walls, showing that their stiffness could be neglected for analysis and design purposes. The overall398 response of the structure is depicted by the continuous red line.399 Fig. 21 to Fig. 24 correspond to a set of deformation stages (amplified 250 times) within the400 analysis process, where the damage variables of the element is also shown. In all cases, images401 on the left represent the confining elements, whereas images on the right represent the masonry402 elements. The selected stages for the displacement have been a) δ=2.00mm,b) δ=6.00mm and c)403 δ=8.20mm, respectively.404 Fig. 21 presents the damage evolution of walls W-01 and W-02, located along constructive axis405 1(rear façade). The evolution of the damage variables is clearly computed as the displacement is406 increased in the expected stress areas – near the joints of the confining elements.407 Fig. 22 and Fig. 23 corresponds to the damage evolution of walls W-05,W-06 and W-07, located408 along constructive axis 3and walls W-08,W-09,W10 and W-07, located along constructive axis 4,409 respectively. Due to the eccentric distribution of the plan, it is observed a stronger evolution of the410 damage variable in axis 3.411 Fig. 24 presents the damage evolution of walls W-15,W-16,W-17 and W-18, located along412 16 Escudero, September 23, 2019 constructive axis 7(front façade). A large value of damage is found in the short masonry walls,413 coinciding with the theory reported in the structural aids for seismic reinforcement buildings.414 For all four cases, the confining elements show some degree of damage through this variable;415 this is interpreted as the cracks and micro-cracks developed in these structural elements as the416 response for the pushover analysis.417 Damage Assessment418 In this section, the damage scenarios the structure B-SSC could undergo in a zone with high419 seismic activity are characterized; therefore, two main assumptions are taking into account:420 1. The capacity spectrum and the bilinear capacity spectrum, used to characterize the damage421 scenarios and depicted in Fig. 25, correspond to the displacement-force response obtained422 from the pushover analysis, previously described.423 2. For comparative purposes, two locations have been selected for the model B-SSC. The424 first location (location L1 from now on) is somewhere in Mexico City within the seismic425 zone ZIIIa according to micro zone described in reference (Gobierno del Distrito Federal426 2004a). The second location (location L2 from now on) is somewhere within the European427 Union where the soil is D, type 1and has a peak ground acceleration equal to 0.20gaccording428 to classification of Eurocode 8 (de Normalisation 2004). The corresponding design spectra429 transformed to a sd −sa are shown in Fig. 28.430 For the computation of the capacity spectrum shown in Fig. 25, two main simplifications to431 model B-SSC had to be performed (Alzate 2013). The first one consists in converting it into a model432 with equivalent stiffness and one-degree-of-freedom per story, where the mass of the corresponding433 story is concentrated. The second simplification sets the mass-concentrated model as a one-degree-434 of-freedom model; this is achieved through a modal decomposition, using the vibrational mode435 1.436 The construction of the bilinear capacity spectrum is based on the three hypothesis described437 next,438 17 Escudero, September 23, 2019 1. The area under the spectral capacity curve must be equal to the equivalent bilinear capacity439 curve.440 2. Coordinates of the point with maximum displacement (Du,Au)match in both curves.441 3. The initial slope in both graphs must be the same.442 which led to the corresponding values for points (Dy,Au)and (Du,Au)shown in equation 4.443 Dy=1.870mm Du=3.640mm Ay=0.375gAu=0.487g (4)444 Once the spectral capacity curve has been identified, the bilinear capacity spectrum is plotted;445 which is essential to estimate the probabilistic damage, since all damage scenarios are based upon446 values of points (Dy,Au)and (Du,Au). Risk-EU (Mouroux et al. 2004) distinguishes four possible447 damage scenarios on a structure, namely, slight (ds1), moderated (ds2), extensive (ds3) and complete448 (ds4), whose definitions depends on points (Dy,Ay)and (Du,Au)of the bilinear capacity spectrum449 according to equation 5.450 ds1=0.7Dy(5) ds2=Dy ds3=Dy+0.25(Du−Dy) ds4=Du Fragility curves represent the probability in which a damage scenario ds on a structure could451 be reached or exceeded, as a function of the parameter that represent the intensity of the seismic452 action. Those values, for the current structure, are given in equation 6, leading to Fig. 26 and Fig.453 27.454 18 Escudero, September 23, 2019 ds1=1.309mm ;βds1=0.280 ds2=1.870mm ;βds2=0.240 ds3=2.313mm ;βds3=0.300 ds4=3.640mm ;βds4=0.350 (6)455 The seismic demand that the structure undergoes, is obtained through its point of performance.456 Its computation is given by a linear equivalent approach. To do so, let us consider figure 28,457 corresponding to the design spectra for location L1 and location L2; both spectra have been plot in458 asd −sa representation in order to link them together with the bilinear capacity spectrum shown459 in Fig. 25.460 Fig. 29 represent the superposition of both, the design and the bilinear capacity, spectra. The461 point of performance corresponding to the location L1 is defined by point A(sdL1=0.89mm),462 which is the crossing of the bilinear capacity spectrum and the design spectrum for L1 – point A463 corresponds to a null damage state, since the seismic demand is low. On the other hand, the crossing464 point of the prolongation of the first segment of the bilinear capacity spectrum, and the design465 spectrum for location L2 correspond to the point of performance for location L2 (sdL2=2.78mm).466 Finally, using the fragility curves depicted in figure 26, and the point of performance shown467 in figure 29, for both locations, it is possible to assess probabilistically the damage scenarios that468 model B-SSC could undergo. Results are summarized in Table 5.469 Table 5 shows a very low probability of occurrence of a low damage scenario, equal to P(ds1)=470 8.65%; whereas the remaining scenarios have no possibility of occurrence, this is obvious, since471 the model B-SSC has been designed to withstand the seismic conditions of location L1. From this472 values, the expected damage can be represented using the damage index DI =0.021.473 For location L2, however, since the seismic demand increases, so the probabilistic damage does,474 which was the reason of using such fictitious location. Results of model B-SSC at location L2475 reports a damage index DI =0.7246, being DI =1.0the index corresponding to the complete476 damage scenario.477 19 Escudero, September 23, 2019 Fig. 30 is a graphical representation of the meaning of fragility curves. It is represented the478 spectral displacement dse=2.78mm, which correspond to the point of performance of structure479 SSC at location L2; the crossing lines represents the probability of occurrence of the given damage480 scenarios, which already have been summarized in table 5.481 COMPUTATIONAL REQUIREMENTS482 The model B-SSC is a real-life masonry structure requiring a great amount of computational483 resources.484 In this section are described the improvements carried out in PLCd to perform analysis of large485 structures in a reasonably short amount of time, and consequently, to be able to analyze structures486 such as model B-SSC.487 Memory-Consuming Based Optimization488 Let us consider model B-OSC, which is fully described in Ref. (Escudero et al. 2017), and model489 B-SSC, herein explored, for comparative purposes in terms of computational resources. Table 6490 shows a comparison among the Random Access Memory (RAM) required for different process in491 the analysis of both models. From left to right: column 2 shows the amount of FE within the used492 mesh; column 3 shows the total amount RAM required; column 4 depicts the required memory493 to store information of the simple materials such as internal variables, stresses and strains; and494 column 5 shows the memory required for the PARDISO solver (Schenk et al. 2010).495 Expressed in term of percentages, the required memory to handle the information of simple496 materials is equal to 45.78%, whereas the solver uses 42.50% of the total. Analyzing such497 percentages makes clear to where should be directed the efforts on optimizing the consumed498 memory.499 From the assumption that not all elements reach a non-linear behavior, the proposed program-500 ming scheme (shown below) was implemented in PLCd.501 •It was stored an internal variable for every non-damaged FE with the same composite502 information.503 20 Escudero, September 23, 2019 •Once a component within a FE has reach a non-linear behavior, then allocation of memory504 to store information of the FE was required (internal variables, stresses and strains).505 •Allocating at every loading step or iteration may be time consuming, however, it will be506 justified for large structures where a considerable amount of RAM would be required.507 •For output and visualization purposes (writing information in GiD format) the information508 of non-damaged finite elements is evaluated using their local displacements, since is the509 only information stored. This would lead to a major drawback.510 The results obtained, once the previously described scheme has been implemented in PLCd,511 are shown in table 7. There is a considerable reduction on the amount of consumed memory;512 however, the 19.00Gb required by model B-SSC seems excessive to manage in a conventional513 desktop computer, and maybe to perform the analysis of such models would be required the node514 of a cluster.515 In this section, it has been carried out a comparison only in terms of consumed memory, see516 Table 8; however, further comparisons in terms of running time while using an iterative solver are517 required, to fit the best alternative.518 Parallelization of PLCD Using OMP directives519 This section covers the parallelization of the module used to solve mechanical problems with520 shell elements, which has been performed using OpenMP (Barney et al. 2010) directives in three521 different sections of the code. Such sections are shown below.522 1. Loop over elements while evaluating the generalized strains ˆ εand the generalized stresses523 ˆ σ. The first section to be parallelized is the loop over elements where the generalized524 strains ˆ εand generalized stresses ˆ σare evaluated. Since the loop in this case is only over525 the elements is less time-consuming than the portion of code parallelized shown in next526 section.527 2. Loop over elements while integrating the constitutive equations (plasticity, damage, etc.).528 3. Loop over elements while writing/reading information to perform a restart operation.529 21 Escudero, September 23, 2019 Results of the implementation of the previously described concepts can be summarized with530 the use of graph on Fig. 31, that reports the time needed to perform 10 steps for each of the 3531 different loading phases, for a total of 30 steps.532 Have to be taken into consideration that the process of reading the input information, and533 allocation of the main variables has not been parallelized, hence, it was also necessary the use of534 graph on Fig. 32, where the time spent along one simple iteration is reported, omitting the time535 it takes to write output information, write information to perform restart operations, allocate the536 variables at the beginning of the analysis process, and so on.537 CONCLUSIONS538 Using the proposed scheme it has been possible to obtain the force-displacement response for539 a confined masonry structure of a six stories high building. Results have been compared with the540 current code regulation for construction in masonry and concrete of Mexico City finding good541 agreement among them. With the results obtained, it enables the characterization of damage542 scenarios that such structures could undergo in high seismic activity zones. To achieve this543 evaluation, it has been used the methodology proposed by Vargas (Alzate 2013) to assess a reliable544 seismic demand, and consequently, to state a probabilistic damage a structure would undergo given545 such seismic demand.546 The implementation of a macro-modeling technique combined with 2-D shells finite elements547 has proven been effective for modeling the full-scale building. On the other hand, the possibility548 of using a large strain formulation approach for the studied structures seems rather inefficient, due549 to the small amount of deformation they undergo when cracking occurs.550 A computational tool was developed in order to reproduce the steel reinforcement pattern551 of real-life constructions. And so, been able to perform a more precisely numerical simulation552 of composite materials, which permit deducting the behaviour of the composites based on the553 constitutive equation of the component materials.554 Due to the computational cost of the model considered in this work, it was necessary to adopt555 a programming strategy that allow to reduce the execution time of analysis and the computational556 22 Escudero, September 23, 2019 resources required in terms of RAM.557 The analysis scheme for confined masonry buildings proposed in this work, stand on the basis558 that it is possible to obtain parameters to define the mechanical and non-linear response of masonry559 elements; this, in order to plot an anisotropic yield function.560 ACKNOWLEDGEMENTS561 This work has been supported by the European Commission, under the Marie Curie program, at562 the IRSES agreement 612607 (TCAINMAND project), by the European Research Council through563 of the Advanced Grant: ERC-2012-AdG 320815 COMP-DES-MAT Advanced tools for computa-564 tional design of engineering materials, by the European Community under grant: NMP-2009-2.5-1565 246067 Multiscale Reinforcement of Semi-crystalline Thermoplastic Sheets and Honeycombs, by566 the Spanish Ministerio de Economia y Competividad through the project MAT2014-60647-R Multi-567 scale and multi-objective optimization of composite laminate structures (OMMC), and by the the568 Mexican government through the grant provided by CONACyT to complete PhD studies. All this569 support is gratefully acknowledged.570 REFERENCES571 Alzate, Y. V. (2013). “Análisis estructural estático y dinámico probabilista de edificios de hormigón572 armado. aspectos metodológicos y aplicaciones a la evaluación del daño..” Ph.D. thesis, Escola573 Tècnica Superior D’Enginyers de Camins, Canals I Ports. Universitat Politècnica de Catalunya,574 Barcelona, España.575 Barney, B. et al. (2010). “Introduction to parallel computing.” Lawrence Livermore National576 Laboratory, 6(13), 10 https://computing.llnl.gov/tutorials/openMP/#Introduction.577 Chaboche, J. (1988a). “Continuum damage mechanics: Part i-general concepts.” Journal of Applied578 Mechanics, 55(1), 59–64.579 Chaboche, J. (1988b). “Continuum damage mechanics: Part ii—damage growth, crack initiation,580 and crack growth.” Journal of Applied Mechanics, 55(1), 65–72.581 CIMNE (1991-2014). PLCd - Non-linear thermomechanic finite element code. International Center582 23 Escudero, September 23, 2019 of Numerical Methods in Engineering, Barcelona, España. Finite element code oriented to PhD583 student education.584 de Normalisation, C. E. (2004). “Eurocode 8–design of structures for earthquake resistance–part585 1: General rules, seismic actions and rules for buildings.” European Standard NF EN, 1.586 Escudero, C., Oller, S., Martinez, X., and Barbat, A. (2017). “Procedures based on composite fem587 technology for the resolution of concrete-framed structures with masonry in-fills: Comparison588 with mexican building code.” Journal of Engineering Mechanics, 143(9), 04017080.589 Escudero, C., Oller, S., Martinez, X., and Barbat, A. H. (2016). “A laminated structural finite590 element for the behavior of large non-linear reinforced concrete structures.” Finite Elements in591 Analysis and Design, 119, 78–94.592 Faria, R., Oliver, J., and Cervera, M. (1998). “A strain-based plastic viscous-damage model for593 massive concrete structures.” International Journal of Solids and Structures, 35(14), 1533–1558.594 G.C. Ingeniería y Diseño, S. C. (2007). “ANEMgcW 3.07 - Análisis de Edificios de Mampostería.595 Gobierno del Distrito Federal (2004a). Normas Técnicas Complementarias para Diseño por Sismo.596 México, D.F.597 Gobierno del Distrito Federal (2004b). Normas Técnicas Complementarias para Diseño y Con-598 strucciones de Estructuras de Concreto. México, D.F.599 Gobierno del Distrito Federal (2004c). Normas Técnicas Complementarias para Diseño y Con-600 strucciones de Estructuras de Mampostería. México, D.F.601 Gobierno del Distrito Federal (2004d). Reglamento de Construcciones del Distrito Federal. México,602 D.F.603 Jason, L., Huerta, A., Pijaudier-Cabot, G., and Ghavamian, S. (2006). “An elastic plastic damage604 formulation for concrete: Application to elementary tests and comparison with an isotropic605 damage model.” Computer methods in applied mechanics and engineering, 195(52), 7077–7092.606 Ju, J. (1989). “On energy-based coupled elastoplastic damage theories: constitutive modeling and607 computational aspects.” International Journal of Solids and Structures, 25(7), 803–833.608 Kupfer, H., Hilsdorf, H. K., and Rusch, H. (1969). “Behavior of concrete under biaxial stresses.”609 24 Escudero, September 23, 2019 ACI Journal Proceedings, Vol. 66, ACI.610 Lopez, J., Oller, S., Oñate, E., and Lubliner, J. (1999). “A homogeneous constitutive model for611 masonry.” International Journal for Numerical Methods in Engineering, 46(10), 1651–1671.612 Lourenço, P. B. (1996). “Computational strategies for masonry structures.” PhD dissertation, Delft613 University, Delft, Netherlands.614 Lourenço, P. B., Milani, G., Tralli, A., and Zucchini, A. (2007). “Analysis of masonry struc-615 tures: review of and recent trends in homogenization techniques..” Canadian Journal of Civil616 Engineering, 34(11), 1443–1457.617 Lubliner, J., Oliver, J., Oller, S., and Oñate, E. (1989). “A plastic-damage model for concrete.”618 International Journal of Solids and Structures, 25(3), 299–326.619 Marques, R. and Lourenço, P. B. (2014). “Unreinforced and confined masonry buildings in seismic620 regions: Validation of macro-element models and cost analysis.” Engineering Structures, 64, 52621 – 67.622 Martínez, X. (2008). “Micro-mechanical simulation of composite materials using the serial/parallel623 mixing theory.” Ph.D. thesis, Escola Tècnica Superior D’Enginyers de Camins, Canals I Ports.624 Universitat Politècnica de Catalunya, Barcelona, Spain.625 Mazars, J. and Pijaudier-Cabot, G. (1989). “Continuum damage theory-application to concrete.”626 Journal of Engineering Mechanics, 115(2), 345–365.627 Meli, R. (1990). “Diseño sísmico de edificios de muros de mampostería. la práctica actual y el628 comportamiento observado.” Sociedad Mexicana de Ingeniería Sísmica, 40, 7–28.629 Mouroux, P., Bertrand, E., Bour, M., Le Brun, B., Depinois, S., Masure, P., and team, R.-E.630 (2004). The European Rusk-EU project: An advanced approach to earthquake risk scenarios,631 with applications to different European towns. The European Commission.632 Oliver, J., Cervera, M., Oller, S., and Lubliner, J. (1990). “Isotropic damage models and smeared633 crack analysis of concrete.” Second international conference on computer aided analysis and634 design of concrete structures, Vol. 2, 945–958.635 Oller, S. (1998). “Un modelo de daño continuo para materiales friccionales.” Ph.D. thesis, Escola636 25 Escudero, September 23, 2019 ds Location L1 Location L2 P(ds1)8.65% 99.65% P(ds2)0.00% 95.02% P(ds3)0.00% 73.04% P(ds4)0.00% 22.12% TABLE 5. Probability of damage occurrence. 32 Escudero, September 23, 2019 Model F.E. Initial Memory Req. S.M. Info Solver B-OSC 30,868 572.70Mb 262.22Mb 243.62Mb B-SSC 1’105,804 28.599Gb 9.82Gb 14.599Gb TABLE 6. Memory comparison between models B-OSC and B-SSC. 33 Escudero, September 23, 2019 Model Initial Memory Req. Final Memory Req. B-OSC 572.70Mb 329.4Mb B-SSC 28.599Gb 19.00Gb TABLE 7. Memory comparison between models B-OSC and B-SSC using the proposed scheme and the direct solver PARDISO. 34 Escudero, September 23, 2019 Model Initial Memory Req. Final Memory Req. B-OSC 572.70Mb 132.51Mb B-SSC 28.599Gb 4.75Gb TABLE 8. Memory comparison between models B-OSC and B-SSC using an iterative solver of library FEMT. 35 Escudero, September 23, 2019 List of Figures674 1 Typical stress-strain relationship of steel. . . . . . . . . . . . . . . . . . . . . . . . 38675 2 Typical stress-strain relationship for concrete. . . . . . . . . . . . . . . . . . . . . 39676 3 Relation between the fictitious isotropic and the real anisotropic space. . . . . . . . 40677 4 Architectural drawing - floor plan (units: cm). . . . . . . . . . . . . . . . . . . . . 41678 5 Architectural Drawing - Section A-A’ (units: cm). . . . . . . . . . . . . . . . . . . 42679 6 Structural drawing - floor plan view (units: cm). . . . . . . . . . . . . . . . . . . . 43680 7 Structural model and its equivalent mass-concentrated model . . . . . . . . . . . . 44681 8 Schematic representation of layer distribution. . . . . . . . . . . . . . . . . . . . . 45682 9 Steel fibers of Axis A,E, G, and K in any level. . . . . . . . . . . . . . . . . . . . 46683 10 Detailed section of columns (units: cm). . . . . . . . . . . . . . . . . . . . . . . . 47684 11 Detailed section of beams (units: cm). . . . . . . . . . . . . . . . . . . . . . . . . 48685 12 Detailed section of slab (units: cm). . . . . . . . . . . . . . . . . . . . . . . . . . 49686 13 Steel fibers of Axis 3.................................. 50687 14 Steel fibers of Axis Aand K.............................. 51688 15 Steel fibers of Axis 1,7and F. ............................ 52689 16 Dead loads - Masonry bearing walls. . . . . . . . . . . . . . . . . . . . . . . . . . 53690 17 Deadloads-Storeysystem............................... 54691 18 Dead loads - Roofing system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55692 19 Isometric view of GiD model for the B-SSC building................. 56693 20 Displacement-force response of the structure B-SSC in X direction. . . . . . . . . . 57694 21 Damage evolution on structural components on constructive axis 1.......... 58695 22 Damage evolution on structural components on constructive axis 3.......... 59696 23 Damage evolution on structural components on constructive axis 4.......... 60697 24 Damage evolution on structural components on constructive axis 7.......... 61698 25 Bilinear representation of the capacity curve displayed in figure 20 transformed to699 spectraldisplacement.................................. 62700 36 Escudero, September 23, 2019 26 Fragility curves for model B-SSC . . . . . . . . . . . . . . . . . . . . . . . . . . . 63701 27 Damage index for model B-SSC . . . . . . . . . . . . . . . . . . . . . . . . . . . 64702 28 Design spectra in sd −sa representation corresponding to ZIIIa and soil Dtype703 1with ag=0.2g. ................................... 65704 29 Point of performance using a design spectra for seismic zone ZIIIa and seismic705 zone Type 1Soil DaccordingtoEC8. ........................ 66706 30 Probability of damage for model B-SSC and seismic zone Type 1Soil D(EC8). . . 67707 31 Computational time improvement obtained for model B-SSC using different number708 of threads running in parallel for a total of 30 steps. . . . . . . . . . . . . . . . . . 68709 32 Computational time improvement obtained for model B-SSC using different number710 of threads running in parallel on one single iteration. . . . . . . . . . . . . . . . . 69711 37 Escudero, September 23, 2019 Fig. 1. Typical stress-strain relationship of steel. 38 Escudero, September 23, 2019 Fig. 2. Typical stress-strain relationship for concrete. 39 Escudero, September 23, 2019 Fig. 3. Relation between the fictitious isotropic and the real anisotropic space. 40 Escudero, September 23, 2019 Fig. 4. Architectural drawing - floor plan (units: cm). 41 Escudero, September 23, 2019 Fig. 11. Detailed section of beams (units: cm). 48 Escudero, September 23, 2019 Fig. 12. Detailed section of slab (units: cm). 49 Escudero, September 23, 2019 Fig. 13. Steel fibers of Axis 3. 50 Escudero, September 23, 2019 Fig. 14. Steel fibers of Axis Aand K. 51 Escudero, September 23, 2019 Fig. 15. Steel fibers of Axis 1,7and F. 52 Escudero, September 23, 2019 Fig. 16. Dead loads - Masonry bearing walls. 53 Escudero, September 23, 2019 Fig. 17. Dead loads - Storey system. 54 Escudero, September 23, 2019 Fig. 18. Dead loads - Roofing system. 55 Escudero, September 23, 2019 4.2. Typical Masonry Building at Mexico City 141 Figure 4.47 Isometric view of GiD model for the B-SSC building.  An uniformed live load equal to 170 kg/m2has to be used in the storey system of residential houses (section 199.V.b). Third Stage Loading Condition: Pushover The third loading stage corresponds to a standard pushover applied xdirection. Hence, a displacement has been imposed at the center of the slab of the sixth storey, whose purpose is to predict the force-displacement response of the structure. Obtained results for model B-SSC are presented in section 4.2.6. 4.2.5 Meshing and Computational Requirements To cover the meshing needs, is selected the use of a pre and post processor for numerical simulations, in this case GiD [33]. GiD is a universal, adaptive and user-friendly pre and post-processor for numerical simulations in science and engineering. It has been designed to cover all the common needs in the numerical simulations field from pre to post-processing: geometrical modelling, effective definition of analysis data, meshing, data transfer to analysis software, as well as the visualization of numerical results. Figure 4.47 on the left, for instance, depicts the GiD visualization from an isometric frontal perspective, whereas figure 4.47 on the right depicts the GiD visualization from the rear perspective. Also the set of figures 4.48 correspond to a isometric perspective of a GiD output that correspond to the constructive axis 3(on the left) and axis Aand K(on the right). Model described on section 4.1 have to be just considered as an example, intended to show the capabilities and scopes of the process followed to perform the structural analysis of a masonry constructions. The model of section 4.1 is small while compared Fig. 19. Isometric view of GiD model for the B-SSC building. 56 Escudero, September 23, 2019 012345678 0 500 1000 1500 2000 2500 Displacement [mm] Force [kN] Overall Response Membrane Walls Shell Walls Initial Stiff. (p. work) Stiffness using ANEM Fig. 20. Displacement-force response of the structure B-SSC in X direction. 57 Escudero, September 23, 2019 012345678 0 0.2 0.4 0.6 0.8 1 Spectral displacement(mm) Damage index Damage index Fig. 27. Damage index for model B-SSC 64 Escudero, September 23, 2019 0 50 100 150 200 250 300 350 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Spectral displacement (mm) Spectral acceleration (g) Design spectrum (ZIIIa) Design spectrum (EC8) Fig. 28. Design spectra in sd −sa representation corresponding to ZIIIa and soil Dtype 1with ag=0.2g. 65 Escudero, September 23, 2019 01234 0 0.1 0.2 0.3 0.4 0.5 0.6 Spectral displacement (mm) Spectral acceleration (g) Bilineal capacity spectrum Design spectrum (ZIIIa) Design spectrum (EC8) Fig. 29. Point of performance using a design spectra for seismic zone ZIIIa and seismic zone Type 1Soil Daccording to EC8. 66 Escudero, September 23, 2019 012345678 0 0.2 0.4 0.6 0.8 1 Spectral displacement(mm) P[ds/sd] Low ds Moderate ds Extensive ds Complete ds Fig. 30. Probability of damage for model B-SSC and seismic zone Type 1Soil D(EC8). 67 Escudero, September 23, 2019 1 2 4 12 20 25 30 35 40 45 50 55 8 Number of Threads Time [min] Fig. 31. Computational time improvement obtained for model B-SSC using different number of threads running in parallel for a total of 30 steps. 68 Escudero, September 23, 2019 1 2 4 12 10 15 20 25 30 35 40 8 Number of Threads Time [Seg] Fig. 32. Computational time improvement obtained for model B-SSC using different number of threads running in parallel on one single iteration. 69 Escudero, September 23, 2019