Rigid block and spring homogenized model (HRBSM) for masonry subjected to impact and blast loading

International Journal of Impact Engineering 109 (2017) 14
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Rigid block and spring homogenized model (HRBSM) for masonry subjected to impact and blast loading TagedPD1X XLuís C. SilvaD2Xa,X *, D3X XPaulo B. Louren¸c oDa4X X , D5X XGabriele MilaniDb6X X TagedP Department of Civil Engineering, ISISE, University of Minho, Azure
m, 4800-058 Guimare aes, Portugal b Department of Architecture, Built environment and Construction engineering (A.B.C.), Technical University in Milan, Piazza Leonardo da Vinci 32, 20133 Milan, Italy a

TAGEDPA R T I C L E

I N F O

Article History: Received 29 December 2016 Revised 3 April 2017 Accepted 18 May 2017 Available online 23 May 2017 TagedPKeywords: Masonry Out-of-plane Homogenization DEM Blast and impact load

TAGEDPA B S T R A C T

In the present study, a simple and reliable Homogenization approach coupled with a Rigid Body and Spring Model (HRBSM) accounting for high strain rate effects is utilized to analyse masonry panels subjected to impact and blast loads. The homogenization approach adopted relies into a coarse FE discretization where bricks are meshed with a few elastic constant stress triangular elements and joints are reduced to interfaces with elasto-plastic softening behaviour including friction, a tension cut-off and a cap in compression. Flexural behaviour is deduced from membrane homogenized stress-strain relationships by on-thickness integration (Kirchhoff
Love plate). Strain rate effects are accounted for assuming the most meaningful mechanical properties in the unit cell variable through the so-called Dynamic Increase Factors (DIFs), with values from literature data. The procedure is robust and allows obtaining homogenized bending moment/torque curvature relationships (also in presence of membrane pre-compression) to be used at a structural level within the HRBS model, which has been implemented in a commercial software. At structural level, the approach resorts to a discretization into rigid quadrilateral elements with homogenized bending/torque non-linear springs on adjoining edges. The model is tested on a masonry parapet subjected to a standardized impact and on a rectangular masonry slab subjected to a blast load. In both cases, a number of previous results obtained by literature models are available for comparison, as well as experimental data. Satisfactory agreement is found between the present results and the existing literature in the field, both experimental and numerical. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction TagedPMasonry is an ancient and still widespread construction material [1]. Unreinforced masonry (URM) buildings are very common in the urban historical centres which allow them to usually foster an intrinsic historic and cultural value [3]. Besides the cultural importance, or at least the historic one, URM buildings can have also an important economic and societal value, being still used as commercial, services or housing buildings. However, these are typically vulnerable to outof-plane failures even for low forces applied [2]. This highlights the importance of their maintenance and conservation, but also the need of mitigating excessive risk by making them more prone to resist dynamic actions such as earthquakes, impacts, explosions or other possible extreme loading cases.

*

Corresponding author. E-mail addresses: [email protected], [email protected] (L.C. Silva), [email protected] (P.B. Louren¸c o), [email protected] (G. Milani). http://dx.doi.org/10.1016/j.ijimpeng.2017.05.012 0734-743X/© 2017 Elsevier Ltd. All rights reserved.

TagedPIn recent decades, a great deal of effort has been made to develop solutions to reduce destructive damage and casualties due to blast loads and impacts (also in light of a major protection of the built heritage against terrorist attacks). Needless to say, masonry structures are in the majority of the cases rather vulnerable to explosions and impacts. In this regard, the scientific community is eager to share advanced numerical studies conceived for a better understanding of the blast/impact structural response of masonry walls [8,11
13]. This is achieved in conjunction with a quantitative insight into the behaviour of the masonry material at high strain rates [4
9], also in regard of an optimal strengthening with innovative materials allowing for a safety increase [10]. TagedPBearing in mind specifically high-strain rate loads, research was conducted to increase the insight of the behaviour of structures when subjected to these extreme cases. The city bombing event in Oklahoma (1995) addressed the importance to carry on such studies, being the target the decrease of potential casualties and buildings damage. Experimentation is still nowadays at a higher level than numerical modelling in this field. Even if more attention was

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TagedPdevoted to concrete [14
16,29] and steel [17,18] structures, some different experimental campaigns are also available for masonry. For instance, some works regarding masonry panels subjected to blast loads [6,19
21], and impacts [22,23] are worth mentioning. TagedPIt is known that URM walls subjected to dynamic loads can resist accelerations higher than their static strength, the so-called dynamic stability [24,25]. Additionally to the fact that the use of static or quasi-statics approaches preclude the consideration of important dynamic phenomena (see [25]), existing research proved that assuming static material properties could lead to an underestimation of the bearing capacity of structures [26,27]. Therefore, according to the applied load strain rate, material properties may exhibit a dynamic enhancement [28] which is of the most importance for blast and impact loads. TagedPThorough experimental campaigns are difficult to carry out due to the involved costs. This brings the need of developing numerical models that may predict with accuracy the dynamic response of structures. Simple models based on a single degree of freedom (SDOF) systems could be found in the literature to study the dynamic behaviour of structures subjected to blast loading [20
21,30]. These are easy to use and practical-oriented, allowing a fast evaluation of the approximate collapse load value. They constitute helpful tools for large structures in case studies, albeit conclusions upon structures damage, deformed shape and debris velocity are not possible. TagedPMore advanced strategies, for instance the typical 1-scale Finite element (FE1) based models allow more accurate solutions, overcoming in part the latter drawbacks. The classical approaches are still macro- and micro- models [31,32]. They differ in the mechanical scale level of analysis consideration. In the former method, masonry is considered as a fictitious homogeneous material without an explicit distinction between units and joints. It is useful to study large structures but it lacks in accuracy at a local scale and demands a considerable number of parameters calibration [33]. In the latter method, the description of the micro-structure of masonry is modelled, meshing bricks and mortar separately. Hence, both constitutive materials are represented with complex models describing the behaviour in the elastic and inelastic range [34]. TagedPMicro-modelling allows obtaining results with great accuracy, but the complexity at the modelling stage and the considerable computational time required at the processing stage makes it more suitable for the analysis of masonry walls with small dimensions [35]. Still, simplified micro-modelling FE approaches are found in literature. As example, Eamon et al. [6] applied such strategy to study concrete masonry walls under one-way bending induced by blast loads. Several simplifications aiming at a reduction of the computational cost regarding mesh size, element type and material properties were undertaken. Despite that, the authors obtained noteworthy results. Likewise, Burnett et al. [26] introduced a simplified micro-modelling approach and implemented into a finite element software. The strategy was based on a simple masonry contact interface model, but proved to reproduce with fair accuracy the response of a masonry wall subjected to an impact load. TagedPHomogenization methods are in-between with these two modelling FE schemes [36
39] and constitute a promising alternative. Homogenization is basically aimed at studying the structural problem at different scales [40]. Research on the topic showed the clear advantages of this process, both at a quasi-static and dynamic range, see for instance [37,41,42] and [7,43,44], respectively. Concisely, a mechanical characterization of masonry at a cell-level is firstly achieved and the resultant information is then transferred as averaged quantities to be used at a structural-level. Hence, according to the intrinsic complexities assumed, masonry texture (orthotropy) may be envisaged, together with the nonlinear behaviour and softening of its constituents, both at tension and compression, without the need of a thorough discrete representation of bricks and mortar joints at a structure level. The homogenization strategy, the so-called

15

TagedP ultilevel, multi-scale finite element method or as FE2 approaches, m appears to be an interesting procedure once non-linear analyses can be conducted with an acceptable trade-off between results accuracy and computational time-cost [41]. TagedPIn this context, the present paper addresses a simple two-step procedure within the scope of homogenization approach. The strategy was validated already for quasi-static purposes in [45]. Aiming to achieve a better insight regarding the behaviour of masonry walls under high-rate loading, the model is herein extended for the outof-plane dynamic analysis range. TagedPThe novelty introduced is focused on the implementation at a meso-scale of a simple procedure, embedded on the homogenization scheme, which accounts for the strain-rate dependency of material properties via a dynamic increase factor approach. The simple bespoke Homogenized rate-dependent model is coupled to a commercial software package (ABAQUS [46]) through a novel Rigid Body and Spring Mass model (denoted hereafter, HRBSM). The procedure directly allows the use of the obtained rate-dependent homogenized curves with material softening by exploiting the commercial software built-in capabilities. TagedPInsomuch, the strategy is intended to be a fast and an accurate solution predictor tool. Thus, the following assumptions are made: (i) the adoption of a simplified micro-modelling FE displacementbased approach for the meso-scale step that limits the computational effort required to find homogenized quantities; (ii) the implementation of a novel HRBSM model at a macro-scale in a commercial software like ABAQUS [46], in which powerful built-in procedures are already at disposal and; (iii) the use of a discrete FE (HRBSM) model whereas the elements that carry the upward-scale transfer of information are linear ones, simplify the procedure and avoid numerical convergence problems. TagedPAlso, the strategy offers the possibility to spread the work to a wide range of potential stakeholders, from researchers to engineering practitioners. Its validation will be achieved through comparison with experimental results on masonry structures subjected to blast and impact loads ([22] and [47], respectively). 2. Meso-scale 2.1. Out-of-plane homogenized model TagedPA multi-scale homogenized
based approach is assumed for the study of masonry panels subjected to different load types. Such strategy lies on the periodicity feature of a given media and it is therefore a suitable strategy for masonry [48]. First, a meso-scale mechanical characterization on a representative volume element (hereafter, RVE) is achieved by solving a boundary value problem (BVP). Then, the macroscopic constitutive response is accomplished through the assemblage of these RVE units. The strategy allows defining the mechanical properties of each material at the unit cell only, obtaining the inelastic stress and strain response by introducing considerations at the component level. Such framework is schematically described in Fig. 1 and poses noticeable advantages. Masonry texture is indirectly represented at a macro-scale by an approach that resorts to a RVE micro-scale description with firstorder kinematics keeping a relatively low computational effort. TagedPThe present out-of-plane homogenization model is based on the initial in-plane identification of an elementary cell. The main features will be explained in what follows and, for further information of the quasi-static approach, the reader is referred to [49] and [45]. Please note that the description will be made for the arrangement of the units (texture) presented in Fig. 2, namely a running bond texture. TagedPIn brief, homogenization consists in deriving the upper-scale properties by introducing averaged quantities for macroscopic strain and stress tensors (E and S, respectively) obtained at a micro-scale

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Fig. 1. Flow-chart of the present two-step approach.

Fig. 2. Micro-mechanical model adopted for the present homogenized model and strength domain for mortar joints reduced to interfaces.

TagedPon the RVE (Y, elementary cell). A FE-scheme is assumed, based on a strain-driven formulation, in order to solve the local boundary conditions of the BVP. Fig. 2 shows a schematic representation of the present RVE. Considering y = [y1,y2,y3] the reference frame for the local description (meso-scale level), v 2 <2 which represent the middle surface of the plate (wall), the Y module is described as Eq. (1) refers [50]: Y ¼ v
t=2; t=2½

ð1Þ

where Y 2 <3 (see Fig. 2). The space for the boundary surface of each RVE is thus limited as described by Eq. (2): @Y ¼ @Y1 [ @ Y3þ [ @ Y
3

ð2Þ

where @Y3 ¼ v  f
t=2; t=2g represents the boundary surfaces in the plate's thickness. The main concept of the homogenization process implies that the macroscopic stress and strain tensors are calculated as Eq. (3). Z Z 1 1 E ¼ 〈ɛ〉 ¼ ɛðuÞdY; S ¼ 〈s 〉 ¼ s dY ð3Þ V V Y

Y

where 〈*〉 is the average operator; ɛ is the local strain value, which is directly dependent of the displacements field u; s is the local stress value and V is the volume of each elementary cell. The latter is

TagedP overned by the Hill-Mandel principle [51,52] that establishes the g energy equivalence between the macroscopic stress power with the micro-scale stress power over the volume of the RVE. All the mechanical quantities are considered as additive functions and periodicity conditions (local periodicity) are imposed on the stress field s and displacement field u [53] so that:

s periodic on @Y and s n antiperiodic on @Y1

ð4Þ

uper periodic on @Y1

ð5Þ

agedPIT n the present model, the RVE is modelled as a continuum FE model, whereas joints are reduced to interfaces with zero thickness and bricks are discretized by means of a mesh constituted by planestress elastic triangles, Fig. 2. The formulation assumes that cracking and all material non-linearity of each RVE are concentrated exclusively on interfaces between adjoining elements, both on brick and joints. This assumption seems in agreement with experimental data, in which crack onset and propagation tend to follow a zigzag along joints and between bricks [54,55]. The latter assumption is specially observed for the cases of strong block masonry structures, i.e. joints present a considerable lower tensile strength and thus are valid for the present study [47,56]. Such damage pattern is also observed in the dynamic range [28]. The elastic domain of joints is bounded by a

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TagedPcomposite yield surface that includes tension, shear and compression failure with softening (see Fig. 2). A multi-surface plasticity model is adopted, with softening, both in tension and compression. The parameters ft and fc are, respectively, the tensile and compressive Mode-I strength of the masonry or mortar
brick interfaces, c is the cohesion, F is the friction angle, and C is the angle which defines the linear compression cap. TagedPThe response of the RVE under out-of-plane actions is obtained subdividing along the thickness the unit cell into several layers (40 subdivisions). A displacement driven approach is adopted, meaning that macroscopic curvature increments Dx11, Dx22, Dx33 are applied through suitable periodic boundary displacement increments. Thus, each layer undergoes only in-plane displacements and may be modelled through plane-stress FEs. TagedPIn this way, homogenized curves are approximated to define the nonlinear behaviour of the interfaces. For each interface at a structural scale, the cross-section equilibrium is iteratively calculated aiming to obtain the M-u homogenized curves. The strategy to derive these curves is based on the macroscopic mode-I and mode-II stresses. Such assumption is plausible because masonry failure mechanisms tend to be mainly governed by joints failure due to its low tensile strength. Bending and torsion moment curves denoted by M ¼ ½ Mx My Mxy T are obtained at each step simply by integration along the thickness of the quantities S ¼ ½Sxx Syy Sxy T described by s in (Eq. (8)). The strategy accounts for potential precompression states. Z 1 M ¼ 〈s y3 〉 ¼ s y3 dY ð6Þ A

cTagedP onsiderations on DIF for the sake of simplicity. According to the strain rate level, the values of the material properties are obtained through the product between the quasi-static property value and DIF, Eq. (7). 8 Eb ¼ DIFEb  Eb > > > < ftm ¼ DIFftm  ftm ð7Þ > Em ¼ DIFEm  Em > > : c ¼ DIFc  c

TagedPThe interface orientations accounted at a meso-scale are guided by the discrete mesh representation at a structural scale. Furthermore, it is important to highlight that the homogenization approach is suited only for running bond masonry. So, the validation of the strategy is limited to masonry structures with such defined texture.

3.1. The rigid body and spring mass model

TagedPTherefore, it is required to define such DIFs by: (i) introducing strain-rate laws, typically logarithmic curves, for each selected parameter; or (ii) using a discrete DIF value, independent from the strain rate level, which is a priori assumed and adopted as a constant value. If the former yields more realistic values, it is also true that the latter is a straightforward, simple and more aligned with normative proposals. TagedPFor the present study, the information proposed by Hao and Tarasov [27] is used to obtain rate dependent homogenised relations. The bending and torsional moment curves may be integrated along the thickness for each strain rate level. The nonlinear curvature-bending moment flexural and torsional behaviours of the interfaces are approximated using holonomic curves, see Fig. 3. The implementation of this information in a finite element package at a macro-scale will allow to represent and study three-dimensional structures due to out-of-plane dynamic actions.

3. Macro-scale

Y

2.2. Strain-rate effect TagedPExisting research proves that the use of static strength properties can lead to unreliable results for the masonry behaviour under fast dynamic actions, see [27,28]. Static strength material properties may exhibit an enhancement according to the strain rate level of the applied load. Research mainly focusing on concrete-like materials can be found in the literature, where assumptions intrinsically related with material effects are present to explain the phenomena, such as the lateral inertial confinement, end support friction and scale-effect [57
59]. TagedPA useful and practical way to numerically represent the material properties change is to define dynamic increase factors (DIFs). Additionally, it is possible to find studies exploring the use of viscosity as a regularization effect [60,61]. Even if it is important to carry more studies aiming at a better insight regarding the causes [59], the use of DIF laws is found suitable to study masonry structures subjected to fast loads application [26,27]. In this way, a homogenized model that may account the latter is relevant. TagedPFocusing on the present homogenized approach, the material model reflects the dynamic characteristics of mortar and brick, and is derived from the static in-plane homogenized model (see also [45]). The values that define the elastic behaviour and the strength envelope of the unit cell, i.e. the parameters that directly rule the plasticity model, are strain-rate dependent. Specifically, the Young's modulus of the brick Eb, Young's modulus of the mortar Em, tensile strength of the mortar ftm, shear modulus of the mortar Gm and cohesion c. Compressive behaviour is in practice scarcely active in out-of-plane loaded periodic masonry and therefore, excluded from the present

TagedPThe adopted two-fold strategy relies into a homogenization approach at a meso-scale and on a discrete FE model at a macroscale level. The strategy can be designated as an up-ward procedure, i.e. information regarding the mechanical characterization at a cell level may be transferred into the structural scale. The dynamic outof-plane analyses of masonry walls are performed using a novel discrete mechanical system at a macro-scale. Fig. 4 presents the model for a clear understanding and was already validated for quasi-static purposes, see [45]. The work by Kawai [62,63] serves as background for its formulation. TagedPBriefly, the system is composed by quadrilateral rigid plates. On the interfaces and connecting these rigid elements, deformable truss and rigid beams are placed on each node. These truss-beam system mimics the presence of flexural and torsional springs, governing the deformation and damage of the equivalent continuum. In other words, the truss-beam system perpendicular to the interface idealizes flexural movements and, on the other hand, torsional behaviour is guaranteed by a suitable truss-beam system parallel to the edge. Additionally, mid-span hinges placed on interfaces allow to fix the axis of rotation for torsion movements without compromising the deformed shape. It is worth noting here that a decoupled characterization of flexural and torsional actions is adopted and such behaviour is ruled by the mechanical and material information derived beforehand at a meso-scale level. Whilst such decoupling is certainly an approximation, it proved to be quite reliable, at least for level of membrane pre-compression far enough for masonry compressive strength. Bearing that the dynamic behaviour is highly influenced by inertial forces, nodal mass elements are lumped on the centre of each rigid plate. These elements concentrate the mass of the equivalent basic cell of the system, see Fig. 4. TagedPTwo issues concerning the structural implementation of the discrete model should be noted. Firstly, that model deformation and material nonlinearity are associated only with linear

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3.2. Material properties: from meso- to macro-scale TagedPThe interface behaviour of each discrete element at a macro-scale is a result of a previous homogenization performed at RVE level that takes into account both the mechanical properties of the constitutive materials and the geometry of the elementary cell. The resultant homogenized material is clearly orthotropic (for instance, the peak bending moment along the horizontal direction is higher than the vertical one because of bricks staggering that make the bed joint work under shear stresses, see Fig. 3). TagedPIn this way, the characterization of the unit cell (RVE) is achieved using a simplified homogenized approach that relies on a model with few degrees of freedom. Thus, the constitutive model defined at the interfaces tries to mimic the material information obtained at a foregoing scale. The model should be capable to allocate the mechanical information and, thus, representing effectively the elastic and inelastic response of the defined masonry RVE. This leads to the associated and fundamental character that each phase has in the two-step framework. TagedPA proper constitutive model able to interpret the latter information obtained from a previous step and scale is essential. Several models can be adopted as, for instance, the smeared crack concrete, the brittle crack concrete and the concrete damage plasticity in ABAQUS. These are all suitable for concrete and other quasi-brittle materials, and eligible to be used to characterize the homogenized trussbeam system. The concrete damage plasticity (hereafter, CDP) model is selected, given the better representation of the inelastic laws. A general overview of its main features and application is presented in what follows, being the reader referred to e.g. [64,65] for further details. TagedPThe model combines a stress-based plasticity with strain-based scalar damage. It is able to assign different yield strengths, different stiffness degradation and recovery effect terms both in tension and compression regimes. Thus, effective stresses tend to govern the plastic part of these models [66]. Moreover, it does consider the latter in the presence of material strain-rate dependency which may successfully idealize both the flexural and torsional interfaces dynamic and/or cyclic loading. The stress
strain relationship is ruled by an isotropic damage scalar affecting the elastic stiffness of the material. According to Eq. (8) the nominal stress s reads:   s ¼ ð1
dÞDel0 : ɛ
ɛpl ð8Þ where Del 0 is the initial elastic stiffness of the material; d is the damage parameter, which defines the stiffness degradation (0 for an undamaged and 1 for a fully damaged material); ɛ is the total strain and ɛpl is the plastic strain part. TagedPRegarding the hardening variables, Eq. (9) describes the law h that express its evolution, in which ɛ_ pl is the plastic multiplier and s is the effective stress.   ~ɛ_ p ¼ h s ; ~ɛ pl :_ɛ pl ð9Þ Fig. 3. Strain-rate dependent homogenized curves used for the study of masonry parapets subjected to impact load [26]: (a) horizontal bending moment; (b) vertical bending moment; and (c) torsional moment.

TagedP(one-dimensional) elements. Thus, these are responsible to catch the onset and propagation of inelastic behaviour, even if not in a continuum description as traditional FE micro- or macro- models. Nevertheless, such consideration permits to avoid numerical problems, related with non-uniqueness of the solution and loss of stability. TagedPSecondly, the discrete element approach is implemented into a commercial finite element software, namely ABAQUS [46]. There are obvious advantages by developing such processing oriented strategy at a structural scale. In fact, ABAQUS [46] allows the user to perform dynamic analyses in the range of high-strain rate loads.

TagedPThe CDP model uses a yield function based on the works of Lubliner et al. [64] and Lee and Fenves [65]. Considering a multi-directional loading, Fig. 5a illustrates the yield surface in the deviatoric plane. The hardening variable Kc controls the meridians shape of the yield shape. If it is assumed a Kc ¼ 2=3 an approximation of the Mohr
Coulomb criterion is obtained. It is stressed that the plasticdamage model assumes a non-associated potential flow. Two other dimensionless parameters may be defined. For instance, the dilation angle f and the eccentricity e which take a value of 10 and 0.1, respectively [64]. It stressed also that such multi-dimensional considerations are not used in the model, which is mono-axial in the HRBSM. TagedPIt may be noted that the strain-rate dependent homogenization model allows obtaining the macroscopic masonry material

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Fig. 4. Description of the novel discrete element system proposed.

TagedPproperties accounting for the strain softening regime. The novel HRBSM model implemented in the finite element package ABAQUS must be able to receive such data to fully represent the inelastic behaviour of masonry through stress-strain curves for axial and shear trusses of the system. But first, an identification of the desired mesh dimensions at a macro-scale and the geometrical characteristics of the masonry structure under study may be performed. This task is required to calibrate the elastic response of the input. TagedPThe procedure described next converts the latter information into valid input data for the FE package used at a structural scale. Bearing in mind that rectangular-square elements are assumed for the sake of simplicity, two different angles are considered for the interfaces: 0 and 90°. The behaviour of the interfaces is obviously orthotropic with softening, because it derives from the aforementioned homogenization strategy. The general form of the strain-rate homogenized bending- and torsional moment-curvature curves of the interfaces are depicted in Fig. 3. Hence, to accomplish this goal, obtaining stress and strain curves for each angle of the interface and

fTagedP or each bending moment direction is mandatory. In other words, the material orthotropy is reproduced at a structural level because the approach offers the possibility to reproduce different input stress
strain relationships according to the trusses plane. The conversion between moment and stress values is achieved by Eqs. (10) and (11):

s axial truss ¼

M  L ðAAxial  tÞ

s torque truss ¼

2xM  L ðATorque  HÞ

ð10Þ ð11Þ

TagedPHere, M is the bending moment per unit of interface length, H the length of each quadrilateral panel, L is the influence length of each truss (equal to half of the mesh size, i.e. H/2), t is the thickness of the wall, AAxial is the axial truss area given by 0.25 £ t £ H and ATorque is the torque truss area given by 0.5 £ e £ H, where e (value of 10 mm) is the gap between the rigid plates, which ideally should be zero but

Fig. 5. (a) Representation of Mohr
Coulomb and Drucker
Prager failure criteria in the deviatoric plane, and (b) General 5-node curve for truss beams implemented in ABAQUS.

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TagedPin practice is assumed small enough to be able to place trusses between elements. TagedPAt last, the stress homogenized input curves may be properly calibrated. An elastic calibration for the stress curves is conducted. The latter is guaranteed separately for both flexural and torsional movements and, therefore, a decoupled behaviour is derived. Briefly, by assuring the energy equivalence between the discrete mechanism and a homogeneous continuous shell element, it can be easily derived that, for both case studies, the Young's moduli of axial (Eflexural) and torque trusses (Etorque) are: Eflexural ¼

Emasonry 1 e Emasonry 1 t4 and Etorque ¼ 2 6 2L þ e ð1
n Þ 3 ð2L þ eÞH2 e ð1 þ nÞ

ð12Þ

TagedPAfter calibration, simplified softening curves as depicted in Fig. 5b are considered for each truss beam. Near the peak strength of the curve, a small plateau is considered. The aim is to circumvent abrupt stiffness loss and, thus, avoid convergence and run time problems. For the simulations, the CDP model in ABAQUS requires that information regarding the post-failure stress
strain behaviour may be introduced for each element that features material nonlinearity, i.e. the truss beams. Specifically, the cracking strain ~ɛt ck must be obtained for each point of the homogenized curve by Eq. (13): ~ɛt ck ¼ ɛt
ɛel o

ð13Þ

where ɛel o is the elastic strain corresponding to the undamaged material and ɛt is the total strain of the holonomic curve. Damage parameters dt should also be introduced, which link the undamaged elastic modulus with that of the damaged material in the unloading phase, as Ed ¼ Eð1
dt Þ, obtained through a plastic strain and total strain relation.

3.3. Implicit solver scheme TagedPAs previously discussed, the use of a commercial software at a macro-scale, such as ABAQUS poses several advantages. The software is robust in solving nonlinear dynamic problems in presence of material softening and it allows the use of implicit or explicit procedures. Explicit schemes need the insertion of density values for all elements, which would turn the modelling more complex for the proposed approach. As seen, the inertial forces are derived from representative nodal masses applied in the gravity centre of each rigid plate. The adopted strategy allows directly defining the mass matrix of the system, which an implicit solver scheme is able to handle. So, conversely to most of the problems conducted in the fast dynamic field, whereas is usually recommended the utilization of an explicit analysis scheme, an implicit solver is adopted here for the dynamic motion equation, which may be degenerated into a quasi-static incremental evolution. TagedPConstant time stepping may lead to solution divergence if a larger step is used, or to excessive and needless computational effort if a small step is defined. In this way, an available automatic time stepping algorithm is used. The Hilber
Hughes
Taylor (HHT) time integration scheme is adopted with a value of ¡0.05 for the a-operator (the reader is referred to [67] for further details of the method). Additionally, a Full-Newton solution technique with a line-search algorithm is used in which the stiffness matrix is formed at each iteration. For each study case, the maximum initial time step control was set to 1/100 of each load amplitude, meaning that at least 100 steps are computed. Additionally, the field equations tolerances and time incrementation controls available in ABAQUS were all taken as the default values. The only exception are the two parameters that defined the number of iterations that control the time increment change if convergence is not reached. In this way, in order to avoid premature cutbacks, these values were set to the recommended in

[TagedP 46] (designated as I0 = 8 and IR = 10), which provided no convergence issues for both studied cases. 4. Numerical model validation 4.1. Overview TagedPThe interest of researchers regarding the dynamic analysis of masonry structures subjected to blast and impacts loads has been growing, but experimental studies on its behaviour are still scarce [8]. The main reasons are the many technical difficulties and the high costs to perform a thorough experimental campaign. TagedPRecognizing the importance to develop further studies on the field, numerical models play an important role. An accurate numerical prediction of masonry behaviour due to blast and impact may decrease research costs as parametric studies may be easily conducted, varying both geometrical and mechanical parameters. In this framework, a discrete homogenized strain-rate sensitive model is proposed here. Its main features were discussed in the previous Sections. Thus, only material and mechanical considerations regarding the model validation will be hereafter assigned and addressed. TagedPTwo sets of experimental campaigns were chosen in the literature. The main goal is to conclude about the capability of the model to provide accurate solutions in the dynamic range. The first experimental set is a ‘stretcher’ bond masonry parapet tested by Gilbert et al. [56]. The wall is subjected to a low velocity impact load that tries to represent the impact of a vehicle. The comparisons will be performed by means of the experimental determined data [56], but also with numerical results collected from the studies of Burnett et al. [26] and Rafsanjani et al. [68]. TagedPThe second example is an infill masonry wall subjected to a blast load [47]. The load is generated by a confined underwater blast wave generator and the numerical results will be validated according to information regarding the response curve in a central control node and the observed failure modes. 4.2. Impact load on a parapet TagedPExperimental data available from the research reported in [56] is used to assess the ability of the present model to represent its behaviour due to a low-velocity impact load. A total of 21 full-scale unreinforced masonry parapets were experimentally tested in [56]. They differ in both wall dimensions (length range between 5.75 and 20.0 m) and brickwork texture (running, English and English garden bonds). Here, only the running bond pattern is considered. TagedPThe selected parapets are designated as C6 and C7 and are replicates, see Fig. 6. Their assemblage was executed with strong concrete blocks and weak mortar (class iii mortar according to BS5628). The walls and brick dimensions are 9150 £ 1130 £ 215 mm3 (length £ height £ thickness) and 440 £ 215 £ 215 mm3 (length £ height £ thickness), respectively. TagedPAt the walls’ base, the surface was coated with epoxy sand to reproduce the roughness of a given street floor. For the lateral supports, two abutment blocks connected to the walls through epoxy mortar were used. Numerically, the boundary conditions are considered to be fixed for each lateral edge and simple supported at the base of the walls. For the sake of simplicity, the base-roughness was not reflected. Aiming to model a car-like impact at both mid-height and length of the walls, a triangular out-of-plane load was applied through a steel plate. The load is idealized as a triangular time history distribution, in which the peak value is equal to 110 kN [56]. The masonry parapets representation as well as the applied force-time history for the numerical simulations are described in Fig. 6. The deformation of the studied parapets was recorded in a node located 580 mm above the base and deviated 250 mm from the centre.

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Fig. 6. Geometry of the running bond masonry parapets C6 and C7 studied by Gilbert et al. [56].

TagedPSeveral studies on the masonry parapets tested by Gilbert et al. [56] are found in the literature. For instance, the same authors proposed in [22] analytical simplified approaches to evaluate the response due to an impact load. This strategy allows a quick analysis but needs the knowledge of the critical failure mechanisms beforehand. Also, the failure modes rely on a macro-block definition and may, by themselves, preclude the analysis of more general modes. Moreover, Burnett et al. [26] presented a discrete-crack model, i.e. a simplified micro-finite element model that represents mortar joints with interface elements. The use of more complex numerical models may overcome the problem and increase the results accuracy. These models allow a greater flexibility in failure definition, having as limitation the increased computational time. Nevertheless, both models considered by the authors are strain-rate independent hence their accuracy is highly dependent on the material properties adopted. The use of static strength properties instead of dynamic ones may mislead the user, i.e. an underestimation of the collapse load may occur. The study of the dynamic properties involves the definition of DIFs, which may turn the response highly subjective if valid experimental data is unavailable. TagedPIn the present model, the numerical simulation of the parapet walls will be accomplished through the aforementioned discrete element homogenized methodology. Firstly, the homogenization step at a meso-scale is performed to mechanically characterize the running bond masonry. The static material properties required are gathered in Table 1 and the rate-dependency issue is addressed. Hao and

Table 1 Static mechanical properties adopted for the homogenization step. Quasi-static range Young's modulus of the brickwork composite (MPa) Young's modulus of the mortar (MPa) Young's modulus of the brick (MPa) Poisson coefficient (¡) Density of the brickwork composite (kg/m3) Shear modulus (MPa) Cohesion, c (MPa) Tensile strength ft (MPa) Compressive strength fc (MPa) Friction angle (f) (degrees) Linearized compressive cap angle (c) (degrees) Mode I fracture energy, GIf (N/mm) Mode II fracture energy, GIIf (N/mm) Kn
axial truss (MPa) Kn
torque truss (MPa)

TagedP arasov [27] studied the experimental dynamic behaviour of a series T of brick and mortar specimens under uniaxial compressive tests through a tri-axial static-dynamic apparatus. The analytical expressions required to describe the value of the DIFs derive from the latter research and are given by Eqs. (14)
(20). It may be noted that the study by Rafsanjani et al. in [68] that covers the current masonry parapets, use the same laws hereby presented to define the strainrate dependency of masonry interfaces using a micro-modelling strategy. TagedPThe regression equations for the ultimate compressive strength of mortar s c0_mortar used are:  DIF ¼ 0:0372ln_ɛ þ 1:4025 for ɛ_ 13s
1 ð14Þ DIF ¼ 0:3447lnɛ_ þ 0:5987 for ɛ_ > 13s
1 TagedPThe regression equation for the strain at ultimate compressive strength for mortar ɛc0_mortar : DIF ¼ 0:15231ln_ɛ þ 2:6479 DIF ¼ 0:7601
0:02272lnɛ_

ð16Þ

TagedPThe regression equations for the ultimate compressive strength of the brick s c0_brick :  DIF ¼ 0:0268lnɛ_ þ 1:3504 for ɛ_ 3:2s
1 ð17Þ DIF ¼ 0:2405ln_ɛ þ 1:1041 for ɛ_ > 3:2s
1 TagedPThe regression equation for the strain at ultimate compressive strength of the brick ɛc0_brick : DIF ¼ 0:0067lnɛ_ þ 1:0876 

20,000 1500 20,000 0.30 2295 600 0.63 0.45 30.0 30.0 50.0 0.012 0.019 174.43 130,449

ð15Þ

TagedPThe regression equation for mortar Young's modulus Emortar:

ð18Þ

TagedPThe regression equations for the brick Young's modulus Ebrick: DIF ¼ 0:0013lnɛ_ þ 1:0174 for ɛ_ 7:3s
1 DIF ¼ 0:3079ln_ɛ þ 0:4063 for ɛ_ > 7:3s
1

ð19Þ

TagedPThe regression equations for the Poisson's ratio of the brick y: DIF ¼ 0:0085lnɛ_ þ 1:1112

ð20Þ

TagedPAs information regarding the strain rate effects on tensile and shear masonry properties is lacking, the DIF regression equations for both tensile ultimate strength s t0_mortar and strain of mortar ɛt0_mortar is assigned to be equal to the compressive one. The implementation of such laws in the homogenized model allows deriving stress
strain rate-dependent homogenized curves. Additionally, moment-curvature relationships are simply derived by the

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Fig. 7. Time history of the out-of-plane displacement for the control node of parapets C6 and C7, with deformed shape for the instant 180 ms.

TagedPintegration of the latter stress
strain curves through the thickness of the wall, see Fig. 3. TagedPFig. 7 reports the displacement magnitude with respect to time. A numerical model with a mesh size of 200 mm was adopted at a macro-scale. Curves resultant from (i) the experimental results [56], (ii) the numerical model by Burnett et al. [26], (iii) the numerical model by Rafsanjami et al. [68] and, the simulation results of the discrete homogenized-based model are depicted. The curve by Burnett et al. [26] leads to excessive displacements (and understiff response) because it considers quasi-static values for the material parameters. Conversely, both the present and the micro-model by Rafsanjami et al. [68] are accurate in predicting the peak displacement, with a relative error of around 10%. Regarding the post-peak behaviour, it is noticeable that the structure displacement restitution of the present numerical model is practically null. Yet, similarly to the experimental results, the latter is not entirely reproduced by the other two numerical models under comparison, presenting both an out-ofplane displacement that slightly decreases in post-peak after the time instant of 180 mm. This is possibly due to the adopted DIFs, which come from a different testing programme and not from the actual materials used in the parapets. The response is still remarkable for such a simplified model. TagedPFig. 8 indicates the observed damage pattern for the present model. Vertical cracks are clear around both the central area and the two supported edges. Also, horizontal cracks spread from the centre along the height of the masonry parapet. As expected, it is evident that damage tends to concentrate on the impact zone. This was implicitly concluded in the experiments tests [56], whereas the lateral supports effect decreases with the increase of the parapet length. Thus, the central localization of the damage leads to a critical plastic deformed area that can explain the results. 4.3. Blast load simulation TagedPIn this section, the ability of the present model to reproduce the response of masonry due to a blast load is assessed. The validation is done using the experimental test conducted by Pereira et al. [47], in which the required material properties are available from [69]. The

cTagedP ase study focuses on an infill-masonry wall in a reinforced concrete frame constructed in a running-bond pattern. The full-scale dimensions (3500 £ 1700 £ 180 mm3) are representative of a 1:1:5 scale. The wall is a single unreinforced masonry leaf, in which the units are hollow clay bricks designated as 30 £ 20 £ 15 (the average dimensions are 293 £ 193 £ 150 mm3 according to [69]) and mortar joints 15 mm thick. The blast load is simulated through a controlled underwater blast wave generator, being the applied pressure load experimentally recorded. The obtained load profile has a peak with a magnitude of 149 kPa at 6 ms, see Fig. 9. For the numerical simulation, as no visible damage is found on the RC frame after the tests [47], the boundary conditions are considered as fixed in the four wall edges. TagedPThe collected quasi-static material properties are presented in Table 2 and come from the experimental sample tested in Pereira [69]. Contrarily to the impact load validation, no expressions for the material dynamic increase factors or data on similar masonry constituents can be found in the literature. Data from existing studies upon the same masonry panel are taken into account [47,68], using the calibrated constant DIFs on a homogeneous model within a macro-modelling approach defined by Pereira et al. [47], see Table 2. TagedPThe model validation is achieved through comparison with both experimental time-history displacement at the central node and observed damage of the masonry wall. Likewise, results of two other existent numerical models, based on a macro-modelling approach, are displayed: (i) a homogeneous model with constant DIF values by Pereira et al. [47] and; (ii) a homogeneous strain-rate dependent model by Rafsanjani et al. [68]. Fig. 10 shows the comparison of the obtained displacement at the central node with respect to time. A mesh size of 100 mm was adopted at a macro-scale. Even if it constitutes a high refinement level for such a homogenization-based approach, the aim is to allow obtaining a more clear displacement field but mainly a better damage map pattern. TagedPSeveral remarks may be posed regarding the out-of-plane displacement time-history analysis. The overall response and behaviour follows the expected one, leading to a good estimation of the peak displacement at 28 ms, with an error lower than 10%. Yet, the model response seems to be slightly stiffer during the pre-peak branch,

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Fig. 8. Damage pattern obtained for the applied load wall side: (a) horizontal cracks and (b) vertical cracks.

TagedPcharacterized by a lower initial curve slope that leads to a minor outof-phase response. The post-peak residual displacement has a 20% of error with the test one. Such differences may be explained by several factors, hereby addressed firstly for the model over-stiffness and then for the residual displacement. The former can be explained with the adoption of a constant DIF value for the material properties instead of a gradual change due to a continuum description. Likewise, the so-called side-effect associated, at a macro-scale, with the considered boundary conditions that totally preclude rotations could have an important role. Also, inertial effects have may play a role in the dynamic range but, even if the adopted density values are not directly taken from [47] but instead from [69], the values are

cTagedP oherent and does not justify by themselves such difference. Regarding the latter, the non-accounting of the resultant negative blast load-phase at the numerical model can explain why the residual displacements are slightly higher than the experimental ones. TagedPThe obtained response is in accordance with the expected behaviour for a plate fixed in its edges and with the experimental one. Particularly, a maximum displacement at the centre of the wall is observed, with the onset of the cracking spreading into the lateral edges, typically expressed by a yield lines, in this case parallel to the longest span. Fig. 11 shows the obtained damage with the present HRBSM model. Considering the aforementioned, severe horizontal cracks are clear in the centre of the wall and moderate damage is

Table 2 Quasi-static mechanical properties adopted for the homogenization step [47,69]. Quasi-static range Young's modulus of the brickwork composite (MPa) Young's modulus of the mortar (MPa) Young's modulus of the brick (MPa) Poisson coefficient (¡) Density of the brickwork composite (kg/m3) Shear modulus (MPa) Cohesion, c (MPa) Tensile strength ft (MPa) Compressive strength fc (MPa) Friction angle (f) (degrees) Linearized compressive cap angle (c) (degrees) Mode I fracture energy, GIf (N/mm) Mode II fracture energy, GIIf (N/mm) Kn - axial truss (MPa) Kn - torque truss (MPa)

Fig. 9. Geometry of the running bond masonry wall studied [69].

3600 1200 2170 0.21 3000 750 0.27 0.47 4.00 30.0 50.0 0.013 0.01 56.81 95,433

Dynamic range Parameter

DIF

Value

Young's modulus of the mortar (MPa) Young's modulus of the brick (MPa) Shear modulus (MPa) Cohesion, c (MPa) Tensile strength ft (MPa) Compressive strength fc (MPa)

3.0 2.0 2.0 3.0 3.0 3.0

3600 4340 2250 0.81 1.41 1.20

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Fig. 10. Out-of-plane time-history displacement for the control node of the masonry wall subjected to a blast load, with the deformed shape for the instant at 28 ms.

TagedPvisible near the top and bottom edges due to wall rotation, see Fig. 11a. Furthermore, Fig. 11b indicates that vertical cracks tend to be more severe along the lateral supports. Also, as previously stated, vertical cracking accumulates in a central band, where the maximum displacements occur, and its width increases with the approximation to the lateral edges. It is noticeable that both horizontal and vertical damages form altogether the expected diagonal yield crack pattern. 4.4. Mesh dependence study and final remarks TagedPA mesh sensitivity study at a macro-scale is conducted for both cases studies. The goal is to evaluate its dependency in terms of accuracy and computational time required. For this purpose, two less refined quadrilateral meshes were considered, i.e. 400 mm and 600 mm for the panel subjected to blast load and 200 mm and 600 mm for the parapet subject to low velocity load. Fig. 12a and b reports the latter information and analysis results. A maximum mesh size up to 600 mm is adopted. Once a uniform quadrilateral mesh pattern is specified, higher mesh dimensions could lead to important geometrical misrepresentations and thus unnecessary systemic errors. TagedPBearing as reference the curves obtained with the maximum refinement level, Fig. 12a and b shows that mesh dependency is moderate from an engineering standpoint. In fact, maximum relative errors of 21% and 25% between peak curves values, for the adopted coarser meshes of the impact and blast test respectively, are found. Alike time-history displacement curves shape are also collected, meaning that the pre- and post-peak behaviour is well captured and not perturbed with the mesh change, at least for the present control nodes. Even so, it is easily observed through the deformed shape presented in Fig. 12, that the displacement field is not so smooth between adjoining discrete elements with the increase of the mesh edge size. The latter is specially observed for the blast load simulation test. Coarser meshes may also lead to an inaccurate

TagedPrepresentation of the damage pattern due to the limited number of elements that govern it (deformable trusses). On the other hand, the computational time required at a macro-scale is significantly decreased with mesh size. TagedPThe authors highlight that a proper mesh size discretization is of relevance. It may minimize the errors due to possible rough geometrical modelling by the use of coarse elements but also, account for the needed output. For instance, if one wants to obtain a clear and accurate structural behaviour, displacement and damage fields, a finer mesh is a better solution. If one is interested in obtaining a quick evaluation of the structure peak displacement, a coarser mesh at a structural scale could be a possibility. TagedPAs final remark, it should be stressed that in typical homogenized full finite element methods, information is transferred between scales through a mapping process by a consistent tensor. The mapping is dependent on a kinematic map, i.e. the location of each Gauss Point of the Finite Element in the fictitious macro-scale continuum model. Several strategies may be found in the literature to overcome scale-effect issues in order to avoid unrealistic results. One is to perform the regularization of post-peak fracture energy between linked micro- RVE and macro- finite element. In this way, the analyses become independent from both the characteristic length of the material and the finite element size used. TagedPThe present strategy is specially oriented for out-of-plane analyses. An ad hoc discrete model is used at a macro-scale that simplifies the problem. Deformable trusses represent the out-of-plane behaviour at a structural scale and so first order movements at a cell level are able to characterize such behaviour. Cauchy stresses are adopted in both scales, an approach which simplifies the problem to a great extent. TagedPAs far as the actual regularization scheme is concerned, this is performed in an implicit way by three steps as shown in Fig. 1. First, the homogenized bending and torsional curves are derived at a meso-scale, and their magnitude is tuned according to a factor dependent on the characteristic length of the unit cell and the mesh

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Fig. 11. Damage pattern obtained for the applied load wall side: (a) horizontal cracks and (b) vertical cracks.

TagedPsize at a macro-scale. Secondly, such curves are converted again in stress
strain relationships through geometrical parameters that identify the macro-scale levels (H,L,e,t), as seen in Eqs. (10) and (11). At last, such curves are regularized by a scaling operator that makes Eq. (12) valid. This means that, through energy equivalence between continuum and the discrete model used, a scaling operator affects the stress
strain curves of each truss in order to hold the out-ofplane elastic energy equivalence. This operator affects, as well, the post peak curve strains and so, in an implicit way, the fracture energy itself. It is worth noting that such strategy turns out to be similar to that recommended in [70] to numerically solve regularization issues during the passage from meso- to macro-scale. 5. Conclusions TagedPA two-step homogenization procedure is presented for the outof-plane dynamic study of masonry walls subjected to impact and blast loads. The approach relies on an up-ward methodology, where a Homogenization scheme links the meso-scale orthotropic mechanical description of a masonry representative volume element with a novel Rigid Body Spring Model (HRBSM) developed at a structural scale.

TagedPAt a meso-scale, a simple FE discretization of the unit cell through constant stress triangle elements, within a plane-stress analysis, and joints reduced to interfaces allows to obtain a good response at a cell level at a very low computational effort. The nonlinearity is concentrated on mortar joints within a plasticity model that accounts for a tension cut-off, a Coulomb friction and compressive cap model. Through a homogenization process, the macroscopic stress and strains are derived and, by considering a Kirchhoff-plate hypothesis, the macroscopic bending- and torsional- moment-curvature are found by integration over the masonry thickness. On the other hand, to account for the material properties enhancement that typically occurs in case of fast dynamic events such as those investigated, a strategy based on dynamic increase factors (DIFs) is chosen. In this way, a procedure based on a simple strain-rate dependent homogenization approach is developed and ready to be implemented at a macro-scale level. TagedPAt the upper-scale, a HRBSM model is implemented into a standard advanced nonlinear finite element code, namely ABAQUS [46]. This strategy has inherent advantages due to the robust numerical procedures available in the software. The discrete model makes use of rigid quadrilateral plates connected at the edges with a system of deformable truss and rigid beams that receives the information

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Fig. 12. Mesh dependence study and registered computational time for (a) the impact test and (b) the blast load simulations.

TagedPtransferred between the different scales. These elements govern the deformation and damage propagation and have an independent constitutive model from the foregoing scale. Additionally, the imposition of both plasticity and damage on one-dimensional elements critically reduces the required computational costs to perform the analyses. It allows easy strain-rate calculations in uniaxial damage plasticity models to control the macro-model response. TagedPFor the model validation, dynamic analyses were performed in masonry panels subjected to an impact [56] and blast load [47]. In both studies, the adopted material properties stand in existent literature. Even if it is often recommended the use of an explicit procedure to solve fast dynamic problems, an implicit procedure that makes use of a Hilber
Hughes
Taylor solution

TagedP ethod is adopted here without any difficulty regarding the m numerical convergence. TagedPThe results show a satisfactory agreement with the ones obtained experimentally for both the cases studied, as far as the peak load and damage patterns are concerned. The considered simplified assumptions, in both scale-levels, make possible a good trade-off between accuracy and computational effort. In fact, the total required time to perform the analyses was 12:38 min and 3:35 min for the impact and blast load studies, respectively. The increase of the time needed for the former is due to the fact that a full strain-rate description is undertaken instead of a constant DIF value approach, as considered for the later. Once the aim of the study was to assess the ability of the model, a finer mesh was herein firstly presented for a better

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TagedPinsight of both the displacement field and damage map. A mesh dependence study was conducted and no significant differences are found from an engineering standpoint. Similar conclusions are given in [45], where the present FE model does not manifests a considerable spurious mesh dependency. TagedPAt last, it is important to address the advantage of the procedure and its efficiency. Using standard commercial FE packages, the effectiveness and robustness of the software to solve problems accounting for the post-elastic behaviour with softening can be used. This also allows the possibility to extend the use of the proposed model at professional level. Specifically, to conduct parametric studies concerning geometrical, material and load properties but also to derive Pressure
Impulse (P
I curves) useful for blast structural design. Acknowledgements TagedPThis work was supported by FCT (Portuguese Foundation for Science and Technology), within ISISE, scholarship SFRH/BD/95086/ 2013. This work was also partly financed by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT
Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER007633. References TagedP [1] Tomazevic M. Earthquake-resistant design of masonry buildings, VOL. 1. World Scientific; 1999. TagedP [2] Griffith M, Magenes G. Evaluation of out-of-plane stability of unreinforced masonry walls subjected to seismic excitation. J Earthq Eng 2003;7:141–69. TagedP [3] Tomazevic M. Heritage masonry buildings and reduction of seismic risk: the case of slovenia. In: Netherlands S, editor. Mater. Technol. Pract. Hist. Herit. Struct.327
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