Refined Rigid Block Model for In-Plane Loaded Masonry

In this work, a refined rigid block model is proposed for studying the in-plane behavior of regular masonry.'e rigid block model is based on an existing discrete/rigid model with rigid blocks and elastoplastic interfaces that already proven its effectiveness in representing masonry behavior in linear and nonlinear fields. In this case, the proposed model is improved by assuming rigid quadrilateral elements connected by one-dimensional nonlinear interfaces, which are adopted both to represent mortar (or dry) joints between the blocks and also to represent inner potential cracks into the blocks. Furthermore, the softening behavior of interfaces in tension and shear is taken into account. Several numerical tests are performed by considering masonry panels with regular texture subjected to compression and shear. Particular attention is given to the collapse mechanisms and the pushover curves obtained numerically and compared with existing numerical and laboratory results. Furthermore, the numerical tests aim to evaluate the applicability limits of the proposed model with respect to existing results.


Introduction
Masonry is a heterogeneous structural material made of natural or artificial blocks connected by dry or mortar joints. Due to the large amount of masonry historical buildings that may be found in Italy and in other south European countries and considering the frequent seismic events in such areas, the assessment of the behavior of masonry and the development of numerical models for masonry structural elements or buildings is an active field of research in Architecture and Civil Engineering.
Among the different numerical strategies that are frequently adopted for studying masonry and that may be found in literature [1][2][3], this work mainly focuses on a rigid block model, together with a finite element (FE) model based on a total rotating strain crack approach (FE-TRSCM) assumed as a comparison for some numerical tests. e first model may be considered a micro or local model, while the latter may be considered a macro or diffused model. A comparative analysis between the numerical models and a comparison with experimental results allow to perform a multiscale analysis of masonry behavior and to validate the new proposed model. e rigid block model adopted here is particularly suitable for studying historical masonry, since it considers blocks as rigid bodies and dry or mortar joints as nonlinear interfaces. It is based on the original numerical model introduced by Cecchi and Sab [16], which follows small displacements hypothesis. Hence, it considers independent elements in contact, but it does not update the contact topology during numerical tests, given that small displacements are assumed. is contribution was extended to the field of material nonlinearity adopting a tension strength criterion and a Mohr-Coulomb yield criterion for restraining interface actions [17]. Such updated model was still based on small displacement hypothesis, which can be considered reasonable for masonry even in case of nonlinear behavior. Furthermore, no contact detection algorithms were proposed, allowing to obtain fast numerical solutions by means of a stiffness matrix approach, which turned out to be an advantage with respect to more general DE models extended to masonry. e resulting nonlinear rigid model turned out to be similar to the rigid block models adopted for performing limit analysis of masonry [18][19][20][21][22]. However, the original nonlinear rigid block model was able to perform pushover tests in load control, but it was not able to reproduce the softening behavior typical of masonry, given that it considers an elastic-perfectly plastic interface behavior, which does not respect the actual response in tension and shear typical of mortar joints. However, this preliminary nonlinear rigid block model was calibrated for modeling inplane loaded masonry panels with dry joints [23][24][25], showing a good agreement between more refined DE models, namely the combined finite element-discrete element (FE-DE) model [26], and laboratory results.
In this contribution, the new aspect introduced for improving the original rigid block model regards the adoption of a softening law in tension and shear for representing the postelastic behavior of mortar joints and the potential cracking of the blocks. In this latter case, a new inner block joint type is introduced, by subdividing each block into two half blocks, in order to simulate the potential cracking of the resisting elements due to tension and shear.
is last aspect was already considered by other authors for better modeling masonry behavior in case of elastic blocks both in-and out-of-plane loaded [8,10,11] and also in case of rigid blocks [27] by performing limit analyses.
Focusing then on FE models for masonry, among the different models that may be found in literature, two main approaches may be considered: micro-and macro-modeling [28]. e first approach envisages the use of standard heterogeneous FE models, with joints and blocks modeled by different materials and specific FEs, or with joints often modeled by means of zero-thickness interface elements [8,29]. e second approach envisages the definition of a continuous material equivalent to the heterogeneous one, modeled by a unique type of FE. In this case, homogenization approaches may be adopted by identifying a periodic cell typical of the masonry material and by adopting local linear and nonlinear mechanical parameters [30][31][32][33], also by performing multiscale analyses [34]. In particular, multiscale approaches are focused on Cauchy or micropolar/nonlocal continuum models [35][36][37][38][39]. Another possibility in the field of continuous modeling is an approach based on the evaluation of the overall properties of a masonry assemblage by fitting experimental tests. Orthotropic and also isotropic damage models are often adopted for this purpose [40]. Focusing on damage models, discrete or smeared crack approaches may be adopted by extending existing models originally introduced for modeling concrete material. e continuous total strain crack model (TSCM) is a proper tool for modeling strengthened masonry structures [41][42][43] and homogenous isotropic materials as rammed earth masonry [44,45]. Moreover, it has also been adopted for modeling unreinforced masonry panels under in-plane loads [40,46,47]. An interesting comparison between several numerical finite element models for the assessment of the behavior of unreinforced masonry has been performed by Bartoli et al. [48], also considering the performance of the total strain crack model. is work is dedicated to the assessment of the nonlinear behavior of masonry panels with regular texture subjected to vertical and horizontal in-plane actions. e aim of this work is to validate the proposed improvements of an existing rigid block model by performing several numerical tests and comparing the results of the updated model with respect to existing laboratory results and other accurate numerical results. e calibration already performed by the authors for rigid, FE/DE, and FE-TRSCM models [49] is extended to the case of the updated rigid block model. Before performing tests on masonry walls, the nonlinear behavior of mortar joints is evaluated for first by performing some numerical tests on two blocks specimens. en, the experimental campaign by Raijmakers and Vermeltfoort [50] on almost square-shaped masonry panels is taken as reference, together with the work of Lourenço and Rots [8], that studied the same panels by means of FE models with interface elements. Such case studies have been recently considered by the authors in a comparison between continuous and discrete models focusing on nonlinear behavior [51] and by other authors in a similar comparison by means of a multiscale/ multimodel approach [15]. Here, the pioneering work of Page [29] is also taken into consideration as a further case study. e paper is organized as follows: the numerical models adopted are introduced, focusing on the updated rigid block model; then, the case studies considered are described, and the results of the numerical tests performed with the adopted models are showed and discussed, accounting also for laboratory results. Some comments and possible future developments are presented in the final part of the work.

Refined Rigid Block Model-Geometry.
A one-leaf masonry panel with regular texture and following the so-called "running bond" pattern is considered (Figure 1(a)). Block dimensions are width b, height a, and thickness s, whereas mortar joint thickness is e, which is assumed to be equal for both head (vertical) and bed (horizontal) joints. A refined rigid block model is defined by introducing rigid elements coincident with one half of each actual resisting element, hence having width b/2 and height a (Figure 1(b)). e resulting discretization is characterized by half blocks aligned both horizontally and vertically and in contact with four neighboring half blocks by means of four interfaces coincident with the four half block edges. Horizontal interfaces are always represented by mortar interfaces, whereas vertical interfaces can be, alternatively, mortar interfaces or inner block interfaces; the latter interface case may represent a potential vertical crack for the original block, which can be obtained in case of a compressive action on the masonry structure without or with a contemporary shear action.
Assuming a two-dimensional coordinate system Oy 1 y 2 , a plane stress state, and considering the hypothesis of rigid elements and small displacements, the in-plane displacement of each half block B i,j is a rigid body motion defined by half block center in-plane translations u ij = {u 1 u 2 } T and half block in-plane rotation ω 3,ij with respect to its center: where Ω ij = skw (ω 3,ij ). Nodal degrees of freedom of a generic element can be collected in a vector q ij � u 1 u 2 ω 3 T ij . e interactions between the half blocks through the interfaces are represented by the resultants of the corresponding contact stresses, which are given by a normal force F n , a shear force F s and a bending moment M: f loc = {F n F s M} T , which are defined considering the local orientation of the interface. e corresponding interface actions in the global coordinate system are f = {F 1 F 2 M} T = R f loc , where R is the rotation matrix of the interface depending on its horizontal or vertical orientation. Such interactions depend on the relative displacements and rotations between adjacent half blocks, defined as d = {d 1 d 2 δ} T , which can be written in the form of local relative displacements d loc = {d n d s δ} T depending on interface orientation, d = {d 1 d 2 δ} T = R d loc .
ese relative displacements can also be written as a function of half block global displacements in vector form by introducing a compatibility matrix H detailed for each block and collecting the distances between each half block center and interface center: with where k 1 = 1, k 2 = 0 for a horizontal interface, and k 1 = 0, k 2 = 1 for a vertical one. It is worth noting that the compatibility matrix H assembled over the entire masonry specimen considered turns out to be coincident with the matrix commonly adopted for performing limit analysis of masonry [20].

Refined Rigid Block Model-Linear and Nonlinear
Behavior. In the elastic field, the constitutive relation that defines the interface actions between adjacent elements is f loc = K loc d loc , where K loc is the local stiffness matrix of the interface collecting the stiffness parameters depending on interface type and geometric dimensions (area A and moment of inertia I). For instance, K loc = {k n k s k r } T = {K n A K s A K n I} T , where K n and K s are, respectively, interface normal and shear stiffness, that in case of mortar joints are defined as the function of mortar elastic modulus E m and Poisson's ratio ] m , together with mortar joint thickness e: Figure 1: Refined rigid block model for regular masonry.

Advances in Civil Engineering
With respect to the original rigid block model proposed by authors [16,17], this work introduces the elastic stiffness parameters of inner block joints, which are determined in the same manner of mortar joints parameters, but they are assumed to be several times larger than those of mortar joints in order to simulate the extremely limited deformability of a block.
It is worth noting that the proposed rigid block model allows to define the elastic stiffness of a masonry assemblage, by assembling the local stiffness matrices over the entire structural element considered. e elastic stiffness matrix of the entire assemblage is not updated during the following numerical tests, namely by adopting a modified Newton-Raphson method based on initial stiffness for studying the nonlinear behavior of masonry specimens, in order to avoid, at each step of the test and for each interface, the evaluation of the secant stiffness or the tangential stiffness, which may lead to a local or global singular stiffness matrix.
e nonlinear behavior of the model is taken into account by adopting a nonlinear part of the constitutive laws after the initial elastic behavior. A tensile strength and a Mohr-Coulomb yield criterion are assumed for defining the elastic limits of tension and shear actions, respectively; whereas a maximum eccentricity criterion is assumed for the bending moment by introducing l as one half of the interface length: Differently, with respect to the work of Lourenço and Rots [8], in this contribution, an elastic-perfectly plastic behavior in compression is assumed for both mortar and inner block interfaces, by introducing a limit in compression F c � −f c A. ese criterions hold for both mortar joints and inner block joints and are based on the corresponding material properties represented by tensile strength f t , compressive strength f c , cohesion c, and friction ratio μ.
Furthermore, in this work, the softening behavior in case of tensile and shear failure, typical of brittle materials, is assumed by introducing the fracture energy G I for the first (tension) and G II for the second (sliding or shear) modes of cracking.
Starting from the elastic limits of Equation (5), the corresponding relative displacements are given by and the softening behavior is defined as follows if the relative normal and shear displacements are larger than the corresponding elastic limits: e above expressions are characterized by an exponential coefficient that tends to zero for increasing relative displacement in normal and tangential direction. For simplicity, the softening behavior in case of bending moment is not directly introduced as a function of relative rotation, given that the limit in terms of bending moment already accounts for the updated normal force of the interface.
In the following numerical tests, the elastic and inelastic parameters for mortar joints and inner block joints will be assumed coincident with those adopted by other numerical tests assumed as reference.

FE-TSRCM.
e FE-TRSCM is a macromodeling approach originally introduced for concrete structures. e model can be extended to masonry by assuming a heterogeneous material as a continuous one, with bricks and mortar considered together as a single homogenous material through a total strain rotating crack model approach [52]. Recently [49], the FE-TRSCM has been adopted by authors for modeling in-plane loaded masonry and compared with a simple rigid block model and a combined FE-DE model. Such a model has been also compared by other authors with other numerical tools for assessing the seismic behavior of unreinforced masonry [48]. e continuous material follows a stress-strain law in compression and tension, where the mechanical properties are assigned in agreement with the information from the experimental test, and they are not determined by means of a proper homogenization procedure. An elastic modulus E and a Poisson's ratio ] are assumed for the continuous material, and a damaged plasticity model is adopted for representing the material nonlinear behavior, based on the incremental plasticity theory and on the concept of isotropic damage elasticity to describe the irreversible damage that occurs during the fracturing process. In particular, incremental plasticity relates plastic stress increments with plastic strain increments by means of a constitutive matrix that adopts a softening postcrack parameter and a shear retention factor. Two failure mechanisms are assumed: cracking in tension and crushing in compression. An exponential softening law and a parabolic law are assumed, respectively, in tension and compression. Both constitutive laws are defined by limit stresses (f t in tension and f c in compression) and by the corresponding fracture energies (G t and G c ), which are considered in terms of dimensionless values with respect to finite element size h [53]: G t � G I /h and G c � G C /h. In particular, G C is obtained by integrating the ellipsoid law for the cap model in 4 Advances in Civil Engineering compression adopted in the study by Lourenco and Rots [8], and h is defined as a function of the area (or the volume) of the finite element adopted in the numerical model. In the following numerical tests, a commercial computer code [54] is adopted, and quadrilateral finite elements in the plane state are assumed, with a regular mesh of almost squareshaped elements is defined; hence, h is assumed to be coincident to the height or the width of the generic finite element.

Introduction to the Tests.
Several numerical tests are performed in order to evaluate the effectiveness of the proposed model for correctly simulating the behavior of masonry panels subject to compression and shear.

Calibration of Mortar Interface Nonlinear Parameters.
e nonlinear behavior of the interfaces in the rigid block model is calibrated for first by performing a set of simple numerical tests involving a few blocks connected by mortar interfaces and subjected to simple tensile and shear actions.
For this purpose, the simple tension and shear tests on specimens made of two blocks (Figure 2(a) and 2(b), respectively), having dimensions b = 250 mm, a = 50 mm, and s = 120 mm, similar but not coincident to those of the work assumed as reference [8], are taken into consideration and reproduced by the proposed refined rigid block model. For simplicity, the mechanical parameters adopted for the shear tests, namely c = 0.87 MPa, μ = 0.73, and G II = 0.058-0.13 σ N/mm, are extended to the tension test by setting f t = c/ 1.4 = 0.62 MPa and G I = 0.012 N/mm. Such mechanical parameters have been determined by Van der Pluijm [55,56] by means of laboratory experimental tests in tension and shear.
Results in terms of normal stresses given by interface normal force F n over interface area A for increasing interface relative normal displacement d n are shown in Figure 3(a), with a softening branch after reaching the tensile strength set as input of the model. Similarly, results in terms of interface shear stresses given by interface shear force F s over interface area A for increasing interface relative tangential displacement d s are shown in Figure 3(b) for three different levels of normal compression. Such a compression level increases the shear strength of the mortar interface and turns out to be close to the residual shear strength attained at the end of each test, which is equal to the contribution given by friction.

Panels Subjected to Compression and Shear.
e case studies considered for the numerical tests are given by the laboratory tests on rectangular masonry panels performed by Raijmakers and Vermeltfoort [50]. e geometric characteristics of the masonry panels are as follows: blocks dimensions are width b � 204 mm, height a � 50 mm, and thickness s � 98 mm. Mortar joint thickness is e � 10 mm. e overall dimensions of the specimen are length L � 990 mm, height H � 1000 mm, and thickness s � 98 mm. ese dimensions are obtained with 4.5 blocks in horizontal direction and 18 blocks in vertical direction (Figure 4). A steel beam at panel base is used for fixing block displacements and rotation, and a similar steel beam at panel top is used for avoiding relative vertical block displacements and for applying different levels of vertical compression (p � −0.30; −1.21;−2.12 MPa) and an increasing horizontal force F. e main mechanical parameters adopted for mortar interfaces and inner block interfaces are listed in Table 1 and were determined by performing laboratory tests on blocks, mortar, and mortar joints. In particular, mortar interface stiffness parameters are obtained starting with a material having E m � 800 MPa, E b � 4672 MPa, and ] m � 0.14, and their nonlinear behavior in shear is characterized by cohesion c � 1.4 f t and friction ratio μ � 0.75, whereas inner block interfaces nonlinear behavior is considered only for tensile and bending failure. e FE-TRSCM adopts E � 3960 MPa, ] � 0.14, f c � 8.8 MPa, and G C � 2 N/mm. e numerical results obtained with the proposed refined rigid block model turn out to be in quite good agreement with experimental results. Focusing on the loaddisplacement curves (Figure 5), the initial stiffness of the panels is correctly reproduced by the numerical model in the three different compression cases. e level of initial failure of the panels is correctly determined with p � 1.21 MPa, whereas it is slightly underestimated with p � 0.30 MPa and overestimated with p � 2.12 MPa. e residual strength of the panels is correctly determined only with p � 0.30 MPa, probably thanks to the small level of compression of the case study. Figure 6 shows the deformed configurations (left column, Figures 6(a) Increasing the level of compression, the interface damage increases and involves also vertical inner block interfaces, which are mainly characterized by bending failure (yellow color in Figures 6(d)-6(f )). e deformed configurations reached at d � 2 mm with p � 1.21 and 2.12 MPa (Figures 6(b) and 6(c)) are similar, but a slightly large vertical deformation can be observed with 2.12 MPa. In these numerical simulations, the compressive strength of mortar joints is never reached; hence, the corresponding representation is not highlighted in the legend of Figure 6.   Advances in Civil Engineering    Advances in Civil Engineering right corner of the panels. However, even if load-displacement curves obtained with the FE-TSRCM are very close to the actual behavior of the panels, the deformed configurations are not able to highlight the local damage of mortar joints or blocks. A further comparison between the results given by the proposed refined rigid block model with respect to the original one is briefly described in appendix in terms of loaddisplacement curves. For further details on the results given by the standard rigid block model, the recent contribution by authors [49] can be taken into consideration.

Masonry Deep Beam Subjected to a Vertical Force.
In order to further evaluate the proposed refined rigid block model, the "Page test" [29] is simulated. Such test is one of the most used in the calibration of numerical models for masonry panels [1,37,57], and it was used by authors for an initial calibration of the original nonlinear discrete/rigid block model [17]. e specimen is made of pressed clay blocks having dimensions b = 122 mm, a = 34 mm, and s = 37 mm, with mortar joint thickness e = 5 mm; overall panel dimensions are L = 757 mm and H = 457 mm. e panel is supported at each side of the base, and it is loaded by a compressive symmetric force P applied by a stiff steel beam long 381 mm. Load and restraint conditions are shown in Figure 8. Brick and mortar linear and nonlinear mechanical parameters are presented in Table 2 where several data are taken from the work of Lourenço [1]. e FE-TRSCM adopts E = 4125 MPa, ] = 0.15, f c = 13 MPa, and G C = 1 N/mm. Figure 9 shows the load-displacement curve obtained with the rigid block and continuous numerical models. e refined rigid block model reaches an ultimate force close to 122.2 kN that turns out to be in quite good agreement with       experimental result equal to 109.2 kN, whereas the FE-TSRCM is in excellent agreement with such results, since it reaches an ultimate force close to 108.2 kN. In this case, the continuous model shows a larger deformability with respect to the rigid one, and the pushover curves are characterized by the same initial stiffness up to 20 kN; then the FE-TSRCM starts to exhibit damage in compression, which causes a stiffness reduction of the model. Figures 10 and 11(a) show, respectively, the deformed configuration of the deep beam and the damage map at P � 100 kN, which correspond to the last load increment considered by Page. e deformed shape of the panel is in agreement with the one proposed by Lourenço [1], and the damage of the panel is characterized by the shear, bending, and tension failure of several vertical joints, in quite good agreement with the numerical results of Page [29] and Lourenço [1]. Figure 11(b) shows the damage map at P � 120 kN, which is characterized by the compression failure of several bed joints (blue color) and the shear failure of two inner block joints (red color, dashed line) close to the upper steel beam, together with the further tension and shear failure of some vertical mortar joints. Figure 12 shows the deformed configuration and the map of normal stresses in y 2 direction obtained with the FE-TSRCM at P � 100 kN. e deformed configuration is in quite good agreement with that obtained with the rigid model, even if the continuous model shows a more diffused deformability along the entire specimen. e map of normal stresses clearly shows the load transfer from the upper steel beam and the lower supports.

Conclusions
In this work, a refined rigid block model has been proposed for studying the in-plane behavior of masonry with regular texture.
e refined rigid block model is based on an existing model with rigid blocks and elastoplastic interfaces that already demonstrated its effectiveness in representing masonry behavior in linear and nonlinear fields, and it was compared with other commercial DE models and laboratory results. However, the original elastic-perfectly plastic law adopted for restraining interface actions of dry or mortar joints was not sufficiently in agreement with actual material behavior; moreover, the model was not able to represent the possible block cracking. For these reasons, the proposed model improves the original one by assuming rigid quadrilateral elements connected by one-dimensional nonlinear interfaces, which are adopted both to represent mortar (or dry) joints between the blocks and also to represent inner potential cracks into the blocks. Furthermore, the softening behavior of interfaces in tension and shear is taken into account. e computational effort required for performing numerical tests with the updated rigid block model is still smaller than that typical of commercial DE and FE models, since the degrees of freedom of the rigid block model are lumped at block centers. Several numerical tests are performed for first by simulating the tension and shear cracking of two blocks connected by a mortar joints and then by considering masonry panels with regular texture subjected to compression and shear. Particular attention is given to the collapse mechanisms and the pushover curves obtained numerically and compared with existing numerical and laboratory results. e numerical simulation of the three shear tests with different compression levels originally performed by Raijmakers and Vermeltfoort [50] turned out to be in quite good agreement with the laboratory test and other numerical results obtained with a FE-TRSCM approach. Furthermore, the numerical simulation of the deep beam test originally performed by Page [29] showed a good agreement with laboratory results both in terms of ultimate load and crack pattern.
Further developments of this work will focus on a more accurate simulation of the behavior of joints in compression, by taking into account, for instance, of the softening behavior typical of the compressed mortar joints, in order to better reproduce the postpeak branch of the pushover tests.
e proposed rigid block model will be further validated and calibrated by simulating other laboratory tests and more complex case studies, such as masonry walls and façades with openings.