In this work, a refined rigid block model is proposed for studying the in-plane behavior of regular masonry. The 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.

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 [

Rigid block models are based on several hypotheses that are typical of the wide field of discrete approaches. In this field, the discrete element (DE) model or method is the most common tool adopted in solid and structural mechanics. In general, DE models are characterized by the definition of elements and contacts between the elements. Following the original definition of the method [

In literature, numerous discrete models based on elastic [

The 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 [

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. This 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 [

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 [

This work is dedicated to the assessment of the nonlinear behavior of masonry panels with regular texture subjected to vertical and horizontal in-plane actions. The 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. The calibration already performed by the authors for rigid, FE/DE, and FE-TRSCM models [

The 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.

A one-leaf masonry panel with regular texture and following the so-called “running bond” pattern is considered (Figure

Refined rigid block model for regular masonry.

Assuming a two-dimensional coordinate system _{1}_{2}, a plane stress state, and considering the hypothesis of rigid elements and small displacements, the in-plane displacement of each half block _{i,j} is a rigid body motion defined by half block center in-plane translations _{ij} = {_{1}_{2}}^{T} and half block in-plane rotation _{3,ij} with respect to its center:_{ij} = skw (_{3,ij}). Nodal degrees of freedom of a generic element can be collected in a vector

The 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 _{n}, a shear force _{s} and a bending moment _{loc} = {_{n}_{s} ^{T}, which are defined considering the local orientation of the interface. The corresponding interface actions in the global coordinate system are _{1}_{2} ^{T} = _{loc}, where _{1}_{2}^{T}, which can be written in the form of local relative displacements _{loc} = {_{n}_{s}^{T} depending on interface orientation, _{1}_{2}^{T} = _{loc}. These relative displacements can also be written as a function of half block global displacements in vector form by introducing a compatibility matrix _{1} = 1, _{2} = 0 for a horizontal interface, and _{1} = 0, _{2} = 1 for a vertical one. It is worth noting that the compatibility matrix

In the elastic field, the constitutive relation that defines the interface actions between adjacent elements is _{loc} = _{loc}_{loc}, where _{loc} is the local stiffness matrix of the interface collecting the stiffness parameters depending on interface type and geometric dimensions (area _{loc} = {_{n}_{s}_{r}}^{T} = {_{n}_{s}_{n}^{T}, where _{n} and _{s} are, respectively, interface normal and shear stiffness, that in case of mortar joints are defined as the function of mortar elastic modulus _{m} and Poisson’s ratio _{m}, together with mortar joint thickness

With respect to the original rigid block model proposed by authors [

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. The 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.

The 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

Differently, with respect to the work of Lourenço and Rots [_{c} = −_{c}

These criterions hold for both mortar joints and inner block joints and are based on the corresponding material properties represented by tensile strength _{t}, compressive strength _{c}, cohesion

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 _{I} for the first (tension) and _{II} for the second (sliding or shear) modes of cracking.

Starting from the elastic limits of Equation (

The 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.

The FE-TRSCM is a macromodeling approach originally introduced for concrete structures. The 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 [_{t} in tension and _{c} in compression) and by the corresponding fracture energies (_{t} and _{c}), which are considered in terms of dimensionless values with respect to finite element size _{t} = _{I}/_{c} = _{C}/_{C} is obtained by integrating the ellipsoid law for the cap model in compression adopted in the study by Lourenco and Rots [

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.

The 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 _{II} = 0.058–0.13 _{t} = _{I} = 0.012 N/mm. Such mechanical parameters have been determined by Van der Pluijm [

Tension (a, c) and shear (b, d) tests on two blocks specimens. Deformed configurations (a, b) and interface damage (c, d).

Results in terms of normal stresses given by interface normal force _{n} over interface area _{n} are shown in Figure _{s} over interface area _{s} are shown in Figure

Load-relative displacement curves for a) tension test, b) shear tests with three levels of compression, on two blocks connected by a mortar joint.

The case studies considered for the numerical tests are given by the laboratory tests on rectangular masonry panels performed by Raijmakers and Vermeltfoort [

Masonry panel considered for the numerical tests [

The main mechanical parameters adopted for mortar interfaces and inner block interfaces are listed in Table _{m} = 800 MPa, _{b} = 4672 MPa, and _{m} = 0.14, and their nonlinear behavior in shear is characterized by cohesion _{t} and friction ratio _{c} = 8.8 MPa, and _{C} = 2 N/mm.

Mechanical properties of mortar joints and inner block joints of the case studies considered.

Interface | _{n} (N/mm^{3}) | _{s} (N/mm^{3}) | _{t} (MPa) | _{c} (MPa) | _{I} (N/mm) | _{II} (N/mm) |

Mortar 0.30 MPa | 82 | 36 | 0.25 | 10.5 | 0.018 | 0.125 |

Mortar 1.21 MPa | 110 | 50 | 0.16 | 11.5 | 0.018 | 0.050 |

Mortar 2.12 MPa | 82 | 36 | 0.16 | 11.5 | 0.018 | 0.050 |

Inner block | 10^{6} | 10^{6} | 2.00 | 50.0 | 0.080 | — |

The numerical results obtained with the proposed refined rigid block model turn out to be in quite good agreement with experimental results. Focusing on the load-displacement curves (Figure

Pushover curves of the numerical shear tests with varying compression. Continuous line for refined rigid block model, dotted line for FE-TSRCM, and dashed line for laboratory test results.

Figure

Deformed configurations (a, b, and c) and interface damage (d, e, and f); continuous lines, mortar interfaces and dashed lines, inner block interfaces) at

Figure

Deformed configuration and shear stresses at

The comparison between the numerical results of the proposed refined rigid block model with respect to laboratory tests and numerical results obtained with FE-TSRCM shows that the behavior of mortar interfaces subjected to compression has to be taken into account into the rigid model, given that the hypothesis of elastic behavior and unlimited strength in compression assumed in the rigid model does not allow to correctly reproduce the softening branch of the pushover tests, together with the compressive failure of some mortar joints close to the top left and bottom 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 load-displacement curves. For further details on the results given by the standard rigid block model, the recent contribution by authors [

In order to further evaluate the proposed refined rigid block model, the “Page test” [_{c} = 13 MPa, and _{C} = 1 N/mm.

Deep beam test configuration.

Mechanical properties of mortar joints and inner block joints of the masonry deep beam test.

Interface | _{n} (N/mm^{3}) | _{s} (N/mm^{3}) | _{t} (MPa) | _{c} (MPa) | _{I} (N/mm) | _{II} (N/mm) |

Mortar | 165 | 70 | 0.29 | 8.60 | 0.018 | 0.050 |

Inner block | 10^{6} | 10^{6} | 2.00 | 50.0 | 0.080 | — |

Figure

Pushover curves of the numerical masonry deep beam tests. Continuous line for refined rigid block model, dotted line for FE-TSRCM, and dashed line for laboratory test results.

Figures

Deformed configuration close to the end of the numerical deep beam tests (

Interface damage close to the end of the numerical deep beam test: (a)

Figure _{2} direction obtained with the FE-TSRCM at

Deformed configuration and vertical normal stresses at

In this work, a refined rigid block model has been proposed for studying the in-plane behavior of masonry with regular texture. The 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. The 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. The numerical simulation of the three shear tests with different compression levels originally performed by Raijmakers and Vermeltfoort [

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. The 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.

The refined rigid block model proposed in this work can be used as the original one by simply avoiding the deformability and potential damage of the inner block interfaces. However, the original rigid block model is characterized by a smaller number of degrees of freedom involved in the numerical tests, namely one half of the overall number of degrees of freedom, with respect to the refined rigid block model. A comparison between the load-displacement curves and the damage maps obtained with the refined and standard rigid block models is showed in the following figure by performing the first three pushover tests considered in Section

As it has already been highlighted in the introduction and in Section

Pushover curves of the numerical shear tests with varying compression. Continuous line for refined rigid block model, dotted line for original rigid block model, and dashed line for laboratory test results.

The data used to support this study are available from the corresponding authors upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was financed by the Research Project PRIN 2017 (grant 2017HFPKZY_002), project “Modelling of constitutive laws for traditional and innovative building materials.”