Edited by: Georgios Eleftherios Stavroulakis, Technical University of Crete, Greece
Reviewed by: Antonio Maria D'Altri, University of Bologna, Italy; Georgios A. Drosopoulos, University of KwaZulu-Natal, South Africa
This article was submitted to Computational Methods in Structural Engineering, a section of the journal Frontiers in Built Environment
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Numerical modeling of masonry structures is nowadays still an active research field, and this is partly due to a number of open issues related to preservation and restoration of historical constructions and the availability of computational tools that have become more and more refined. This work focuses on the analysis of settlement-induced failure patterns characterizing the in-plane response of two-dimensional dry-joints masonry panels, which differ in terms of texture, geometry, and settlement configuration. Brick-block masonry, interpreted as a jointed assembly of prismatic particles in dry contact, can be modeled as a discrete system of rigid blocks interacting through contact surfaces with no tensile strength and finite friction, modeled as zero thickness elasto-plastic Mohr-Coulomb interfaces. Different approaches and numerical models have been adopted herein: Limit Analysis (LA), a discrete model DEM, and a continuous Finite Element Model (FEM). Limit Analysis is able to provide fast and reliable results in terms of collapse multiplier and relative kinematics. In this work, a standard LA procedure was coded through Linearized Mathematical Programming to take into account sliding mechanisms through dilatant joints. Discrete models are particularly suitable to study historical masonry materials, where rigid bodies interact between contact and friction. Here, a combined Finite/Discrete Element approach (FEM/DEM) is adopted. Finally, analyses are conducted through the Finite Element approach, resorting to a continuum anisotropic elastic perfectly plastic constitutive model. Some selected case studies have been investigated and found to have adopted the above mentioned models, and numerical results have been interpreted to highlight the capability of the approaches to predict failure patterns for various geometrical features of the structure and settlement configurations.
Masonry is one of the most ancient structural materials and constitutes a vast majority of the World's architectural heritage. It is a composite and heterogeneous medium, resulting from the assemblage of natural or artificial blocks by means of mortar layers or dry joints. Being characterized by an internal structure, which is reflected in a complex mechanical response, masonry and its constitutive behavior still represent a challenging research field. Thanks to the availability of more and more powerful computational resources, over the last decades, a large number of numerical applications have been developed, many of which have resorted to using different constitutive assumptions and solution algorithms. Nonetheless, it is not possible to argue that each of these models suit just any structural problem their applicability needs to be evaluated case by case on the basis of geometrical features, the extent of the structure, and the boundary conditions. Among the available numerical modeling techniques for masonry structures, a broad distinction can be made between micromechanical, micromechanical, and multiscale models defined as follows. A significant classification can be found in D'Altri et al. (
According to micromodeling strategy, the constituents, namely, the units, mortar (if present), and unit/mortar interfaces, are separately modeled, and each part is assigned a properly calibrated constitutive law. This approach is particularly suitable if the response of the assemblage needs to be accurately described (Lotfi and Shing,
Following a macromechanical approach, the heterogeneous medium is modeled as a continuum, and the constitutive behavior is usually described through phenomenologically based mathematical relations in which degrading phenomena are the product of damage or friction variables. In this case, macroscopic mechanical properties are more easily derived from standard experimental tests on small masonry specimens. These models are, if compared to micromechanical ones, more efficient from a computational point of view (Del Piero,
Multiscale, i.e., micro-macro, continuum models represent a very promising approach for the analysis of masonry structures since they can accurately keep track of the mechanical and geometrical properties of the material at the microstructure with a reduced computational cost if compared to a fully micromechanical model (Masiani and Trovalusci,
Due to their characteristics, masonry constructions have proven to be particularly vulnerable not only to earthquakes but also to structural settlements. Cracking and damage, in fact, often occur as a consequence of ground movements. In urban areas, this phenomenon is related to the realization of underground infrastructures or, more generally, to anthropic triggering factors, while natural hazards (e.g., slow-moving landslides, liquefaction, or consolidation processes) are more likely to interfere with masonry constructions in rural areas. In both cases, the understanding of the phenomenon and its description are fundamental to identifying the causes as well as preventing the effects with appropriate protection measures, both for modern and historical constructions. Several approaches to the prediction of settlement-induced damage that consider masonry either as an assembly of discrete blocks (DeJong,
In this work, three modeling techniques have been adopted to describe settlement induced crack patterns in masonry panels characterized by different geometrical configurations and boundary conditions, reproducing the ground movement as a downward moving rigid block. Equations governing systems of rigid blocks interacting though no tension and frictional interfaces formally correspond to those of perfect plastic systems with non-associative flow rule (Fichera,
The results of the analyses, performed on panels, walls with openings, and facades, are compared with the ones obtained through an FEM/DEM approach (Baraldi et al.,
The first adopted model is framed within the Limit Analysis (LA) theory, taking into account the presence of friction. The model considers as a system of
In order to provide the mechanical details of the model, consider two simple blocks represented in
The static variables are the internal forces acting at each
The kinematic variables, or generalized strain, are the relative displacement rates at joints: normal displacement ξ
The kinematic compatibility for the whole system is expressed by equation
where
The equilibrium of the whole structure is defined by the equation
The generalized yield domain of the system can be written as
where
The flow rule expresses the vector
The plastic behavior of contact surfaces is defined through the complementarity condition
Furthermore, the non-negative work of the live loads which cause the collapse mechanism is defined by the following equation
In Baggio and Trovalusci (
As reported in Pepe et al. (
Here, the optimization problem presented in Pepe et al. (
assumed in a compact form as
where, for the whole structure, the matrix
where introducing
The equation (
Following the idea of Portioli and Cascini (
Another difference introduced into the model concerns the definition of the loads applied to the blocks. Indeed, in that case, every block of the structure is subjected only to its dead load, while live load is applied only on the block that simulates the settlement. In detail, for that block, dead load is an upward force assumed to be proportional to an admissible base reaction without foundation settlement, considering a uniform distribution of vertical loads in the structure, and it is denoted as
It is consequently possible to particularize the movement of the support block
The equation
A discrete model, made by means of a combined Finite-Discrete Element Model (FEM/DEM) approach, has been adopted here.
Discrete Element Models (DEM) (Cundall,
In order to describe the deformability of the elements, simple FE discretizations have been proposed since the beginning of the development of DE Method (Cundall et al.,
The FEM/DEM approach adopted here has been successfully adopted for masonry structures by some of the authors (Reccia et al.,
Numerical analyses are performed through the open source computer codes Y2D/Y-GUI (Mahabadi et al.,
The specimens were discretized through a triangular CST FEs mesh under plane stress hypotheses. As rigid and cracking can only occur in the joints, masonry units were modeled as zero-thickness interfaces based on a Mohr-Coulomb strength criterion with no cohesion holds. Since the aim of the work was to compare the results of different models with those obtained with LA, which implicitly considers the blocks non-deformable, in order to avoid cracks inside the bricks, the FEM/DEM model units were characterized by a very high value of Young's Modulus.
The third approach adopted to analyze failure mechanisms characterizing the response to settlements of masonry constructions is a continuum finite element (FE) one. The constitutive model, implemented in the FE code
JMM bed and head joint plane orientation.
In the proposed examples, only two planes (head and bed joints) are activated, while a third plane might be considered in the case of walls with double facing. Yield functions are defined, for each orientation, in terms of local stress components according to Coulomb's and tensile criterion:
where i = 1,2, and 3 is the plane
The tensile strength and cohesion on the head joints are thus
and the modified strength criterion is reported in
Modified Mohr-Coulomb criterion.
In the analyses performed the following model parameters have been assumed: all the tests are characterized by a unit weight γ = 12 kN/m3, σ
The models have been validated using, as a benchmark, the experimental and numerical results obtained by Portioli and Cascini (
After this first validation, some geometries of masonry walls with openings or façades were analyzed under settlement conditions involving a portion of the structure.
Two identical dry joints masonry walls, named Panel 1 and Panel 2, both characterized by the presence of an opening, have been considered. The panels are constituted by 55 blocks of the dimensions 50 × 100 × 50 mm3, and they are 600 mm wide, 50 mm thick, and 550 mm high; the opening dimensions are 300 × 250 mm2. They differ in terms of length of the settling area, namely, 200 × 50 mm2 for Panel 1 and 300 × 50 mm2 for Panel 2; the effect of this feature is here investigated with the three approaches.
In the following analyses, the effect of settlement configuration is investigated together with the influence of wall height and presence of openings. Three slender panels, characterized by different height and number of openings, named Panel 3, Panel 4, and Panel 5, have been studied. The dimensions of the blocks are 50 × 100 × 50 mm3 for all the specimens, and the openings are 100 × 250 mm2 wide. Panel 3 and Panel 4 are both made of 73 blocks and have the same dimensions: 400 mm width, 50 mm thickness, and 900 mm height, but they differ because of the extension of the portion involved in the settlement. The fictitious downward moving block has dimension of 100 × 150 × 50 mm3 for Panel 3 and dimension 100 × 250 × 50 mm3 for Panel 4. Panel 5 is taller and characterized by the presence of three openings. It is 400 mm wide, 100 mm thick, and 1,300 mm high and is made of 105 blocks. The dimensions of the fictitious block are the same adopted in the Panel 4 case.
For Panel 3 and Panel 4, as expected, results show that the collapse mechanism is affected by the extent of the settling portion of the structure.
The collapse mechanism of two-story and three-story masonry façades characterized by different geometries has been investigated. Façade 1 and Façade 2 have the same geometry and are characterized by the presence of four openings, two doors at the ground level, and two windows at the first floor which have the same dimensions (200 × 350 mm2). Each façade is 1,500 mm wide, 50 mm thick, and 1,200 mm high, and they are made up of 320 blocks with dimensions 50 × 100 × 50 mm3. The dimension of the fictitious block is 100 × 300 × 50 mm3 for both the analyzed cases. Façade 1 is affected by a lateral settlement, while Façade 2 is affected by a central settlement.
In conclusion, a three-story structure, named Façade 3, shown in
This work presents the comparison of failure patterns characterizing the response of dry joints masonry walls subjected to settlements. Three numerical formulations have been adopted: a Limit Analysis as well as an FEM/DEM and FEM approach. The models have been briefly described and gone through preliminary validation, referring to experimental and numerical literature results. A comparative study has then been performed, varying the main factors affecting the response of masonry structures in the case of settlements, namely, wall dimensions, the presence of openings, and the extension of the settling area. All the models have proven to be very efficient from a computational point of view and able to reproduce the collapse mechanisms, and they can thus be considered a useful tool with which to back-analyze real-scale problems in order to identify the causes of observed crack patterns or to predict the damage distribution when a settlement is expected to occur, as in the case of underground excavation or in case of natural triggering factors. The adoption of micro-models, i.e., Limit Analysis and FEM/DEM, implies that the structure is described with its real texture, taking into account block dimensions and the
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.
MPe and MPi developed a novel feature of ALMA 2.0 that takes into account the effect of foundation settlement and performed the analysis with the LA approach. ER and MS carried out the analysis with the FEM/DEM and FEM approach, respectively. All authors equally contributed to the common sections and took care of the description of the adopted models. PT and GF supervised the whole work during the entire process.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.