Elsevier

Computers & Structures

Volume 86, Issues 21–22, November 2008, Pages 1958-1973
Computers & Structures

Numerical analysis of structural masonry: mesoscale approach

https://doi.org/10.1016/j.compstruc.2008.05.007Get rights and content

Abstract

This paper is focused on modeling of progressive failure in brick masonry. A mesoscale approach is adopted, in which the structural behavior is examined at the level of constituents, i.e., brick and mortar. An advanced constitutive model, capable of addressing both pre and post-localization behavior, is developed and implemented in a commercial finite element package. The performance of this model is verified by simulating a series of experimental tests reported in the literature. Those include the tests conducted by van der Pluijm [Pluijm R van der. Shear behavior of bed joints. In: Hamid AA, Harris HG, editors. Proceedings of the 6th North American Masonry Conference, Drexel University, Philadelphia, PA, USA; 1993. p. 125–36], Atkinson et al. [Atkinson RH, Amadei BP, Saeb S, Sture S. Response of masonry bed joints in direct shear. J Struct Eng ASCE 1989;115(9):2276–96], and Page [Page AW. The strength of brick masonry under biaxial tension–compression. Int J Masonry Constr 1983;3(1):26-31]. In latter case, the whole masonry panel is discretized and the directional strength characteristics for different loading scenarios are obtained. Later, a numerical homogenization is implemented to study the same problem and the results based on these two approaches are compared.

Introduction

Over the last few decades the numerical analysis of masonry structures has become the main focus of many research programs, leading to a significantly better understanding of the complex behavior of this composite material. This has enabled engineers to develop more rational methods for practical analysis and design while altering the traditional perception of masonry as a decorative material. In the context of numerical modeling, the research has focused mainly on two distinct approaches, that is, at meso- and macroscale. Mesomodels incorporate the properties of the constituents, for example, brick and mortar, as well as the details of the architectural arrangement and therefore can be used to study the interaction of the constituents and the damage propagation pattern under different loading histories. Also, they can be effectively implemented to assess the average mechanical properties of masonry, which can later be used in the formulation of macromodels [38], [2], [3], [46], [22].

The finite element analysis at the mesoscopic level requires, in general, the discretization of both the units and the mortar joints. Such an approach gives the most detailed information about the structural behavior; however, given the enormity of the models emerging in real engineering applications, it is not feasible to implement it for large-scale structures. Simplified approaches are derived based on the observation that, for most loading histories, the cracks initiate in the joints while the blocks remain intact. Considering this fact, only the masonry units are discretized using continuum elements, whereas the joints are modeled as weakness planes by employing interface elements [26], [16], [13]. The main limitation of this approach is the fact that the interaction of the joints with the brick units cannot be properly captured. In fact, due to the noticeable differences between the mechanical properties of bricks and mortar joints, significant lateral stresses will develop in the regions adjacent to the joints, which cannot be described in this formulation. This approach has also difficulties in capturing the out-of-plane effects, which can be of great importance in biaxial compressive loading. Moreover, the size of the generated models for real masonry structures is still extremely large, so that this methodology cannot be considered as a pragmatic approach for real engineering applications.

The analysis of large masonry structures should best be conducted at a macro-level. In this case, the masonry can be described as a continuum whose average properties are identified at the level of constituents taking into account their geometric arrangement. Over the last few decades, a number of different approximations have been developed for assessing the average properties of structural masonry. Those include the micropolar Cosserat continuum models as well as the estimates based on the theory of homogenization for periodic media.

In micropolar continuum approach [41], [21], the Cosserat theory of elasticity is employed. In this case, the equations of traditional elasticity are enriched by incorporating additional degrees of freedom related to the rotation at each point. As a consequence, couple stress components (torque per unit area) are added to the stress tensor which becomes non-symmetric. In view of this, some additional material parameters appear in the constitutive relation, which is believed to provide a more accurate representation of the material behavior. However, developing a systematic methodology towards the identification of equivalent continuum properties in this approach cannot be easily achieved. Consequently, there have been virtually no attempts to solve practical engineering problems using this methodology.

In recent years a significant research effort has been devoted to the application of theory of homogenization for periodic media in assessing the equivalent properties of brick masonry. Given the complexity of the problem as well as restrictions imposed by inelastic behavior of constituents, several simplified methodologies have been developed incorporating various explicit kinematic/static constraints. Relying upon these simplifying assumptions, some authors [20], [30] derived simplified nonlinear constitutive laws for the homogenized material. Pietruszczak and Niu [30] employed a two-step homogenization approach, in which the bed and head joints were introduced in two successive steps. Pande et al. [29] derived the equivalent elastic properties of brick masonry in terms of the elastic properties of brick and mortar by introducing a stacked brick–mortar system consisting of a set of parallel layers which behaved elastically. This approach was then extended so that masonry consisting of two sets of bed and head joints could be represented by an equivalent homogeneous orthorhombic elastic material.

Even though the simplified methods provide efficient and attractive methodologies for the implementation in numerical analysis, a more accurate representation is often desirable. The latter may be obtained using numerical homogenization. One of the most rigorous homogenization procedures has been that introduced by Anthoine [2], [3], where the global elastic properties of masonry were estimated as a function of elastic properties of both constituents (brick and mortar) as well as the finite thickness of the joints. Here, the solution for the unit cell was obtained numerically using the finite element technique. A similar attempt was also made by Piszczek et al. [34], who used a 3D numerical homogenization.

Although the former methodologies prove to be very useful in the analysis of masonry structures, they are not, in general, applicable in the context of inelastic analysis. The extension to elastoplastic regime can only be achieved by introducing suitably defined macroscopic yield/failure surfaces. The latter issue has been addressed in the works of Alpa and Monetto [1] and de Buhan and de Felice [10]. Other applications of homogenization theory for estimating the conditions at failure as well as macroscopic properties include the works of Luciano and Sacco [19], Lourenço and Zucchini [18], Miliani et al. [23] and Cluni and Gusella [9]. While the former two deal with periodic microstructures, the latter one is concerned with nonperiodic masonry works.

Finally, significant work has also been undertaken with regards to the development of phenomenologicaly-based macroscopic failure criteria for structural masonry. Examples include the studies of Andreaus [4], Zhuge et al. [45], Lourenço et al. [17], Raffard et al. [37], and Ushaksaraei and Pietruszczak [43]. In this approach, the major problem is the identification of material parameters incorporated in the mathematical framework. The latter usually requires a large number of specially designed experiments, which can be quite expensive and are often subject to strong influence of boundary conditions, so that the scatter in data can be significant. The issue of specification of strength parameters in a continuum formulation has been recently addressed in the article by Kawa et al. [14]. The procedure advocated there was to employ various homogenization techniques (including those based on limit analysis and numerical homogenization) to generate the anisotropic strength characteristics of masonry and to identify the properties on the macroscale.

The present paper is focused on the mesoscale approach and it is organized as follows. In Section 2, the formulation of an advanced constitutive model is presented. The model addresses all distinct stages of the deformation process including pre- and post-localization response. This framework is subsequently implemented as a user-defined constitutive module within COSMOS 2.7 finite element package. Section 3 deals with the numerical implementation of the model in the context of analysis of masonry structures. First, the existing experimental data on the strength characteristics of brick masonry are used for the purpose of verification. This includes simulations of a series of shear tests conducted by van der Pluijm [36] and Atkinson et al. [6]. Subsequently, the model is implemented to study the in-plane behavior of brick masonry panels. Page [28] experiments are used as a benchmark, and the whole panel is analyzed under different loading scenarios. Later, the numerical homogenization is employed to study the orientation-dependent strength characteristics and the results are compared with those obtained for the whole panel.

Section snippets

Mathematical formulation

A realistic solution to problems involving mechanical response of masonry structures depends to a large extent, on the choice of the adopted constitutive model. In this section a material model is proposed which is capable of describing the behavior of brick masonry constituents, i.e. brick and mortar, under different stages of deformation. All three distinct phases in the material response; namely elastic, elastoplastic and brittle/softening, are invoked in the formulation. The onset of

Numerical simulations

The constitutive model introduced in the preceding section has been implemented in COSMOS 2.7 FE package as a user-defined material module. The performance of the model has been verified against some of the available experimental data on brick masonry. The latter include the shear tests conducted by Pluijm [36], the direct shear tests performed by Atkinson et al. [6] and the biaxial tension–compression tests carried out by Page [28]. As mentioned earlier, the analysis has been carried out at

Summary and discussion

An advanced constitutive model has been presented which is capable of addressing all distinct stages of deformation, that is elastic, elastoplastic, and softening. Subsequently, some experimental tests reported in the literature were used to verify the performance of the proposed model. Specifically, Pluijm direct shear tests [35] and Atkinson shear tests [6] were studied. Later, Page biaxial tension–compression tests [28] were examined and the directional strength characteristics of the brick

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