A micro-mechanical model for the homogenisation of masonry

Abstract

Masonry is a composite material made of units (brick, blocks, etc.) and mortar. For periodic arrangements of the units, the homogenisation techniques represent a powerful tool for structural analysis. The main problem pending is the errors introduced in the homogenisation process when large difference in stiffness are expected for the two components. This issue is obvious in the case of non-linear analysis, where the tangent stiffness of one component or the tangent stiffness of the two components tends to zero with increasing inelastic behaviour. The paper itself does not concentrate on the issue of non-linear homogenisation. But as the accuracy of the model is assessed for an increasing ratio between the stiffness of the two components, the benefits of adopting the proposed method for non-linear analysis are demonstrated. Therefore, the proposed model represents a major step in the application of homogenisation techniques for masonry structures. The micro-mechanical model presented has been derived from the actual deformations of the basic cell and includes additional internal deformation modes, with regard to the standard two-step homogenisation procedure. These mechanisms, which result from the staggered alignment of the units in the composite, are of capital importance for the global response. For the proposed model, it is shown that, up to a stiffness ratio of one thousand, the maximum error in the calculation of the homogenised Young's moduli is lower than five percent. It is also shown that the anisotropic failure surface obtained from the homogenised model seems to represent well experimental results available in the literature.

Homogenised limit analysis of masonry walls, Part II: Structural examples

“…As it can be noticed, the provided moduli are reliable in a large range of E b /E m ratios, even for the most simple model with constant stresses tensor (P0). Nevertheless, Figure 4-a shows that a progressively reduced accuracy of the P0 model can be noticed for the A 1111 module, due to the presence of shear stresses in the bed joint (as already discussed in [14]). Finally, in Figure 5 the relative error, in terms of energy norm, arising from a comparison with F.E.…”

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches

“…Most of the methods of homogenization simplify the geometry of the basic unit with a 2 step introduction of vertical and horizontal joints and thus without taking into account the regular offset of vertical mortar joints. However, this kind of approach results in significant errors when applied to non-linear analysis Micromechanical homogenization, derived independently by van der Pluijm [135], Lopez et al [68] and Zucchini and Lourenço [137], is based on the detailed finite element analysis of the elementary cell and overcomes the approximations introduced by the 2 steps simplified method. Recent advances in terms of sophisticated analysis homogenization tools are discussed in Lourenço et al [76] and include the polynomial stress field expansion approach of Milani et al [89] and the mesoscopic approach of Massart et al [85], Calderini and Lagomarsino [19], and Shieh-Beygia and Pietruszczak [130].…”

“…In a first step, Zucchini and Lourenço [137] introduced homogenization in the elastic field by deriving the mechanical properties of masonry from a suitable micro-mechanical model taking into account the staggered alignment of the units. The homogenization model has been subsequently extended to non-linear problems in the case of a masonry cell failure under tensile loading parallel to the bed joint, Zucchini and Lourenço [138], or under compressive loading perpendicular to the bed joint, Zucchini and Lourenço [139].…”

Numerical prediction of the behavior, strength and elasticity of masonry in compression

“…Finally, threedimensional periodic unit cells have been used for the verification of the results of homogenization methods for masonry under triaxial normal and shear stress, Cecchi et al [18], Zoucchini & Lourenço [19].…”