Modelling of masonry walls under seismic loadings

Developed. The method is based on the two-dimensional micropolar continuum theory and makes use of the kinematic approach of limit analysis in conjunction with a rigorous homogenization technique. The method is introduced in a general way, with regard to the genericclass of discrete periodic media made of particles of the same type. The case of masonry is presented as application. The homogenised strength domains of masonry columns and walls are retrieved in terms of the generalized stresses and couple stresses of the Cosserat continuum. The formulation of the method based on the Cosserat continuum enables the investigation of the influence of the relative rotation of the particles on the strength of the discrete medium. This influence is illustrated by the application to masonry structures, in comparison with other methods presented in the literature. The development of the homogenisation method continues with its extension to discrete periodic media made of particles disposed along three directions and showing three periodicity vectors. In this case, the approach relies on the three-dimensional micropolar theory. This enables to capture the three-dimensional effect of the relative translations and rotations of the particles constituting the discrete medium. The application to masonry columns and walls shows how the in-plane and out-of-plane actions result coupled in the assessment of masonry strength. The relative rotation of the blocks accentuates this effect, which consistently diminishes the in-plane strength. Masonry walls are finally ascribed to homogenised plates with Cosserat kinematics. A finite element formulation for Cosserat plate models is next developed. The formulation is first presented for elasticity and dynamics. The validation of a specific finite element is made by means of numerical benchmarks and patch tests. The actual use of the element is presented in an application to masonry structures. The natural frequencies of a masonry panel modelled by discrete elements are computed and compared with those given by a homogenisation model implemented in the element. This allows to investigate the role of the in-plane rotations of the blocks and to show their implication towards seismic analyses of masonry structures. The finite element formulation is next extended to the elastoplastic framework. The implementation of the multisurface plasticity theory into the Cosserat finite element is presented. The implementation of this theory is based on a projection algorithm. An important limitation of the classical implementation of this algorithm prevents its use in the framework of multisurface plasticity in efficient way. This limitation is discussed and a solution strategy is proposed. The finite element for Cosserat plate models is finally validated through numerous numerical benchmarks. In conclusion, three different modelling approaches for masonry are proposed and comviipared. A continuum model based on the Cosserat continuum is first presented. The model isconstructed by implementing the homogenised yield criteria computed based on the proposed analytical method into the developed finite element. A homogenisation model based on Cauchy continuum is next introduced. This model is constructed by selecting appropriate constitutive laws and yield criteria from the literature. The performance of those homogenisation models in representing the elastoplastic response of a masonry panel is discussed, based on the comparison with a third analogue discrete elements model. The capability of the three models in predicting the scale effect in the formation of failure mechanisms is investigated in a practical application to masonry structures