Keywords: unreinforced masonry, failure criterion, homogenisation, numerical simulation, proportional loading, out-of-plane behaviour, anisotropy.

Masonry is a composite material exhibiting a complex nonlinear behaviour. Mostly, plane structural elements like shear walls are used in order to bear lateral loads and are subjected to biaxial stress states. To permit an ultimate limit state design and to assess the capacity of the structure the whole in-plane failure criterion of masonry is required. But failure modes and strength characteristics of masonry are highly sensitive to the orientation of the principal stresses with respect to the joint direction. This anisotropy is mainly due to the relative weakness of the mortar when compared with the units and it may be emphasized if the units exhibit anisotropic strength properties because of perforations. Thus, to define masonry failure, a three-dimensional surface in terms of the two principal stresses and their respective orientation to the bed joint is required.

This paper presents a general procedure in order to prognosticate numerically such failure criterion of masonry derived directly from the material parameters of the constituents. It is not based of any phenomenological macroscopic failure criteria which may be applied only to a special kind of brickwork and cannot be copied to others. No other parameters need to be identified subject to the condition that the nonlinear material behaviour of the brick and mortar are known. Since the procedure is independent of the used material law of each constituent it can be applied to each kind of masonry.

First, the homogenisation theory for two-dimensional periodic media is briefly discussed. On the basis of a 2-d unit cell an homogeneous macroscopic field is prescribed by enforcing the displacement fields to be strain-periodic [1]. Hereon, a method is presented which allows a proportional macroscopic loading, that means a constant relation of the principal stresses, imposed in force but controlled in displacement and in dependence of the orientation with respect to the bed joints. This method provides the opportunity to observe the post-peak behaviour of masonry on the meso scale level if the material behaviour of each constituent is adequate represented. This study is focused on the supply of the failure surface of homogenised running bond pattern masonry which may be used as input for a non-linear smeared macro model of masonry like the one of Vratsanou [4]. Therefore, the use of an elastoplastic material behaviour combined with a hyperbolic rounded Mohr-Coulomb criterion is adequate in order to obtain an upper limit of the failure surface for the composite. The numerical results will be compared with experimental data from biaxial in-plane compression tests of Page [2,3].

Finally, an extension of the homogenisation theory is given for additional out-of-plane loading which may induced by a lateral drift in the case of an earthquake. Great importance is attached to the boundary conditions of the 3-d unit cell with which the homogeneous strain-periodic displacement field is prescribed. Therewith, it is possible to derive the in-plane as well as the out-of-plane characteristics in one step. In the linear case the study shows that it's quite a good approach to assume masonry as an transversal orthotropic material described by four independent parameters where the out-of-plane characteristics may be derived from the in-plane characteristics. Since these parameters are determined by means of a 2-d model the use of a 3-d model is avoidable. Nevertheless, in the nonlinear case, the strength is load-path dependent. By means of using master points it's possible to follow the load path displacement controlled in order to investigate the failure process for any kinds of loading.

1

Anthoine, A., "Derivation of the in-plane elastic characteristics of masonry through homogenization theory", Int. J. Solids Structures, Vol. 32, No. 2, pp. 137 - 163, 1995. doi:10.1016/0020-7683(94)00140-R

2

Dhanasekar, M., Page, A.W., Kleemann, P.W., "The failure of brick masonry under biaxial stresses", Proceedings of Institution of Civil Engineers, 79, June, pp. 295 - 313, 1985. doi:10.1680/iicep.1985.992

3

Page, A.W., "The biaxial compressive strength of masonry", Proceedings of Institution of Civil Engineers, 71, Sept., pp. 893 - 906, 1981. doi:10.1680/iicep.1981.1825

4

Vratsanou, V., "Das nichtlineare Verhalten unbewehrter Mauerwerksscheiben unter Erdbebenbeanspruchung", Dissertation, University of Karlsruhe, 1992.

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