Keywords: finite element method, non-linear analysis, masonry, homogenization, plasticity, smeared crack model.

It is often important to provide a numerical analysis of masonry structures that is more precise than methods reccomended by current technical standards (including the Eurocodes). There is a large number of possible approaches. One of these possibilities is a detailed non-linear modelling of masonry structure with respect of the geometry and of ther different material properties of the bricks and of the mortar.

If the structure is large, then it is more practical to replace a detailed material (that respects individual materials) model with homogenized material properties. The homogenized material properties can be obtained from experimental tests of parts of structure or from a detailed numerical analysis of a typical part of the structure. The homogenized material properties can be both elastic and inelastic.

It is usually useful to combine both mentioned approaches. The paper discusses the elastoplastic material model for masonry and homogenization procedures for masonry (for elastic properties only).

The finite element method [5] is used for the computations. We assume that analysed have a non-linear behaviour but the behaviour of structures studied is geometrically linear. Masonry consists of two main compoments with different properties: bricks (or stones) and a mortar.

The material of bricks is modeled as linear isotropic. The mortar is modelled as a non-linear material with behaviour similar to concrete. The mortar model is based on a combination of elastoplastic behaviour and a smeared crack model (for modelling of mortar with cracks). Steel reinforcements can also be included in a masonry structure. It is modeled as a smeared material and its behaviour is elastoplastic.

For large structures a homogenization of material properties can be used. It allows to make computational models of relatively large structures. It is possible to obtain both linear and non-linear homogenized material properties from a detailed numerical model of a small part of the structure.

We use isoparametric finite elements for both 2D and 3D problems. Four-node isoparametric elements [4] are used for 2D solution and eight-node isoparametric elements (derived from the formulation in the [3]) have been used for 3D problems.

The initial behaviour of mortar is elastoplastic. Because the behaviour of mortar is different under different loads it is necessary to use a special plasticity condition. There are several plasticity and failure conditions for concrete that were developed by different authors, for example Willam and Warnke, Kupfer or Chen and Chen [2].

The Chen-Chen criteria has been selected for our work. It can be used for both 2D and 3D problems and uses relatively easy obtainable input data (strength in uniaxial and biaxial compression and strength in uniaxial tension ).

The smeared crack model [1] is used for the modelling of mortar with tension cracks. The orthotropic behaviour of the cracked material is assumed.

In many cases it is impossible to do a detailed numerical analysis of a complete masonry building that use (for example) the constitutive model discussed in this article. It is not only because of computer limitations (that can be - at least partially - solved by the use of high performance computing) but also because of too complicated creation of detailed computational models.

We use a simple method for computing of homogenized material properties. A detailed model of small parts of the masonry structure is created and simple material tests are simulated. These tests include compression tests in the , and directions and shear tests.

1

Z. P. Bazant, J. Planas, "Fracture and Size Effect in Concrete and Other Quasibrittle Materials", CRC Press, Boca Raton, 1998.

2

A. C. T. Chen, W. F. Chen, "Constitutive Relations for Concrete", Journal of the Engineering Mechanics Division ASCE, 1975.

3

V. Kolar, J. Kratochvil, F. Leitner, A. Zenisek, "Finite element analysis on planar and spatial structures" (in Czech: "Vypocet plosnych a prostorovych konstrukci metodou konecnych prvku"), SNTL, Prague, 1976

4

D. R. J. Owen, E. Hinton, "Finite Elements in Plasticity", Pineridge Press Ltd., Swansea, 1980

5

O. C. Zienciewicz, "The Finite Element Method in Engineering Science", McGraw-Hill, London 1971

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