Keywords: masonry, homogenization, limit analysis, out-of-plane loads, lower bound, upper bound.

The prediction of the ultimate load bearing capacity of brickwork panels out-of-plane loaded is a key issue for the design of masonry. Out-of-plane failures are mostly related to seismic and wind loads and earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of masonry (see for instance [1]).

Another important aspect is that masonry structures are usually subjected simultaneously to in-plane compressive vertical loads and out-of-plane actions. As experimentations show, in-plane loads increase both the ultimate out-of-plane strength and the ductility of masonry in terms of out-of-plane loads / out-of-plane deflections.

Many laboratory tests, conducted on brick masonry walls subjected to lateral loads, have demonstrated that failure takes place along a definite pattern of lines, so suggesting approximate analytical solutions based on the yield line theory [2]. Probably for their theoretical simplicity, yield-line approaches have been adopted by many codes, as for instance BS 5628 [3] and EC 6 [4]. Nevertheless, codes of practice employ only out-of-plane horizontal and vertical masonry strengths (which are experimentally available directly), leading to a strong theoretical approximation connected to the fact that torsion contribution is neglected.

In this paper, a novel micro-mechanical model combining homogenization and limit analysis and able to take into account torsion effects is presented. In a different way with respect to a classical finite element (FE) discretization of the unit cell, a linear programming problem with a very limited number of variables is derived. The elementary cell is subdivided along the thickness in several layers. For each layer, fully equilibrated stress fields are assumed, and a-priori selection of the polynomial expressions for the stress tensor components in a finite number of sub-domains.

In order to obtain a statically admissible stress field on the unit cell, continuity of the stress vector on the interfaces between adjacent sub domains is imposed, as well as internal equilibrium inside each sub-domain. In the framework of lower bound homogenization [5], further anti-periodicity conditions on the boundary surface are imposed. Admissibility conditions for the constituent materials (bricks and mortar) are finally imposed on a regular grid of points.

In this way, a strong reduction of the total unknown stress parameters is obtained and a (non) linear optimisation problem with few variables is derived in order to obtain masonry out-of-plane homogenized failure surfaces. The non linearity of the problem is essentially due to the admissibility conditions of the constituent materials and can be easily avoided both by means of classic linearization procedures [6] or non-standard recursive algorithms [7].

Out-of-plane failure surfaces so recovered are then implemented in the FE limit analysis codes (both upper and lower bound) for the homogenized limit analysis of entire panels loaded out-of-plane. The lower bound approach is based on the equilibrated triangular element by Hellan [8] and Herrmann [9], whereas the upper bound is based on the triangular element by Munro and Da Fonseca [10].

Meaningful structural examples are treated for two panels loaded out-of-plane [11]. The comparisons with experimental evidences reported in the paper (in terms of collapse loads and failure mechanisms) show the reliability of the proposed model.

1

R. Spence, A. Coburn, "Strengthening building of stone masonry to resist earthquakes", Meccanica, 27, 213-221, 1992. doi:10.1007/BF00430046

2

B.P. Sinha, "A simplified ultimate load analysis of laterally loaded model orthotropic brickwork panels of low tensile strength", J. Struct. Eng. ASCE, 56B(4), 81-84, 1978.

3

British Standard Institution, "BS 5628 part I: Use of Masonry 1978; BS 5628 part II: Use of Masonry 1985; BS 5628 part III: Use of Masonry 1985"

4

EN 1996. Euro Code 6: Design of masonry structures.

5

P. Suquet, "Analyse limite et homogeneisation", Comptes Rendus de l'Academie des Sciences - Series IIB - Mechanics, 296, 1355-1358, 1983.

6

E. Anderheggen, H. Knopfel, "Finite element limit analysis using linear programming", International Journal of Solids and Structures, 8, 1413-1431, 1971. doi:10.1016/0020-7683(72)90088-1

7

G. Milani, P.B. Lourenço, A. Tralli, "Homogenized limit analysis of masonry walls. Part I: failure surfaces", Submitted Computers & Structures, 2005.

8

K. Hellan, "Analysis of elastic plates in flexure by a simplified finite element method", Acta Polytech. Scand., Trondheim, Ci 46, 1-28, 1967.

9

L.R. Herrmann, "Finite element bending analysis for plates", J. Eng. Mech. Div. ASCE, 93, 13-26, 1967.

10

J. Munro, A.M.A. Da Fonseca, "Yield-line method by finite elements and linear programming", J. Struct. Eng. ASCE, 56B, 37-44, 1978.

11

E.A. Gazzola, R.G. Drysdale, A.S. Essawy, "Bending of concrete masonry walls at different angles to the bed joints", Proc. 3th North. Amer. Mas. Conf., Arlington, Texas, USA, Paper 27, 1985.

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