Keywords: masonry, instability, brick walls, nonlinear constitutive law, no-tension material, eccentric loads, finite differences method.

The adoption of a nonlinear elastic models in the description of the stress-strain constitutive law of masonry represents a step forward compared to the use of the linear elastic model since it allows better approximation of the material's experimental curve. As a consequence, in the case of the study of elastic instability phenomena one arrives at theoretical results that describe the real behaviour of structures with excellent approximation. The various proposals that have thus far been formulated and adopted in theoretical studies differ to a certain extent (Frish-Fay [1,3], Powell et al [2], Priestley et al [4], Mura [5], La Mendola et al [6], Eurocode 2 [7]). Such studies call for the adoption of quite different constitutive laws. The dispersion of experimental results and the deformation curves subsequently adopted derive from an objective factor: they are essentially attributed to the dispersion of the characteristics of the different kinds of masonry involved. One model describing the realistic material behaviour for concrete is given in a standardised material law in [7]. In general, two barriers can be set: linear-elastic material behaviour (theory of elasticity) as the lower limit; or rigid-plastic material behaviour (theory of plasticity) as the upper limit. The actual material behaviour of brickwork is nonlinear and lies within these two barriers for all masonry unit and mortar combinations. The schematization of the constitutive law with a second-degree parabolic trend and the vertex corresponding to the maximum strength value appears to be the most generalized one. Such a schematization indeed describes the behaviour of brick walls and concrete walls (see [2,4]).

Examined herein is the behaviour of load-bearing masonry walls or piers with no reinforcement subject to eccentric loads, made of a no-tension material and whose stress-strain law is nonlinear (a second-degree parabolic trend). For a fixed free-ended column subject to axial load with constant initial eccentricity along the axis, the complete critical path is predicted by a computer program. As a preliminary step, the stress and strain distribution is derived for a rectangular section of width and thickness , assuming that the plane section remains plane after deformation. In particular, with the variation in the intensity and geometric eccentricity with which the load is applied, the conditions of cracking and failure are defined by reaching the maximum compression tension allowed by the material.

Then the differential equations governing the problem of equilibrium stability are formulated. Since analytic integration of this type of nonlinear differential equations turn out to be quite complex, the recourse is to program the automatic calculation optimized for the purpose of performing numerical integration with the finite difference method. The results of the numerical investigation will be illustrated in detail, both through examination of the load-deflection curves obtained directly and through illustration of the diagram deduced from them, which supplies the reduction coefficient of the load by eccentricity and slenderness . The stability curves show that failure of the structure may come about both through instability in elastic equilibrium and through the reaching of the maximum compression strength allowed by the material in the fixed section (under the greatest stress). The failure modalities cited above may be reached both with the structure reacting entirely and after partial or total cracking of the element. Stability curves have been calculated considering the maximum deformation value equal to =0.004, which leads to sufficiently conservative results for practical applications. Finally, it will be shown that with the same admissible stress of the material, the failure load increases in such a way that in some cases it can even be conspicuous with the reduction in deformation .

1

R. Frish-Fay, "Stability of masonry piers", Int. J. Solids Struct., 11, 187-198, 1975. doi:10.1016/0020-7683(75)90052-9

2

B. Powell and H.R. Hodgkinson, "The determination of stress/strain relationship of brickwork", Proc. 4th Int. Brick Masonry Conference, Brugge, 26-28 April, 1976.

3

R. Frish-Fay, "Quasi-Analytical Method for the Analysis of a Masonry Column with a Non-Linear Stress-Strain Law", Int. J. of Masonry Construction, 2, 41-46, 1981.

4

M.J.N. Priestley and D.M. Elder, "Stress-Strain Curves for Unconfined and Confined Concrete Masonry", ACI Journal, 80(3), 192-201, 1983.

5

I. Mura, "Stabilità di murature pressoinflesse non resistenti a trazione aventi legge costitutiva elastica non lineare", Convegno CNR-GIS 6-8 giugno, Milano. Atti della Facoltà di Ingegneria, Univ. Cagliari, Vol 27 Ott. 1985.

6

L. La Mendola and M. Papia, "Stability of masonry piers under their own weight and eccentric load", Journal of Structural Engineering, ASCE, 119(6), 1678-1693, 1993. doi:10.1061/(ASCE)0733-9445(1993)119:6(1678)

7

ENV 1992-1-1: Eurocode 2: Design of concrete structures - Part 1: General rules and rules for building. January 2001.

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