Keywords: masonry, instability, double-hinged walls, nonlinear constitutive law, no-tension material, combined load, finite differences method.

Examined herein is the post buckling of unreinforced load-bearing masonry walls or piers, hinged at the ends, subject to a combined load consisting of a vertical uniformly distributed axial load and a concentrated eccentric top load. The study presents different causes of nonlinearity. The nonlinearity derived from the constitutive law of the material is added to the nonlinearity of a geometric nature derived from second-order effects and the possible reduction of reacting sections due to cracking. Because of this, the stiffness of the material is reduced with the increase in the intensity of the acting stress.

Various nonlinear stress-strain laws have thus far been adopted in theoretical studies. The differences originate in the experimental results. The actual material behaviour of brickwork is nonlinear and lies within two barriers (linear-elastic and rigid-plastic material behaviour) 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. No tension material with a second-degree parabolic stress-train law in compression is considered in this paper.

Taking into consideration only walls or piers to which a concentrated load at the top and a vertical load uniformly along the axis are applied simultaneously, it appears that all the different causes of nonlinearity have rarely been considered together.

The present paper extends the theoretical formulation used in the study [1] to hinged masonry piers. The piers are considered to have an eccentric load applied at the top and a uniformly applied vertical load along the axis. The piers are also presumed to be made of a no-tension material whose stress-strain law in compression is nonlinear (a second-degree parabolic trend).

The integro-differential formulation supporting the problem of the stability of the equilibrium has been formulated extremely carefully and solved numerically with the finite difference method (FDM). To find the load that produces collapse (due to elastic instability or crushing of masonry) two different methods of calculation were used: the bisection method and the simulated annealing method.

The results of the numerical investigation are illustrated by examining first of all the load-deflection curves. Then the diagrams deduced from them are illustrated to obtain the value of the external load applied at the top that produces failure with variation of the non-dimensionalized distributed axial load and of the value assigned to parameter m which characterizes the eccentricity of the top load. On commenting on such curves, the great influence of the uniformly distributed axial load both on the collapse load and on the way in which collapse occurs is pointed out. Finally an extensive numerical comparison of the results obtained using the two methods of calculation is presented. The results have practically insignificant differences, but the calculating times appear much more favourable when the bisection method is used.

1

Mura I., "Stability of nonlinear masonry members under combined load", Computers & Structures, 2008. doi:10.1016/j.compstruc.2008.01.003

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