A homogenized Love-Kirchhoff model for out-of-plane loaded random 2D lattices: Application to "quasi-periodic" brickwork panels

A homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a masonry obtained by a random perturbation of the periodic running bond masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610–625]. A numerical evaluation of the scatter between the discrete models and the 2D Love–Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610–625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show (both by means of a discrete model and a homogenized model) that the running bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction.