Keywords: three-dimensional rigid body, monolithic objects of art, overturning, pulse-type excitation.

The study of the behaviour of rigid blocks has had a continuous evolution in time. In recent years, the complexity of the models has been widely increased and different phenomena have been discovered. However only a few three-dimensional models, which were mainly dedicated to bodies of circular shape, have been studied. The same authors of the present work extensively investigated the behaviour of rigid blocks, representative of a monolithic object of art, with two-dimensional models, both in the case of a free-standing block and in the case of an isolated block with different kinds of constraint [1,2].

In the present paper a three-dimensional model has been developed and the rigid body has been taken as non-symmetric and of rectangular base, thus rocking can occur on a side or on a vertex of the base. The rigid body is assumed able only to rock since its slenderness does not allow occurrence of sliding between the base and the ground. Exact nonlinear equations of motion describing the rocking motion are obtained from the general balance principle. Impacts between the base of the body and the ground have been modelled taking into account the conservation of the angular momentum. Results have been obtained by using the program Mathematica as the main strategy and Fortran language for the numerical integration, where particular attention has been devoted to the choice of the algorithm of integration and of the minimum time step.

The effects of a one-sine pulse excitation, have been studied as in [3]. Several analyses have been carried out in order to highlight the influence of different geometrical parameters on the motion. The direction of the input has been varied continuously to analyse the effects of the eccentricity and of the slenderness of the rigid body on the minimum acceleration amplitude that needs to overturn the rigid block. Great attention has been also devoted to the role of the period of the impulsive excitation. The analyses conducted on near-square based non-symmetric bodies (human-like statues) highlighted the existence of particular directions of the input, where the amplitude of the overturning excitation is lower than the one obtained for a classical two-dimensional model. Therefore it seems that a three-dimensional model is needed to evaluate the overturning of a rigid body in favour of safety.

1

A. Di Egidio, A. Contento, "Base isolation of sliding-socking non-symmetric rigid blocks subjected to impulsive and seismic excitations", Engineering Structures, 31, 2723-2734, 2009.

2

A. Di Egidio, A. Contento, "Seismic response of a non-symmetric rigid block on a constrained oscillating base", Engineering Structures, 32, 3028-3039, 2010. doi:10.1016/j.engstruct.2010.05.022

3

N. Makris, C.J. Black, "Dimensional analysis of bilinear oscillators under pulse-type Excitations", Journal of Engineering Mechanics (ASCE), 130(9), 1019-1031, 2004. doi:10.1061/(ASCE)0733-9399(2004)130:9(1019)

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