Numerical analysis of rocking of unanchored bodies subjected to seismic load using Finite Element analyses

Rocking is a highly nonlinear phenomenon, which cannot be completely described analytically. Therefore, the rocking motion of a rigid body under various conditions will be analysed analytically and numerically by numerical integration of the equations of motion and by using the Finite Element Method. It is shown that a finite element program with an implicit time integration method can solve the impact problem. By using the Finite Element Method it is possible to investigate the influence of the properties of the base, the contact conditions and slight deviations from the rectangular corner geometry on the motion of the body and especially on the coefficient of restitution. In practice, most unanchored bodies are not slender and therefore this paper studies a compact body with an aspect ratio of 2. A main topic is the discussion of the coefficient of restitution. It is shown that more realistic finite element models, which take into account the above mentioned influences, lead to higher coefficients than those derived analytically from simplified models. For a stiff base like steel or concrete the agreement between results from finite element analyses and the numerical integration of the equations of motion for two rotation centres at the corners of the body is very good if the coefficients of restitution are based on best estimate values from finite element analyses. The low stiffness of a 10 mm thick anti-slip pad has the effect that the use of a model with two fixed rotation centres is not possible. The dynamic investigations consider an excitation by sine, triangle or rectangle vibrations and time histories from earthquake ground accelerations. It is remarkable that in some cases the amplitudes will increase significantly with time until overturning.