Modeling of the 3D rocking problem

The rocking motion of a rigid rectangular prism on a moving base is a complex three dimensional phenomenon. Although, with very few exceptions, the previous models in the literature make the simplified assumption that this motion is planar, this is usually not true since a body will probably not be aligned with the direction of the ground motion. Thus, even in the case where the body is fully symmetric, the rocking motion involves three dimensional rotations and displacements.In this work, a three dimensional formulation is introduced for the rocking motion of a rigid rectangular prism on a deformable base. Two models are developed: the Concentrated Springs Model and the Winkler Model. Both sliding and uplift are taken into account and the fully non-linear equations of the problem are developed and solved numerically.The models developed are later used to examine the behavior of bodies subjected to general ground excitations. The contribution of phenomena neglected in previous models, such as twist, is stressed.

► Two models, the CSM and WM, are developed for the 3D Rocking problem. ► The CSM uses concentrated vertical springs and dampers, and the WM distributed. ► Euler angles are used to describe rotations and Lagrange's equations are used to derive the equations of motion. ► Sliding, uplift and geometric non-linearities are included in the behavior of the models. ► Examples illustrate the ability of the models to capture 3D rocking.