Modeling of the 3D rocking problem

Abstract

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.

Introduction

The rocking motion of a body is a highly non-linear phenomenon, that has attracted a significant number of researchers, since it was first formulated by Housner [1] in 1963. The majority of the studies on rocking have only considered the problem as two dimensional, i.e. the planar rocking of a body, and the associated complex properties. However, rocking is in general a three dimensional phenomenon and thus planar rocking is only a limited aspect of it.

In fact, even in the simplest case of a symmetric body, one cannot expect the rocking motion to be planar, unless the ground excitation is in parallel with one of the axes of symmetry of the body. The latter, however, is rarely true and thus one should expect the motion of the body to include 3D displacements and rotations. Despite this fact, the literature for the full three dimensional formulation of the problem is rather limited in comparison with the work done on the planar aspect of the phenomenon.

Housner proposed the first model, the inverted pendulum model [1], to study the planar rocking motion of a body. Expansions of this model in assemblies of two stacked rigid bodies were presented in [2], [3]. While the inverted pendulum model neglected sliding, uplift and the deformability of the support medium, a series of papers gradually incorporated these effects into the 2D rocking models. Sliding and uplift, for a rigid support medium, were included with some additional assumptions, in the work of [4], [5]. Psycharis and Jennings [6] proposed a model, considering the flexibility of the ground, with either concentrated springs and dampers at the corners, or a Winkler type of foundation, which however was geometrically linear and ignored sliding and uplift. Koh et al.'s model [7] also included a Winkler type of foundation model with geometrically non-linear equations, however still the effects of friction and uplift were not incorporated. The latter model was used to investigate the response to random input in [8]. Finally, fully geometrically non-linear CSM and WM models that included the effects of uplift and sliding were proposed in [9].

On the other hand, the literature on the modeling of the 3D rocking problem is rather limited. Konstantinids and Makris in [10] proposed a 3D rocking model for rigid symmetric prisms on a rigid foundation, however the analysis was stopped right before an impact occurred, while sliding and uplift was also neglected. 3D rocking models for rigid cylindrical objects on a rigid base were presented in [11], [12], with the assumption of no loss of energy during impacts, while the motion of a cylinder on a Winkler foundation was examined under the assumptions of no slipping or twisting in [13]. Apart from these models, it has been suggested in [14] that the 3D rocking problem can be examined with 2D approximations, an approach which as the authors state would require the existence of a constant rocking plane, i.e. the plane perpendicular to the vector of rotation. In addition, researchers [15], [16], [17] have used models based on the DEM theory as developed by Cundall [18], or general FEM models such as those proposed in [19], [20]. These models, however, were developed for 3D contact problems in general and are thus not optimized for the rocking problem.

In the current paper, two 3D rocking models are developed by expanding the benefits of the CSM and WM, proposed in [9], into the 3D rocking case. The suggested models examine rigid body prisms on a deformable base and take geometric non-linearities, sliding and uplift fully into account. The models are later used in suitable examples, where the complex 3D character of rocking is highlighted.