Investigating the rocking dynamics of rigid right prisms with regular polygonal bases

Seismic activity affects many regions of the world and many types of objects; not only typical structures studied by engineers such as buildings and bridges, but also objects unattached to their support. Such objects, when slender, are prone to rocking about one or multiple points which can result in failure of the object by toppling or by fracture when high-energy impacts between the object and the support occur during rocking. Objects prone to rocking are not limited to rectangular prisms and simple geometries but include a variety of irregular or curved geometries such as museum artifacts and cylinders. These objects can have irregular locations for their center of mass, more than four corners and four edges about which they can rock, and other peculiarities such as holes or concave geometry. To better understand the dynamics of such objects, a new model has been developed extending the inverted pendulum model for rocking introduced by Housner (1963) to 3-D objects with irregular geometries. The model determines the three-dimensional rocking response of a slender object with any polyhedral geometry during ground accelerations. Objects with irregular or curved geometries are often approximated as rectangular blocks in the literature. However, objects with curved geometry exhibit motions that are defined as rolling about their edges; a motion which does not result in loss of energy as is seen during transitions between vertices of a straight edge at impact. Therefore, the importance of using an accurate geometric representation of objects with curved geometry is examined. Examples of cylinders and other objects with curved geometries will be presented along with their equivalent polyhedral approximations with varying precisions. The level of accuracy at which solutions converge for the curved geometry and the polyhedral approximation will also be examined to determine the level of geometric detail required to estimate the seismic response of a curved object. The 3-D analysis of these objects will be important to their study as the rolling of a cylinder around its base cannot be limited to a model with only two degrees of freedom. Furthermore, ground accelerations should not be limited to one direction as studying multiple directions can produce a substantial change in the movement of the body, particularly when the object is curved on its base and therefore lacking resistance to 3-D rocking or rolling motions. The 3-D response of objects with irregular geometry will be compared to results from planar analysis to highlight the differences between the results and the importance of using a 3-D model for earthquake analysis. The presentation of these findings will be enhanced using interesting figures and animations of objects moving during earthquakes. Conclusions will be derived regarding the necessity of 3-D versus 2-D analysis for such objects. This work will assist in determining the rocking and failure risk of objects previously understudied by earthquake engineers