Free Rocking of Prismatic Blocks

This paper investigates both experimentally and theoretically the free rocking of a prismatic block supported by a stationary , horizontal foundation: the block is tilted, almost to the point of overturning, and released from this position. It is shown that the standard mathematical model for this problem is often inaccurate. A critical review of the implicit assumptions behind the standard model reveals that the free-rocking response of short blocks depends crucially on bouncing after each impact; out-of-plane effects are significant in very short blocks. The response of slender blocks is found to be easier to predict. Rocking has been observed during earthquakes in structures that consist of fairly rigid, unbonded elements, e.g. stacks of graphite blocks in nuclear reactors. and ancient Greek columns, and also in slender structures with foundations unable to resist uplift. INTRODUCTION: SIMPLE ROCKING MODEL (SRM) This paper investiga'tes the free-rocking response of a prismatic block supported by a stationary, horizontal foundation, as shown in Fig. 1. The problem we are interested in can be described, in slightly simplified terms and referring to Fig. 2, as follows. The block is rotated through a small angle < a about A and then released: initially, it rotates about A until it becomes vertical; at this point, B suddenly comes into contact with the foundation, while A loses contact. Then, the block continues to rotate in the same sense, but about B; its angular velocity decreases gradually, until it becomes zero, at which point the reverse motion begins. This cycle comes to an end when the block becomes vertical and starts to rotate again about A. Because some energy is dissipated in each impact, the amplitude of the motion is gradually reduced, until the block comes to rest after a series of cycles. This problem has practical relevance because a broadly similar response has been observed, during earthquakes, in slender water tanks and petroleum cracking towers (Housner 1956) , ancient Greek and Roman stone temples (Fowler and Stillwell 1932), and stacks of graphite blocks in the core of nuclear reactors (Olsen et al. 1976). Of course, during an earthquake the foundation does not remain stationary, and hence earthquake-induced oscillations are more complex; a good understanding of the free -rocking response is essential before going on to forced rocking, as we shall discuss. In this paper we show that the deceptively simple response just described is actually quite hard to model accurately. Following Housner (1963), let us consider the uniform block shown in Fig. 1, with mass M and moment of inertia I about its center of mass G, subject to gravity. In the simple rocking model (SRM) it is assumed that the motion of the block is essentially two-dimensional, as in the foregoing 'Engr., Murray-North Ltd., 106 Vincent St., Auckland, New Zealand. 2Lect., Dept. of Engrg., Univ. of Cambridge, Trumpington St., Cambridge, England, CB2 1PZ. Note. Discussion open until December 1, 1993. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 1, 1991. This paper is part of the Journal of Engineering Mechanics, Vol. 119, No. 7, July, 1993. ©ASCE, ISSN 0733-9399/93/0007-1387/$1.00 + $.15 per page. Paper No. 3038.