CRITERIA FOR INITIATION OF SLIDE, ROCK, AND SLIDE-RoCK RIGID-BODY MODES By Harry W. Shenton 111, 1 Associate Member, ASCE ABSTRACT: The motion of a plane rigid body that rests unrestrained on a rigid foundation that is accelerating is described by five modes of response: rest, slide, rock, slide-rock, and free flight. Determining the correct initial mode of response is essential to the analysis of the generalized behavior. In the past it has been generally assumed that only a slide or rock mode can be initiated from rest. It is demonstrated in the present paper that a slide-rock mode can also be initiated from rest. Criteria governing the initiation of the slide, rock, and slide- rock modes are derived. Results show that it is incorrect to assume that a rock mode ensues if static friction is simply greater than the width-to-height ratio of the body; a slide-rock mode is initiated when friction is greater than the width-to-height ratio, but less than a critical value that is a function of the ground acceleration. The friction required to initiate a rock mode increases with ground acceleration, as would be expected. The results demonstrate a more natural transition of the governing mode of response, from sliding to slide-rock and finally to pure rocking, as friction is increased for a given foundation acceleration. The analysis is restricted to a rectangular block, assuming Coulomb friction and horizontal ground acceleration. 0' 2H n " Yf:- FIG. 1. Definition Diagram a peculiar result that a slide-rock mode could only be initiated for objects with a width-to-height ratio greater than 1 and static friction greater than 1. A detailed derivation of the criteria that govern the initiation of the slide, rock, and slide-rock modes from rest is presented in the following section. The analysis is restricted to a rigid, rectangular block and horizontal foundation acceleration. All of the conditions that must be satisfied for the initiation of a mode from rest are examined in detail. Results are presented and compared to the often used criteria for establishing slide and rock modes. INITIATION OF SLIDE, ROCK, AND SLIDE-ROCK MODES 28 Assume a symmetric rigid block is at rest relative to a mov- ing foundation, as shown in Fig. 1. The block has mass m, weight W, centroidal mass moment of inertia I, width 2B, height 2R, and aspect ratio "y = RIB. The bottom of the block is flat so that contact is made with the foundation along the base of the body. The normal and horizontal reactions at the base are denoted by h and Ix. The horizontal reaction force is governed by Coulomb friction: Static and dynamic coefficients of friction are denoted by IJ.. and IJ.b respectively, where IJ.k s IJ. •. Displacements of the block relative to the foundation are denoted by x and y; angular rotations are denoted bye, positive in the counterclockwise direction. The ground accel- eration is assumed to be described by a function that is piece- wise-continuous and builds from zero (Le., the foundation is INTRODUCTION A free-standing rigid body on an accelerating foundation can undergo a wide range of motions. In two dimensions, the behavior is described by five "modes" of response: rest, slide, rock, slide-rock, and free flight. A fundamental first step in the dynamic analysis of such a system is to determine which of the four nontrivial modes of response is initiated from rest, for a given foundation acceleration. This is important, since the initial mode is often assumed to be the exclusive mode of response. Simple criteria that govern the initiation of a sliding motion or a rocking motion have been in use for some time; criteria governing the initiation of a slide-rock mode from rest have not previously been determined; these are derived herein. Analysis of the response of rigid bodies to base excitation has applications in a number of areas, including earthquake hazard mitigation and the transportation of goods. Numerous studies have been conducted on the seismic response of free- standing rigid bodies, recent examples include Hogan (1994), Shenton and Jones (1991b), Nigbor et al. (1994), and Chopra and Zhang (1991). The stability of free-standing objects or packages that are being shipped or transported by truck, rail, or conveyor, is also governed by the problem considered here. In both the seismic and transport problems, it is generally as- sumed (either explicitly or implicitly) that the object will either slide or rock from a rest mode. In the past, the mode that governed had been determined by comparing the available static friction to the width-to-height ratio of the object, irre- spective of the magnitude of the ground acceleration. The in- itial mode depends not only on the width-to-height ratio and static friction, but also on the magnitude of the base acceler- ation. Sinopoli (1987) derived an equation for the minimum fric- tion required to sustain a free-rocking motion (i.e., foundation at rest). In a discussion of a paper by Shenton and Jones (1991a), Sinopoli (1993) presented criteria for the initiation of a slide-rock mode from rest for a block on an accelerating foundation. Details of the derivation of the criteria were not presented or referenced; furthermore, the discussion suggested 'Asst. Prof., Dept. ofCiv. and Envir. Engrg.• Univ. of Delaware, New- ark. DE 19716. Note. Associate Editor: John L. Tassoulas. Discussion open until De- cember I, 1996. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this technical note was submitted for review and possible publication on August 3. 1995. This technical note is part of the Journal of Engineering Mechanics, Vol. 122, No.7, July. 1996. ©ASCE, ISSN 0733-9399/96/ 0007-0690-06931$4.00 + $.50 per page. Technical Note No. 11307. 690/ JOURNAL OF ENGINEERING MECHANICS / JULY 1996 J. Eng. Mech., 1996, 122(7): 690-693 Downloaded from ascelibrary.org by Universita di Bologna on 08/14/16. Copyright ASCE. For personal use only; all rights reserved.