Rocking bodies with arbitrary interface defects: Analytical development and experimental verification

Abstract

SummaryThe rocking response of a rigid, freestanding block in two dimensions typically assumes perfect contact at the base of the block with instantaneous impacts at two distinct, symmetric rocking points. This paper extends the classical two-dimensional rocking model to account for an arbitrary number of rocking points at the base representing geometric interface defects. The equations of motion of this modified rocking system are derived and presented in general terms. Energy dissipation is modeled assuming instantaneous point impacts, yielding a discrete angular velocity adjustment. Whereas this factor is always less than unity in the classical model, it is possible for this factor to exceed unity in the presented model, yielding a finite increase in the angular velocity at impact and a markedly different rotational response than the classical model predicts. The derived model and the classical model are numerically integrated and compared to the results of recent shake table tests. These comparisons show that the new model significantly enhances agreement in both peak angular displacement and motion decay. The equations of motion and the energy dissipation of the presented model are further investigated parametrically considering the size of the defect, the number of rocking points, and the aspect ratio and size of the block. | INTRODUCTIONThe original motivation for studying freestanding and rocking structures stems from the observed overturning of small, slender structures compared to the apparent stability of certain larger, slender structures following strong earthquake excitations. 1 More recently, motivation has shifted to the intentional design of rocking structures in an effort to harness their inherent energy dissipation and self-centering mechanisms (e.g., Ref [ 2 ], and references therein). Additionally, the class of freestanding and rocking structures includes water towers, gravestones, nuclear radiation shields, mechanical and electrical equipment, and culturally significant statues and columns. Since many of these components are critical to a facility's post-earthquake functionality or are culturally significant, accurate prediction of the seismic response is necessary.The well-known and broadly utilized classical rocking model was introduced in the pioneering work of Housner. 1 This rocking model considers a two-dimensional, symmetric, rectangular, rigid block that oscillates about two rocking points at its base atop a rigid foundation. The model further assumes that there is sufficient friction to prevent sliding and that the block transitions smoothly between rocking points through an inelastic impact with conservation of angular momentum. The response of this block is modeled as a set of piecewise equations of motion and an instantaneous reduction of angular velocity at the moment of impact. This velocity reduction is typically known as a coefficient of restitution and is strictly a function of geometry

Is rocking motion predictable?

“…26 Further investigation of contact surface stiffness and imperfection effects on rocking motion are available in ElGawady et al 44 and Wittich and Hutchinson. 52…”

Free‐rocking tests of a freestanding object with variation of center of gravity

“…[ 56 ] The main cause was the imperfections at the interface of the rocking system. [ 33,42,43,45 ] This phenomenon was discovered in the experimental and analytical works of Lipscombe and Pellegrino (1993), ] 32 ] ElGawady et al. (2011), [ 33 ] Wittich and Hutchinson (2016), [ 46 ] and Kalliontzisa and Sritharan (2018).…”

Rigid Impacts of Three-dimensional Rocking Structures

“…can be anywhere in the interval r ∈ (r Housner , 1). Wittich and Hutchinson [37] examined a model with arbitrary geometric imperfection and demonstrated how the corresponding value of r can be determined under assumptions H1, H4a, H5. Chatzis et al [10] seems to be the first one to notice that if assumptions H1, H4a, H5 are combined with any type of infinitesimally small geometric imperfection, the emerging discrete or continuous distribution of linear momentum can be replaced by an instantaneous resultant momentum [P x , P y ] T acting in a single point.…”