Toward a unified model for elastoplastic structural analysis

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28028/1/0000467.pd

Dimensional analysis of the earthquake-induced pounding between adjacent structures

In this paper the dynamic response of two and three pounding oscillators subjected to pulse-type excitations is revisited with dimensional analysis. Using Buckingham's Π-theorem the number of variables that govern the response of the system is reduced by three. When the response is presented in the dimensionless Π-terms remarkable order emerges. It is shown that regardless of the acceleration level and duration of the pulse all response spectra become self-similar and follow a single master curve. This is true despite the realization of finite duration contacts with increasing durations as the excitation level increases. All physically realizable contacts (impacts, continuous contacts, and detachments) are captured via a linear complementarity approach. The study confirms the existence of three spectral regions. The response of the most flexible among the two oscillators amplifies in the low range of the frequency spectrum (flexible structures); whereas, the response of the most stiff among the two oscillators amplifies at the upper range of the frequency spectrum (stiff structures). Most importantly, the study shows that pounding structures such as colliding buildings or interacting bridge segments may be most vulnerable for excitations with frequencies very different from their natural eigenfrequencies. Finally, by applying the concept of intermediate asymptotics, the study unveils that the dimensionless response of two pounding oscillators follows a scaling law with respect to the mass ratio, or in mathematical terms, that the response exhibits an incomplete self-similarity or self-similarity of the second kind with respect to the mass ratio

Experimentation on Analogue Models

Summary Analogue models are actual physical setups used to model something else. They are especially useful when what we wish to investigate is difficult to observe or experiment upon due to size or distance in space or time: for example, if the thing we wish to investigate is too large, too far away, takes place on a time scale that is too long, does not yet exist or has ceased to exist. The range and variety of analogue models is too extensive to attempt a survey. In this article, I describe and discuss several different analogue model experiments, the results of those model experiments, and the basis for constructing them and interpreting their results. Examples of analogue models for surface waves in lakes, for earthquakes and volcanoes in geophysics, and for black holes in general relativity, are described, with a focus on examining the bases for claims that these analogues are appropriate analogues of what they are used to investigate. A table showing three different kinds of bases for reasoning using analogue models is provided. Finally, it is shown how the examples in this article counter three common misconceptions about the use of analogue models in physics

Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure

In an earlier report (TAM Report No. 80), the authors considered the buckling and post-buckling behavior of an ideal elastic cylindrical shell loaded by uniform external pressure on its lateral surface, and by an axial compressive force. Assumptions were introduced which reduced the shell to a system with one degree of freedom. The present investigation is a generalization and a refinement of this theory. The shell is treated as a system with 21 degrees of freedom. By the imposition of constraints on the 21 generalized coordinates, various end conditions can be realized; for example, simply supported ends with flexible end plates (no axial constraint), simply supported ends with rigid end plates, and clamped ends. Also, effects of reinforcing rings have been incorporated in a more general way than in TAM Report No. 80. The restrictive assumption that the centroidal axis of a ring coincides with the middle surface of the shell has been eliminated. A pressure-deflection curve for an ideal cylindrical shell that is loaded by external pressure has the general form shown in Figure 1. The falling part of the curve (dotted in the figure) represents unstable equilibrium configurations. Also, the continuation of line OE (dotted) represents unstable unbuckled configurations. Actually, the shell snaps from some configuration A to another configuration B, as indicated by the dashed line in Figure 1. Theoretically, point A coincides with the maximum point E on lhe curve, but initial imperfections and accidental disturbances prevent the shell from reaching this point. Point E is the buckling pressure of the classical infinitesimal theory (called the "Euler crítical pressure", since Euler applied the infinitesimal theory to columns). To some extent, point A is indeterminate, but it is presumably higher than the minimum point C unless the shell has excessive initial dents or lopsidedness. In TAM Report No. 80. a hypothesis of Tsien was used to locate point A. In the present investigation, point A is not considered. Rather, attention is focused on the development of a theory that will determine the en tire load-deflection curve. For short thick shells, such as the inter-ring bays of a submarine hull, the Euler critical pressures, determined by TAM Report No. 80, are too high, presumably because the assumption that the shell buckles without incremental hoop strain is inadmissible in this range. The present report corrects this error. Numerical data on the Euler critical pressures of shells with simply supported ends and flexible end plates have been obtained with the aid of lhe Illiac, an electronic digital computer. The data are tabulated at the end of this reporto For short shells without rings, the buckling pressures are appreciably lower than those determined by von Mises' theory. The numerical data for the Euler buckling pressures of sheUs with uniformly spaced reinforcing rings are sufficiently extensive to permit interpolation to estimate effects of various ring sizes. Some exploratory numerical investigations of post-buckling behavior have been conducted with the Illiac. lt is not feasible, at the present time, to handle nonlinear equilibrium problems for systems with 18 degrees of freedom. Consequently, for the numerical work, some higher harmonics were discarded so that the system was reduaed to 7 degrees of freedom. Even then, the numerical problem is formidable. The calculations were confined principally to the determination of the minimum point C on the post-buckling curve (Figure 1). The pressure at point C is the minimum pressure at which a buckled form can exist. It is found that the ordinate of point C, determined by TAM report No. 80, is somewhat too high. The two theories are compared by a table anq curves at the end of this report

Dimensional response analysis of bilinear systems subjected to non-pulselike earthquake ground motions

The maximum inelastic response of bilinear single-degree-of-freedom systems when subjected to ground motions without distinguishable pulses is revisited with dimensional analysis by identifying time scales and length scales in the time histories of recorded ground motions. The characteristic length scale is used to normalize the peak inelastic displacement of the bilinear system. The paper adopts the mean period of the Fourier transform of the ground motion as an appropriate time scale and examines two different length scales which result from the peak ground acceleration and the peak ground velocity. When the normalized peak inelastic displacement is presented as a function of the normalized strength and normalized yield displacement, the response becomes self similar and a clear pattern emerges. Accordingly, the paper proposes two alternative predictive master curves for the response which involve solely the strength and yield displacement of the bilinear SDOF system in association with either the peak ground acceleration or the peak ground velocity, together with the mean period of the Fourier transform of the ground motion. The regression coefficients that control the shape of the predictive master curves are based on 484 ground motions recorded at rock and stiff soil sites and are applicable to bilinear SDOF systems with post-yield stiffness ratio equal to 2% and inherent viscous damping ratio equal to 5%