The stability of elastic equilibrium
01 Feb 1970-
TL;DR: In this article, a general theory of elastic stability is presented, augmented by an investigation of the buckled structure in the immediate neighborhood of the bifurcation point, which explains why some structures, such as a flat plate supported along its edges and subjected to thrust in its plane, are capable of carrying loads considerably above the buckling load, while other structures, e.g., an axially loaded cylindrical shell, collapse at loads far below the theoretical critical load.
Abstract: : A general theory of elastic stability is presented. In contrast to previous works in the field, the present analysis is augmented by an investigation of the behavior of the buckled structure in the immediate neighborhood of the bifurcation point. This investigation explains why some structures, e.g., a flat plate supported along its edges and subjected to thrust in its plane, are capable of carrying loads considerably above the buckling load, while other structures, e.g., an axially loaded cylindrical shell, collapse at loads far below the theoretical critical load.
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TL;DR: In this article, the authors present a unified, general presentation of the basic theory of buckling and post-buckling behavior in a form suitable for application to a wide variety of special problems.
Abstract: Publisher Summary The general theory of buckling and post-buckling behavior of elastic structures has led to a considerable amount of research in this field. A comprehensive survey provides a very useful bibliography, together with an overview of the achievements, status, and goals of post-buckling theory. The chapter provides a unified, general presentation of the basic theory in a form suitable for application to a wide variety of special problems. This will be done with the help of the succinct notation of functional analysis, which turns out to be remarkably appropriate for the purpose. Simple conceptual models can illustrate with remarkable verisimilitude many of the essential characteristics of the buckling and post-buckling behavior of more complicated structural systems. Before undertaking a general analysis of arbitrary elastic structures, such models are exploited in order to expose basic concepts of bifurcation buckling, snap buckling, imperfection-sensitivity, load-shortening relations, and stability.
486 citations
TL;DR: In this paper, a set-theoretical, convex description of uncertainty is discussed in detail, where uncertainty is described as a set of constraints unlike the classical probabilistic approach, and instead of conventional optimization studies, where the minimum possible responses are sought, here an uncertainty modeling is developed as an "anti-optimization" problem of finding the least favorable response under the constraints within the set theoretical description.
272 citations
TL;DR: In this paper, a model of a strut-on-a-wool structural system with a subcritical post-buckling response is presented, with localized buckles first forming and then locking up in sequence.
Abstract: A long structural system with an unstable (subcritical)post-buckling response that subsequently restabilizes typically deformsin a cellular manner, with localized buckles first forming and thenlocking up in sequence. As buckling continues over a growing number ofcells, the response can be described by a set of lengthening homoclinicconnections from the fundamental equilibrium state to itself. In thelimit, this leads to a heteroclinic connection from the fundamentalunbuckled state to a post-buckled state that is periodic. Under suchprogressive displacement the load tends to oscillate between twodistinct values. The paper is both a review and a pointer tofuture research. The response is described via a typical system, asimple but ubiquitous model of a strut on a foundation which includesinitially-destabilizing and finally-restabilizing nonlinear terms. Anumber of different structural forms, including the axially-compressedcylindrical shell, a typical sandwich structure, a model of geologicalfolding and a simple link model are shown to display such behaviour. Amathematical variational argument is outlined for determining the globalminimum postbuckling state under controlled end displacement (rigidloading). Finally, the paper stresses the practical significance of aMaxwell-load instability criterion for such systems. This criterion,defined under dead loading to be where the pre-buckled and post-buckledstate have the same energy, is shown to have significance in the presentsetting under rigid loading also. Specifically, the Maxwell load isargued to be the limit of minimum energy localized solutions asend-shortening tends to infinity.
237 citations
Cites background from "The stability of elastic equilibriu..."
...where∇4 is the two-dimensional bi-harmonic operator, x ∈ R is the axial andy ∈ [0,2πr) is the circumferential coordinate, w is the outward radial displacement measured from an unbuckled state, and φ is a stress function [36]....
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...Experimentally a well defined number, s, of periodic waves is observed circumferentially [36, 37] in the buckled deformation, corresponding to an invariance under rotation of 2 π/s....
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TL;DR: The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytical and numerical methods to study the post-bifurcation behavior of periodic solutions as mentioned in this paper.
Abstract: The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytical and numerical methods to study the post-bifurcation behaviour of periodic solutions. It is shown that wrinkling is highly unstable owing to the nonlinear interaction among the multiple modes associated with the critical compressive state. Concomitantly, wrinkling is sensitive to exceedingly small initial imperfections that significantly reduce the compressive strain at which the instability occurs. The study provides insight into the connection between wrinkling and an alternative surface mode, the finite amplitude crease or sulcus. The shape of the critical combination of wrinkling modes has the form of an incipient crease, and a tiny initial imperfection can trigger a wrinkling instability that collapses into a crease.
210 citations
Cites background or methods from "The stability of elastic equilibriu..."
...The type of nonlinear coupling among simultaneous modes in wrinkling is rare but it is similar to that in two structural problems that also have multiple buckling modes and are notoriously imperfection-sensitive—the elastic buckling of cylindrical shells under axial compression (Koiter 1945; van der Heijden 2009), and spherical shells under external pressure (Hutchinson 1967)....
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...In §3, Koiter’s postbifurcation (Koiter 1945; van der Heijden 2009) approach is presented as relevant to wrinkling—the finite deformation of a nonlinear elastic solid with multiple modes associated with the critical bifurcation stress....
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...Only the lowest order influence of the imperfections is sought following an approach similar to that of Koiter (Koiter 1945; van der Heijden 2009)....
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...…it is similar to that in two structural problems that also have multiple buckling modes and are notoriously imperfection-sensitive—the elastic buckling of cylindrical shells under axial compression (Koiter 1945; van der Heijden 2009), and spherical shells under external pressure (Hutchinson 1967)....
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TL;DR: In this article, a new deterministic approach is presented for determining the lower bound of the buckling load of thin-walled cylindrical composite shells, which is derived from phenomenological test data.
Abstract: Thin-walled shell structures like circular cylindrical shells are prone to buckling. Imperfections, which are defined as deviations from perfect shape and perfect loading distributions, can reduce the buckling load drastically compared to that of the perfect shell. Design criteria monographs like NASA-SP 8007 recommend that the buckling load of the perfect shell shall be reduced by using a knock-down factor. The existing knock-down factors are very conservative and do not account for the structural behaviour of composite shells. To determine an improved knock-down factor, several authors consider realistic shapes of shells in numerical simulations using probabilistic methods. Each manufacturing process causes a specific imperfection pattern; hence for this probabilistic approach a large number of test data is needed, which is often not available. Motivated by this lack of data, a new deterministic approach is presented for determining the lower bound of the buckling load of thin-walled cylindrical composite shells, which is derived from phenomenological test data. For the present test series, a single pre-buckle is induced by a radial perturbation load, before the axial displacement controlled loading starts. The deformations are measured using the prototype of a high-speed optical measurement system with a frequency up to 3680 Hz. The observed structural behaviour leads to a new reasonable lower bound of the buckling load. Based on test results, the numerical model is validated and the shell design is optimized by virtual testing. The results of test and numerical analysis indicate that this new approach has the potential to provide an improved and less conservative shell design in order to reduce weight and cost of thin-walled shell structures made from composite material.
204 citations