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Computational Technology Reviews
ISSN 2044-8430
Computational Technology Reviews
Volume 2, 2010
Sensitivity Analysis of the Postbuckling Behavior of Elastic Structures
H.A. Mang, G. Höfinger and X. Jia

Institute for Mechanics of Materials and Structures, Vienna University of Technology, Austria

Full Bibliographic Reference for this paper
H.A. Mang, G. Höfinger, X. Jia, "Sensitivity Analysis of the Postbuckling Behavior of Elastic Structures", Computational Technology Reviews, vol. 2, pp. 1-22, 2010. doi:10.4203/ctr.2.1
Keywords: imperfection insensitivity, Koiter's postbuckling analysis, consistently linearized eigenvalue problem, bifurcation buckling, hilltop buckling, zero-stiffness postbuckling.

Summary
The purpose of sensitivity analysis of the postbuckling behavior of elastic structures is to convert imperfection-sensitive into imperfection-insensitive structures through minor changes of the original design. The motivation for such a conversion is that imperfect structures corresponding to imperfection-insensitive perfect structures experiencing loss of stability of the prebuckling path by means of bifurcation buckling are characterized by monotonically increasing load-displacement paths. Imperfect structures corresponding to imperfection-sensitive perfect structures fail by snap-through at a lower load level than the load at which bifurcation buckling occurs for the perfect structure. Optimization restricted to the buckling load disregarding the postbuckling behavior can produce highly imperfection-sensitive structures and is thus unrewarding.

In order to study the postbuckling behavior in the vicinity of the bifurcation point, Koiter’s initial postbuckling analysis is employed. The basic idea of this method is to find a series expansion for the secondary path in terms of an independent path parameter [1]. A structure is imperfection insensitive if the coefficient of the linear term of the load along the secondary path vanishes and the coefficient of the quadratic term is positive.

The consistently linearized eigenvalue problem represents a linearized version of the nonlinear stability problem formulated for arbitrary points on the primary path. The results are eigenvalue functions and eigenvector curves, which, in the vicinity of the true stability limit, become estimates of the buckling load and the buckling mode, respectively. The eigenvector curve is used to distinguish between two classes of problems. The first class is the general one for which a conversion from imperfection sensitivity into imperfection insensitivity is not always possible. The second class aggregates all examples for which the eigenvector curve forms a straight line. In this class, stiffening the system by increasing a design parameter causes the coefficient of the quadratic term of the load in Koiter's postbuckling analysis to increase monotonically, eventually rendering the structure imperfection insensitive [2].

Hilltop buckling, i.e. the coincidence of a bifurcation point with a snap-through point, is always imperfection sensitive. Zero-stiffness postbuckling, characterized by a constant load level along the secondary path, represents the most favorable form of transition from imperfection sensitivity to insensitivity because all coefficients with an even subscript in the expansion of the load along the secondary path become positive. Zero-stiffness as such is imperfection insensitive.

References
[1]
W.T. Koiter, "On the stability of elastic equilibrium", (Translation of "Over de stabiliteit van het elastisch evenwicht", 1945), Technical Report, Polytechnic Institute Delft, H.J. Paris Publisher Amsterdam, NASA TT F-10, 833, 1967.
[2]
H.A. Mang, G. Hoefinger, X. Jia, "On the interdependency of primary and initial secondary equilibrium paths in sensitivity analysis of elastic structures", Computer Methods in Applied Mechanics and Engineering, 200(13-16), 1558-1567, 2011. doi:10.1016/j.cma.2010.12.025

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