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Review Articles

# Asymptotic and Probabilistic Approach to Buckling of Structures and Materials

[+] Author and Article Information
Kiyohiro Ikeda

Department of Civil Engineering, Tohoku University, Sendai 980-8579, Japanikeda@civil.tohoku.ac.jp

Kazuo Murota

Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan

The importance of nonlinearity in a physical behavior was recognized at the beginning of the 20th century through the development of classical nonlinear mathematics (2-4).

The word “asymptotic” means that all results are local, valid only for sufficiently small values of initial imperfection parameters and in a sufficiently close neighborhood of the critical point under consideration. Asymptotic theory appears in various fields of research (see, e.g., Refs. 9-10).

This reduction procedure is called the Liapunov–Schmidt reduction(17,23), the Liapunov–Schmidt–Koiter reduction(24), or the elimination of passive coordinates(25).

The transcritical and pitchfork points are often called asymmetric and symmetric (cusp) points of bifurcation, respectively (25).

The expansion of the potential was employed, for example, in Refs. 40-41,57.

We aim at presenting the main ideas for engineers without sacrificing the mathematical rigor. To this end we restrict ourselves to finite-dimensional equations. For a thorough treatment, the reader is referred to Refs. 17,23.

The piecewise linear law originally found in Thompson and Schorrock (63) corresponds to the case of $B0001ϵ=0$ in Eq. 20.

Details of the numerical analyses are given in Okazawa et al. (228).

Large circles are formed as an assemblage of a number of deformed hexagonal cells and such circles are regularly arranged in space to form this pattern.

We restrict ourselves to a finite-dimensional system.

Appl. Mech. Rev 61(4), 040801 (Jun 30, 2008) (16 pages) doi:10.1115/1.2939583 History: Published June 30, 2008

## Abstract

The general theory of elastic stability invented by Koiter (1945, “On the Stability of Elastic Equilibrium  ,” Ph.D. thesis, Delft, Holland) motivated the development of a series of asymptotic approaches to deal with the initial postbuckling behavior of structures. These approaches, which played a pivotal role in the precomputer age, are somewhat overshadowed by the progress of computational environment. Recently, the importance of the asymptotic approaches has been revived through the extension of their theoretical background and the combination with the framework of finite element method and with group-theoretic bifurcation theory in nonlinear mathematics. The approaches serve as an efficient and insightful strategy to tackle probabilistic scatter of critical loads. We review, through the perspective of theoretical engineers, the historical development and recent revival of the asymptotic approaches for buckling of imperfection-sensitive structures and materials.

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## Figures

Figure 1

Solution curves in the neighborhood of simple critical points expressed by the leading terms of the bifurcation equation. —: path for the perfect system; – – –: path for an imperfect system; thick line: stable path; thin line: unstable path.

Figure 2

Imperfection sensitivity of structures expressed in terms of fc∕fc0 versus ϵ relationships (fc0 is the critical load for the perfect system)

Figure 3

Truss tower

Figure 4

Top two layers of the truss tower. (1)–(5) express node numbers; 1–22 express member numbers.

Figure 5

Figure 6

A hilltop bifurcation point: ○

Figure 7

Probability density function (Eq. 21) of f̃c

Figure 8

Normalized load-displacement curves of steel specimens with L∕H=2,4,6,8,10. P: applied load; u: axial displacement; L: member length; Young’s modulus E=200GPa; Poisson’s ratio ν=0.333; yield stress σY=400MPa; yield strain eY=σY∕E=1∕500.

Figure 9

(a) Imperfection patterns imposed on members and (b) comparison of histograms and theoretical probability density functions

Figure 10

General view of imperfection sensitivities of plates under compression for various width-thickness ratios [(○): bifurcation point; (●): maximum point; (△): yield point]

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