Postbuckling behavior and imperfection sensitivity of elastic structures by the Lyapunov-Schmidt-Koiter approach

Abstract

Beginning with the work of Koiter in 1945, valuable insights into the postbuckling behavior of structures have been gained by Lyapunov-Schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point. Here this methodology is generalized to include nonlinear prebuckling behavior, as well as multiple, not necessarily coincident buckling modes. The expansion of the reduced equilibrium equations is performed about a reference point (which need not coincide with any of the bifurcation points), and applies no matter whether the modes are coincident, closely spaced, or well separated. From a variety of possible decompositions of the admissible space of displacements, two are incorporated into a finite element program. Theoretical considerations, and numerical examples in which asymptotic results are compared to 'exact' results, indicate that one of the decompositions has some important advantages over the other. Examples include a shallow arch, and a beam on elastic foundation problem exhibiting symmetry-breaking modal interaction.

Buckling of a compliant hollow cylinder attached to a rotating rigid shaft

“…The first-order correction t 1 is the linear mode from Section III. The asymptotic post-buckling expansion method [12][13][14][15][16][17][18] proceeds by inserting the expansion in Eqs.19-20 into the non linear equilibrium written earlier in Eq.5 as…”

Selection of hexagonal buckling patterns by the elastic Rayleigh-Taylor instability

“…This expansion method is known under different names, such as 'Koiter's method'-although asymptotic expansion method of this kind have been developed well before Koiter by Lyapunov, Schmidt and others. It has been successfully used to analyze post-bifurcation in a variety of elastic structures in the past (Koiter, 1945;Hutchinson, 1967;Hutchinson and Koiter, 1970;Budiansky, 1974;Peek and Triantafyllidis, 1992;Peek and Kheyrkhahan, 1993). There exist several variants of Koiter's method: we use the one without imperfection and with multiple linear modes.…”

Analysis of post‐buckling branches at multiple symmetric bifurcations

“…The method to examine the singular points used here is based on a LyapunovSchmidt decomposition of the solution space and an asymptotic expansion method where the displacements and loads as well as the equilibrium equations are expanded into Taylor series around the singular point. Other work on asymptotic expansion and Lyapunov-Schmidt decomposition relevant to this article have been done by Thompson and Hunt [7], Triantafyllidis and Peek [8], Peek and Kheyrkhahan [9] and Kouhia and Mikkola [10].…”