Keywords: buckling, mode interaction, snaking, thin-walled components, nonlinearity.

In this paper, recent work on the overall and local buckling mode interaction in I-beams under pure bending [1] is extended to the classical case of I-section struts under axial compression. An analytical model that includes Timoshenko bending theory and is based on variational principles, which describes the nonlinear buckling behaviour, is presented. The inclusion of shear deformability has been shown in the literature [2] to be necessary for modelling the interaction between local and global modes of buckling. A system of nonlinear ordinary differential equations and integral equations that govern the equilibrium response is derived from minimizing the total potential energy of the system. The equations are solved numerically using the numerical continuation package AUTO [3].

The classical Euler buckling mode, which in the current model is assumed to occur first, subsequently interacts with plate buckling, which is observed in the more compressed flanges, and the resulting mechanical response shows an initially flat response under overall buckling. The response is subsequently destabilized at a secondary bifurcation point that marks the onset of the mode interaction with plate buckling. Beyond the secondary bifurcation, both local and overall modes grow in magnitude, but the interaction causes a series of snap-backs to be observed with the plate buckling profile showing a progressive reduction in wavelength arising from the inherent stability of plate buckling being counteracted by the inherent instability of the interactive buckling. This type of progressive destabilization and restabilization sequence in the response has been termed in the literature as cellular buckling [4], although this phenomenon has been observed in physical experiments [5], as far as the authors are aware, this is the first analytical model that has been found to be capable of reproducing it for the I-section strut.

Linear eigenvalue studies demonstrate that the overall mode is triggered at the theoretical Euler load that is corrected for finite shear stiffness. In terms of simulating the nonlinear behaviour, highly encouraging results emerge both in terms of the mechanical destabilization exhibited and the deflection profile of the post-buckling deformation. This demonstrates that the fundamental physics of this system is captured by the analytical approach. A discussion is also presented on how the current model could be enhanced to allow for the case where local buckling is critical for future studies.

1

M.A. Wadee, L. Gardner, "Cellular buckling from mode interaction in I-beams under uniform bending", Proc. R. Soc. A, 468, 245-268, 2012. doi:10.1098/rspa.2011.0400

2

G.W. Hunt, M.A. Wadee, "Localization and mode interaction in sandwich structures", Proc. R. Soc. A, 454(1972), 1197-1216, 1998. doi:10.1098/rspa.1998.0202

3

E.J. Doedel, B.E. Oldeman, "AUTO-07P: Continuation and bifurcation software for ordinary differential equations", Technical report, Department of Computer Science, Concordia University, Montreal, Canada, 2009. http://indy.cs.concordia.ca/auto

4

G.W. Hunt, M.A. Peletier, A.R. Champneys, P.D. Woods, M.A. Wadee, C.J. Budd, G.J. Lord, "Cellular buckling in long structures", Nonlinear Dyn, 21(1), 3-29, 2000.

5

J. Becque, K.J.R. Rasmussen, "Experimental investigation of the interaction of local and overall buckling of stainless steel I-columns", ASCE J. Struct. Eng, 135(11), 1340-1348, 2009. doi:10.1061/(ASCE)ST.1943-541X.0000051

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