Advanced postbuckling and imperfection sensitivity of the elastic-plastic Shanley-Hutchinson model column

The postbuckling behavior and imperfection sensitivity of the Shanley-Hutchinson plastic model column introduced by Hutchinson in 1973 are examined. The study covers the initial, buckled state and the advanced postbuckling regime of the geometrically perfect realization as well as its sensitivity to geometric imperfections.

In Section 1, which is concerned with the perfect structure, a new, simple explicit upper bound for all solutions to the problem is found when the tangent modulus at bifurcation vanishes compared to the linear elastic (unloading) modulus. The difference between the upper bound and the solution to an actual problem is determined by an asymptotic expansion involving hyperbolic trial functions (instead of polynomials) which fulfill general boundary conditions at bifurcation and infinity. The method provides an accurate estimate of the maximum load even if it occurs in an advanced postbuckling state. Finally, it is shown that the maximum load is often considerably larger than the bifurcation load.

Section 2 presents a new asymptotic expansion which is utilized to study the imperfection sensitivity of the Shanley-Hutchinson elastic-plastic model column. The method is mainly characterized by three novel features. Firstly, unlike other expansions which are performed around one or maybe two points, ours takes the total postbuckling path of the geometrically perfect structure as its basis, that is, the equilibrium of an imperfect path is written as the postbuckling path of the perfect structure plus an asymptotic contribution. Secondly, the expansion parameter is chosen as the buckling mode amplitude minus its value at initiation of elastic unloading. In this connection, the asymptotic expansion of initiating elastic unloading to the lowest order given by Hutchinson serves as a kind of boundary value for the asymptotic expression. Thirdly, a new and more suitable set of base functions is introduced to enhance the accuracy of the asymptotic expansion for large imperfection levels without compromising the asymptotic behavior for small imperfections. If an asymptotically exact postbuckling solution for the perfect structure around the maximum load has been obtained by some method, be it numerical or asymptotic, then the prediction of the imperfection sensitivity is asymptotically correct.