Element behavior in post-critical plane frame analysis

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Depending on the type of uni-modal (cuspoid) catastrophe which governs the local behavior of a structure at a simple critical point, correct prediction of the post-critical path requires an accurate estimation of a number of terms in the Taylor series expansion of the total potential function at the critical point. This imposes certain conditions on the accuracy of the adopted non-linear elements. These conditions are mainly related to the quality of the kinematic assumptions which underlie the strain definition used in the model. If these assumptions are not accurate enough, the fourth and higher order terms in the Taylor expansion of the total potential function will not be correctly represented. As a consequence, the model will fail to predict correctly even the initial post-buckling behavior whenever the criticality at hand is more complex than the fold (limit point, asymmetric bifurcation). This proves to be the case for the so-called ‘technical’ beam models. This inability is inherited by any constructed finite element model, regardless of the interpolation function used in its definition and of the number of elements used in the discretization. An improved non-linear model based on the treatment by Antman (Bifurcation problems in non-linearly elastic structures, in: P.H. Rabinowitz, ed., Application of Bifurcation Theory, Academic Press, NY, 1977), is adopted, and several finite elements are developed on the basis of this model. These elements are tested for a number of problems for which the critical behavior is governed by fold, cusp and butterfly singularities. The numerical results outline the importance of the ‘small’ kinematic terms, especially in conjunction with the occurrence of higher-order uni-modal singularities. Some risks for numerical locking are pointed out and remedied. The recommended element leads to a computational cost, which is comparable to an implemented ‘shallow arch’ element.