Asymptotic techniques in elastic-plastic analysis of structures

Abstract

Elastic-plastic structures can nowadays be analyzed with the powerful numerical procedures of the finite element method. Nevertheless, in many engineering applications, analytical expressions capable of predicting with sufficient accuracy the stress distributions, the extent of the plastic zones and the load displacement behavior could be of great practical value. For simple structures and loading stages not too far from the elastic limit, such analytical expressions may be obtained by using perturbation methods and asymptotic expansions. A small dimensionless parameter ϵ is defined as the ratio of a length characterizing the extent of the narrow plastic zone, to a conveniently chosen typical dimension of the structure. Stresses and displacements are formally expanded as asymptotic series in terms of powers of ϵ. For each order of magnitude, the exact basic relations lead to a separate set of simplified differential equations which can be integrated analytically or numerically by using standard procedures. The method is very general and can be applied to several classes of plastic behaviour and of structural problems. Three examples of very simple structures are chosen in particular to illustrate the applicability of the perturbation method to engineering problems: (1) For a simply supported beam under a concentrated load, ϵ is chosen as the ratio of the largest axial extension of the plastic zone to twice the beam length. (2) The case of a long thin cylindrical shell subject to a gradually increasing radial ring of load is algebraically more involved but can be treated similarly. (3) For a simply supported circular plate under a ring of load, ϵ is defined as the thickness ratio of the central plastic zone which is assumed to be sufficiently thin in the first stage of loading beyond the elastic limit. Both Tresca's and von Mises' yield criteria with the associated flow rule (rate theory) can be used. The method can also be applied to more involved geometries (in conjunction with linear FE-procedures) and more complex material behaviour.