Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

Abstract

This work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner-Mindlin thick plates in shear-bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner-Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second-order cone programming (SOCP) problem and is solved using the industrial software package Mosek. A comprehensive study of the shear-locking phenomenon is also conducted and it is shown that discontinuous elements avoid such a phenomenon quite naturally, whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so-obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates.

A numerical approach to the yield strength of shell structures

“…presenting an infinite strength towards shear forces compared to membrane forces and bending moments. As intensively discussed in (Bleyer et al, 2015a), this assumption imposes that γ = 0 and [[w]] = 0 as kinematic constraints, reducing to the classical Love-Kirchhoff conditions in the case of a plate.…”

“…In the following, the shell will be discretized as an assembly of N E triangular planar facets (or plates). As a result of this discretization, the decoupling between membrane and bending equilibrium and strain compatibility equations in each facet makes it possible to use a combination of membrane and plate bending finite elements which have been developed in (Bleyer and de Buhan, 2014) (lower bound) and (Bleyer et al, 2015a) (upper bound). The practical implementation being very similar to the one described in these papers, only particular aspects related to the shell model will be highlighted for the sake of concision.…”

“…As regards the upper bound kinematic approach, the w6-d plate bending finite element presented in (Bleyer et al, 2015a) is combined with an upper bound membrane element. Hence, a quadratic interpolation of the virtual velocity U and a linear interpolation of the virtual rotation velocity Θ are considered in each element.…”

“…Hence, a quadratic interpolation of the virtual velocity U and a linear interpolation of the virtual rotation velocity Θ are considered in each element. As in (Bleyer et al, 2015a), the considered interpolation may be discontinous from one element to another, the degrees of freedom being attached to the element itself and not to a geometrical node shared by several adjacent elements. Including potential discontinuities of the virtual velocity fields in upper bound kinematic approaches have already proved to be not only more efficient in terms of accuracy of the produced estimates, but also to avoid in a natural way the shear locking phenomenon in the thin plate limit (Bleyer et al, 2015a).…”

A computational homogenization approach for limit analysis of heterogeneous materials

“…In recent years, thanks to the development of such an efficient primal‐dual interior point algorithm, implemented in a commercial optimization software Mosek , limit analysis problems have gained increasing attention. In fact, most commonly used yield criteria can be cast in the form of conic or semi‐definite constraints, and optimization problems involving such constraints can be solved efficiently by such an algorithm, evidenced by recent works .…”