Shakedown and Safety Assessment

Abstract

The main subjects of this chapter are the variational formulations for the computation of a safety factor in shakedown analysis, the numerical procedures available to implement this analysis, and an overview on the wide field of applications and conceptual extensions. In structural engineering and solid mechanics, the word shakedown , introduced in the 1940s by Prager, became, since then, synonymous with elastic adaptation in the presence of variable loadings, a safe stabilized phenomenon, often due to some limited plastic deformation (or equivalently, due to the residual stress distribution associated with inelastic strains). Concerning the failure analysis of ductile structures, the terms alternating plasticity (AP) and incremental collapse (IC), besides plastic collapse , are also widely used to identify failure modes under variable loadings. A structure undergoes AP when the fluctuating loading program produces, after any arbitrarily large time, some plastic deformation as well as a subsequent vanishing of the net plastic deformation. This induces failure due to low‐cycle fatigue. Likewise, the structure fails by IC when plastic deformations accumulate in the form of a compatible strain distribution that leads to excessive inelastic deformation. Shakedown analysis allows working under the realistic assumption that only the range of variable loadings is known, unlike the usual prescription of a particular loading history. That is, we deal in this chapter with direct methods that are based solely on the knowledge of a range of load variations (or a reference loading, in limit analysis). It is worth emphasizing that the differences between incremental and direct methods go beyond the computational advantages frequently obtained with the latter ones. In fact, the fundamental questions about whether or not critical loads or cycles do exist, independently from loading histories, can only be answered by the theories of direct methods. The mathematical foundations of shakedown theory strongly rest on optimization theory and the related areas of convex analysis and mathematical programming. This chapter contains simplified derivations of the static, kinematic, and mixed variational principles of shakedown theory by means of the basic techniques of convex analysis. Moreover, numerical procedures suitable to solve the shakedown analysis problem are discussed here in the framework of mathematical programming techniques combined with common procedures in the field of finite element methods. Applications in fundamental examples are then presented, namely (i) a restrained block under thermomechanical loadings; (ii) the classical Bree problem of a tube under variable pressure and temperature, with a more realistic model and including an exact solution; (iii) a plate with a circular hole subjected to variable tractions; (iv) a plate with simulated imperfections showing critical mechanisms with strongly localized deformations; and (v) a pipe elbow under internal pressure and in plane bending moment.

An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

“…For the load λL with λ > ζ * , no solution of the primal and dual problems exists; the body is unable to sustain the loading and collapses. Also, we note the similarity between (1.1) and the shakedown analysis problem (see [33] and the references therein).…”