Shakedown analysis of bounded kinematic hardening engineering structures under complex cyclic loads: Theoretical aspects and a direct approach

There exist two basic formulations of static shakedown theorem for bounded kinematic hardening (KH) model in literature. Whether the two formulations are equivalent, whether the conditions of loading-path-independence are satisfied, and how to employ the shakedown theorem to obtain the safety margin of realistic engineering structures under cyclic loading are some key issues to be solved and addressed. This paper presents theoretical aspects of the static shakedown theorem of KH and proposes a direct approach for shakedown analysis of KH structures. The presented direct approach allows for both the two shakedown formulations and considers the simultaneous influences of multiple load vertices. The two formulations are suitable for plastic models with different hardening laws and determine different shakedown load boundaries in the region of incremental collapse. Instead of dealing with a complicated problem of mathematical programming, the presented approach performs several linear elastic finite element iterative calculations, which ensures its high computational efficiency. Furthermore, the direct approach is implemented into Abaqus software platform so that it becomes a general tool to handle large-scale shakedown problems of engineering structures with complex geometry and loading conditions. An illustrative sample with analytical solution and three numerical examples from power plant engineering are adopted to validate these theoretical aspects and to present the potential of the direct approach.