Keywords: shakedown analysis, n-dimensional loading space, limited kinematical hardening, interior-point algorithm, convex optimization, nonlinear programming.

Determining the load bearing capacity is essential for the design of engineering structures, but poses a challenging task in the case of varying thermo-mechanical loading beyond the elastic limit. In general, if the loading varies with time, the limit state of the elasto-plastic structure considered is defined by: instantaneous collapse, alternating plasticity, or incremental collapse; often referred to as ratcheting in the case of cyclic loading. However, if none of these failure mechanisms occur, the system shakes down and thus is designated "safe". To compute the corresponding maximum loading factor by means of the conventional step-by-step methods generally leads to cumbersome computations. Moreover, the whole loading history needs to be given deterministically, which is not realistic in many technical applications. As an alternative, shakedown analysis constitutes an appropriate tool to avoid these problems.

For most engineering structures, it is inevitable to adequately reflect the material behaviour, which is required for taking into account limited kinematical hardening. Furthermore, complex loading situations arise in most technical applications, which can only be represented by considering n-dimensional loading spaces. Admittedly, if kinematical hardening is considered, shakedown analysis is, as yet, restricted to either one or two independently varying loads. As a consequence, the aim of this paper is to incorporate both limited kinematical hardening and complex loading situations at the same time.

For this, the statical shakedown theorem by Melan was extended to limited kinematical hardening by means of a two-surface model. Therein, kinematical hardening was regarded as movement of the yield surface within the stress space, limited by a second surface, the bounding surface. Both surfaces were described by the von Mises criterion. Then, this extended theorem was transferred to a convex nonlinear optimization problem in a generalized manner, such that arbitrary numbers of loadings could be included. Subsequently, the resulting optimization problem was solved using the interior-point algorithm IPSA, recently developed by the author.

Concluding, the first three-dimensional shakedown domain accounting for hardening is presented for a flanged pipe subjected to thermo-mechanical loading, which illustrates the potential of the proposed method. Since shakedown analysis has been restricted to special cases, up to now, its operational area was mainly confined to academic problems. The new approach presented in this paper eradicates these restrictions and thus facilitates closing the gap between academic research and practical application.

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