Determination of post-shakedown quantities with the Simplified Theory of Plastic Zones including second-order geometric effects

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The service life of a structural system, operated beyond the elastic limit and subjected to varying loads, depends on whether a so-called ratcheting mechanism develops, which leads to a one-way increase of strain in successive load cycles (progressive deformation). This process may eventually come to a halt when the state of shakedown is achieved.For life assessment of structures, the maximum value of the accumulated strain encountered is to be compared with predetermined strain limits, and a fatigue analysis is to be performed, which requires the strain range in the shakedown condition as an input quantity. These quantities are usually calculated using incremental elastic-plastic analyses of a finite element model of the structure over many load cycles. The required computational effort may correspond to ten thousand times a linear elastic analy-sis.The Simplified Theory of Plastic Zones (STPZ) on the other hand allows the approximate determination of the structural behavior in the shakedown condition in a simplified manner, without having to look at the sequence of load cycles in detail. With only a few linear elastic analyses an estimate of the accumu-lated strain is obtained, as well as the strain range, the displacements, etc.To date the STPZ exists only for calculations within first order theory, where the deformations have to be so small that it is sufficient to formulate the equilibrium conditions for the undeformed system. However, it is obvious that in structures that may develop a ratcheting mechanism, second order geo-metric effects may arise due to the progressive deformation, which make it necessary for the equilibri-um conditions to be formulated at the deformed system (second order theory). Under the proposed project therefore the STPZ of second order is to be developed.First positive experiences, already existing for simple structures with uniaxial stress conditions and lim-ited to a few degrees of freedom allow the project to seem promising. However, they are based solely on ad hoc developed approaches that must be generalized and placed on a systematic basis so that more complex structures with multiaxial stress states and many degrees of freedom can be treated.This makes it possible to detect reduction of buckling loads due to ratcheting. This phenomenon has so far received little attention in the scientific literature of simplified methods.These goals are very important because no other mechanically sound simplified calculation methods are known that can quantify the strains in the shakedown condition of hardening materials, especially not considering effects of second order.