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

结构系统在超出弹性极限并承受不同载荷的情况下的使用寿命取决于所谓的棘轮机制是否发展，这种机制在连续的载荷循环中导致应变单向增加（渐进变形）。这一过程最终可能会在达到安定状态时停止。对于结构的寿命评估，需要将遇到的累积应变最大值与预定的应变极限进行比较，并进行疲劳分析，这需要将安定状态下的应变范围作为输入量。这些量通常是通过对结构的有限元模型进行多次荷载循环的增量弹塑性分析来计算的。所需的计算量可能相当于线性弹性分析的一万倍。另一方面，简化塑性区理论（STPZ）允许以简化的方式近似确定安定条件下的结构行为，而不必详细查看荷载循环的顺序。仅通过一些线性弹性分析，就可以获得累积应变的估计，以及应变范围，位移等。迄今为止，STPZ仅存在于一阶理论内的计算中，其中变形必须很小，足以为未变形的系统制定平衡条件。然而，很明显，在可能发展棘轮机制的结构中，由于渐进变形可能会产生二阶几何效应，这使得有必要在变形系统（二阶理论）上制定平衡条件。因此，根据拟议的项目，将开发二级STPZ。第一个积极的经验，已经存在于单轴应力条件下的简单结构，并且受限于几个自由度，使该项目看起来很有希望。然而，它们完全基于特别开发的方法，必须加以推广并置于系统的基础上，以便能够处理具有多轴应力状态和多个自由度的更复杂的结构。这使得检测由于棘轮引起的屈曲载荷减少成为可能。迄今为止，这种现象在简化方法的科学文献中很少受到关注。这些目标是非常重要的，因为目前还没有其他机械上合理的简化计算方法可以量化硬化材料安定状态下的应变，特别是不考虑二阶效应。

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.