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

结构系统的使用寿命，超过弹性极限运行，并承受变化的载荷，取决于是否出现所谓的棘轮机制，导致应变在连续的载荷循环中单向增加（渐进变形）。在结构寿命评估中，需要将累积应变的最大值与预先确定的应变限值进行比较，并进行疲劳分析，这就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。所需的计算工作量可能相当于线弹性分析的一万倍。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，而不必详细查看载荷循环的顺序。仅用少量的线弹性分析就可以得到对应变的估计，以及应变范围、位移等。迄今为止，STPZ仅存在于一阶理论的计算中，在一阶理论中，变形必须很小，足以制定未变形系统的平衡条件。然而，很明显，在可能发展棘轮机制的结构中，由于渐进变形，可能会出现二阶几何效应，这使得有必要在变形系统处制定平衡条件（二阶理论）。因此，在拟议的项目下，将开发二阶STPZ。首先，对于具有单轴应力条件的简单结构已经存在的积极经验和有限的几个自由度使该项目看起来很有前途。然而，这些方法都是基于专门开发的方法，这些方法必须被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自由度的更复杂的结构，这使得检测由于棘轮效应引起的屈曲载荷的降低成为可能。这一现象至今在简化方法的科学文献中很少受到注意，这些目标是非常重要的，因为还没有其他机械上可靠的简化计算方法可以量化硬化材料安定条件下的应变，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.