Shakedown Analysis of Axisymmetric Nozzles Under Primary and Secondary Cyclic Loads

In this paper, the finite element method is used to develop the lower bound limit for the elastic shakedown analysis of axisymmetric nozzles under periodic loading conditions. The Nonlinear Superposition Method is employed to calculate the lower bound shakedown loads by quoting Melan’s theorem in a nonlinear finite element analysis. The calculation is divided into two separate iterations which are blended with a technique that matches the elastic-plastic part of the analysis with the linear part. In the first part of the calculation, the cyclic load is applied as a static load to generate an elastic stress field in the structure. The same cyclic load is subsequently combined with the constant fraction of the load in the second part of the calculation, and the total load is applied in an elastic-plastic analysis that exceeds the yield limit. For each solution increment, the residual stress is generated from the superposition of the elastic stress field scaled through the applied cyclic load and the shakedown stress field calculated from the nonlinear analysis. The results obtained from the lower bound method are compared with the full cyclic loading analyses based on nonlinear material properties, and this paper discusses the choice of the global shakedown in terms of the radial strain, and the local through thickness shakedown defined by the hoop strain. Furthermore, this paper presents the development of a generic model that emulates the behavior of the finite element model under cyclic loads in a simplified form, with the statistical representation based on a sampling of base-model data for a variety of test cases. The probabilistic method takes variations of the geometrical dimensions, nonlinear material properties, and pressure load as the input parameters, whereas the response variable is defined in terms of the lower bound of the shakedown loads.