Piping Elbow Cyclic Analyses for Shakedown Verification

Six Cyclic elasto-plastic Finite Element Analyses are being performed in this paper. The Finite Element Model considered here is a Piping Elbow with a straight section of pipe on each side. This Model consists of shell elements. In each cyclic analysis, the internal pressure is kept constant, and is in fact the same for all six analyses. The various Finite Element Analyses are performed using different types of boundary conditions and different types of cyclic loading (thermal expansion in the elbow and applied rotation at the free end of the model). The main purpose of this paper is to address both the global shakedown and the through-thickness shakedown. The global shakedown is defined as the stabilization of the elbow, as far as the global in-plane displacements are concerned. The through-thickness shakedown is defined as the stabilization of the radial displacements on the side of the elbow. The main results from these cyclic analyses are the displacements. The decisions whether shakedown occurs, or not, are based on the displacements and are shown on the x-y Diagram, also called the “shakedown vs. ratcheting” Diagram. An evaluation of this Diagram leads to the justification of the revised elbow C₂ stress index, at least for the global in-plane displacements of the piping elbow. The variations of the mid-thickness strain values and of the maximum strain values, as a function of the cycles, have not been evaluated at this time. This should be done at a later date.