Shakedown Limit Load Determination for a Kinematically Hardening 90 deg Pipe Bend Subjected to Steady Internal Pressures and Cyclic Bending Moments

A simplified technique for determining the lower bound shakedown limit load of a structure, employing an elastic–perfectly plastic (EPP) material model, was previously developed and successfully applied to a long radius 90 deg pipe bend (Abdalla et al., 2006, “Determination of Shakedown Limit Load for a 90 Degree Pipe Bend Using a Simplified Technique,” ASME J. Pressure Vessel Technol., 128, pp. 618–624). The pipe bend is subjected to steady internal pressure magnitudes and cyclic bending moments. The cyclic bending includes three different loading patterns, namely, in-plane closing, in-plane opening, and out-of-plane bending moment loadings. The simplified technique utilizes the finite element (FE) method and employs a small displacement formulation to determine the shakedown limit load without performing lengthy time consuming full elastic-plastic (ELPL) cyclic loading FE simulations or conventional iterative elastic techniques. In the present research, the simplified technique is further modified to handle structures employing an elastic-linear strain hardening material model following Ziegler’s linear kinematic hardening (KH) rule. The shakedown limit load is determined through the calculation of residual stresses developed within the pipe bend structure accounting for the back stresses, determined from the KH shift tensor, responsible for the rigid translation of the yield surface. The outcomes of the simplified technique showed an excellent correlation with the results of full ELPL cyclic loading FE simulations. The shakedown limit moments output by the simplified technique are utilized to generate shakedown diagrams (Bree diagrams) of the pipe bend for a spectrum of steady internal pressure magnitudes. The generated Bree diagrams are compared with the ones previously generated employing the EPP material model. These indicated relatively conservative shakedown limit moments compared with the ones employing the KH rule.