Bounds to shakedown loads

The shakedown analysis of structures under variable loads is considered, and the corresponding variational principles are presented. Separate formulations for elastic (or classical) shakedown, plastic shakedown and incremental collapse are compared. Two upper bounds for classical safety factors are presented and interpreted as the result of separate analysis concerning alternating plasticity and prevention of simple mechanisms of incremental collapse. As a ®rst application, exact solutions are obtained for a closed tube under variable pressure and temperature. This example is the Bree problem with logarithmic temperature variation across the wall. A ®nite element procedure for shakedown analysis of tubes is then presented. This numerical procedure is validated by comparison with the exact solutions for the closed tube. Finally, the numerical method is applied to a restrained tube under variable pressure and logarithmic temperature.