The Bree problem with primary load cycling in-phase with the secondary load

Shakedown and ratcheting behaviour is derived for uniaxial primary membrane and thermal bending stresses both cycling strictly in-phase. The complete analytic solution for perfect plasticity is presented, including the shakedown/ratcheting diagram and the ratchet strains or cyclic plastic strains where relevant. This cyclic loading is shown to be considerably more benign than that of the original Bree problem in which the primary membrane stress does not cycle.

► Ratcheting is analysed for uniaxial primary membrane and thermal bending stresses.
► The complete analytic solution is given for both loads cycling in-phase.
► This cyclic loading is more benign than that of the original Bree problem.
► The ratchet boundary is at least one yield stress higher than for the Bree problem.
► Algebraic expressions for the cyclic plastic and ratchet strains are given.