Solution of the ratchet-shakedown Bree problem with an extra orthogonal primary load

•A complete shakedown and ratcheting solution is derived analytically.•The problem is Bree-like but with an extra orthogonal primary load.•The ratchet diagram can be expressed in a form applicable to any orthogonal load.•Tensile ratcheting can occur if the primary load exceeds half the orthogonal load.•Compressive ratcheting can occur for smaller primary loads.

The complete shakedown and ratcheting solution is derived analytically for a flat plate subject to unequal biaxial primary membrane stresses and a cyclic secondary bending stress in one in-plane direction (x). The Tresca yield condition and elastic-perfectly plastic behaviour are assumed. It is shown that the results can be expressed in the form of a “universal” ratchet diagram applicable for all magnitudes of orthogonal load. For sufficiently large cyclic bending stresses, tensile ratcheting can occur in the x direction if the x direction primary membrane stress exceeds half that in the orthogonal direction. Conversely, for sufficiently large cyclic bending stresses ratcheting in the x direction will be compressive if the x direction primary membrane stress is less than half that in the orthogonal direction. When the x direction primary membrane stress is exactly half that in the orthogonal direction ratcheting cannot occur however large the cyclic secondary bending stress.