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.