The Bree problem with the primary load cycling out-of-phase with the secondary load

Ratchet and shakedown boundaries are derived analytically for the Bree problem with both primary and secondary loads cycling, but out of phase. Ratchet strains and cyclic plastic strains are derived in terms of the X,Y position on the ratchet diagram. The ratchet boundary lies between the Bree ratchet boundary and that for both loads cycling in-phase (or in exact anti-phase), but generally lies closer to the former. This provides a warning for ratcheting analyses with two loads cycling: employing the simplifying assumption of cycling exactly in-phase (or exactly in anti-phase) may be substantially non-conservative.