Incipient deformation mechanisms of a semi-circular cantilever beam subjected to an out-of-plane step-loading

The incipient dynamic deformation mechanisms for a rigid-plastic semi-circular cantilever beam subjected to an out-of-plane step-loading applied at its tip depend on the non-dimensional force f and a structural parameter η, which is the ratio of the fully plastic torque and the fully plastic bending moment of the cross-section of the beam. When η > 1, mechanisms with a single plastic hinge provide complete solutions for the incipient dynamic response. For semi-circular beams for which η lies in the range 0.316 < η < 1 (solid circular and square cross-sections, and thin-walled circular tubular sections all fall into this range), as the magnitude of the applied load f is increased, the incipient deformation mechanism changes from a single-hinge one to a double-hinge one and finally to a triple-hinge one. The last of these contains an interior combined torsion-bending (T-M) hinge, an interior pure-torsion hinge and a hinge at the root. While the triple-hinge provides a complete solution, the single-hinge mechanism gives a good approximate solution to the problem, provided the load f is sufficiently large. The case when η < 0.316 while f is very large remains to be fully analysed although some incipient deformation mechanisms for this case are also studied herein.