On the behaviour of infinitely long rigid-plastic beams under transverse concentrated load

This paper is concerned with the behaviour of an infinitely long uniform beam, made of a non-hardening plastic-rigid material, under the action of a single transverse concentrated load. The effects of transverse shearing deformation and rotatory inertia are assumed negligible, and the yield condition is assumed to depend only upon the bending moment. This problem is solved subject to the condition that the ratio of the impact velocity of the concentrated load to the time is non-increasing in time. The impact velocity is allowed to vary with the time. The associated problem concerning the motion of the beam following the removal of the concentrated load is also solved. Hitherto the only problem of the present type whose solution has been published is that involving constant impact velocity, but even there the solution of the unloading problem was not found. The general analysis of this paper is applied to solve completely the constant impact velocity problem, and some detailed results for other cases are also given. The motion, subject to the above restriction upon the impact velocity, is such that there are always a central, fixed yield-hinge and two lateral, outwards-moving yield-hinges, and is also such that the beam does not come to rest in any finite time.