The object of this paper is to show that in certain circumstances the plastic deformation of beams, made of a material with linear strain hardening and subject to dynamic transverse loading, can be determined by the techniques used in solving elastic problems. In particular, the differential equation of motion for such beams is in some instances of the same form as in the corresponding linear elastic case, and so any of the methods employed for solving elastic-beam problems, such as the normal-mode method, Laplace-transform method, or Boussinesq’s solutions for infinite beams, can be used. Because of this linear character of the differential equation of motion encountered in the analysis presented here, it also is shown that some initial-motion problems for beams undergoing large plastic deformations due to transverse loading can be solved by superposing solutions. In these problems the disturbance part of the solution is obtained by some elasticity technique and is then superposed on the initial motion of the beam. The method of solution is demonstrated by means of several examples involving finite beams. The first example is an initial-motion problem and illustrates the method of superposition. The second example is an initial-stress problem for a simply supported beam. Again, the method of superposition is used. The last example is a free-boundary problem for a cantilever beam. This problem is solved by an inverse method whereby the form of the solution is assumed and the physical problem associated with this solution is then determined.