This paper puts forward a simplified method for assessing the failure modes of rigid-plastic beams by solving the governing partial differential equation with defined initial conditions based on five previously developed transverse velocity profiles. It is found that there are twelve different response patterns possible for a rigid-plastic beam subjected to a triangular pulse shape impulsive load. As developed in this paper, the P-I diagram is based on the derived deformation pattern such that selected failure criteria can be used to identify and differentiate various failure patterns. The shear-to-bending ratio, the boundary conditions, and the load pulse shape, all influence the response of the rigid-plastic beams, and are discussed in detail. Although the peak pressure and the shear-to-bending ratio together determine the response pattern, it is also found that the combination of peak pressure, impulse, and shear-to-bending ratio define the failure type and sequence. Fully clamped beams are relatively resistant to bending failure; however, they are more vulnerable to shear failure at the supports than simply supported beams. Compared with the triangular-shape impulsive load, the rectangular-shape impulsive load is more vulnerable in the pressure and impulse combination-controlled zone. The explicit solutions are compared with results obtained by using the Single-Degree-of-Freedom (SDOF) approach and are found to be exactly the same whether the beams have a small shear-to-bending ratio (υ ≤ 1) or a large such ratio (υ ≥ 1.5). But for beams with medium shear-to-bending ratio (1 ≤ υ ≤ 1.5), only the presented approach derived solutions.