In this paper, a continuous fourth-order ordinary differential equation Shanley-type model is suggested for analytical analysis of the problem of elastic-plastic beam dynamics. A co-dimension three bifurcation problem and its simplified case, an incomplete co-dimension two bifurcation of a pair of pure imaginary eigenvalues and a simple zero eigenvalue are presented and analyzed, and the normal form analysis and the unfoldings of 2-jet and 4-jet cases of the incomplete normal forms are provided. Since elastic-plastic beam dynamics are of great non-linear complexity and the vector fields are multiple degeneracies, small differences of physical parameters cause dramatic essential changes of behavior of the motion. Through these results the rich dynamical behaviors of the elastic-plastic beam dynamics, including the counterintuitive behavior and its sensitivity to small parameters of this problem, can be illustrated.