Abstract: The entire elasto-plastic process of loading and deformation of a beam with one degree of indeter-minacy under a concentrated load is analyzed. Based on the deformation feature, the up-loading process can be divided into four stages: the elastic stage, the second stage when the beam region near the fixed end is in the plastic propagation state, the third stage when both the beam regions near the fixed end and near the point where the concentrated load acts are in the plastic propagation state, and the fourth stage when the fixed end behaves as a plastic hinge while the plastic region near the fixed end is unloaded and the plastic region near the mid-point is propagating to the emergence of the second plastic hinge. In the elastic stage, the relation between the moment and the load is proportional, so is that between the deflection and the load. In the second and third stages, the relation between the moment and the load is complicated and nonlinear, so is that between the deflection and the load. In the fourth stage, the relation between the moment and the load is linear but not proportional, while the relation between the deflection and the load is much complicated and nonlinear. The formulas of the moment and the deflection at an arbitrary point at each stage are derived. The formulas can be applied to the engineering structure design.