The elasto-plastic problem relevant to a constrained rigid body system, equipped with lumped elastic-perfectly plastic devices, is addressed. Simplifying hypotheses are introduced, according to which: (i) the loads are all proportional to a monotonically increasing parameter \(\lambda \) , and (ii) elastic returns are excluded. Two different analyses are carried out. The first, said evolution analysis, is aimed to describe the response of the structure during the whole loading process, starting from the initial application of the loads until the collapse. It allows defining the ductilityDuctility of the structure as the ratio \(\delta :=q_{u}/q_{e}\) between the ultimate \(q_{u}\) and elastic limit \(q_{e}\) values of a significant displacement q. The second analysis, said limit analysis, is aimed at determining the ultimate load multiplier \(\lambda _{u}\) without following the evolution of the structure. Since the exact evaluation of \(\lambda _{u}\) is cumbersome, approximate methods are applied, in which just a part of the equations ruling the problem is satisfied, while the remaining part is ignored. The approaches are: (a) the static method, in which just equilibrium and yield conditions are satisfied, but kinematics and flow law are generally violated; (b) the kinematic method, in which just kinematics and the flow law are satisfied, but equilibrium and yield conditions are generally violated. Two fundamental theorems state that the static method determines a lower bound, while the kinematic method an upper bound to the exact ultimate multiplier. Moreover, it is proved that \(\lambda _{u}\) is unique, and coincides with the smallest kinematic and the largest static multipliers. All these concepts are preliminary illustrated referring to an example; then, they are rigorously discussed for general systems.