Extending Castigliano’s Theorems to Model the Behavior of Coupled Systems

Extensions of the Castigliano theorems are developed in the context of modeling the behavior of both discrete coupled linear systems and various coupled beams. It is shown that the minimization of the displacement of a parallel system determined from Castigliano’s second theorem can be used to formally define the apportionment of loads among the system elements, while the minimization of the load in a series system determined from Castigliano’s first theorem can be used to formally define the apportionment of the displacement among the system elements. These extensions provide a means for apportioning loads in coupled continuous systems, as will be shown for the cases of coupled cantilever Timoshenko beams supporting discrete and continuous loads.