Consistent loading in structural reduction procedures for beam models

In multiphysics problems, a solid body is in interaction with various three-dimensional fields that generate complex patterns of rapidly varying distributed loading on the solid. Since three-dimensional computation requires excessive resources, methods of reduction to structural models are traditionally exploited in mechanics for the analysis of slender bodies. Although such procedures are well established, the reduction of loads is often performed in an ad hoc manner, which is not sufficient for many coupled problems. In the present work, we develop rigorous structural reduction (SR) procedures by using a variational framework to consistently convert three-dimensional data to the form required by structural representations. The approach is illustrated using the Euler-Bernoulli and Timoshenko beam theories. Some of the loading terms and boundary conditions of the four resulting structural problems (namely, tension, torsion, and two bending problems), which are formulated in terms of the original three-dimensional problem, could not be derived by ad hoc considerations. Numerical results show that the use of the SR procedures greatly economizes computation and provides insight into the mechanical behavior while preserving a level of accuracy comparable with the fully three-dimensional solution.