On exact rod/beam/shaft-theories and the coupling among them due to arbitrary material anisotropies

We prove the existence of an exact one-dimensional representation of the three-dimensional theory of linear elasticity for a structural member with constant, two-fold symmetric cross-section, without the use of any a priori assumptions. We show that the general problem of three-dimensional elasticity decouples into four independent one-dimensional subproblems for isotropic material: a rod-, a shaft- and two orthogonal beam-problems. A unique decomposition of any three-dimensional load case with respect to the direction and the symmetries of the load is introduced. It allows us to identify each part of the decomposition as a driving force for one of the four one-dimensional subproblems. Furthermore, we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials.