On the foundations of anisotropic interior beam theories

This study has two main objectives. First, we use the Airy stress function to derive an exact general interior solution for an anisotropic two-dimensional (2D) plane beam. Second, we cast the solution into the conventional form of 1D beam theories to clarify some basic concepts related to anisotropic interior beams. The derived general solution provides the exact third-order interior kinematic description for the plane beam and includes the Levinson/Reddy-kinematics as a special case. By applying the Clapeyron's theorem, we show that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy-based considerations related to the interior beam in order to avoid artificial end effects. Exact 1D interior beam equations are formed from the general 2D solution. Finally, we develop an exact interior beam finite element based on the general solution. With full anisotropic coupling, the stiffness matrix of the element becomes initially asymmetric due to the interior nature of the plane beam. By redefining the generalized nodal axial forces of the element, the stiffness matrix takes a symmetric form.