Exact finite element formulation in generalized beam theory

This paper presents the formulation of exact stifness matrices applied in linear generalized beam theory (GBT) under constant and/or linear loading distribution in the longitudinal direction. Also, the assortment of the correct exact stifness matrix and the corresponding shape function are presented based on main transversal deformation mode, which can be divided into: (1) dominant distortion mode; (2) dominant torsion mode; (3) and critical distortion–torsion mode. Special attention is given to the hyperbolic–trigonometric shape functions, which are organized in a system of vector in function of longitudinal direction and a coefcient matrix obtained from the completeness requirement. This approach has the beneft of compacting the terms of the stifness matrix and systematizing the boundary conditions of an element by applying the completeness coefcient matrix as a transformation matrix. As a result, in linear analysis, a single element can represent the stress and displacement felds. Moreover, due to the higher-order continuous derivatives properties of hyperbolic–trigonometric shape functions, the generalized internal shear is obtained without the typical discontinuity of Hermitian shape functions. A full and detailed example, applied in a thin-walled circular hollow cross section, provides not only an illustration of the presented approach, but also a quick introduction point in GBT.