Exact Finite Element for Nonuniform Torsion of Open Sections

A finite-element formulation that captures both the St. Venant and warping torsional effects of open sections is developed. The formulation is applicable to prismatic thin-walled open sections commonly used in steel construction and is able to represent general torsional and warping end restraints. The element has two end nodes, each having two degrees of freedoms, and featuring continuity of the first derivative of the angle of twist (C1 continuity). A family of hyperbolic interpolation functions is adopted in the derivation in lieu of conventional polynomial displacement interpolation schemes. The stiffness matrix and the associated energy equivalent load vector are demonstrated to yield nodal generalized displacements and forces in exact agreement with classical solutions for any externally applied twisting moment distribution. Consequently, the formulation allows the adoption of coarser subdivisions than those needed in other finite-element formulations while preserving the exactness of the results. The formulation is simple and provides a basis to assess the quality of results of other approximate finite-element solutions. The implications of the exactness of the formulation are discussed and illustrated through examples.