An effective finite element for torsion of constant cross-sections including warping with secondary torsion moment deformation effect

In this paper an effective finite element of constant stiffness is developed for torsion with warping of both open- and closed-shape cross-sections. The secondary torsion moment deformation effect has been taken into account. The local element stiffness matrix that describes torsion with warping of both open- and closed-shape cross-sections has been derived using an analogy between torsion with warping (including the secondary torsion moment deformation effect) and the 2nd order beam theory (including the shear force deformation effect). The deformation effect of the secondary torsion moment must be taken into account first of all when dealing with closed-shape cross-sections. The warping part of the first derivative of the twist angle has been considered as an additional degree of freedom in each node at the element ends. This can be regarded as part of the twist angle curvature caused by the warping moment. Numerical results are presented to demonstrate the efficiency and accuracy of this new beam finite element. The necessity of including the deformation effect of the secondary torsion moment into the solution is demonstrated on the torsion of closed cross-sections. The results are compared with those obtained using other special commercial codes.