An accurate nonlinear 3d Timoshenko beam element based on Hu-Washizu functional

• An accurate and robust 3d mixed beam finite element is developed.
• Interaction of axial force, shear forces, bending moments and torsion is considered.
• Proposed element is superior to displacement based elements in nonlinear analysis.
• Effect of axial force is important for both solid and hollow circular sections.
• Effect of torsion coupling is more pronounced on the response of hollow sections.

An accurate 3d mixed beam element that is efficient especially in nonlinear analysis is presented in this paper. The mathematical theory is based on Hu-Washizu principle that uses three-fields in the variational form. The composition of the variational form ensures independent selection of displacement, stress and strain fields. Timoshenko beam theory is extended to three dimensions for deriving strains from displacement field. Numerical integration of stress–strain relations along control sections is carried out for numerical analysis. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the control sections of the beam. This form of the element permits coupling of the stress resultants and eliminates necessity of displacement components along the beam element except at the nodes. Consequently the element is free from shear-locking. Numerical examples on uniform and tapered structural members with solid and hollow circular sections demonstrate the nonlinear interaction between axial, shear force, bending moments and torsion, where the results compare well with closed form solutions and available data in literature. Furthermore, the proposed element has superior performance in both linear and nonlinear analysis compared to a locking-free higher order displacement based 3d Timoshenko beam element.