A novel mixed-variable beam element and its application on the free vibration of Timoshenko beam

Abstract In the framework of mixed-variable finite element method, this paper proposes a new 2-node 8-degree-of-freedom Timoshenko beam element. The nodal degrees of freedom of this novel element are constructed based on the transverse displacement of the beam, the cross-sectional rotation, and the internal forces, namely the shear force and bending moment. Though the transverse displacement field and the cross-sectional rotation field are mutually independent in the Timoshenko beam, two novel displacement field expressions that respectfully correspond to the transverse displacement and cross-sectional rotation of the Timoshenko beam element is proposed based on the cubic Hermite interpolation function. The two displacement fields are synergistically used to describe the vibration characteristics of the beam element, then the stiffness and mass matrices of the element are derived and the natural frequencies of a Timoshenko beam are solved. Free vibration analyses are conducted by using the proposed beam element with different meshes under three typical boundary conditions. Through comparison with the existing results, the advantage of the proposed finite element in computational accuracy is verified, whose error is normally half of or less than that of the conventional Timoshenko beam element with the same interpolation order of the shape function. Further, the internal forces at the nodes are directly obtained together with the displacements in the solving process as the shear force and bending moment are naturally and reasonably composed into the nodal freedoms. The precision of the internal force by the proposed element is also high in that the distribution of the shear force and the bending moment obtained is consistent with the analytical results. The numerical analysis demonstrates the efficiency of the presented novel element together with excellent precision for nodal internal forces. The predicted results by the proposed element are expected to serve as benchmarks for evaluating the performance of other numerical methods.