Novel Approach to Finite Element Simulation of the Timoshenko-Ehrenfest Beam under Static and Dynamic Loads

The problem of Timoshenko-Ehrenfest beam bending under static and dynamic loading is considered in the finite element formulation. In the static formulation, the shear stiffness matrix is derived from the condition of stationarity of the shear deformation energy, in so doing linear shape functions and the corresponding rows of the bending stiffness matrix are used. As a numerical example, the bending of a simply supported beam made of fiber-reinforced concrete is considered. When constructing a dynamic finite element model, the shear damping matrix is derived from the requirement for the stationary state of its shear deformation energy dissipation. The results of the beam vibrations modeling are presented in comparison with the solution of a similar problem for the Euler-Bernoulli beam.