Nonlinear free vibrations of Timoshenko–Ehrenfest beams using finite element analysis and direct scheme

In this work, nonlinear free vibrations of fully geometrically exact Timoshenko–Ehrenfest beams are investigated. First, the exact strong form of the Timonshenko–Ehrenfest beam, considering the geometrical nonlinearity, is derived, and the required formulations are obtained. Since the strong forms of governing equations are highly nonlinear, a nonlinear finite element analysis (FEA) is employed to obtain the weak form. The FEA is utilized to compute natural frequencies and mode shapes; the direct scheme is adopted to solve the eigenvalue problem which is obtained by eliminating nonlinear terms. Then, each eigenvector is normalized, and the nonlinear stiffness matrix is derived and the nonlinear free vibration analysis is carried out. A recursive procedure is adopted to proceed until the convergence criterion is satisfied. Finally, the applicability of the proposed formulation is provided with some examples and results are compared with those available in the literature.