An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach
This study presents an analytical solution to the free vibration analysis of a uniform Timoshenko beam. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. A material law combining bending, shear, curvature, and natural frequency is presented. This complete study is based on this material law and closed-form solutions are found. The free vibration response of single-span systems, as well as that of spring-mass systems, is analyzed. Closed-form formulations of matrices expressing the boundary conditions are presented; the natural frequencies are determined by solving the eigenvalue problem. First-order dynamic stiffness matrices in local coordinates are determined. Finally, second-order analysis of beams resting on an elastic Winkler foundation is conducted.