Out-of-plane bending and torsional resonance frequencies and mode shapes of symmetric cross-ply laminated beams including shear deformation and rotary inertia effects

In this study, the stiffness method solution based on the first order shear deformation theory is used for the purely out-of-plane (out-of-plane bending and torsional) free vibration analysis of symmetric cross-ply laminated beams. A beam element has six degrees of freedom. The influence of the rotary inertia terms is included in the solution. Accounting for the shear deformation effects, the exact numerical element stiffness matrix is obtained based on the transfer matrix method. The element inertia matrix consists of the concentrated masses. The sub-space iteration and Jacobi's methods are employed in the solution of the large-scale general eigenvalue problem. The out-of-plane vibration mode shapes associated with the first eight natural frequencies of the vibration composite beam with several boundary conditions are illustrated