Effect of the longitudinal to transverse moduli ratio on the in-plane natural frequencies of symmetric cross-ply laminated beams by the stiffness method

Abstract

In this study, the stiffness method is used for the solution of the purely in-plane free vibration problem of symmetric cross-ply laminated beams. The rotary inertia, axial and transverse shear deformation effects are considered in the mathematical model by the first-order shear deformation theory. A total of six degrees of freedom, four displacements and two rotations are defined for an element. The exact in-plane element stiffness matrix of 6×6 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. After verifying the accuracy of the present formulation, the in-plane mode shapes associated with the first eight natural frequencies of the vibrating beam with fixed-free, fixed-simple, and fixed-fixed boundary conditions are illustrated. Finally, the effect of the longitudinal to transverse moduli ratio on the first three in-plane natural frequencies is investigated for different length/thickness ratios (5 and 20) and boundary conditions (fixed-free and fixed-fixed).