Solutions to natural frequencies of laminated composite beams based on the scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is further extended to compute the natural frequencies of laminated composite beams. In the proposed method, the beam is simplified into a one-dimensional model. Only the displacement components along x and z directions are selected as the fundamental unknowns. Starting with the fundamental equations of elasticity and built on the scaled boundary coordinate, the principle of virtual work and the dual vector technique, the first-order ordinary differential scaled boundary finite element dynamic equation for composite beams is obtained, whose general solution is the analytical matrix exponential function. The Padé expansion is utilized to solve the matrix exponential function, and the dynamic matrix of each beam lamina can be acquired. According to the principle of matching degrees of freedom, the global stiffness and mass matrices of the laminated beam are gained. Solving the eigenvalue equation results in the vibration frequencies of laminated composite beams. This method is widely applicable, and there is no limitation on the layer number and boundary conditions. Comparisons with natural frequencies of three-, four- and ten-layered beams as well as the stepped beam, the accuracy, high efficiency and fast convergence of the introduced SBFEM are validated.