Elasticity solution for free vibration of laminated beams

Based on the two-dimensional theory of elasticity, a new approach combining the state space method and the differential quadrature method is presented in this paper for freely vibrating laminated beams. Applying the differential quadrature method to the state space formulations along the axial direction of the beam, new state equations about state variables at discrete points are obtained. Using matrix theory, the solution can be easily derived, which can very conveniently deal with the continuity conditions. Frequency equation governing the free vibration of laminated beams is then derived and the natural frequencies are obtained. No other assumption on deformations and stresses along the thickness direction is introduced, so that the present method is efficient for laminated beams with arbitrary thickness. It also can cope with arbitrary boundary conditions without applying Saint-Venant’s principle. Numerical examples of multi-layered beams and sandwich beams are performed. Results are verified by comparing them with the published results obtained from various finite element methods and shear beam theories.