Free vibration of unsymmetrically laminated beams having uncertain ply orientations

Three models were developed to predict randomness in the free vibration response of unsymmetrically laminated beams: exact Monte Carlo simulation, sensitivity-based Monte Carlo simulation, and probabilistic finite element analysis (FEA). It is assumed that randomness in the response is only caused by uncertainties in the ply orientations. The ply orientations may become random or uncertain during the manufacturing process. A new 16-degree-of-freedom beam element, based on the first-order shear deformation beam theory, is adapted for use in studying the probabilistic nature of the natural frequencies. With the use of variational principles, the element stiffness matrix and mass matrix are obtained through analytical integration. With the use of a random sequence, a large data set following normal distribution is generated, containing possible random ply orientations. The sensitivity derivatives are numerically calculated through an exact analytical formulation. The eigenvalues are expressed in terms of deterministic and probabilistic quantities, which allows the determination of how sensitive they are to variations in ply angles. The predicted mean value and coefficient of variation of the natural frequencies for sensitivity-based Monte Carlo simulation and probabilistic FEA are in good agreement with the exact Monte Carlo simulation. Results show that variations of ±5 deg in ply angles have little effect on the lower mode natural frequencies of unsymmetrically and symmetrically laminated beams.