Spectral method to determine the Tsai-Wu probability of failure of a composite laminate subjected to random vibrations

This work presents a spectral-based methodology for the probabilistic failure analysis of composite laminates subjected to zero-mean, stationary Gaussian random vibrations. The proposed approach efficiently computes both the probability density (PDF) and cumulative distribution functions (CDF) of nonlinear failure indices, using the Tsai–Wu criterion as a case study. By integrating the frequency response functions (FRFs) of the stress tensor components with the excitation’s power spectral density (PSD), the method constructs the complete covariance matrix of the stress field. This enables the definition of a joint multivariate Gaussian distribution of stresses, from which Monte Carlo sampling is performed to evaluate any nonlinear failure function.The methodology is general and can be applied to any nonlinear stress function, such as von Mises stress, provided the structure remains linear and the excitation Gaussian. Validation is performed through finite element (FE) simulations of a composite plate with a central hole. The RMS von Mises stresses predicted by the proposed method closely match those obtained with the Segalman spectral approach implemented in ANSYS, confirming its correctness. Further comparison with time-domain transient simulations demonstrates excellent agreement in failure probabilities while achieving a computational speedup exceeding two orders of magnitude.