A hybrid approach is proposed to evaluate the probability of unacceptable performance with respect to uncertain parameters. The evaluation of structural reliability and the solution of maximum vibration response are performed simultaneously. A constrained optimization problem is deduced for which several techniques have been developed to obtain the reliability index. The nonlinear equality constraints of the optimization problem are constructed based on the harmonic balance equations, the optimality condition of the maximum vibration response with respect to the vibration frequency and the limit state failure function. With the nonlinear equality constraints imposed on the harmonic balance equations and the derivative of the maximum vibration response with respect to the vibration frequency, the inner loop for solving the maximum vibration response is avoided. The sensitivity gradients are derived by virtue of the adjoint method. The original optimization formulation is then solved by means of the sequential quadratic programming method (SQP) method. Finally, the developed approach has been verified by comparison with reference values from Monte Carlo simulation (MCS). Numerical results reveal that the proposed method is capable of predicting the failure probability of nonlinear structures with random uncertainty.

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