An integrated approach for time-dependent reliability-based design optimization of vibratory systems with random parameters under stationary excitation is presented in this paper. The time-dependent probability of failure is computed using an integral equation which involves up-crossing and joint up-crossing rates. The total probability theorem addresses the presence of the system random parameters and a sparse grid quadrature method calculates the integral of the total probability theorem efficiently. The sensitivity derivatives of the time-dependent probability of failure with respect to the design variables are computed using direct differentiation combined with finite differences. The Modified Combined Approximations (MCA) reanalysis method is used to reduce the overall computational cost from repeated evaluations of the system frequency response or equivalently impulse response function. The method is applied to the shape optimization of a vehicle frame under stochastic loading. The novelty of the proposed work lies in the integration of the joint up-crossing rate method for time-dependent reliability, sparse grid approach for efficient evaluation of multi-dimensional integrals, reanalysis methods for efficient vibratory analysis, and sensitivity analysis for time-dependent Reliability-based Design Optimization (RBDO).