In spite of the benefits, one of the most challenging issues for implementing optimization under uncertainty, such as the use of robust design approach, is associated with the intensive computational demand of uncertainty propagation, especially when the simulation programs are computationally expensive. In this paper, an efficient approach to uncertainty propagation via the use of metamodels is presented. Metamodels, created through computer simulations to replace expensive simulation programs, are widely used in simulation-based design. Different from existing techniques that apply sample-based methods to metamodels for uncertainty propagation, our method utilizes analytical derivations to eliminate the random errors as well as to reduce the computational expenses of sampling. In this paper, we provide analytical formulations for mean and variance evaluations via a variety of metamodels commonly used in engineering design applications. The benefits of our proposed techniques are demonstrated through the robust design for improving vehicle handling. In addition to the improved accuracy and efficiency, our proposed analytical approach can greatly improve the convergence behavior of optimization under uncertainty.