In reliability-based design optimizations, surrogate model has been widely used to replace expensive physical-based simulations in computationally intensive engineering applications. Due to the lack of data, surrogate model uncertainty is involved since surrogate models cannot perfectly represent actual simulations or experiments. To ensure a reliable optimal design, surrogate model uncertainty needs to be considered in design under uncertainties when the training data is limited. In this work, we presents an equivalent reliability index (ERI) approach to handle both parameter variations and surrogate model uncertainty in RBDO. The Gaussian process (GP) modeling technique is first employed for building surrogate models and Monte Carlo simulations are used for reliability assessment with the consideration of input randomness. In detail, a Gaussian mixture model (GMM) is constructed based on the GP model, and the reliability is estimated by an equivalent reliability index that calculated based on the first and second statistical moment of the GMM. To facilitate the RBDO process, the sensitivity of the ERI is analytically derived without incurring extra computational costs. Two case studies are used to demonstrate the effectiveness and robustness of the proposed approach.