This paper presents a reliability-constrained Bayesian optimization framework for structural design under uncertainty, addressing challenges in stochastic optimization where the objectives and constraints are defined implicitly by potentially expensive numerical models. Our approach explicitly accounts for parameter uncertainty using results from Bayesian quadrature for uncertainty propagation in Gaussian process surrogate models. The method accommodates arbitrary probability distributions and employs gradient-based optimization for acquisition function maximization, strategically selecting sample points to minimize numerical model evaluations. We demonstrate our algorithm’s superior performance over random search and conventional Bayesian optimization through both an analytical test function and a prestressed tie-beam design case study, showing its practical applicability to structural optimization problems.