Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semiinfinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parametrized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including first-order reliability and second-order reliability methods and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder.