The planning and design of buildings and civil engineering concrete structures constitutes a complex problem subject to constraints, for instance, limit state constraints from design codes, evaluated by expensive computations such as finite element (FE) simulations. Traditionally, the focus has been on minimizing costs exclusively, while the current trend calls for good trade-offs of multiple criteria such as sustainability, buildability, and performance, which can typically be computed cheaply from the design parameters. Multi-objective methods can provide more relevant design strategies to find such trade-offs. However, the potential of multi-objective optimization methods remains unexploited in structural concrete design practice, as the expensiveness of structural design problems severely limits the scope of applicable algorithms. Bayesian optimization has emerged as an efficient approach to optimizing expensive functions, but it has not been, to the best of our knowledge, applied to constrained multi-objective optimization of structural concrete design problems. In this work, we develop a Bayesian optimization framework explicitly exploiting the features inherent to structural design problems, that is, expensive constraints and cheap objectives. The framework is evaluated on a generic case of structural design of a reinforced concrete (RC) beam, taking into account sustainability, buildability, and performance objectives, and is benchmarked against the well-known Non-dominated Sorting Genetic Algorithm II (NSGA-II) and a random search procedure. The results show that the Bayesian algorithm performs considerably better in terms of rate-of-improvement, final solution quality, and variance across repeated runs, which suggests it is well-suited for multi-objective constrained optimization problems in structural design.