This paper presents a discretization strategy, based on the concept of consistent approximations, for certain optimal beam design problems, where the beam is modeled using Euler-Bernoulli beam theory. It is shown that any accumulation point of the sequence of the stationary points of the family of resulting approximating problems is a stationary point of the original, infinite-dimensional problem. The construction of approximating problems requires the development of a relaxation of constraints to ensure existence of solutions. The numerical solution of the approximating problems, by means of nonlinear programming algorithms that are not scale invariant, must be preceded by a change of variables to guard against deterioration of performance. The use of such approximating problems, in conjunction with a diagonalization strategy, is illustrated by a numerical example.