This work addresses the use of the topology optimization approach to the design of continuum structures with failure constraints under the hypothesis of uncertainties in the spatial distribution of Young’s modulus. To this end, the first order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to represent the random field. The objective is the minimization of the amount of material used in the design, subjected to local stress constraints under uncertainties. The probability of failure is bounded by the one-sided Chebychev inequality, since the exact probability distribution function of the stress constraints is not known in advance. The effective probability of failure of the obtained optimal designs is validated with the use of the Monte Carlo Simulation approach, indicating that the probability of failures of the topologies obtained with the stochastic approach is within the bounds provided by the one-sided Chebychev inequality. The optimization problem is solved by means of the augmented Lagrangian method, in order to address the large number of constraints associated to this kind of formulation. It is shown that the correlation length and the number of standard deviations considered in the formulation play an important role in both the obtained topology and effective probability of failure.