The present paper is dedicated to a problem of shape optimization where the external loads applied to the structure are subject to uncertainties. The objective functional or the constraints can be written as the expected value of a polynomial functional of degree m. We provide a deterministic expression of the expectation of the polynomial as a function of the first m moments of the random variables modeling the uncertainties, as well as a method to compute its shape derivative according to Hadamard. In particular, no further assumptions on the distribution of the random variables are required, and the method presented in this article is not based on computationally expensive sampling techniques. The proposed method can be applied in different contexts, like the study of the variance of a quadratic functional, or the optimization of a functional approaching the L ∞-norm of a quantity in the structure.