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This work pertains to the stochastic stress-based topology optimization of frame structures considering uncertainty. Specifically, this paper presents an investigation of the second deviatoric stress invariant, J 2, as the measure of stress in the domain where the objective is to minimize the maximum of the expected values of the stress invariant. Analytical expressions for the expected value of the J 2 invariant are developed employing the perturbation approach. The premise being that the J 2 invariant eliminates the square root operator that would otherwise be present if using the von Mises stress, and by eliminating this operator, the nonlinearity in the functional mapping between the random input (magnitude and direction of external forces) and uncertain output (expected values of stress) is reduced. Hence, an improved accuracy can be achieved with an analytical approximation of a given order. The analytical …