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Stress related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. A new and effective topological optimization method of continuum structures with the objective function being the structural volume and the constraint functions being stresses has been presented in this paper. First, in the proposed method，in order to representing stresses highly nonlinear dependence on the design variables, stress gradients and a similar normal density function are introduced to build some structural stress measure functions as gathering stress constraints to control stress concentrations and all local stresses，and a new approximate continuum structural topological optimization model with stress constraints is constructed，being incorporated with the solid isotropic material with penalization （SIMP），a penalty function for considering a discrete condition of density variables to generate black/white designs，and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon.

Second, a set of stress quadratic explicit approximations is constructed, based on stress sensitivities and the Method of Moving Asymptotes (MMA). A set of algorithm for the one level optimization problem is proposed for the inequality constrained nonlinear programming problem with simple bounds, based on the primal-dual theory, in which the non-smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function.

Third, a two-level optimization design scheme with varying constraint limits, is proposed to deal with the gathering stress constraints that always are of loose constraint features in the conventional structural optimization method.

Finally, a new structural topological optimization method with stress constraints and its algorithm are formed，and examples are provided to demonstrate that the proposed method is feasible, very effective and efficient for solving the continuum structural topological optimization problems with the objective function being the structural volume and the constraint functions being stresses.

Topological optimization; Stress constraint; Continuum structure; stress gradient ; normal density function; varying stress limits