Abstract: Most existing study on topology optimization have concentrated on maximizing the system stiffness. Especially, the minimization of static compliance subject to the volume fraction is widespread in the formulation. From the engineering point of view, structural strength design is of vital importance. Past study on stress constraint have shown that an amount of numerical difficulties with the stress-constrain topology optimization exist including the so-called singularity, vast of stress constraints, highly nonlinear behavior and so on. To achieve the topological design under stress constraint requirement, the normalized stress measure using p-norm function is adopted for the reduction of stress constraints. Following the modeling manner of independent continuous mapping method, the reciprocal function of relative density is regarded as the design variables. The sensitivities of stress constraint and volume objective with respect to the design variable are derived, and their explicit expressions are formulated based on the first-order and second-order Taylor approximation respectively. By setting up the sub-problem in the form of a quadratic program, the original topology optimization problem is efficiently solved using the sequential quadratic programming approach. The difference between stiffness and strength design, as well as the effect of various upper bounds of stress value on the optimized results for stress constraint are investigated in 2D numerical examples. Through the comparison of the proposed method and traditional variable density method, the feasibility and effectiveness of the proposed optimization approach in stress constrained problems are verified. The results also demonstrate that the consideration of stress constraint in continuum structure is indispensable.