This paper proposes an approach for the topological design of continuum structures with global stress constraints considering self-weight loads. The rational approximation of material properties is employed to describe the material distribution for overcoming the parasitic effect for low densities. The structure volume is used as the objective function to be minimized. The local stress constraints for all elements are aggregated into a global stress constraint using the improved P-norm method. A model for the stress-constrained topology optimization of continuum structures considering the self-weight loads is established. The projection filtering method is adopted to avoid numerical instability, and the topology optimization problems are solved using the method of moving asymptotes. Several numerical examples are presented to demonstrate the validity of the proposed method. The structures obtained by the proposed method can have better performance. The effects of different norm parameters, stress constraints and mesh densities on the topological structures are analyzed.