Summary: Plane strain problem of a thick-walled cylindrical pipe under internal and external pressures, made of elastic-perfectly plastic material with a radial dependence of the yield stress (produced, e.g., by one-sided accelerated cooling) described by a polynomial function is considered in the framework of small deformations theory, using the von Mises yield criterion. It is found that, at the fixed wall thickness and strength gradient, the gain in the ultimate loading capacity depends on the magnitude of the gradient but its dependence on the gradient direction, as well as pipe radii is rather weak. At the same time, the onset of yielding strongly depends on all the mentioned parameters. It is also shown that one-sided accelerated cooling enhances the loading capacity of the pipes, as compared to the normalized state whereas comparison with the steel that has experienced quenching with tempering may produce diverse results.