An approximate solution of the plane problem of the theory of plasticity for main pipelines, cylindrical vessels, and shell structures subjected to biaxial loading is proposed which allows one to perform theoretical analysis aimed at evaluation of strength and stressed state of asymmetric mechanically inhomogeneous welded joints with incomplete fusion at the center of the weld in the case of ductile fracture. Regulating the degree of mechanical inhomogeneity and thickness of the soft interlayer, it is possible to get the uniform strength of welded joints with incomplete fusion and shift fracture processes to the base metal under biaxial tension. We also determine the range of sizes of the domain of incomplete fusion for which the uniformity of strength is attained for the joint with asymmetric mechanical inhomogeneity of the butt weld with incomplete fusion.