Single parameters such as the stress intensity factor K or J-integral in traditional fracture mechanics depend strongly on the geometry and loading conditions. Therefore a second parameter like the T-stress measuring the stress constraint is additionally needed to characterize general crack-tip fields. While many research works have been done to verify the J– T description of elastic–plastic crack-tip stress fields in plane strain specimens, a limited amount of work (especially for bimaterials) has been performed to describe the structural surface crack-front stress fields with two parameters. On this basis, via detailed 3D finite element (FE) analyses for surface-cracked plates and pipes of homogeneous materials and bimaterials, we investigate the extended validity or limitation of the two parameter approach. We here first develop a full 3D mesh-generating program for semi-elliptical surface cracks, and calculate elastic T-stress from the obtained FE stress field for “purely elastic” joint with modulus mismatch. We then validate the J– T methodology in characterizing the surface crack-front fields of welded plates and pipes under various loadings by comparing the J– T predictions to the “elastic–plastic” 3D FE stress fields.