A method has been carried out for the determination of the welding residual stress, assuming as experimental data the four global parameters of the welding inherent deformations: the longitudinal and transverse shrinkages and bendings. Referring to a thin plate butt-welded joint, the longitudinal and transverse inherent strains have been applied on the weld bead as concentrated Volterra or Somigliana distortions , according to their distributions; then a functional  representation has  been given of the gradient tensor of the generated displacement field,  and of the divergence tensor of the corresponding elastic stress tensor in the Cauchy equation; finally an energetic criterion has been adopted for reducing the δ (Dirac functional) and δ’ (distributional derivative of δ) components of the divergence tensor to surface and volume unit loads.

The approximation degree related with the concentration of the weld distortions on the longitudinal section of the weld or on the transversal middle section of the welded joint, has been evaluated by a series of numerical simulations carried out by a parametric finite element model of the thin plate butt-welded joint. The displacement and stress fields that arise in the welded plates, when diffused and concentrated distortions are applied, have been determined for several values of the sizes of the welded plates and width of the HAZ. 

The analysis of the numerical results shows that, when a diffuse distorsion is substituted by a concentrated one having the same intensity, the displacement field is unchanged in most of the two regions of the plates outside the HAZ. Small differences between the two fields are observable only in the zones close to the weld ends, where the welding residual stress field is of limited amplitude.