An Eulerian thermomechanical elastic-viscoplastic model with isotropic and directional hardening is used to analyse the residual mechanical state resulting from the arc welding of a multi-pass weld. Details of the weld test plate, weld filler material, and numerical implementation of the model are provided, including integration algorithms and consistent tangent modulus. For the computational welding mechanics analyses, the austenitic ASME stainless steel grade 316L was considered so that no phase transformations of solid states needed to be considered. The maximum residual stresses were found to be about 500-600 MPa, which is of the order of the yield stress of the base material. Variations in the heat input and the resulting weld cooling time had a significant influence both on the residual stress state and on the resulting geometry of the weld. The predicted stress levels were compared to the experimental results. Overall, the proposed Eulerian framework seems to be a promising tool for analysing melting/solidification processes and residual mechanical states.