This study presents a scaling analysis of thermally induced stresses and strains produced during welding and additive manufacturing of thin structures such as plates or walls. The Order of Magnitude Scaling (OMS) technique was used to develop an appropriate dimensionless formulation, to obtain asymptotic expressions, and to determine the limits of validity. Non-linear finite element simulations of welding procedures were performed to validate the asymptotic model; plasticity and temperature dependent materials properties for structural steel, stainless steel, and aluminum were considered. Thermal stresses cause plasticity when the temperature reaches a critical temperature termed the first yield temperature. The model developed is valid when the first yield isotherm is elongated, which is the case for most welding and metal additive manufacturing applications. A rigorous novel expression for the criterion of applicability is presented and utilized for prediction of the width of the plastic zone surrounding a weld or additive manufacturing bead. Extrapolations beyond the region of applicability show a consistent trend, which is captured in the form of a general dimensionless empirical expression. This work establishes the foundation for the estimation of forces and distortions induced by welding or additive manufacturing processes, and also the incorporation of effects of departure from idealizations.