Manufacturing processes involving moving heat sources include additive manufacturing, welding, laser processing (cladding and heat treatment), machining, and grinding. These processes involve high local thermal stresses that induce plasticity and result in permanent residual stress and distortion. The residual stresses are typically calculated numerically at great computational expense despite the fact that the inelastic fraction of the domain is very small. Efforts to decouple the small plastic part from the large elastic part have led to the development of the tendon force concept. The tendon force can be predicted analytically for the case of infinitely rigid components; however, this limitation has prevented the broader use of the concept in practical applications. This work presents a rigorous mathematical treatment using dimensional analysis, asymptotics, and blending which demonstrates that the effect of geometric compliance depends on a single dimensionless group, the Okerblom number. Closed-form expressions are derived to predict the effect of compliance without the need for empirical ad-hoc fitting or calibration. The proposed expressions require input of only material properties and tabulated process parameters and are thus ideally suited for use in metamodels and design calculations, as well as incorporation into engineering codes and standards.