A closed-form eigenstrain framework for classical springback in inelastically bent beams

Abstract Springback of prismatic beams following inelastic bending is a classical problem in engineering plasticity, with benchmark closed-form solutions available for canonical cross sections under elastic–perfectly plastic assumptions. Here, the same bending–unloading problem is reformulated systematically within the eigenstrain framework by treating the plastic strain accumulated during loading as a retained inelastic strain source after unloading. This recasts the post-unloading configuration as a residual stress problem driven by retained inelastic strains, yielding a unified analytical route to residual curvature, residual elastic strain, and self-equilibrated residual stress distributions. Closed-form expressions are derived for rectangular, circular, and I-shaped sections and implemented in MATLAB to compare moment–curvature response, springback magnitude, and residual-field characteristics across geometries. The closed-form results are verified against an independent classical beam-plasticity benchmark based on piecewise elastic–perfectly plastic stress blocks integrated over the section and elastic unloading via sectional stiffness, showing excellent agreement in both the elastic and post-yield regimes. The formulation embeds classical springback results within an eigenstrain-based residual-stress viewpoint and provides a transparent analytical baseline for geometry comparisons and future extensions through more general eigenstrain inputs.