This is a 2D companion to the 3D V-die bending simulator, but it computes springback a genuinely different way. Instead of the 3D sim's closed-form Gardiner cubic formula, this model slices the sheet's cross-section into thin layers through the thickness, works out the bending strain and elastic-perfectly-plastic stress in each layer, and numerically integrates those stresses into a real bending moment — the same "moment-curvature" method used to seed FEA springback predictions.
strain(y) = κ·y y = distance from neutral axis, κ = 1/(Ri + t/2)
stress(y) = clamp(E·strain(y), −σy, +σy) elastic-perfectly-plastic
M′(κ) = ∫ stress(y)·y dy (numerical layer sum, per unit width)
Δκ = M′ / (E·I′) I′ = t³/12, per unit width — elastic unloading
κf = κi − Δκ θf = θi·(κf / κi) (arc length conserved)
The stress ribbon on the right shows exactly this: a linear elastic core near the neutral axis flanked by saturated plastic bands (flat at ±σy) once the outer fibers yield. Only the elastic core's stored strain energy drives the springback — the plastic bands contribute a moment but relax with the same curvature ratio, since no further plastic flow occurs during unloading.
- Elastic core fraction — the fraction of the cross-section still below yield at full bend. A thin sliver (tight R/t) means almost the whole section yielded; a wide core means the bend barely reached yield at all, so the part springs back almost flat.
- Higher σy/E (spring steel) → the elastic strain that must relax at unload is larger relative to the same curvature → more springback, same conclusion as the 3D sim's Gardiner formula, reached here from the stress integral instead of a curve fit.
- Larger R/t → smaller bending strain everywhere → thicker elastic core → the moment needed to hold the bend is a smaller share of the fully-plastic moment → more of the curvature is elastic and un-does itself on release.
Real-world relevance: production die designers rarely trust a single formula — this layer-integration method is exactly what a spreadsheet or simplified FEA pre-check does before cutting an over-bent die, because it captures the actual stress distribution instead of a curve-fit approximation.