Press-Brake Metal Bending (2D): Section View of Plastic Bend and Springback

What you see

A bench press brake: a V-die (two shoulders a distance S apart) and a punch with a rounded tip push a flat metal strip into a bend. The strip is drawn exactly as the model computes it, with no exaggeration. Colours show the bending stress in each fibre (blue compression on the inside, red tension on the outside, orange where the metal has gone past its yield stress). When the punch rises, the elastic part of the bend springs back and only the plastic part stays.

The model

Plane sections stay plane, so a fibre at height y from the mid-thickness has strain e = kappa y for a local curvature kappa. The metal is elastic up to the yield strain e_y = sigma_y / E and then hardens linearly with tangent modulus E_t. Integrating the stress over the thickness gives the moment in closed form: elastic M = E I kappa up to the yield curvature kappa_y = 2 sigma_y / (E t), and beyond it, with the elastic core half-depth y_p = e_y / kappa, M = 2 b [E kappa y_p^3 / 3 + sigma_y (1 - E_t/E)(c^2 - y_p^2)/2 + E_t kappa (c^3 - y_p^3)/3] with c = t / 2. For a rectangular section the fully plastic moment is 1.5 times the first-yield moment M_y = sigma_y b t^2 / 6 (the shape factor).

In three-point bending by a punch force P the moment is M(x) = P x / 2 between the die shoulders (|x| below S/2). The curvature at each section follows from the inverse of M(kappa); the strip centre sinks by delta = integral from 0 to S/2 of x kappa(x) dx and each side turns by theta = integral from 0 to S/2 of kappa dx. The page finds the force that gives the punch depth you chose, then draws the strip by integrating the curvature along its length (the parts beyond the shoulders stay straight and tilt with the end rotation).

Springback

Unloading is elastic: every section recovers the curvature M / (E I), so kappa_released = kappa(M) - M / (E I) and the residual angle is theta_res = theta - P S^2 / (16 E I). The recovered part is larger for a high yield stress, a low modulus, a wide die opening and a thin strip: hardened spring steel recovers a large part of the bend, soft annealed copper very little. The fibres left behind are in self-equilibrium (zero net force and moment); the residual stress at the surface is shown. For an elastic-perfectly plastic strip the relative recovery of curvature is 1.5 (kappa_y/kappa)(1 - (kappa_y/kappa)^2 / 3).

Honest simplifications

• Small rotations: the die span is taken as the horizontal distance between the shoulders. For bends beyond about 20 degrees per side the strip also slides over the shoulders and the span changes; real "air bending" tables include this.
• Bilinear stress-strain with the same curve in tension and compression and no Bauschinger effect; no strain-rate dependence; no thickness change or neutral-axis shift at small radii; constant elastic modulus.
• A narrow strip (plane stress). A very wide plate bends in plane strain and is stiffer by 1/(1 - nu^2), about 10 %.
• Friction at the shoulders and the punch tip is ignored, and the contact is a point at the centre and at each shoulder; real punches have a tip radius that sets the inside radius of the bend.
• Material numbers are typical handbook values (mild steel S235, 304 annealed stainless, 5052-H32 aluminium, annealed copper, hardened spring steel). Hardening moduli are rough bilinear fits, so use the page for trends (thickness, die opening, yield stress, springback), not to set up a real machine.

Checks

Four boxes under the controls run the page's own functions: the closed-form moment-curvature against 4,000-fibre integration; the perfectly plastic limit and the shape factor 1.5, with the elastic die-span formula P S^3 / (48 E I); the residual stress balance after release; and the load-deflection integral against a 20,000-point Simpson reference together with the energy balance (area under the curve equals P delta / 2 in the elastic range, and the plastic work is positive above yield).