Sheet-Metal Stamping & Press-Brake Forming, Gardiner Elastic-Plastic Springback Law
Sheet-Metal Bending Elastic Springback & Overbend Formula
Calculate Gardiner Springback Factor Ks, Unloaded Inside Bend Radius Rf (mm), Required Press-Brake Overbend Angle (deg), and Elastic Springback Angle Recovery (deg).
Switch from mild steel to Grade 5 Ti-6Al-4V titanium on a 12 mm radius punch and press-brake springback explodes 12x from 1.4° to 16.4°!
Every sheet-metal bend retains an elastic core straddling its neutral axis: the instant the press-brake ram retracts, elastic unloading relaxes curvature so **Loaded Inside Radius `R_i`** (`mm`) springs open to a larger **Final Unloaded Radius `R_f`** (`mm`) while **Included Bend Angle `theta_f`** (`deg`) opens back by **`Delta_theta` degrees**! **F. J. Gardiner’s (`1957`, `ASME`) Canonical Cubic Springback Equation** governs pure elastic-perfectly-plastic bending from dimensionless ratio **`u = (R_i * sigma_y) / ((E * 1,000) * t)`** (across Sheet Thickness `t` mm, Yield Strength `sigma_y` MPa, and Young’s Modulus `E` GPa): **`K_s = (R_i + t/2) / (R_f + t/2) = 4 * u^3 - 3 * u + 1`**! Because neutral-axis arc length `(R + t/2) * theta` is conserved during unloading, hitting **Target Finished Bend Angle `theta_{target}`** (`deg`) requires **Loaded Press-Brake Overbend Angle `theta_{punch} = theta_{target} / K_s`** (`Springback Loss Delta_theta = theta_{punch} - theta_{target}`) and leaves **Unloaded Inside Radius `R_f = ((R_i + t/2) / K_s) - t/2`** (`mm`)!
Verify on a `2.0` mm HSLA `340`-MPa Steel Bracket bent to `90.0°`—Loaded_Inside_Radius `R_i` = 6.0 mm (B2), Sheet_Thickness `t` = 2.0 mm (B3), Yield_Strength `sigma_y` = 340 MPa (B4), Youngs_Modulus `E` = 205 GPa (B5), and Target_Bend_Angle = 90.0° (B6): Gardiner Factor `K_s` equals `0.9851` (E2), Unloaded Inside Radius grows slightly to `6.106` mm (E3), Required Ram Overbend equals `91.36°` (E4), compensating `1.36°` elastic springback (E5)!
| Gardiner Press-Brake Springback KPI (t=2.0 mm, Target=90.0°: HSLA Steel vs Ti-6Al-4V) | Formula | Verified Output |
|---|---|---|
| Gardiner Springback Factor Ks | 4 * ((Ri*sy / (E*1000*t))^3) - 3*u + 1 | 0.9851 steel (0.8458 titanium!) |
| Unloaded Inside Radius Rf (mm) | ((Loaded_Ri_mm + t_mm/2) / Ks) - t_mm/2 | 6.106 mm (12.0 -> 14.370 mm growth!) |
| Required Punch Overbend (deg) | Target_Bend_Angle_deg / Springback_Ks | 91.36° (106.41° deep V-die needed!) |
| Elastic Springback Recovery (deg) | Required_Overbend_deg - Target_Angle_deg | 1.36° steel (16.41° titanium!) |
Gardiner (1957) ASME Elastic-Core Cubic Springback Ratio Formula
Validate `Inside_Radius R_i > 0.2` mm, `Thickness t > 0.1` mm, `Yield_Strength` in `[30, 2,200]` MPa, `Modulus E` in `[40, 420]` GPa, plastic-yielding regime `u = (R_i * sigma_y) / (E * 1,000 * t) < 0.45` (`above u = 0.50` the sheet stays 100% elastic and springs completely flat!), and `Target_Angle` in `[5°, 175°]`!
=LET(ri_m, B2, ts_m, B3, sy_m, B4, em_g, B5, ta_d, B6, uv_r, (ri_m * sy_m) / ((em_g * 1000) * ts_m), ks_f, 4 * (uv_r ^ 3) - 3 * uv_r + 1, IF(OR(ri_m <= 0.2, ts_m <= 0.1, sy_m < 30, sy_m > 2200, em_g < 40, em_g > 420, uv_r >= 0.45, ta_d < 5, ta_d > 175), "Invalid Springback Input", ROUND((ta_d / ks_f) - ta_d, 2)))Aerospace Ti-6Al-4V Titanium (`R_i = 12` mm, `sigma_y = 980` MPa, `E = 114` GPa) 16.4° Springback Check
Form `2.0` mm `Ti-6Al-4V` (`B4 = 980` MPa high yield + `B5 = 114` GPa half-modulus of steel!) over a generous `B2 = 12.0` mm (`6t`) punch radius to prevent outer-fiber cracking: `K_s` E2 drops to **`0.8458`**, Inside Radius E3 blooms `+2.37` mm to `14.370` mm, and Springback E5 explodes **`12x` to `16.41°`**—demanding a `73.6°` included (`106.41°` swept overbend in E4) acute V-die!
=IF(ISTEXT(E5), E5, LET(uv_r, (B2 * B4) / ((B5 * 1000) * B3), ks_f, 4 * (uv_r ^ 3) - 3 * uv_r + 1, ROUND(((B2 + B3 / 2) / ks_f) - (B3 / 2), 3)))Try the example
Use the original values or change them to test your understanding.