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Plastic Forming Simulator

Spring-Back Simulator — Elastic Recovery in Sheet Metal Bending

Spring-back ratio K = 4X^3 - 3X + 1 with elastic parameter X = sigma_y R_i / (E t), unloaded inside radius R_f = R_i / K and 90 deg return angle delta_theta = 90 (1 - K) for sheet metal pure bending. Inputs are yield stress sigma_y, Young's modulus E, inside bend radius R_i and thickness t. The loaded sheet (blue) and the spring-back shape after release (red) are drawn together with the K-X curve, so the scaling of elastic recovery in plate bending is easy to internalise.

Parameters
Yield stress sigma_y
MPa
Young's modulus E
GPa
Inside bend radius R_i
mm
Sheet thickness t
mm

Defaults are sigma_y = 300 MPa, E = 210 GPa (mild steel), R_i = 50 mm, t = 2.0 mm. Larger X (higher sigma_y or R_i, or smaller E or t) gives a smaller K and more spring-back. Thick-plate cases with small X stay near K = 1 with almost no recovery.

Results
—
Spring-back ratio K
—
Unloaded radius R_f
—
90 deg return angle
—
Elastic parameter X
Sheet bending schematic — loaded vs unloaded

Punch (yellow) and die (grey) bracket the sheet. Blue arc = loaded state at radius R_i. Red dashed arc = unloaded state at R_f = R_i / K. The red arc always opens wider, showing the elastic recovery direction.

K-X curve (elastic parameter vs return ratio)

Horizontal axis: X = sigma_y R_i / (E t). Vertical axis: K = 4X^3 - 3X + 1. K = 1 at X = 0 (no recovery), monotonically decreasing with X. Yellow marker = current operating point. Reference bands at K = 0.9 and K = 0.7 show the visual scale of recovery.

Theory & Key Formulas

For pure bending of an idealised elastic-perfectly plastic sheet, the Boothroyd / Marciniak closed-form spring-back ratio is:

$$K = \frac{R_i}{R_f} = 4X^3 - 3X + 1, \quad X = \frac{\sigma_y\, R_i}{E\, t}$$

$K$ is the ratio of unloaded to loaded curvature, $X$ is a dimensionless elastic parameter, $\sigma_y$ is the yield stress, $R_i$ is the inside bend radius, $E$ is Young's modulus and $t$ is the sheet thickness. Smaller $X$ (thicker plate, stiffer material, lower yield) pushes $K$ towards 1 and the recovery towards zero.

$$R_f = \frac{R_i}{K}, \quad \Delta\theta_{90} = 90^\circ\,(1 - K)$$

The unloaded inside radius $R_f$ scales as 1/K and the return angle on a 90 deg bend is linear in $(1 - K)$. To compensate, the punch angle should be set to $\theta / K$ — the classical over-bend rule.

$$\sigma = E\,\varepsilon \;(\varepsilon \le \varepsilon_y),\quad \sigma = \sigma_y \;(\varepsilon > \varepsilon_y)$$

Underlying constitutive law: linear up to the yield strain $\varepsilon_y = \sigma_y / E$, then perfectly plastic. Real materials work-harden, so the measured K is typically closer to 1 than the prediction of this simple model.

About the Spring-Back Simulator

🙋
When I bend a sheet to 90 degrees on a press brake and then release the punch, it usually springs back to around 88 degrees. Why does that happen? Does the metal "remember" its original shape?
🎓
Good observation — that is spring-back. When you bend a sheet, fibres outside the neutral plane are in tension and fibres on the inside are in compression. The fully plastic portion stays as permanent strain, but a thin layer near the neutral plane only deforms elastically. When the tool releases, that elastic strain springs back and reduces the curvature. With the tool defaults (sigma_y = 300 MPa, E = 210 GPa, R_i = 50 mm, t = 2.0 mm, basically mild steel) you get K = 0.893, an 11% recovery, and a 90 deg bend becomes about 80.4 deg.
🙋
The tool calls X = sigma_y R_i / (E t) the "elastic parameter". What does it physically mean?
🎓
Think of X as roughly the elastic strain at the outer fibre divided by the total bending strain. The outer fibre strain of a sheet of thickness t bent to a radius R_i is about t / (2 R_i), and the yield strain is epsilon_y = sigma_y / E. Their ratio looks like sigma_y R_i / (E t), so when X is large the elastic zone dominates and spring-back is big; when X is small the section is almost entirely plastic and spring-back is small. Try sliding t from 2 to 8 mm in the tool — X drops to a quarter and K climbs near 0.99, so the return angle almost vanishes.
🙋
People in the shop say high-strength steel "springs back more". Why is that?
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Advanced high-strength steels have large sigma_y (590, 780 or 1180 MPa grades) but the same Young's modulus as mild steel (about 210 GPa), so X grows by a factor of 2 to 4. Push sigma_y to 780 MPa in the tool and you get X = 0.0929, K = 0.722 and a 25 deg return angle. That is why automotive structural panels need very careful over-bend and die compensation. Aluminium alloys are a different story: their yield is moderate, but E is only about a third of steel's, so again X is large and recovery is roughly three times that of mild steel for the same geometry.
🙋
So how do real shops compensate for it? Do they calculate the over-bend angle every time?
🎓
Three approaches are used in practice. (1) Angle compensation: set the punch angle to theta_target / K — for K = 0.893 that is 100.8 deg for a 90 deg target. (2) Radius compensation: also reduce the punch radius to R_target * K so the unloaded inside radius hits the target. (3) Bottoming or coining: drive the punch into the die and force plasticity through the full thickness, which mechanically suppresses spring-back at the price of 3-10x the press force. This tool handles (1) and (2) analytically; bottoming needs its own model, but the K and R_f numbers here are still a useful first cut for tooling design and try-out planning.

FAQ

The expression follows from the moment-curvature relationship for an elastic-perfectly-plastic plate in pure bending. With an elastic core of half-thickness c the bending moment is M = sigma_y b (t^2/4 - c^2/3)/2, with c = E t / (2 sigma_y R_i). Elastic recovery on unloading changes the curvature by an amount proportional to M*6/(E b t^3); combining these expressions yields K = R_i / R_f = 4X^3 - 3X + 1 with X = sigma_y R_i / (E t). The derivation is presented in detail in classical sheet metal forming textbooks by Boothroyd and Marciniak.
For the idealised case of pure bending, linear elastic-perfectly plastic behaviour and no work hardening, the theory guarantees K <= 1, with K = 1 meaning no spring-back at all. K > 1 would require negative X, which is impossible for physical material parameters. The tool clamps K to [0.01, 1] for display robustness. In real shops, an apparent K > 1 is sometimes observed when work hardening, Bauschinger effects or combined bending and stretching modify the elastic recovery — those scenarios fall outside the assumptions of this simple model and require explicit FE simulation.
K = R_i / R_f is a curvature ratio that is independent of the bend angle, so it applies at any angle. The angular spring-back scales linearly with the bend angle: delta_theta = theta_target * (1 - K). For example, with K = 0.893 a 30 deg bend recovers by 3.2 deg and a 120 deg bend by 12.8 deg. The tool displays the 90 deg case for reference; just scale by theta_target / 90 for other targets. Extreme small angles (below 10 deg) and very large angles (above 170 deg) deviate from the pure-bending assumption because of friction and contact geometry.
The most common pattern is that the measured K is smaller than the theoretical value, i.e. the simple model under-predicts spring-back. Typical sources are (1) work hardening, which raises sigma_y during deformation, (2) the Bauschinger effect, which lowers the compression-side yield, (3) friction at the die shoulder that introduces a tensile component, (4) thickness reduction during bending, and (5) anisotropy of the elastic modulus. In thick-plate cases the opposite trend can appear when plane strain breaks down. Use the tool value as a starting point and refine through try-outs and FE simulation.

Real-world applications

Automotive body panel pressing: Modern car bodies use 590-1180 MPa advanced high-strength steels and 6000-series aluminium for weight reduction, all of which spring back 2-5 times more than mild steel. Increasing sigma_y to 590 MPa in the tool gives K = 0.79 and a return angle of 18.5 deg, which is hard to compensate by angle correction alone. Centre pillars on cars like the Audi A8 or Tesla Model 3 use bottoming combined with multi-stage bending to hold the final angle within +/- 0.5 deg.

Press brake V-bending: The everyday workhorse of sheet metal shops produces V- and L-bends where the difference between the target angle and the formed angle drives most quality complaints. A 1.0 mm mild-steel sheet (sigma_y = 250 MPa) bent on a small R_i = 1.0 mm punch yields X = 0.00119 and K = 0.996 — essentially no recovery. Switching to t = 3.0 mm and R_i = 8 mm typical of air bending raises K to about 0.99 and the return angle to about 0.85 deg; with sigma_y = 780 MPa it grows to 2.4 deg, which a modern CNC press brake stores in its angle-correction table.

Roll forming profiles: In continuous bending of steel strip for solar panel frames or building cold-formed sections, the cumulative angle from each stand drives the final cross-section accuracy. The K from each stand sets the over-bend angle 1/K. For sigma_y = 350 MPa structural strip, t = 2.3 mm and R_i = 5 mm the tool gives K = 0.978 and a 2 deg recovery per stand, so each roll is set to 92 deg to land 90 deg after release.

Precision bending of aircraft skins: Aluminium alloys (2024-T3, 7075-T6) used for ribs and stringers have E = 70 GPa and sigma_y from 300 to 500 MPa, so X is large. With sigma_y = 400 MPa, E = 70 GPa, R_i = 10 mm and t = 1.6 mm the tool returns X = 0.0357 and K = 0.893 — essentially the same as the mild-steel default. Stretch forming, where tension is applied during bending, brings K above 0.95; that case is outside the scope of this tool but the pure-bending numbers serve as a useful upper bound on the recovery.

Common misconceptions and caveats

The most frequent misconception is that "K is a material constant". The formula shows that K depends on the dimensionless group X = sigma_y R_i / (E t), so the same material returns very different K values for different geometries. For example SPHC steel (sigma_y = 300 MPa, E = 210 GPa) with R_i = 10 mm and t = 4 mm gives X = 0.00357 and K = 0.989 — almost no recovery — but R_i = 100 mm and t = 1 mm gives X = 0.143 and K = 0.583, a dramatic spring-back. Use the sliders to see how the same material behaves completely differently as geometry changes.

The second misconception is that over-bend alone is enough. In practice the tooling design must combine (a) angle compensation, where the punch angle is set to theta_target / K, (b) radius compensation, with R_punch = R_target * K, and (c) optionally bottoming. Angle compensation by itself usually leaves the unloaded inside radius too large. Use the R_f reading from the tool to size the punch radius too. Once you move into bottoming territory the simple closed-form is no longer valid and explicit FE codes such as PAM-STAMP, AutoForm or LS-DYNA take over.

The third pitfall is extending the plane-strain plate formula to three-dimensional forming. This tool assumes pure bending of an infinitely wide sheet with no in-plane strain transverse to the bend. That is a reasonable model for press-brake V-bending or roll-formed flanges where the bend length is much greater than the part width, but it fails for deep drawing, flanging and hemming, where in-plane tension and compression couple to bending. Use the tool for first-pass estimates only and rely on physical try-outs and full 3D CAE for complex geometries.

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