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V-die vs U-die bending force: how the equations compare
The force needed to bend a sheet depends on the tool as much as on the material. A V-die supports the sheet on two shoulders and the punch pushes at the middle; a U-die forms both legs of a channel against the punch, usually with a pad under the bottom. Different books give different equations for each case. This article puts them side by side: what each one assumes, where they agree and where they do not.
Scope: peak force estimates only. Springback, blank length and bend radius limits are not covered. To estimate the V-die air bending force from your own inputs, use the bending force calculator.
Calculate the force with your own data →
The equations side by side
| Case | Equation | Variables and conditions | Source |
|---|---|---|---|
| V-die (empirical) | F = Kbf·TS·w·t²/D | Kbf = 1.33 for V-bending. F in N with TS in MPa and w, t, D in mm. Based on simple beam theory. | Groover, Eq. 20.8 |
| V-die (empirical) | L = l·t²·k·S/s | US units: L in tonf, l and t in in., S in tsi, s = die opening in in. k goes from 1.2 (s = 16t) to 1.33 (s = 8t). 90° bends, no bottoming. | ASM Handbook 14B, Eq. 1 |
| V-die (empirical) | Fb = 1.2·b·s²·Rm/dw | N, mm, N/mm². dw = 5·ri (7·ri at the minimum radius). | Tschaetsch, 16.7.1 |
| V-die, air bending (moment-based) | F = 4M / [lk − 2(Rk + Rt + T)·sin(φ/2)] · cos²(φ/2) | M = bending moment (M = n·UTS·b·T²/4, with n = 1.6 to 1.8, Eq. 5.6a); lk = die opening; φ = bend angle; Rk = die radius; Rt = punch radius; T = thickness. | Boljanovic, Eq. 5.9 |
| V-die, bottoming | Fbbot = n·Fb | n = 2 for ri/s ≥ 0.7, 2.5 at 0.5 and 3.5 at 0.35. | Tschaetsch, p. 200 |
| V-die, bottoming | F1 = 2·p·b·c·cos(φ/2) | Extra force for the final reinforcement of the bend. p = specific pressure (Table 5.1), b = width, c = length of the straight end. | Boljanovic, Eq. 5.9a |
| U-die, no pad | Fb = 0.4·s·b·Rm | Without a backing pad the bottom bulges. | Tschaetsch, 16.7.2 |
| U-die, with pad | FbT = 0.5·s·w·Rm | Total force, about 1.25·Fb. Pad force FPP = 0.25·Fb. | Tschaetsch, 16.7.3 |
| U-die (moment-based) | F = (2M/l)·(1 + sin φ), l = Ri + Rk + T | Same M as above. With an ejector the force must increase by about 30%. | Boljanovic, Eq. 5.7, 5.7a |
| U-die, planished bottom | F = p·A | p = specific pressure (Table 5.1), A = area of the bottom. | Boljanovic, Eq. 5.7b |
Variable names and units follow each source and are not harmonized: the same quantity appears as t, s or T for thickness and as b or w for width. TS, S, Rm and UTS are all the tensile strength.
What they have in common
- All of them are static estimates. Boljanovic states that the bending force is obtained by treating the process as a simple bending beam, and Groover’s equation is also based on a simple beam.
- In the empirical and moment-based equations the force grows with the tensile strength and with the width of the part.
- The V-die equations scale with thickness squared divided by the die opening, which is what a beam moment balance gives. Tschaetsch’s U-die equation scales with thickness to the first power.
- A pressure pad adds roughly 25% to 30% of the force: 25% in Tschaetsch (FbT ≈ 1.25·Fb) and about 30% in Boljanovic.
- Both Boljanovic (Fig. 5.9, V-die) and Tschaetsch (Fig. 16.10, U-die) show a force–displacement curve with a rise, a hump, a dip and a steep final rise when the tool bottoms.
Where they differ
- Die opening. The V-die forces depend on the die opening (F ∝ 1/W); the empirical U-die equations contain no span.
- Process regime. The empirical V-die equations give the force before bottoming: the ASM Handbook states “without bottoming” explicitly, and Tschaetsch treats bottoming as an extra factor. Bottoming needs more force: Tschaetsch multiplies by n = 2 to 3.5, and Boljanovic adds a separate term, Eq. 5.9a. The ASM Handbook says the force for bottoming is difficult to estimate.
- Coefficient. The empirical coefficient is 1.33 in Groover, 1.2 to 1.33 depending on the die opening in the ASM Handbook, and a fixed 1.2 in Tschaetsch. Misumi, a secondary source, lists 1.5 for a die opening of about 5t.
- Moment-based versus empirical. Boljanovic’s equations need radii and the bend angle, which the empirical equations absorb into a single coefficient.
Worked example
The part comes from Tschaetsch’s own example (Section 16.11): thickness s = 2 mm, width b = 35 mm, tensile strength Rm = 400 N/mm², bend radius ri = 10 mm, so the die width is dw = 5·ri = 50 mm. All values below use Tschaetsch’s equations only, so every number traces to one source.
| Case | Equation used | Force |
|---|---|---|
| V-die (before bottoming) | 1.2·35·2²·400/50 | 1,344 N (the result printed in the book) |
| V-die, bottoming (n = 2) | 2 × 1,344 | 2,688 N |
| U-die, no pad | 0.4·2·35·400 | 11,200 N |
| U-die, with pad (total) | 0.5·2·35·400 | 14,000 N (pad force 0.25 × 11,200 = 2,800 N) |
The V-die value is the one printed in the book. The other three are our own arithmetic with the book’s equations. Dividing the book’s two equations gives FU/FV = dw/(3·s), which is 8.3 here. That ratio is our derivation, not a statement from the book.
The U-die force is much larger than the V-die force because the two equations describe different situations: the V-die value is bending over a wide opening (here 25 times the thickness) before bottoming, and the U-die value is the force to form a channel against the die. The empirical coefficient also matters: with k = 1.33 (Groover) the same V-die case gives 1,490 N, and with k = 1.5 it gives 1,680 N. Note that a die opening of 25t is wider than the 8t to 16t range for which the ASM Handbook gives k, so the V-die value is less certain than the others.
Limits
- These are estimates from textbooks and a handbook, not guaranteed values. Confirm the result against your tooling manufacturer’s data before choosing a press.
- The equations come from different books and conventions. The table compares their form; the numbers are not interchangeable.
- Boljanovic defines φ in Eq. 5.7 only in the figure, not in the text, so we quote the equation as printed.
- Radii, friction, lubrication and real variation in material properties are not modeled explicitly in the empirical equations.
Sources
- Groover, M. P., Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, 3rd ed., John Wiley & Sons, 2007. Chapter 20, Eq. (20.8), pp. 450–451.
- ASM Handbook, Volume 14B: Metalworking: Sheet Forming, S. L. Semiatin (ed.), ASM International, 2006. “Press-Brake Forming”, pp. 307–318, Eq. 1 (p. 308) and sample calculation (p. 309), DOI 10.1361/asmhba0005177; and “Bending of Sheet Metal”, pp. 295–306 (bottoming force, p. 297).
- Tschaetsch, H., Metal Forming Practise: Processes – Machines – Tools, Springer, 2006 (translated by A. Koth from the 8th German edition, Vieweg, 2005). Chapter 16, Sections 16.7.1 to 16.7.3 (pp. 199–200), Fig. 16.10 (p. 202). ISBN 978-3-540-33216-9.
- Boljanovic, V., Sheet Metal Forming Processes and Die Design, Industrial Press, New York, 2004. Sections 5.4.1 to 5.4.3, Eq. 5.7 to 5.9a (pp. 51–54), Fig. 5.9 (p. 54). ISBN 978-0-8311-3182-1.
- Misumi Tech Central, “Press forming force (3): Bending force” (secondary source for the coefficient at 5t).
Last reviewed: November 2026.