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Bulk Metal Forming, Open-Die Upset Forging, Barreling Friction Hill & Press Tonnage

Open-Die Upset Forging Friction-Hill & Tonnage Formula

Calculate Compressive Logarithmic True Strain, Constant-Volume Upsetted Diameter Df (mm), Siebel Friction-Hill Shape Multiplier C_f, and Peak Forging Press Force (kN).

As a forged pancake gets thinner, radial platen friction builds a conical "friction hill" that pushes press tonnage far above Yf * Area!

Compressing a cylindrical steel billet of initial diameter `D_0` (mm) and height `h_0` (mm) down to final pancake height `h_f` (mm) induces Logarithmic True Height Strain `epsilon = LN(h_0 / h_f)` while plastic incompressibility (`A_0 * h_0 = A_f * h_f`) expands Instantaneous Diameter to `D_f = D_0 * SQRT(h_0 / h_f)`! Because metal must flow radially outward against top and bottom flat-die Coulomb friction `mu`, radial inward shear restraint creates barreling and an axisymmetric conical pressure peak ("Friction Hill") at the billet center! Erich Siebel integrated the slab equilibrium equation to obtain the Forging Shape Multiplier `C_f = 1 + (2 * mu * (D_f / 2)) / (3 * h_f) = 1 + (mu * D_f) / (3 * h_f)`—so Peak Forging Force is `F = (Y_f * C_f * A_f) / 1,000` kN!

Verify on a Hot-Forged 4140 Gear-Blank Pancake—Initial_Billet `D_0` = 90.0 mm (B2), Initial_Height `h_0` = 135.0 mm (B3), Final_Forged_Height `h_f` = 54.0 mm (B4 -> True Strain `epsilon = 0.916` in E2 & Final Diameter `D_f = 142.30` mm in E3), Hot_Flow_Stress `Y_f` = 165.0 MPa (B5), and Graphite-Lubricated_Die_Friction `mu` = 0.24 (B6): Siebel Friction-Hill Factor equals `1.211` (E4), requiring `3,177.4` kN (`324` metric tons) of hydraulic press force (E5)!

Open-Die Upset Forging KPI (D0=90 mm, h0=135 -> hf=54 mm, Yf=165 MPa, mu=0.24)FormulaVerified Output
Compressive True Strain epsilonLN(Initial_h0_mm / Final_hf_mm)0.916
Upsetted Final Diameter Df (mm)D0_mm * SQRT(h0_mm / hf_mm)142.30 mm
Siebel Friction-Hill Factor Cf1 + (mu * Df_mm) / (3 * hf_mm)1.211 (1.395 @ dry unlubricated 0.45!)
Peak Forging Press Force (kN)(Yf_MPa * Cf * ((PI()/4)*Df^2)) / 10003,177.4 kN (3,661.5 kN @ dry die!)

Incompressible Area Expansion & Siebel Friction-Hill Upset Forging Formula

Validate `D_0 > 0` mm, `0 < h_f < h_0` (check `h_0 / D_0 <= 2.5` in practice so slender billets do not Euler-buckle before upsetting!), `Flow_Stress Y_f > 0` MPa, and `Die_Friction mu` in `[0, 0.577]` (von Mises sticking shear ceiling `1 / SQRT(3) ≈ 0.577`)!

=LET(dz_m, B2, hz_m, B3, hf_m, B4, yf_m, B5, mu_d, B6, IF(OR(dz_m <= 0, hf_m <= 0, hz_m <= hf_m, (hz_m / dz_m) > 3, yf_m <= 0, mu_d < 0, mu_d > 0.577), "Invalid Forging Input", LET(df_m, dz_m * SQRT(hz_m / hf_m), af_m, (PI() / 4) * (df_m ^ 2), cf_s, 1 + (mu_d * df_m) / (3 * hf_m), ROUND((yf_m * cf_s * af_m) / 1000, 1))))

Dry Scale-Covered Die Lubricant Breakdown Check (Set Friction mu B6 = 0.45)

Lose graphite die lubrication so hot oxide scale raises `mu` B6 from `0.24` to `0.45`: Final Diameter E3 stays `142.30` mm, yet Siebel Friction-Hill Factor E4 climbs from `1.211` to `1.395`, adding `+484.1` kN (`3,661.5` kN in E5) and severely barreling the billet perimeter!

=IF(ISTEXT(E5), E5, LET(df_m, B2 * SQRT(B3 / B4), ROUND(1 + (B6 * df_m) / (3 * B4), 3)))

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