Powertrain & Off-Highway Machine Elements, ISO 4156 / ANSI B92.1 Involute Spline
SAE / ISO 4156 Involute Spline Flank & Shear Formula
Calculate Total Pitch-Circle Tangential Force (kN), Load-Shared Tooth Flank Compressive Stress (MPa), Minor-Root Torsional Shear Stress (MPa), and % of 120 MPa Case-Hardened Limit.
Assuming all 24 involute spline teeth share torque 100% equally misses Dudley’s Km = 0.50–0.75 index-error factor—fretting short misaligned PTO couplings into red iron-oxide dust!
**ISO `4156` and ANSI `B92.1` (`30°` Pressure-Angle) Involute Splined Shafts** transmit `30%–50%` more torque than straight-sided keys because self-centering `30°` involute flanks pilot under load—yet real index spacing errors and angular shaft misalignment mean only **Load-Distribution Factor `K_m` (`0.75` fixed aligned grease-packed hub; `0.50` sliding/misaligned PTO sleeve!)** of the **`N` teeth** carry **Pitch-Circle Tangential Force `F_t = (2,000 * T) / D`** (`N` at Torque `T` N·m and Pitch Diameter `D` mm)! Checking **Compressive Flank Bearing Stress `sigma_b = F_t / (N * h * L_e * K_m)`** (`MPa` across Radial Tooth Contact Height `h` mm and Effective Hub Engagement `L_e` mm, kept `<= 120` MPa on carburized `58–62` HRC splines or `<= 30` MPa sliding!) alongside **Solid Minor-Root Torsional Shear `tau_r = (16,000 * T) / (pi * D_{re}^3)`** (`MPa` at Minor Diameter `D_{re}` mm) prevents both flank fretting wear and torsional twist-off!
Verify on a `24`-Tooth, `45` mm Pitch-Diameter Carburized `8620` Axle Hub Spline—Peak_Torque `T` = 1,450 N·m (B2), Pitch_Diameter `D` = 45.0 mm (B3), External_Minor_Root `D_{re}` = 41.5 mm (B4), Tooth_Count `N` = 24 (B5), Radial_Contact_Height `h` = 1.65 mm (B6), Engaged_Length `L_e` = 38.0 mm (B7), and Aligned_Load_Sharing `K_m` = 0.75 (B8): Total Pitch Tangential Force equals `64.44` kN (E2), Flank Compressive Stress equals **`57.10` MPa** (E3), Minor-Root Shaft Shear equals `103.32` MPa (E4), utilizing **`47.6%` of the `120` MPa hardened flank ceiling** (E5)!
| ISO 4156 / Dudley Involute Spline KPI (24 Teeth, D=45 mm, Dre=41.5 mm, h=1.65 mm) | Formula | Verified Output |
|---|---|---|
| Pitch Tangential Force Ft (kN) | ((2000 * Torque_Nm) / Pitch_D_mm) / 1000 | 64.44 kN (117.78 kN shock torque) |
| Flank Bearing Stress sb (MPa) | Ft_N / (Teeth_N * Height_h * Le * Km) | 57.10 MPa (270.38 MPa crushing!) |
| Minor-Root Shaft Shear (MPa) | (16000 * Torque_Nm) / (PI() * (Dre_mm ^ 3)) | 103.32 MPa (188.83 MPa root twist) |
| % of 120 MPa Hardened Limit | (Flank_Bearing_MPa / 120) * 100 | 47.6% SAFE (225.3% FRETTING STRIP!) |
ISO 4156 / Dudley Involute Spline Flank Contact & Polar Root Shear Formula
Validate `Torque T > 0` N·m, `Pitch_Diameter D` in `[8, 500]` mm, `Minor_Root D_{re}` in `[0.60 * D, D)`, `Teeth N` in `[6, 120]`, `Contact_Height h` in `[0.3, 15.0]` mm, `Engaged_Length L_e` in `[5, 500]` mm, and `Sharing_Factor K_m` in `[0.25, 1.00]`!
=LET(tq_n, B2, dp_m, B3, dr_m, B4, nt_s, B5, hc_m, B6, le_m, B7, km_f, B8, sb_m, ((2000 * tq_n) / dp_m) / (nt_s * hc_m * le_m * km_f), IF(OR(tq_n <= 0, dp_m < 8, dp_m > 500, dr_m < 0.6 * dp_m, dr_m >= dp_m, nt_s < 6, nt_s > 120, hc_m < 0.3, hc_m > 15, le_m < 5, le_m > 500, km_f < 0.25, km_f > 1), "Invalid Spline Input", ROUND((sb_m / 120) * 100, 1)))Half-Engaged Misaligned Sliding Sleeve (`T = 2,650` N·m, `L_e = 22` mm, `K_m = 0.50`) Strip Check
Fail to seat a snap ring so hub engagement slips back to `B7 = 22.0` mm under angular misalignment (`B8 = 0.50`) during a `B2 = 2,650` N·m driveline shock: While Minor-Root Shear E4 (`188.83` MPa) still survives in alloy steel, **Flank Bearing Stress E3 jumps `4.7x` to `270.38` MPa** (`225.3%` of limit in E5)—spalling carburized cases and stripping internal hub teeth smooth!
=IF(ISTEXT(E5), E5, ROUND(((2000 * B2) / B3) / (B5 * B6 * B7 * B8), 2))Try the example
Use the original values or change them to test your understanding.