Shaft Torsion Calculator: Shear Stress, Twist Angle & Keyseat Kt
Torsional shear stress in a circular shaft is τeff = Kt × T / Zp, and the elastic angle of twist is θ = (T × L) / (G × J), where polar moment J = π(do4 − di4) / 32. A solid Ø40 mm 4140 shaft carrying 450 N·m across 600 mm with a profiled keyseat (Kt = 1.60) develops 57.30 MPa effective stress (SF = 2.88) and twists 0.769°.
Solid & Hollow Shaft Torsional Shear Stress, Twist Angle & Keyseat K_t Calculator
Compute polar moment of inertia J = π(do4 − di4)/32, nominal and keyseat-concentrated shear stress τeff = Kt·T/Zp, torsional safety factor, and elastic twist angle θ = T·L/(G·J).
How Do Polar Moment of Inertia (J) and Keyseat Stress Concentration (K_t) Govern Shaft Strength?
When a drive shaft transmits torque T (in N·m, per NIST SI Units Reference and NIST Special Publication 811), elastic shear stress varies linearly from zero at the neutral axis to a nominal maximum τnom = T / Zp at the outer surface, where polar section modulus is Zp = 2J / do = π(do4 − di4) / (16do). Because the core material near the center carries minimal shear stress, boring a 40.0 mm solid shaft out to a 24.0 mm hollow tube (di/do = 0.60) removes 36.0% of the weight while increasing effective shear stress by only 14.9% (from 57.30 MPa to 65.83 MPa, still holding SF = 2.51).
In real machinery, shafts rarely fail on smooth cylindrical sections; over 90% of torsional fatigue fractures initiate at keyseats (ASME B17.1 (Keys and Keyseats)), shoulder steps, or retaining-ring grooves. According to DIN 743, Peterson's stress concentration factors, and Machinery's Handbook (31st Ed., Industrial Press), an end-milled profile keyseat (DIN 6885 Form A) applies a fatigue notch factor of Kt = 1.60 (raising effective shear stress on our Ø40 mm shaft from 35.81 MPa to 57.30 MPa), whereas a sled-runner keyseat cut with a disc mill (Form B) blends smoothly at the ends and lowers Kt to 1.30 (46.55 MPa, SF = 3.54). Material strength limits derive from ISO 6892-1 (Metallic Tensile Testing), matching standard references such as Engineering ToolBox and MechaniCalc.
| Shaft Diameter do | Polar Moment J | Section Modulus Zp | Nominal τnom | Keyed Peak τeff (Kt=1.6) | Twist θ (600 mm / deg·m−1) |
|---|---|---|---|---|---|
| Ø20 mm | 15,708 mm&sup4 | 1,571 mm³ | 286.5 MPa | 458.4 MPa (SF=0.36) | 12.311° (20.518°/m) |
| Ø25 mm | 38,350 mm&sup4 | 3,068 mm³ | 146.7 MPa | 234.7 MPa (SF=0.70) | 5.042° (8.404°/m) |
| Ø30 mm | 79,522 mm&sup4 | 5,301 mm³ | 84.9 MPa | 135.8 MPa (SF=1.21) | 2.432° (4.053°/m) |
| Ø35 mm | 147,324 mm&sup4 | 8,418 mm³ | 53.5 MPa | 85.5 MPa (SF=1.93) | 1.313° (2.188°/m) |
| Ø40 mm | 251,327 mm&sup4 | 12,566 mm³ | 35.8 MPa | 57.3 MPa (SF=2.88) | 0.769° (1.282°/m) |
| Ø50 mm | 613,592 mm&sup4 | 24,544 mm³ | 18.3 MPa | 29.3 MPa (SF=5.62) | 0.315° (0.525°/m) |
| Ø60 mm | 1,272,345 mm&sup4 | 42,412 mm³ | 10.6 MPa | 17.0 MPa (SF=9.72) | 0.152° (0.253°/m) |
| Ø75 mm | 3,106,311 mm&sup4 | 82,835 mm³ | 5.4 MPa | 8.7 MPa (SF=18.98) | 0.062° (0.104°/m) |
Why Must Long Drive Shafts Be Checked for Twist Angle Per Meter (θ / L)?
Because polar moment J scales with the fourth power of diameter (do4) while section modulus scales with the cube (do³), long or small-diameter shafts frequently pass stress checks while suffering excessive angular deflection θ = T·L / (G·J). Furthermore, heat-treating a steel shaft from 25 HRC to 52 HRC raises allowable stress τallow but does not change the shear modulus of rigidity G (~79.3 to 80.0 GPa for all carbon and alloy steels)—meaning only a larger diameter reduces elastic twist!
| Shaft Alloy | Shear Modulus G | Allowable τallow | Plain Shaft (Kt=1.0) | Profiled Keyseat (Kt=1.6) | Twist Angle (L=600 mm) |
|---|---|---|---|---|---|
| SAE 1045 steel | 79.3 GPa | 110 MPa | 35.8 MPa (SF=3.07) | 57.3 MPa (SF=1.92) | 0.776° (1.294°/m) |
| SAE 4140 Q&T steel | 80.0 GPa | 165 MPa | 35.8 MPa (SF=4.61) | 57.3 MPa (SF=2.88) | 0.769° (1.282°/m) |
| 304 stainless steel | 77.0 GPa | 85 MPa | 35.8 MPa (SF=2.37) | 57.3 MPa (SF=1.48) | 0.799° (1.332°/m) |
| 6061-T6 aluminum | 26.0 GPa | 65 MPa | 35.8 MPa (SF=1.82) | 57.3 MPa (SF=1.13) | 2.367° (3.946°/m) |
| Ti-6Al-4V grade 5 | 44.0 GPa | 180 MPa | 35.8 MPa (SF=5.03) | 57.3 MPa (SF=3.14) | 1.399° (2.332°/m) |
Worked Example: Complete Strength & Stiffness Audit
For our Ø40.0 mm solid 4140 Q&T shaft (τallow = 165 MPa, G = 80.0 GPa) transmitting 450.0 N·m across L = 600.0 mm, polar moment is J = 251,327.4 mm&sup4, polar modulus is Zp = 12,566.4 mm³, and nominal shear stress is 35.81 MPa. With a profiled keyseat (Kt = 1.60), effective stress is 57.30 MPa (SF = 2.88) and twist angle is 0.769° (1.282°/m). Size your keyway on our ANSI B17.1 keyway calculator and keyway broaching force calculator, compare keyless shrink fits on the press-fit interference calculator, check circlip grooves on the retaining ring groove chart, or upgrade to splines with the DIN 5480 involute spline calculator.
Cite this page
ToolSpecVault Engineering Data Team. (2026, October 29). Shaft Torsion Calculator: Shear Stress, Twist Angle & Keyseat Kt. ToolSpecVault. https://toolspecvault.com/shaft-torsional-shear-stress-angle-of-twist-calculator/