Engineering
Shaft Torsional Shear Stress (16T / pi*d^3) & Twist Angle (TL / GJ) formulas
Calculate elastic stress magnitude and signed twist for an ideal uniform solid circular shaft under pure torsion.
Use the solid circular shaft model within its assumptions
Assume a uniform solid circular shaft, constant applied torque along its length, linear elastic material and small deformations. Use torque in N*m, diameter and length in m and shear modulus in Pa. J=PI()*d^4/32 applies to this circular section.
Maximum surface shear magnitude uses ABS(T); twist retains the torque sign to indicate direction. Reversing torque changes twist direction but not stress magnitude. Doubling diameter divides ideal stress magnitude by 8 and twist by 16.
These formulas do not size a real drive shaft by themselves. Hollow or noncircular sections, keyways, stress concentrations, fatigue, combined bending and critical speeds need additional analysis. No allowable material stress or service limit is supplied.
| Drive Shaft Metric | Formula / Cell | 40 mm Steel Shaft (500 N*m) | Engineering Result |
|---|---|---|---|
| Torque T & Diameter d | B2 = 500 N*m, B3 = 0.04 m | Length B4 = 1.2 m, G B5 = 79.3 GPa | Solid steel drive shaft |
| Max Shear Stress tau_max | ((16 * B2) / (PI() * B3^3)) / 1E6 | 8,000 / (PI() * 6.4E-5) Pa | 39.79 MPa |
| Twist Angle (Degrees) | DEGREES((B2 * B4) / (B5 * J)) | 600 / (79.3E9 * 2.513E-7) rad | 1.72 deg |
| Double Diameter (80 mm) | B3 set to 0.08 m | 39.7887 / 8 | 4.97 MPa (8x lower stress!) |
Maximum outer-surface torsional shear stress tau_max in MPa
Divides Pascals by 1,000,000 to report shaft shear stress in MPa.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(((16 * ABS(B2)) / (PI() * (B3 ^ 3))) / 1000000, 2)))Elastic shaft twist angle in degrees via DEGREES((T * L) / (G * J))
Substitutes J = (PI() * B3^4) / 32 directly into (T * L) / (G * J) and converts radians to degrees.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(DEGREES((B2 * B4) / (B5 * ((PI() * (B3 ^ 4)) / 32))), 2)))Try the example
Use the original values or change them to test your understanding.