What this calculator does
Torsional stiffness describes how much a shaft resists twisting under an applied torque: a stiffer shaft twists through a smaller angle for the same torque. It depends on the material's shear modulus, the cross-section's resistance to twisting, and how long the shaft is.
This calculator handles the common case of a solid circular shaft, where the cross-sectional resistance to twisting, the polar moment of inertia, can be worked out directly from the diameter. A longer shaft is always less torsionally stiff than a shorter one of the same material and diameter, since the same twist is spread over more length.
The formula
Torsional stiffness k equals the shear modulus G multiplied by the polar moment of inertia J, divided by the shaft length L. For a solid circular cross-section, J equals pi times the diameter to the fourth power, divided by 32. A larger diameter increases stiffness very quickly, since it enters the formula to the fourth power.
| Term | Meaning |
|---|---|
| k | Torsional stiffness: the torque needed to twist the shaft through one radian, in newton-metres per radian. |
| G | Shear modulus of the shaft material, a measure of its resistance to shear deformation. |
| J | Polar moment of inertia of the cross-section: for a solid circular shaft, pi times diameter to the fourth power, divided by 32. |
| L | The length of the shaft over which the twist is measured. |
The inputs explained
| Field | What to enter |
|---|---|
| Shear modulus (G) (GPa) | The shear modulus of the shaft material. Steel is commonly around 79 to 81 GPa, aluminium around 26 GPa. |
| Shaft diameter (mm) | The diameter of the solid circular shaft. |
| Shaft length (L) (m) | The length of the shaft between the two points where the twist is being measured. |
When to use it
Comparing shaft materials at the same size
Swapping the shear modulus for different materials at the same diameter and length shows directly how much stiffer a steel shaft is than an aluminium one of identical dimensions.
Seeing why diameter matters so much
Because J scales with diameter to the fourth power, even a modest increase in shaft diameter produces a large jump in torsional stiffness, which this calculator makes easy to see across a range of diameters.
Checking the effect of extending a shaft
Comparing torsional stiffness at a few different lengths for the same shaft shows how quickly stiffness falls off as a shaft is made longer.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How torsional stiffness changes with shaft diameter
A 1 metre steel shaft (79 GPa) at a range of diameters.
| Shaft diameter | Torsional stiffness (k) | Polar moment of inertia (J) |
|---|---|---|
| 20 mm | 1,240.93 N·m/rad | 15,707.96327 mm4 |
| 30 mm | 6,282.20 N·m/rad | 79,521.56404 mm4 |
| 40 mm | 19,854.87 N·m/rad | 251,327.4123 mm4 |
| 50 mm | 48,473.79 N·m/rad | 613,592.3152 mm4 |
| 75 mm | 245,398.58 N·m/rad | 3,106,311.095 mm4 |
| 100 mm | 775,580.69 N·m/rad | 9,817,477.042 mm4 |
How torsional stiffness changes with shaft length
The same 50mm diameter steel shaft at a range of lengths.
| Shaft length | Torsional stiffness (k) |
|---|---|
| 0.5 m | 96,947.59 N·m/rad |
| 1 m | 48,473.79 N·m/rad |
| 1.5 m | 32,315.86 N·m/rad |
| 2 m | 24,236.90 N·m/rad |
| 3 m | 16,157.93 N·m/rad |
| 4 m | 12,118.45 N·m/rad |
Questions
Does this work for a hollow shaft?
No, this calculator assumes a solid circular cross-section, where J = pi x d^4 / 32. A hollow shaft has a different, smaller polar moment of inertia that also depends on the inner diameter, so it needs a different formula.
Why does diameter affect stiffness so much more than length?
Length enters the formula only in the denominator, so stiffness is inversely proportional to it. Diameter enters through J to the fourth power, so small changes in diameter produce much larger changes in stiffness than the same proportional change in length.
What is the difference between torsional stiffness and shear modulus?
Shear modulus is a fixed material property. Torsional stiffness combines that material property with the specific shaft's geometry, diameter and length, so two shafts of the same material can have very different torsional stiffness.
What units does the result come out in?
With shear modulus in gigapascals, diameter in millimetres and length in metres, the torsional stiffness result is in newton-metres per radian, the standard engineering unit for this quantity.
For a related measure of a column or strut's resistance to buckling rather than twisting, see the slenderness ratio calculator. For a linear spring's stiffness rather than a shaft's, see the spring constant calculator.