Torsional Stiffness Calculator

Enter a round shaft's dimensions and shear modulus to find its polar moment of inertia and torsional stiffness (k = GJ/L), plus the twist angle under an applied torque.

Quick Facts

Torsional stiffness formula
k = G · J / L
Torque per radian of twist; G is shear modulus, J is polar moment of inertia, L is shaft length.
Solid shaft polar moment
J = πD⁴/32
For a hollow shaft with inner diameter d, use J = π(D⁴ − d⁴)/32.
Typical shear modulus
Steel ≈ 79-80 GPa
Aluminum ≈ 26 GPa, titanium ≈ 44 GPa, brass ≈ 40 GPa — confirm with your material spec sheet.
Diameter sensitivity
k ∝ D⁴
Because J scales with the fourth power of diameter, a small diameter increase produces a large stiffness gain.

Your Results

Calculated
Torsional Stiffness
-
k = G × J / L, in N·m per radian
Polar Moment of Inertia
-
J = π(D⁴ − d⁴)/32, in mm⁴
Stiffness per Degree
-
Torque needed to twist the shaft by 1°
Twist Angle at Applied Torque
-
θ = T / k, for the torque you entered

Ready

Enter shaft dimensions, shear modulus, and torque, then press Calculate.

Formula and Method for Torsional Stiffness

Torsional stiffness (also called torsional rigidity) measures how much torque is required to twist a shaft through a given angle. It depends on the shaft's material, through the shear modulus G, and its cross-sectional geometry, through the polar moment of inertia J, combined with its length L: k = G × J / L, where k is expressed in newton-meters per radian (N·m/rad). This calculator finds J for a solid or hollow round shaft, applies the stiffness formula, and — if you provide an applied torque — reports how far the shaft twists.

How the calculation works

Enter the shaft's outer diameter, inner diameter (leave at 0 for a solid shaft), length, dimension unit, and the material's shear modulus in gigapascals. The calculator first finds the polar moment of inertia for the circular cross-section, J = π(D⁴ − d⁴)/32, converts your dimensions to meters, then multiplies the shear modulus (converted to pascals) by J and divides by the length to get the torsional stiffness k = GJ/L in N·m/rad. Multiplying k by π/180 converts it to N·m per degree, which is often more intuitive for small twist angles. If you also enter an applied torque T, the tool solves θ = T/k for the resulting angle of twist, reported in degrees.

Common mistakes

  • Confusing shear modulus with Young's modulus: torsion uses the shear modulus G, not the tensile modulus E — using E in place of G overstates stiffness by roughly a factor of 2.5 for most metals.
  • Forgetting the fourth-power relationship: doubling the diameter increases J — and therefore k — by a factor of 16, not 2, so small measurement errors in diameter have an outsized effect on the result.
  • Mixing units: diameter and length must be entered in the same unit before the calculator converts them to meters; treat inches and millimeters as interchangeable and the stiffness will be wrong by orders of magnitude.

Real-world applications

  • Drive shafts and axles: engineers size shaft diameter to keep torsional twist within tolerance under peak engine or motor torque.
  • Torsion bar springs: suspension and torsion-bar systems rely on k = GJ/L to tune ride stiffness.
  • Rotating machinery shafts: coupling alignment and vibration analysis both depend on knowing a shaft's torsional stiffness relative to attached components.
  • Servo and drive-train design: shaft torsional stiffness affects backlash and control-loop response in precision motion systems.

Frequently Asked Questions

What is the formula for torsional stiffness?
Torsional stiffness is k = GJ/L, where G is the material's shear modulus (Pa), J is the polar moment of inertia of the cross-section (m⁴), and L is the shaft length (m). The result, in N·m per radian, is the torque required to twist the shaft through one radian.
How do I calculate the polar moment of inertia for a shaft?
For a solid round shaft of diameter D, J = πD⁴/32. For a hollow shaft with outer diameter D and inner diameter d, J = π(D⁴ − d⁴)/32. Both formulas assume a circular cross-section; non-circular sections such as rectangular bars or I-beams use different, section-specific torsion constants.
Why does shaft diameter affect stiffness so much more than length?
Torsional stiffness is proportional to diameter raised to the fourth power (through J) but only inversely proportional to length. Increasing diameter by 20% raises J — and stiffness — by roughly 107%, while increasing length by 20% reduces stiffness by only about 17%.
What shear modulus should I use for common materials?
Typical shear modulus values are about 79-80 GPa for steel, 26 GPa for aluminum, 44 GPa for titanium, and 40 GPa for brass. Always confirm with the exact alloy's datasheet, since heat treatment and composition shift these values.

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