What is Torsional Stiffness?
Torsional stiffness (also called torsional spring rate or torsional rigidity) is the resistance of a structural member to angular deformation when subjected to a torsional load (torque). It is the rotational analog of the spring constant in linear systems — described by Hooke's law — but applied to twisting instead of stretching or compressing.
In engineering, torsional stiffness is critical for shafts, drive axles, beams, and torsion springs, where excessive twist can lead to failure, misalignment, or poor performance.
The Formulas for Torsional Stiffness Calculation
There are two approaches to calculating torsional stiffness:
Method 1: Experimental Formula — k = T / φ
The general definition of torsional stiffness is the ratio of the applied torque to the resulting angle of twist:
k = T / φ
- k — Torsional stiffness (N·m/rad, lbf·ft/rad, or lbf·in/rad)
- T — Applied torque (N·m, lbf·ft, or lbf·in)
- φ — Angle of twist (radians)
This formula is universal — it applies to beams, torsion springs, and any system that resists torsional loads. For torsion springs, φ is called the angular deflection of the coil body.
Method 2: Geometric Formula — k = GJ / L
By substituting the angle of twist formula for beams (φ = TL/JG) into the equation above, we get a more practical formula that uses material and geometry properties:
k = G·J / L
- k — Torsional stiffness (N·m/rad)
- G — Shear modulus of the material (Pa)
- J — Polar moment of inertia of the cross-section (m⁴)
- L — Length of the beam or shaft (m)
This formula applies only to straight beams with uniform cross-sections. For non-circular cross-sections, use the torsional constant (Jt) instead of the polar moment of inertia.
Torsional Stiffness Units
The unit of torsional stiffness is torque per unit angle. The most common units are:
| Result Unit | G unit | L unit | J unit | T unit |
|---|---|---|---|---|
| N·m/rad (SI) | Pa (N/m²) | m | m⁴ | N·m |
| lbf·ft/rad | lbf/ft² | ft | ft⁴ | lbf·ft |
| lbf·in/rad | psi (lbf/in²) | in | in⁴ | lbf·in |
Polar Moment of Inertia (J)
The polar moment of inertia measures a cross-section's resistance to torsion. For common shapes:
- Solid circular shaft: J = π·d⁴/32, where d is the diameter
- Hollow circular shaft: J = π(do⁴ − di⁴)/32
- Non-circular sections: Use the torsional constant (Jt), which is generally smaller than the polar moment
Shear Modulus (G)
The shear modulus (also called the modulus of rigidity) is a material property that describes its resistance to shear stress. Typical values:
- Steel: G ≈ 79–80 GPa (11.5–11.6 × 10⁶ psi)
- Aluminum: G ≈ 26 GPa (3.8 × 10⁶ psi)
- Copper: G ≈ 48 GPa (7.0 × 10⁶ psi)
- Titanium: G ≈ 41 GPa (6.0 × 10⁶ psi)
FAQs
How do I calculate the torsional stiffness?
There are two ways:
- Using the applied torque (T) and angle of twist (φ): k = T/φ
- Using material and geometry: k = G·J/L
For example, a beam experiencing a torque of 80,000 N·m that causes a 0.02037 rad twist: k = 80,000 / 0.02037 ≈ 3,927,344 N·m/rad.
What is the difference between torsional stiffness and torsional rigidity?
Torsional stiffness (k = GJ/L) depends on the beam length and represents the stiffness of a specific member. Torsional rigidity (GJ) is a cross-section property that is independent of length — it is the product of shear modulus and polar moment of inertia.
Does a longer beam have higher or lower torsional stiffness?
Lower. Since k = GJ/L, stiffness is inversely proportional to length. A longer beam twists more for the same applied torque, making it less stiff.
Can I use this calculator for torsion springs?
Yes, using Method 1 (k = T/φ). For torsion springs, the "angle of twist" is the angular deflection of the spring body, and the result is the spring's torsional spring rate.