Torsional Stiffness Calculator
Enter the shaft geometry and material shear modulus to calculate torsional stiffness (k = GJ/L), angle of twist, maximum shear stress, and power transmission. Works for solid and hollow circular cross-sections in both metric and imperial units. Results update as you type.
Formula
Worked example
A solid steel shaft (G = 80 GPa) with d = 30 mm and L = 500 mm: J = pi x 30^4 / 32 = 79,522 mm^4. Stiffness k = 80,000 MPa x 79,522 mm^4 / 500 mm = 12,723 N*m/rad. Under T = 200 N*m the shaft twists theta = 200 / 12,723 = 0.0157 rad = 0.90 deg. Max shear stress tau = 200,000 N*mm x 15 mm / 79,522 mm^4 = 37.7 MPa, well below the yield shear stress of mild steel (~120 MPa).
What is torsional stiffness?
Torsional stiffness (k) is the resistance of a shaft or structural element to angular deformation under an applied twisting moment. It is defined as the ratio of applied torque T to the resulting angle of twist theta: k = T / theta, with units of N*m per radian (or lbf*in/rad in imperial). A higher stiffness means the shaft twists less under the same load, which is important for precise power transmission, resonance avoidance, and dimensional accuracy in machine tools and drivetrains.
How torsional stiffness is calculated
For a shaft made from a linear-elastic material, torsional stiffness is expressed as k = G * J / L, where G is the material shear modulus (resistance to shear deformation), J is the polar moment of inertia of the cross-section (a measure of how the area is distributed about the centroidal axis), and L is the effective shaft length. For a solid circular shaft, J = pi * d^4 / 32. For a hollow circular shaft, J = pi * (d_o^4 - d_i^4) / 32. Doubling the diameter increases J by a factor of 16, so shaft diameter has by far the greatest influence on stiffness. Halving the length doubles stiffness. Switching from steel (G ~80 GPa) to aluminium (G ~26 GPa) reduces stiffness by about two-thirds.
Angle of twist and maximum shear stress
Once torsional stiffness is known, the angle of twist under any torque T follows immediately: theta = T / k (radians). Converting to degrees: theta_deg = theta x 180 / pi. The maximum shear stress occurs at the outermost fibre of the shaft: tau = T * r / J, where r is the outer radius. This stress must stay below the material yield stress in shear (approximately 0.577 times the tensile yield stress for ductile metals, from the von Mises criterion). A safety factor of 2 to 4 is standard in rotating machinery to account for fatigue, stress concentrations at keyways or notches, and dynamic overloads.
Power transmission
A rotating shaft transmitting torque T at speed n (RPM) delivers mechanical power P = 2 * pi * n * T / 60 (in watts when T is in N*m). In imperial units, dividing by 745.7 converts watts to horsepower. The power formula links the structural problem (how stiff must the shaft be?) to the performance requirement (how much power must it carry?), and both must be satisfied simultaneously in drivetrain design. Higher power at the same speed means higher torque, which in turn demands a stiffer and stronger shaft.
Typical shear modulus values for common engineering materials
| Material | G (GPa) | G (ksi) | Typical use |
|---|---|---|---|
| Carbon steel (mild) | 79-82 | 11,500-11,900 | Shafts, axles, fasteners |
| Alloy steel (4340) | 80 | 11,600 | High-strength drivetrain |
| Stainless steel (304) | 73 | 10,600 | Corrosion-resistant shafts |
| Aluminium (6061-T6) | 26 | 3,770 | Lightweight aerospace structures |
| Titanium (Ti-6Al-4V) | 42 | 6,090 | Aerospace, medical implants |
| Copper | 44 | 6,380 | Electrical connectors, springs |
| Brass (70/30) | 37 | 5,370 | Bearings, bushings |
| Cast iron (gray) | 41 | 5,950 | Machine bases, housings |
| Nylon (PA66) | 0.9 | 130 | Light-duty gears, cams |
Use these values in the Shear Modulus (G) field. Values are approximate and may vary by alloy and condition.
Frequently asked questions
What is the difference between torsional stiffness and torsional rigidity?
Torsional rigidity is the product G * J (units of N*m^2 or Pa*m^4), representing the intrinsic resistance of a cross-section and material combination independent of length. Torsional stiffness is k = G * J / L, which divides by length to give the actual spring rate of a specific shaft. A short, fat shaft has high stiffness even if its rigidity is modest.
Why does shaft diameter matter so much?
The polar moment J scales with the fourth power of diameter. Doubling the diameter increases J by 16x and therefore stiffness by 16x for the same material and length. This makes diameter by far the most effective lever for improving torsional stiffness, but it also increases mass and cost rapidly, so the two must be balanced.
When should I use a hollow shaft instead of a solid shaft?
A hollow shaft delivers a much better stiffness-to-mass ratio than a solid shaft of the same outer diameter, because the central material contributes little to J (the integrand r^3 dr is small near the axis). In weight-critical applications such as aerospace or motorsport, a hollow shaft can save 30-50% mass while retaining 85-95% of the stiffness of a solid shaft with the same outer diameter.
What units should I use for G in this calculator?
Select metric and enter G in GPa (gigapascals). Typical values: mild steel 79-82 GPa, aluminium alloys 25-28 GPa, titanium alloys 40-45 GPa, copper alloys 37-44 GPa. In imperial mode enter G in ksi (thousands of pounds per square inch). The calculator converts internally to SI before computing.
How do I check if my shaft will yield under torsion?
Compare the computed maximum shear stress tau to the shear yield stress of your material. For ductile metals, the shear yield stress is approximately 0.577 x tensile yield stress (von Mises yield criterion). For example, mild steel with a tensile yield of 250 MPa has a shear yield of about 144 MPa. Apply a safety factor of at least 2 for static loads and 3-4 for rotating or fatigue-prone applications.