Calculate torsional stiffness from shaft geometry and material properties or from measured torque and twist with consistent unit handling.
Advanced options
How to use our Torsional Stiffness Calculator
- Choose Calculation mode: use geometry when you know dimensions and material, or use test mode when you measured torque and twist.
- In geometry mode, choose Cross-section type: solid circular shaft, hollow circular shaft, or direct entry of Torsion constant J (selected unit^4).
- Enter Member length (selected unit) and, in geometry mode, enter Shear modulus G (selected unit).
- If you picked a circular shape, enter Outer diameter (selected unit); for a tube also enter Inner diameter (selected unit). If you picked direct J, enter the section property directly.
- In test mode, enter Torque T (selected unit) and Twist angle theta (selected unit), then make sure Angle input unit matches your angle value.
- Open Advanced options if needed to set Unit system, Torque input unit, Display format, or to enable Also compute twist for a given torque.
- Click Calculate to see torsional stiffness per radian and per degree. In geometry mode you will also see Torsional rigidity GJ and Section torsion constant J.
- Sanity-check the result: a longer Member length (selected unit) should lower stiffness, while a larger diameter or larger Shear modulus G (selected unit) should raise it.
Definitions
Calculation mode: Chooses whether stiffness is found from geometry and material data or from measured torque and twist.
Shear modulus G: A material property that tells how strongly the material resists shear, which is the sideways deformation caused by twisting.
Torsion constant J: A section property that depends on shape and size. For circular shafts here, it is computed from diameter or entered directly.
Torsional rigidity GJ: Material resistance times section resistance. It is not the final member stiffness until you divide by length.
Torsional stiffness k: Torque needed for a certain amount of twist over the entered length. Higher k means harder to twist.
Twist angle theta: The angular rotation caused by torque, entered in degrees or radians.
Torque T: The twisting moment applied to the member.
Common mistakes and quick fixes
Mistake: Entering Twist angle theta (selected unit) in degrees while Angle input unit is set to radians.
Fix: Change Angle input unit to match your measurement before calculating Torsional stiffness k (per radian) .
Mistake: Using hollow-shaft dimensions where Inner diameter (selected unit) is equal to or larger than Outer diameter (selected unit) .
Fix: For Cross-section type set to hollow, make sure outer diameter is greater than inner diameter.
Mistake: Typing a material value in GPa into Shear modulus G (selected unit) while Unit system is set to SI with Pa.
Fix: Either switch Unit system to metric engineering or enter the full value in pascals so Torsional rigidity GJ is correct.
Mistake: Entering zero for Twist angle theta (selected unit) in test mode.
Fix: Use a nonzero measured twist angle because Torsional stiffness k (per radian) is undefined when angle is zero.
Mistake: Confusing Torsion constant J (selected unit^4) with Torsional rigidity GJ .
Fix: J is a section property only; GJ multiplies that property by material stiffness, so check both outputs separately.
Mistake: Forgetting that Member length (selected unit) is part of the geometry formula.
Fix: If your Torsional stiffness k (per degree) looks too high or too low, recheck the actual twisting length used in the setup.
Limitations & Key Assumptions / Boundary Conditions
- These results assume a uniform member with constant cross section and linear elastic torsion.
- The built-in shape formulas are only for solid and hollow circular shafts. For other shapes, use direct entry of Torsion constant J (selected unit^4) only if you already know a valid J value.
- Torsional stiffness k (per radian) from geometry uses k = GJ/L, so errors in length, modulus, or diameter can strongly change the result.
- Test mode assumes the measured Twist angle theta (selected unit) and Torque T (selected unit) come from the same length and loading setup.
- Very large or very small values may be shown in scientific notation even when decimal display is selected, to keep numbers readable.
- This tool does not model yielding, plastic deformation, stress concentrations, nonuniform shafts, joints, or temperature effects.
Methodology
Core equations
The calculator uses one of two standard torsion relations depending on the selected mode.
k = GJ / L
Here, k is torsional stiffness, G is shear modulus, J is torsion constant for the section, and L is member length.
k = T / θ
In test mode, T is torque and θ is twist angle in radians. If the user enters degrees, the calculator converts degrees to radians first.
Section property for circular shafts
For circular sections, the calculator can compute J automatically.
J = π d^4 / 32
J = π (d_o^4 - d_i^4) / 32
The first line is for a solid circular shaft with diameter d. The second is for a hollow circular shaft with outer diameter d_o and inner diameter d_i.
Related outputs
The calculator also reports torsional rigidity so users can separate section-and-material resistance from whole-member stiffness.
GJ = G * J
When the optional reference load is enabled in geometry mode, the twist angle is also computed.
θ = T L / (GJ)
Stiffness per degree is derived from stiffness per radian.
k_per_deg = k_per_rad * (π / 180)
Unit handling
Internally, the calculator keeps angle math consistent by converting input twist to radians before using k = T / θ. It also converts selected torque, pressure, and length units so the formulas stay consistent.
Worked mini-example
Suppose a solid shaft has length 1 m, shear modulus 80,000,000,000 Pa, and diameter 0.05 m.
J = π * 0.05^4 / 32 = 6.1359 x 10^-7 m^4
GJ = 80,000,000,000 * 6.1359 x 10^-7 = 49,087.4 N m^2
k = GJ / L = 49,087.4 / 1 = 49,087.4 N m/rad
k_per_deg = 49,087.4 * (π / 180) = 856.7 N m/deg
This means the shaft needs about 856.7 N m of torque for each degree of twist over that 1 m length.
Assumptions behind the result
The result is most reliable for uniform circular members in the elastic range. Real parts can differ if the shaft is stepped, has keyways or joints, uses noncircular open sections, or is loaded outside small-twist linear behavior.