Angle of Twist Calculator

Related Calculators

Solve for: The quantity the calculator will find for you, such as twist, diameter, torque, or length.

Shaft type: The cross-section choice. This tool only supports solid round and hollow round circular shafts.

Shear modulus G: A material stiffness value that tells how strongly the material resists shear deformation. Higher G means less twist for the same shaft and load.

Polar moment of inertia J: A geometry value that shows how well a circular section resists torsion. Larger J means a stiffer shaft in twist.

Torsional rigidity GJ: The combined twist resistance from the material and the shaft shape.

Angle of twist: How far one end of the shaft rotates relative to the other end under torque. The calculator shows it in both radians and degrees.

Maximum outer shear stress: The shear stress at the outside surface of the shaft, where torsional stress is largest for these circular sections.

Allowable twist check: A comparison between the computed twist and your optional Allowable twist (deg) limit.

Mistake: Entering Diameter (mm) as if it were in meters, such as typing 0.03 instead of 30.
Fix: Type diameters in millimeters exactly as the label shows, so a 30 mm shaft should be entered as 30.

Mistake: Filling in Inner diameter (mm) when Shaft type is solid, or making Inner diameter (mm) equal to or larger than Diameter (mm) for a hollow shaft.
Fix: Leave Inner diameter (mm) for hollow shafts only, and make sure it is smaller than Diameter (mm) .

Mistake: Leaving Allowable twist (deg) blank when Solve for is set to Required diameter , Torque , or Length .
Fix: Enter a positive target twist in Allowable twist (deg) before calculating those solve-for modes.

Mistake: Typing 79.3 in Shear modulus G (GPa) but thinking the field uses Pa.
Fix: Keep the value in GPa exactly as labeled. For a steel example, 79.3 means 79.3 GPa, not 79.3 Pa.

Mistake: Reading Angle of twist in radians as if it were degrees, or the other way around.
Fix: Check both Angle of twist outputs and use Output angle unit to make your main display match the unit you want to interpret.

Mistake: Using the Maximum outer shear stress result as a full pass-fail design check.
Fix: Treat Maximum outer shear stress as a quick companion result only, and use a separate material or code check if your design needs one.

• This calculator uses the elastic torsion model for uniform circular shafts only. It does not apply to noncircular sections.
• The shaft is assumed to be homogeneous and linear-elastic, so the formulas are intended for behavior before yielding or other nonlinear effects become important [2].
• Diameter (mm) sizing in Solve for = Required diameter is provided only for a solid round shaft. Hollow-shaft diameter sizing is not solved here with a closed-form result.
• The formulas assume one constant cross-section and one constant material over the entered Length (m). Stepped shafts, mixed materials, or varying diameter need segment-by-segment analysis.
• Results use positive magnitudes for torque, twist, and outer shear stress. Direction of rotation and sign convention are not modeled.
• Maximum outer shear stress is a basic mechanics result, not a full design-code check for fatigue, stress concentrations, keys, splines, temperature, or safety factors.
• If Allowable twist (deg) is used, the limit check compares against that entered value only. Real projects may also need stress, deflection, vibration, and serviceability checks.

Core equations

The calculator uses the standard circular-shaft torsion relation for elastic twist [1][2].

θ (angle of twist in rad) = T*L/(J*G)

Here, T is torque, L is shaft length, J is polar moment of inertia, and G is shear modulus.

J = π*d^4/32

This solid-shaft formula uses outside diameter d in meters.

J = π*(d_o^4 - d_i^4)/32

This hollow-shaft formula uses outside diameter d_o and inner diameter d_i in meters, with d_o > d_i >= 0 [1].

τ_max (maximum outer shear stress) = T*c/J

The outer radius c is d/2 for a solid shaft or d_o/2 for a hollow shaft. Stress is displayed in MPa.

Solve-for modes

When you choose a target twist, the calculator first converts Allowable twist (deg) to radians, because the torsion equations use radians.

d = (32*T*L/(π*G*θ))^(1/4)

This closed-form diameter equation is used only for Required diameter with a solid round shaft.

T = θ*J*G/L

This is used for Required torque.

L = θ*J*G/T

This is used for Required length.

Unit conversions

The calculator converts Shear modulus G (GPa) to pascals by multiplying by 1,000,000,000. It converts Diameter (mm) and Inner diameter (mm) to meters by multiplying by 0.001. It also converts radians to degrees using 180/π for the alternate angle output.

Mini example

For a solid shaft with T = 120 N*m, L = 2 m, G = 79.3 GPa, and d = 30 mm, the calculator first finds J, then uses the twist equation.

J = π*(0.03)^4/32 = 7.952156404e-8 m^4

θ = 120*2/(7.952156404e-8 * 79.3e9) = 0.0475768183 rad

θ in degrees = 0.0475768183 * 180/π = 2.725796843 deg

That means the shaft twists about 2.73 degrees over the 2 m length for that load.

Practical interpretation

If Polar moment of inertia J or Torsional rigidity GJ increases, twist goes down for the same torque and length. If torque or length increases, twist goes up. A hollow shaft can still resist twist well when much of its material stays far from the center because J depends strongly on diameter.

Input checks used

The calculator blocks nonpositive Shear modulus G (GPa). It also blocks invalid geometry such as Inner diameter (mm) greater than or equal to Diameter (mm) for hollow shafts, and it requires a positive Allowable twist (deg) in solve-for modes that need a target twist.