Angle of Twist Calculator
Enter the applied torque, shaft dimensions, length, and material to find the angle of twist and maximum shear stress. Choose solid or hollow shafts, switch between metric and imperial units, pick a material preset, or type your own shear modulus. The step-by-step panel shows exactly how each number is derived.
What is the angle of twist?
When a torque is applied along the axis of a shaft, every cross-section rotates slightly relative to the adjacent one. The total rotation from one end to the other is the angle of twist (usually written phi or theta). It is a measure of how much the shaft winds up under load. A small angle of twist is generally desirable in precision drives - excessive twist causes timing errors, vibration, and fatigue failure at stress concentrations. The governing equation from linear elastic torsion theory is: phi = T L / (G J), where T is the applied torque, L is the shaft length, G is the shear modulus of the material, and J is the second polar moment of area of the cross-section.
How to use this calculator
Select metric or imperial units first, then choose solid or hollow shaft. For a hollow shaft an inner diameter field appears. Enter the torque, shaft length, outer (and inner) diameter, and pick a material preset. If you know the exact shear modulus of your alloy, choose "Custom" and type it in. The allowable shear stress field is optional: it triggers a stress safety factor and shows the maximum torque the shaft can carry before reaching that limit. Results update instantly. The step-by-step panel below the result card shows every intermediate value so you can verify each calculation yourself.
Solid vs. hollow shafts
A hollow shaft is more efficient than a solid one of the same outer diameter because most of the torsional resistance comes from the outer annulus of material. Removing the lightly loaded core reduces weight with only a modest reduction in J. For the same mass, a hollow shaft always has a higher outer diameter and therefore a higher J than its solid equivalent, making it the preferred choice whenever weight and stiffness are both important, as in automotive prop shafts and bicycle frames. The polar moment for a hollow section is J = (pi/32)(D^4 - d^4), while for a solid section it simplifies to J = (pi/32) D^4.
Shear stress and the safety factor
Torsion produces a maximum shear stress at the outer surface of the shaft: tau_max = T (D/2) / J. This is the same formula as the flexure formula but for torsion. If this stress exceeds the material yield strength in shear (roughly half the tensile yield strength for most metals), the shaft yields plastically. The design practice is to set an allowable shear stress - typically 40-60 percent of the shear yield strength for static loading, lower for fatigue - and keep the safety factor (allowable / actual) above 1. For rotating shafts subject to cyclic loading, a safety factor of at least 1.5 to 2 is common.
Polar moment of inertia and how to maximise torsional stiffness
The polar moment of inertia J (mm^4 or in^4) measures how spread out the cross-sectional area is from the shaft axis: the further material sits from the centre, the higher J and the stiffer the shaft in torsion. Doubling the diameter increases J by a factor of 16 (it scales as D^4), so diameter is the most powerful lever for reducing twist. Lengthening a shaft increases twist proportionally, so designers keep torsionally sensitive shafts as short as practical, or add intermediate support bearings. Choosing a higher-modulus material (for example, steel at 79 GPa instead of aluminum at 26 GPa) reduces twist by about three times for the same geometry.
Shear modulus of common engineering materials
| Material | Shear Modulus G (GPa) | Typical application |
|---|---|---|
| Carbon / alloy steel | 79 | Drive shafts, fasteners, structural members |
| Stainless steel | 77 | Corrosion-resistant shafts, food/medical equipment |
| Aluminum alloy | 26 | Lightweight frames, aerospace components |
| Copper | 44 | Electrical bus bars, heat exchangers |
| Brass | 37 | Bushings, decorative hardware, valve stems |
| Titanium alloy | 41 | Aerospace and biomedical implants |
| Cast iron | 41 | Machine bases, engine blocks |
| Magnesium alloy | 17 | Ultra-light aerospace/automotive housings |
Representative values at room temperature. Exact values vary by alloy and heat treatment.
Frequently asked questions
What does the angle of twist formula phi = TL / (GJ) mean?
The formula says the twist angle is directly proportional to torque (T) and length (L), and inversely proportional to shear modulus (G) and polar moment of inertia (J). A longer shaft twists more, a higher torque twists it more, but a stiffer material or a larger/hollow cross-section resists twist. All four variables have a one-to-one linear relationship with the result.
What is the polar moment of inertia and why does it matter?
The second polar moment of area J (mm^4 or in^4) represents how the cross-sectional area is distributed around the shaft centre. Material far from the axis contributes much more than material near it (it scales with radius to the fourth power). A larger J means less shear stress and less twist for a given torque. For a solid circle, J = (pi/32) D^4; for a hollow circle, J = (pi/32)(D^4 - d^4).
What shear modulus value should I use for steel?
For carbon and alloy steel (the most common shaft material), use approximately 79-80 GPa. Stainless steels are slightly lower at about 77 GPa. These values assume room-temperature operation; at elevated temperatures G decreases. The material preset in this calculator uses 79 GPa as the default for steel.
Why is the angle of twist expressed in both radians and degrees?
The formula naturally produces radians, which is the SI unit for angle and is used in subsequent calculations. Degrees are more intuitive for most engineers when judging whether a shaft is appropriately stiff - 0.5 degrees is clearly small, whereas 0.00873 rad requires mental conversion. This calculator gives you both.
How do I reduce the angle of twist if it is too large?
The four levers are: (1) increase the outer diameter - most effective because J scales as D^4; (2) use a hollow section to move material to the outer radius; (3) choose a higher shear modulus material such as steel instead of aluminum; (4) shorten the shaft by adding a support bearing or redesigning the power path. Doubling the diameter is typically the fastest fix.
What is the difference between torsional stiffness and torsional strength?
Stiffness refers to how much a shaft resists angular deformation (small phi for a given T) and is controlled by G and J. Strength refers to how much torque the shaft can carry before yielding or fracturing and is governed by the allowable shear stress and J. A shaft can be stiff but weak (short but thin-walled) or strong but flexible (long, thick solid bar). This calculator addresses both: the angle of twist covers stiffness and the safety factor addresses strength.