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Angle of Twist Calculator
Calculate angle of twist, shear stress, and torsional stiffness for solid or hollow shafts under torque. Compare materials side by side.
Material Presets
Angle of Twist⧉
1.1775°
0.020552 rad — twist over the full shaft length
Polar Moment J⧉
613,592.32 mm⁴
Second polar moment of area of the cross-section
Max Shear Stress⧉
20.37 MPa
Shear stress at the outer surface
Torsional Stiffness⧉
24,328.94 N·m/rad
Torque required per radian of twist
Power @ 1000 RPM⧉
52.36 kW
Transmitted power at 1 000 RPM
Weight per Metre⧉
15.41 kg/m
Approximate mass per unit length (steel)
Twist vs Diameter
Material Shear Modulus Reference
| Material | G (GPa) | Twist (°) @ current load |
|---|---|---|
| Steel | 79.3 | 1.1775 |
| Aluminum | 26 | 3.5914 |
| Copper | 44 | 2.1222 |
| Brass | 39 | 2.3943 |
| Titanium | 41 | 2.2775 |
| Cast Iron | 41 | 2.2775 |
Twist vs Length
| Length (m) | Twist (°) | Twist (rad) |
|---|---|---|
| 0.5 | 0.2944 | 0.005138 |
| 1 | 0.5888 | 0.010276 |
| 2 | 1.1775 | 0.020552 |
| 3 | 1.7663 | 0.030827 |
| 5 | 2.9438 | 0.051379 |
| 8 | 4.7101 | 0.082207 |
Planning notes, formulas, and examples
About the Angle of Twist Calculator
The **Angle of Twist Calculator** computes how much a circular shaft rotates under an applied torque. This is one of the most fundamental calculations in mechanical and structural engineering — used every time a drive shaft, axle, or structural tube is designed.
Given the torque, shaft geometry (solid or hollow circular cross-section), length, and material shear modulus, the calculator finds the total twist angle in degrees and radians, plus the maximum shear stress at the outer surface. It also reports the torsional stiffness and the power the shaft can transmit at any given speed.
Choosing between a solid and hollow shaft is a critical design trade-off: hollow shafts are lighter but may buckle or have larger deflections. The built-in material presets — steel, aluminum, copper, brass, titanium, and cast iron — let you compare twist angles across materials. Use the reference tables to explore how twist varies with length and across materials.
When This Page Helps
Knowing the twist angle and shear stress prevents shaft failures, excessive vibration, and misalignment in power-transmission systems. This calculator replaces tedious manual calculations with direct results and material comparisons.
How to Use the Inputs
- Choose a material preset or enter a custom shear modulus G in GPa.
- Select the cross-section type: solid or hollow circular.
- Enter the applied torque in Newton-metres.
- Set the shaft length in metres and outer diameter in millimetres.
- For hollow shafts, also enter the inner diameter.
- Read the angle of twist, max shear stress, torsional stiffness, and power output.
- Use the comparison tables to optimise diameter and material choice.
Formula used
Angle of Twist: θ = T L / (G J)
Polar Moment (solid): J = π D⁴ / 32
Polar Moment (hollow): J = π (D⁴ − d⁴) / 32
Max Shear Stress: τ = T r / J
where T = torque, L = length, G = shear modulus, D = outer diameter, d = inner diameter, r = outer radius.Example Calculation
Result: 0.586° twist, 40.7 MPa max shear stress
A 50 mm solid steel shaft (G = 79.3 GPa) carrying 500 N·m over 2 m twists about 0.586° with a peak shear stress of 40.7 MPa — well within typical steel yield limits.
Tips & Best Practices
- Hollow shafts with the same outer diameter weigh less but have slightly more twist — check both.
- If twist exceeds about 1° per metre, consider increasing the diameter.
- For stepped shafts, calculate each segment separately and sum the angles.
- Always verify the shear stress against the material allowable stress.
- Use SI units consistently: N·m, metres, Pa.
When To Use This Calculator
Calculate angle of twist, shear stress, and torsional stiffness for solid or hollow shafts under torque. Compare materials side by side. Use it when you need a repeatable calculation in the physics / general category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
Last updated:
Frequently Asked Questions
It is the angular deformation of one end of a shaft relative to the other when a torque is applied along its axis.
The formula yields radians; multiply by 180/π to convert to degrees.
J quantifies the cross-section's resistance to torsion. Larger J means less twist and lower shear stress for the same torque.
Hollow shafts save weight with only a modest increase in twist. They are preferred in aerospace, automotive, and bicycle frame design.
Typical allowable shear stress for structural steel is 0.4 × yield strength, roughly 100–150 MPa for common steels.
Yes. The shear modulus decreases at elevated temperatures, increasing the angle of twist.
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