What is the relationship between torque and angle of twist?

For a cylindrical shaft that is subjected to constant torque and whose cross-section does not change, the following applies: Angle of twist = phi = T*L/(G*J), where T is the internal torque, L is length, G is the shear modulus and J is the polar moment of inertia of the cross-section. The angle of twist increases with increasing torque and length, while it decreases with higher material stiffness or larger cross-section. The result is given in radians.

How do you convert degrees of twist?

The formulas always give values in radians. To get the value in degrees multiply by 180 and divide by pi, since pi radians is equal to 180 degrees. In this calculator you can simply switch the unit of the result for the angle between radians, degrees and percent degrees.

What is the difference between a solid bar and a hollow bar?

They use the same formula to calculate torque but the moment of inertia is different. For a solid round bar it is: J = (pi/32)*D^4, while for a hollow shaft it is given by: J = (pi/32)*(Do4 - Di4). As a hollow shaft has material removed from the middle of it, it only needs to be a fraction of the weight of a solid shaft to achieve nearly the same stiffness. This is also why axles and drive shafts are often made as tubes.

How to calculate maximum twist angle before flowing?

First calculate the torque at which the shear modulus limit strain is reached and then substitute this value into the torsion formula. For a solid material shaft the elastic limit torque is equal to the shear modulus limit strain multiplied by the cube of the shaft radius, multiplied by (pi/2). Enter this torque along with length, shear modulus and cross section in this calculator to get the angle of twist at which flow begins.

Is it possible to measure the shear modulus of a material with this calculator?

Yes, in a torsion test a known torque is applied and the angle of twist read off. Enter the torque, length, angle of twist and cross section area and leave the shear modulus blank. The calculator will rearrange the formula to give G = T*L / (J*phi). Averaging several data points on the linear section of the torque-twist curve will give a more reliable value.

Angle of Twist Calculator

Shaft cross-section

Pick the shape so the polar moment of inertia is computed for you, or enter it directly for a known or non-circular section.

Solid circular shaft

Hollow circular shaft

Enter the polar moment directly

Shaft diameter (D)

Torque, length and material

Applied torque (T)

Shaft length (L)

Shear modulus (G)

Angle of twist (φ)

Set the cross-section, then enter the torque, length, and shear modulus and leave the angle of twist blank to solve it. To work backward, fill the angle in and leave torque, length, or shear modulus blank instead.

Results

Polar moment of inertia (J)

Please input values into the fields

Polar moment of inertia (J): cm⁴

How the torsion formula works

The angle of twist follows the elastic torsion formula

φ = T L / (G J)

. Torque and length twist the shaft more; a stiffer material or a fatter section resists it. For a solid round bar J = (π/32) D⁴, so doubling the diameter cuts the twist by a factor of sixteen.

Torsional twist matters for drive shafts, axles, drill strings, and torsion bars. Too much of it throws off timing and gear alignment and sets up vibration, which is why engineers size a shaft for an allowable twist as well as an allowable stress.

When one end of a shaft is subjected to torque and the other end is fixed, the shaft experiences slight rotation along its length. This relative twist between the two ends is called the angle of twist. This calculator computes the angle of twist using the formula for elastic torsion. Conversely, the same relationship can be used to determine the torque, length or shear modulus of the material.

After you have defined the cross section, the calculator automatically calculates the moment of inertia for the cross section. Enter three arbitrary values for torque, length, shear modulus and angle of twist, and the calculator will calculate the fourth value. It also outputs the maximum shear stress and torsional stiffness of the shaft.

What is the twist angle?

The twist angle is denoted by the Greek letter φ and measures the amount that one end of a shaft rotates relative to the other end when a torque acts along the length of the shaft. It is an angular quantity, whose calculated result will be given in radians; however it is often converted into degrees for ease of understanding.

Every real shaft has a small torsion which is normal. Excessive torsion can cause problems. In drive shafts, excessive torsion causes gears and couplings to move out of centerline, backlash increases, and vibration and noise may occur. Therefore, the sizing of shafts is designed so that the torsion stays below a limit while meeting load requirements.

Formula for the twist angle.

For a cylindrical bar of uniform cross-section subjected to a constant torque, the angle of twist is given by:

\phi = \frac{T\,L}{G\,J}

Each symbol represents a physical quantity.

Symbol	Quantity	Typical unit
φ	Angle of twist	radian (rad)
T	Internal torque	newton-metre (N·m)
L	Shaft length	metre (m)
G	Shear modulus of the material	pascal (Pa)
J	Polar moment of inertia of the section	metre to the fourth (m⁴)

The torsional stiffness modulus G and the polar moment of inertia J both give information about resistance to twisting. G is a material property while J is a geometric property that depends on the shape and dimensions of the cross section. The result from the formula will always be in radians. To convert it into degrees, multiply by 180 and divide by pi, or change the unit of the result directly.

Backward calculation for any variables:

Since these four properties are contained in the same equation, knowing three of them allows you to determine the fourth. Leave a field blank and the calculator will re-arrange the formula accordingly.

T = \frac{G\,J\,\phi}{L} \qquad L = \frac{G\,J\,\phi}{T} \qquad G = \frac{T\,L}{J\,\phi}

The determination of G is done through a shear test, where the material properties are measured by the shear modulus of proportionality. A known torque is applied to a specimen and the angle of twist is read off, then the formula is rearranged. By using multiple data points (torque-angle of twist) from the linear region of a graph and averaging them, more stable values can be obtained.

The moment of inertia for a circular cross-section

The area moment of inertia indicates the distribution of the cross-sectional area relative to the center of gravity. For a solid round bar with diameter D:

J = \frac{\pi}{32}\,D^4

For a hollow cylindrical shaft or tube with an outside diameter of Do and inside diameter of Di, the area of the void must be subtracted:

J = \frac{\pi}{32}\left(D_o^4 - D_i^4\right)

Since J (the polar moment of inertia) increases in proportion to the fourth power of the diameter, even small changes in dimensions have a large effect. For a given torque, doubling the polar moment of inertia will approximately double the angle of twist, while halving it. Hollow shafts eliminate material near the center where there is little stress, making them significantly lighter while providing nearly identical stiffness. For non-circular or composite cross-sections, please specify the polar moment of inertia (more precisely, the torsional constant) directly.

Example calculation:

Let's take a massive aluminum bar with length of 3 meters, diameter of 100 mm, shear modulus of 80 GPa and torque of 10 kN·m. First we calculate the cross-sectional moment of inertia:

J = \frac{\pi}{32}\,(0.100\ \text{m})^4 \approx 9.82 \times 10^{-6}\ \text{m}^4

These values are then entered into the torsion formula:

\phi = \frac{(10\,000\ \text{N}\cdot\text{m})(3\ \text{m})}{(80 \times 10^{9}\ \text{Pa})(9.82 \times 10^{-6}\ \text{m}^4)} \approx 0.0382\ \text{rad} = 2.19^\circ

The torsion angle between the two ends of the axle is approximately 2.19 degrees. The calculator will give you the same result and provide it in the unit that you select.

Shear stress and torsional rigidity

When a shaft is subjected to torsion, shear stresses also occur. The maximum shear stress occurs at the outer surface and gradually reduces to zero towards the center. For a circular cross-section, the maximum value is:

\tau_{\max} = \frac{T\,c}{J}

where c is the outer radius. This stress should be well below the shear strength of the material; for normal engineering applications safety factors of two or three are used. Another important parameter is torsional stiffness, i.e., the torque required to produce a given angular deformation:

k_t = \frac{G\,J}{L}

High rigidity means that the shaft hardly twists under load, which is essential for precise positioning and stable power transmission.

Shear strength module of common materials

If the measured value for the shear modulus (G) is not available, typical data for these engineering materials can be used as a good starting point. The shear modulus can also be calculated from the Young's modulus (E) and Poisson's ratio (ν) using the following formula: G = E / [2(1 + ν)].

Material	Shear modulus G
Structural steel (A36)	79 GPa
Stainless steel 304	77 GPa
Cast iron (gray)	40 to 45 GPa
Titanium Ti-6Al-4V	44 GPa
Copper	45 GPa
Brass	37 to 41 GPa
Aluminium 6061-T6	26 GPa
Magnesium alloy	17 GPa

In case of changes in torque or cross-section along the axis

The simple formula assumes that there is only one torque and cross-section over the entire length L. If the torque changes abruptly at a point with a gear or if the diameter changes, then the shaft should be divided into multiple sections where each section remains constant. The angle of twist for each section is calculated separately and then added symbolically:

\phi = \sum \frac{T_i\,L_i}{G_i\,J_i}

Determine the sign of each internal torque using the right hand rule as opposite directions will partially cancel out. Enter each section one at a time into this calculator and sum up the results.

How to use this calculator.

First select the cross section. For a solid or hollow cylinder the calculator will calculate the polar moment of inertia after you have entered the diameter, otherwise select the direct input option and enter J.

Then enter the torque, length and shear modulus, leave the twist angle blank, and the calculator will calculate it. Conversely you can input an allowable twist angle and leave the torque, length or shear modulus blank to calculate the corresponding limits. The units are freely combinable; each field has its own unit selection option.

This tool is for learning and initial sizing. The final design should consider applicable codes and safety factors as well as check combined loading, fatigue, and stress concentrations.

Frequently asked questions

What is the relationship between torque and angle of twist?: For a cylindrical shaft that is subjected to constant torque and whose cross-section does not change, the following applies: Angle of twist = phi = T*L/(G*J), where T is the internal torque, L is length, G is the shear modulus and J is the polar moment of inertia of the cross-section. The angle of twist increases with increasing torque and length, while it decreases with higher material stiffness or larger cross-section. The result is given in radians.
How do you convert degrees of twist?: The formulas always give values in radians. To get the value in degrees multiply by 180 and divide by pi, since pi radians is equal to 180 degrees. In this calculator you can simply switch the unit of the result for the angle between radians, degrees and percent degrees.
What is the difference between a solid bar and a hollow bar?: They use the same formula to calculate torque but the moment of inertia is different. For a solid round bar it is: J = (pi/32)*D^4, while for a hollow shaft it is given by: J = (pi/32)*(Do4 - Di4). As a hollow shaft has material removed from the middle of it, it only needs to be a fraction of the weight of a solid shaft to achieve nearly the same stiffness. This is also why axles and drive shafts are often made as tubes.
How to calculate maximum twist angle before flowing?: First calculate the torque at which the shear modulus limit strain is reached and then substitute this value into the torsion formula. For a solid material shaft the elastic limit torque is equal to the shear modulus limit strain multiplied by the cube of the shaft radius, multiplied by (pi/2). Enter this torque along with length, shear modulus and cross section in this calculator to get the angle of twist at which flow begins.
Is it possible to measure the shear modulus of a material with this calculator?: Yes, in a torsion test a known torque is applied and the angle of twist read off. Enter the torque, length, angle of twist and cross section area and leave the shear modulus blank. The calculator will rearrange the formula to give G = T*L / (J*phi). Averaging several data points on the linear section of the torque-twist curve will give a more reliable value.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Hibbeler, R. C. -- Mechanics of Materials
Standard reference derivation of the elastic torsion and angle-of-twist formulas.
Wikipedia: Polar moment of inertia
Definitions and formulas for the polar moment of solid and hollow circular sections.
Encyclopaedia Britannica: Shear modulus
Overview of the modulus of rigidity used in the torsion formula.