Angle Of Twist - Solid and Hollow Shaft Torsion

An angle of twist calculator solves for the rotation one end of a shaft makes relative to the other when a torque is applied along its length. The result follows directly from the torsion equation theta = T times L divided by G times J, where T is the applied torque, L is the shaft length, G is the shear modulus of the material, and J is the polar moment of inertia of the cross-section. Engineers and students use this calculation to size drive shafts, predict wind-up in long rotating members, and verify deformation stays inside design code limits.

• • Drive shaft sizing: Predict the twist of a motor-to-gearbox shaft so the angular misalignment at the coupling stays within the manufacturer's tolerance.
• • Torsion homework and exams: Solve a mechanics of materials problem step by step for a solid or hollow circular shaft in radians and degrees.
• • Propeller and rotor shaft design: Estimate the twist of marine propeller shafts, helicopter rotor masts, and turbogenerator shafts to confirm torsional vibration stays clear of resonance.
• • Quality control checks: Compare the measured twist of a test bar to the theoretical value when validating the shear modulus of a new material batch.

The rotation is usually a small number, often a fraction of a degree for stiff steel shafts, but it can climb into many degrees for long, thin shafts made of low-modulus alloys, so reporting the answer in both radians and degrees keeps the result usable for any course or report convention.

When the same shaft also carries bending loads, our Beam Bending Stress Calculator covers the normal-stress side of the same design check.

The shaft torsion calculation uses the torsion equation from elementary mechanics of materials, applied to a uniform circular shaft with constant torque. The polar moment of inertia is computed from the outer and inner diameters before the rotation is solved.

• theta: Angle of twist at the far end of the shaft relative to the fixed end, in radians.
• T: Applied torque acting along the shaft axis, in newton-metres (N·m).
• L: Length of the shaft over which the torque is transmitted, in metres (m).
• G: Shear modulus (modulus of rigidity) of the shaft material, in pascals (Pa).
• J: Polar moment of inertia of the cross-section, in metres to the fourth power (m^4).

For a hollow circular shaft the polar moment of inertia becomes pi times the difference between the fourth powers of the outer and inner diameters divided by 32, so the same torsion equation covers both cross-sections once J is known.

According to Wikipedia Torsion (mechanics), the resulting rotation of a circular shaft equals T*L/(G*J), where J is the polar moment of inertia of the cross-section, equal to pi*d^4/32 for a solid shaft and pi*(d_o^4 - d_i^4)/32 for a hollow shaft.

According to Omnicalculator Angle of Twist, the rotation in radians for a circular shaft is the torque multiplied by shaft length divided by the product of the shear modulus and the polar moment of inertia, and the result can be converted to degrees by multiplying by 180 divided by pi.

For a related geometric angle used in granular material and bulk solids problems, Angle of Repose Calculator covers the slope at which a pile of particles comes to rest.

Four core ideas drive the torsion equation: the torque input, the polar moment of inertia of the cross-section, the shear modulus of the material, and the torsional stiffness that bundles G and J together.

Keep the units consistent when substituting values: use newton-metres for torque, metres for length, pascals for shear modulus, and metres to the fourth power for J. Mixing units will give an answer that is off by a power of ten.

When the same component acts as a torsional spring, the approach in Spring Constant & Deflection Calculator for axial springs translates directly to a torsional stiffness GJ per radian.

Enter the torque, length, shear modulus, and shaft geometry to get the rotation, polar moment of inertia, and torsional stiffness in one step.

1. 1 Pick the shaft type: Choose Solid for a solid circular shaft, or Hollow to enable the inner diameter input for tube-style shafts.
2. 2 Enter the applied torque: Type the torque carried by the shaft in newton-metres. Convert pound-feet by multiplying by 1.355818 first.
3. 3 Enter the shaft length: Type the active length over which the torque is transmitted in metres. Convert inches by multiplying by 0.0254.
4. 4 Choose a material preset or custom G: Select a material from the preset list to fill the shear modulus in gigapascals, or leave Custom and type your own G value.
5. 5 Enter the diameter or diameters: Type the outer diameter in millimetres. For a hollow shaft, also enter the inner diameter; leave it at 0 for a solid section.
6. 6 Read the rotation: Check the primary result for the twist in radians, then read the same value in degrees along with the polar moment of inertia and torsional stiffness.

Try a 40 mm steel shaft, 1 m long, loaded by 500 N·m. The polar moment of inertia should be about 251327 mm^4, the torsional stiffness about 19930 N·m^2, and the rotation about 0.025 rad or 1.44 degrees.

Before picking a diameter, plot the bending and torque diagrams with Shear Force and Bending Moment Calculator so the torsion calculation lines up with the most heavily loaded segment.

A focused torsion tool gives answers in both radians and degrees and keeps every variable editable so you can iterate on a design quickly.

• • Solves both solid and hollow shafts: Switch between solid and hollow cross-sections without leaving the page, and the polar moment of inertia updates automatically.
• • Includes material presets: Common materials ship with their shear modulus preset, so you do not have to look up G in a table before each calculation.
• • Reports radians and degrees: Read the rotation in the unit your course or report uses without opening a separate conversion calculator.
• • Surfaces G*J and J alongside theta: Get the polar moment of inertia and the torsional stiffness together with the rotation, so you can compare candidate shafts in the same view.
• • Catches invalid hollow-shaft inputs: Detect an inner diameter that is not strictly less than the outer diameter before the rotation is reported, so a wrong cross-section does not silently produce nonsense.

Use this calculator as a quick check before running a finite-element torsion analysis or as a stand-alone tool for homework problems where a closed-form answer is expected, since the outputs match the formulas in standard mechanics of materials textbooks.

To turn the torque used here into the power and rotational speed for the same drivetrain, run the inputs through Torque, Power & Speed Calculator.

Four material and geometry factors control how much a shaft twists, and a few common assumptions limit how closely this linear-elastic formula matches a real part.

• • This calculator assumes uniform torque, a uniform cross-section, and linear elastic behaviour. Stepped shafts, multiple torques, or torques near yield need a segment-by-segment calculation.
• • Stress concentrations at shoulders, keyways, splines, and fillets are ignored. Real shafts twist a bit more than this prediction in those regions.

For a long, slender shaft the rotation can become large enough that the small-angle assumption behind the torsion equation no longer holds, so treat the result as a lower bound once the rotation exceeds roughly 15 to 20 degrees.

According to Wikipedia Shear modulus, the shear modulus of structural steel is about 79.3 GPa, aluminum 6061-T6 is 26 GPa, and cartridge brass is 39 GPa, and these modulus values are commonly used in torsion calculations.

Long, slender shafts that twist easily often have low torsional natural frequencies, so cross-check the twist rate against Vibration Natural Frequency Calculator to confirm you are not landing near a resonance.