Shaft Torsion Calculator

How torsion affects a solid circular shaft

Circular shafts are widely used to transmit torque in rotating machinery (drive shafts, couplings, gear trains) and in structures where twisting occurs. When a torque T acts on a circular shaft, the material develops shear stress that increases linearly from the center (zero) to the outer surface (maximum). At the same time the shaft twists by an angle θ over its length L. This calculator applies the classical (Saint-Venant) torsion relationships for a solid circular shaft with constant diameter and a linearly elastic material.

Inputs and units (to avoid common mistakes)

• Shaft diameter is entered in mm (the calculator converts to meters internally).
• Shaft length is entered in m.
• Applied torque is entered in N·m.
• Shear modulus G is entered in GPa (the calculator converts to Pa internally).

Typical shear modulus values: steel is often around 79–82 GPa, aluminum alloys around 26–28 GPa, and many polymers are far lower. Using the correct G is essential for a realistic twist prediction.

Key formulas used

For a solid circular shaft with radius r (where r = d/2) the polar moment of inertia is:

Plain-text equivalents (for readability/accessibility):

• J = π r^4 / 2
• τ_max = T r / J
• θ = T L / (J G)

The calculator reports:

• Maximum shear stress at the outer surface, τ_max (commonly shown in MPa).
• Angle of twist over the entered length, θ (in radians and degrees).

Interpreting the results

1) Shear stress check (strength)

Compare the computed τ_max to an allowable shear stress for your material and design standard. Allowables depend on yield/ultimate strength, safety factor, temperature, fatigue requirements, keyways/notches, and whether the loading is steady or cyclic. If τ_max is too high, common fixes are increasing diameter (very effective), reducing torque, using a higher-strength material, or switching to a hollow shaft sized for higher torsional stiffness/strength per weight.

2) Angle of twist check (stiffness / serviceability)

Even if stress is acceptable, excessive twist can cause alignment issues, poor gear meshing, vibration, control problems, or reduced positioning accuracy. Acceptable twist is application-dependent; many drivetrain designs target relatively small twists (often expressed as degrees per meter), but you should use the limits appropriate to your system (manufacturer guidance, internal requirements, or applicable codes).

Why diameter matters so much

Because J scales with r^4, small diameter increases can dramatically reduce both stress and twist. For example, increasing diameter by 10% increases radius by 10% and increases J by about 1.1⁴ ≈ 1.46, so stress and twist drop by roughly 32% for the same torque and length.

Worked example (using the default inputs)

Given: diameter d = 50 mm, length L = 2 m, torque T = 1000 N·m, shear modulus G = 79 GPa.

1. Convert diameter to meters and compute radius: d = 0.05 m, r = 0.025 m.
2. Polar moment for a solid circle: J = π r^4 / 2 ≈ 6.14 × 10⁻⁷ m⁴.
3. Maximum shear stress: τ_max = T r / J ≈ (1000 × 0.025) / (6.14×10⁻⁷) ≈ 40.7 MPa.
4. Angle of twist: θ = T L / (J G) ≈ 0.0413 rad ≈ 2.37° over 2 m.

If your calculator outputs are close to ~40.7 MPa and ~2.37° for these defaults, it is behaving consistently with the standard torsion formulas (small differences can come from rounding).

Solid vs hollow shafts (quick comparison)

This page calculates solid shafts. Hollow shafts often provide higher torsional stiffness/strength per unit mass. For a hollow circular shaft with outer radius r_o and inner radius r_i:

J = π(r_o⁴ − r_i⁴)/2 (used in the same stress and twist equations).

Assumptions & limitations (applicability)

• Linear elastic material: G is constant and the shaft does not yield. If stress approaches yield or the material is nonlinear (some plastics), results will differ.
• Homogeneous, isotropic material: A single shear modulus represents the whole shaft (not composites or direction-dependent materials).
• Uniform, circular cross-section: Diameter is constant along the length and the cross-section is circular. Non-circular sections can warp and require different analysis.
• Saint-Venant torsion (no significant warping restraint): The formulas assume free warping typical for circular shafts. End constraints, flanges, or complex geometry can change stiffness/stress.
• Uniform applied torque along the length: If torque varies with position (multiple loads, distributed effects), the twist should be integrated segment-by-segment.
• Small deformations: Angle of twist is assumed small enough that geometry changes do not materially affect results (common for many metal shafts under service loads).
• Stress concentrations not included: Keyways, splines, shoulders, holes, and notches can raise peak stress above the nominal τ_max. Use appropriate concentration factors and fatigue design methods when relevant.
• Temperature and time effects ignored: Elevated temperature, creep, or viscoelasticity can alter G and long-term twist.

FAQ

Is the reported shear stress the maximum?

Yes. For a solid circular shaft in elastic torsion, shear stress varies linearly with radius and is maximum at the outer surface.

Why does the calculator ask for shear modulus G, not Young’s modulus E?

Torsional twist depends on the shear modulus G. If you only have E and Poisson’s ratio ν, you can estimate G = E / (2(1+ν)) for isotropic materials.

Can I use this for a hollow shaft?

Not directly. The stress and twist formulas stay the same, but you must replace J with the hollow-shaft expression using inner and outer radii.

What if torque changes along the shaft?

Compute twist in segments: for each segment, use its local torque and length, then sum the angles of twist.