Shaft Torsion Calculator — Shear Stress & Angle of Twist
Calculate maximum nominal shear stress and elastic angle of twist for solid, hollow, or known circular shaft sections, with optional stress and twist utilization checks.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Calculation mode
Torque
Shaft length
Material
Strength limit — optional
Twist limit — optional
Circular shaft cross-section
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
For a circular shaft subjected to torque, the calculator uses the classical elastic torsion relationships for maximum nominal shear stress and total angle of twist.
Maximum shear stress
τmax = T · c / J
where:
- τmax — maximum nominal torsional shear stress at the outer radius,
- T — applied torque magnitude,
- c — outer radius of the shaft section,
- J — polar moment of inertia of the circular cross-section.
The stress calculation is used in Maximum shear stress and Shaft check — stress + twist modes.
Angle of twist
θ = T · L / (J · G)
where:
- θ — total elastic angle of twist over the entered shaft length,
- T — applied torque magnitude,
- L — uniform shaft length,
- J — polar moment of inertia,
- G — shear modulus of the material.
The twist calculation is used in Angle of twist and Shaft check — stress + twist modes.
Polar moment of inertia
For a solid circular shaft of diameter D:
J = π · D4 / 32
For a hollow circular shaft with outer diameter D and inner diameter d:
J = π · (D4 − d4) / 32
For both geometries, the outer radius used for maximum stress is:
c = D / 2
If Known circular-section properties is selected, the entered value of J is used directly. The outer radius c is additionally required whenever shear stress is calculated.
Utilization against optional limits
If an allowable shear stress is entered:
Stress utilization = τmax / τallow
If an allowable angle of twist is entered:
Twist utilization = θ / θallow
A utilization of 1.0 represents the entered limit. Values below or equal to 1.0 satisfy that individual criterion; values above 1.0 exceed it.
When to use this calculator
Use this calculator to evaluate the elastic torsional response of a solid or hollow circular shaft subjected to an applied torque. It can calculate maximum nominal shear stress, total angle of twist, or both in a combined shaft check.
Typical applications
- checking an existing drive shaft or transmission shaft under torque,
- estimating maximum nominal torsional shear stress,
- checking angular deformation over a specified shaft length,
- comparing calculated stress with a user-defined allowable shear stress,
- comparing shaft twist with a user-defined allowable angle,
- comparing solid and hollow circular shaft geometries,
- evaluating a circular section when its polar moment of inertia J is already known.
Calculation modes
- Maximum shear stress — requires torque and circular-section properties. Shaft length and shear modulus are not required.
- Angle of twist — requires torque, shaft length, polar moment of inertia, and shear modulus.
- Shaft check — stress + twist — calculates both maximum nominal shear stress and total angle of twist and can evaluate both optional limits.
This calculator is intended for a uniform circular shaft segment operating within the assumptions of linear elastic torsion. It is not a complete shaft-design or failure-analysis tool.
Use a more complete analysis when the shaft contains keyways, shoulders, splines, grooves, large stress concentrations, changing diameters, combined bending or axial loads, fatigue loading, significant dynamic effects, or material behavior outside the elastic range.
How to interpret the result
The calculator reports up to two primary torsion results: maximum nominal shear stress τmax and total angle of twist θ. The available results depend on the selected calculation mode.
Maximum nominal shear stress
τmax represents the nominal torsional shear stress at the outer radius of the circular shaft section. For a fixed geometry, increasing torque increases the calculated stress proportionally.
Increasing the polar moment of inertia J reduces the calculated shear stress. Because circular-shaft J depends strongly on diameter, changes in shaft diameter can have a large effect on torsional response.
Angle of twist
θ represents the total elastic angular deformation over the entered uniform shaft length. Twist increases with torque and shaft length and decreases as either the polar moment of inertia J or shear modulus G increases.
Stress utilization
If an allowable shear stress is entered, the calculator reports:
Stress utilization = τmax / τallow
- ≤ 1.0 — the calculated nominal shear stress remains at or below the entered allowable limit.
- > 1.0 — the entered allowable shear-stress limit is exceeded.
Twist utilization
If an allowable twist is entered, the calculator reports:
Twist utilization = θ / θallow
- ≤ 1.0 — the calculated angle of twist remains at or below the entered allowable limit.
- > 1.0 — the entered allowable twist limit is exceeded.
Combined-mode status
In Shaft check — stress + twist mode, exceeding either entered limit produces an unsuccessful result for the evaluated criteria.
If both allowable limits are provided and both utilization values are at or below 1.0, the evaluated stress and twist criteria are satisfied.
If only one of the two limits is provided and that limit is not exceeded, the calculator returns a warning rather than treating the complete combined check as satisfied. If no allowable limits are entered, the calculated values are informative results rather than acceptance criteria.
A positive status does not establish complete shaft safety. Additional failure modes and design requirements may still require verification.
Calculation example
Example — solid shaft checked for stress and twist
A solid circular shaft transmits a torque of 1000 N·m. The shaft diameter is 50 mm, the evaluated length is 1000 mm, and the shear modulus is 80 GPa.
The design limits entered for the check are:
- allowable shear stress: 60 MPa,
- allowable total angle of twist: 1.5°.
1. Convert the quantities to a consistent unit set
T = 1000 N·m = 1,000,000 N·mm
L = 1000 mm
G = 80 GPa = 80,000 N/mm2
D = 50 mm
c = D / 2 = 25 mm
2. Calculate the polar moment of inertia
J = π · D4 / 32
J = π · 504 / 32 = 613,592 mm4
3. Calculate maximum nominal shear stress
τmax = T · c / J
τmax = 1,000,000 · 25 / 613,592 = 40.74 MPa
The corresponding stress utilization is:
40.74 / 60 = 0.679
4. Calculate the angle of twist
θ = T · L / (J · G)
θ = (1,000,000 · 1000) / (613,592 · 80,000) = 0.02037 rad
Converting to degrees:
θ = 1.167°
The twist utilization is:
1.167 / 1.5 = 0.778
5. Interpret the result
Both utilization values are below 1.0, so the entered shear-stress limit and twist limit are satisfied for this calculation.
This result applies only to the nominal elastic torsion criteria evaluated here. It does not verify fatigue, stress concentrations, combined loading, yielding under a multiaxial failure criterion, or other possible shaft failure modes.
Assumptions and limitations
- The evaluated member is a circular shaft: solid, concentric hollow, or represented by valid known circular-section properties.
- The shaft segment is treated as straight and prismatic over the evaluated length.
- Torque is treated as a torsional load acting about the shaft axis and is constant over the evaluated segment.
- The material response is assumed to remain within the linear elastic range.
- The entered or material-derived shear modulus G is assumed to represent the actual material condition relevant to the calculation.
- The angle-of-twist equation assumes constant T, J, and G over the entered shaft length.
- The maximum stress result is a nominal torsional shear stress. Local stress concentrations are not included.
- Keyways, shoulders, fillets, grooves, splines, cross-holes, notches, and similar discontinuities are not explicitly modeled.
- The calculator does not evaluate fatigue life, alternating torque, impact loading, shock factors, or torsional vibration.
- Combined bending, axial loading, transverse shear, and multiaxial yield criteria are outside this calculation.
- The calculator does not determine material yield strength or automatically derive an allowable shear stress from a design code. Any allowable stress must be selected by the user for the applicable material, loading condition, safety philosophy, and governing requirements.
- The allowable angle of twist is also user-defined. Appropriate limits depend on the machine, alignment, stiffness, accuracy, coupling, gearing, and service requirements.
- Temperature effects are not calculated directly. Their influence is represented only if the entered shear modulus and allowable values already reflect the relevant operating temperature.
- For a custom section, the entered J and, where required, outer radius c must correctly represent the evaluated circular section.
- For stepped shafts or shafts with varying torque, geometry, or material properties, separate segments should be evaluated rather than applying one uniform set of properties to the entire shaft.
- Final shaft design may require additional verification for strength, stiffness, fatigue, stress concentrations, critical speed, connections, bearings, manufacturing details, and applicable design standards.