Torsion Shaft Calculator — Maximum Shear Stress & Angle of Twist Validator
Calculate maximum shear stress and angle of twist using torque, shaft geometry, length, and shear modulus.
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
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
Maximum shear stress formula:
τmax = T · c / J
Angle of twist formula:
θ = T · L / (J · G)
where:
- T — Torque (torque)
- c — Outer radius (length)
- J — Polar moment of inertia (area moment)
- L — Shaft length (length)
- G — Shear modulus (stress)
- τmax — Maximum shear stress (stress)
- θ — Angle of twist (angle)
When to use this calculator
When to use this calculator:
- Calculate maximum shear stress for a shaft loaded by torque.
- Check shaft stress utilization against allowable shear stress.
- Calculate angle of twist from torque, shaft length, polar moment of inertia, and shear modulus.
- Evaluate torsion response for a solid shaft, hollow shaft, or custom cross-section.
- Compare how shaft geometry changes polar moment of inertia and torsional response.
How to interpret the result
Maximum shear stress is defined as the torsional shear stress at the outer radius of the shaft section.
Maximum shear stress depends on torque, outer radius, and polar moment of inertia. Increasing torque or outer radius increases maximum shear stress. Increasing polar moment of inertia decreases maximum shear stress.
Angle of twist is defined as the torsional rotation produced by torque over shaft length. It depends on torque, shaft length, polar moment of inertia, and shear modulus. Increasing torque or shaft length increases angle of twist. Increasing polar moment of inertia or shear modulus decreases angle of twist.
- Safe — stress utilization is less than or equal to 1.000 when allowable shear stress is provided.
- Warning — stress utilization is greater than 1.000 and less than or equal to 1.200 when allowable shear stress is provided.
- Unsafe — stress utilization is greater than 1.200 when allowable shear stress is provided.
- Info — no allowable shear stress is provided, or the angle of twist mode is used.
- Invalid — required torque, shaft length, shear modulus, polar moment of inertia, outer radius, or supported section properties are missing or not greater than zero.
The result is used to evaluate torsional stress level, stress utilization, and torsional rotation for the selected shaft section.
Calculation example
Example:
A user wants to check the maximum shear stress in a custom shaft section under a known torque.
- T = 500 N·m
- J = 8.0e-6 m⁴
- c = 0.04 m
- τallow = 3.0e6 Pa
τmax = 2.5e6 Pa, utilization = 0.833.
Assumptions and limitations
- The shaft is evaluated using torsion relationships based on torque, polar moment of inertia, outer radius, shaft length, and shear modulus.
- Custom sections require direct input of polar moment of inertia and outer radius.
- Generated sections must provide polar moment of inertia and outer radius through the section property model.
- Stress utilization is calculated only when allowable shear stress is provided and greater than zero.
- Angle of twist is calculated only when shaft length and shear modulus are provided and greater than zero.