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Torsion Calculator

Estimate torsional shear stress, angle of twist, torque, power, and circular shaft size using SI-normalized elastic torsion formulas.

Torsion inputs

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.

Calculation mode

Display precision

Material properties

Material property mode

Material preset

Young's modulus

Young's modulus unit

Shear modulus

Shear modulus unit

Yield strength

Yield strength unit

Ultimate strength

Ultimate strength unit

Generic steel values. Use grade-, standard-, temperature-, and process-specific data for real design.

Shaft geometry and loading

Shaft type

Torque

Torque unit

Diameter

Diameter unit

Length

Length unit

Allowable shear stress

Allowable shear stress unit

Allowable angle of twist

Radians.

Hollow diameter ratio

Inner diameter divided by outer diameter for sizing mode.

Power

Power unit

Rotational speed

Rotational speed unit

Length output unit

Stress output unit

Result section

warningshear-stress

Maximum torsional shear stress

40.7437 MPa

Secondary results

Maximum torsional shear stress: 40.7437 MPa
Angle of twist: 2.03718e-5 rad
Angle of twist: 0.00116722 deg
Polar second moment of area: 6.13592e-7 m^4
Outer radius: 0.025 m
Torsional rigidity GJ: 49,087.4 N*m^2
Torque: 1,000 N·m
Power: 104.72 kW
Rotational speed: 1,000 rpm
Solid diameter for entered allowable shear: 46.7018 mm
Hollow outer diameter for entered ratio: 48.9133 mm

Unit conversion summary

Unit-normalization summary: 1,000 N*m; 6.13592e-7 m^4; 8.00000e+10 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Material properties vary with grade, condition, temperature, manufacturing process, loading direction, heat treatment, applicable standard, and data source. Verify all properties for the actual material.
These calculations are preliminary scalar mechanics checks and do not certify design adequacy.
No keyway, shoulder, notch, warping, plastic torsion, fatigue, or combined loading assessment is included.

Assumptions

Circular prismatic shaft, uniform material, elastic Saint-Venant torsion.
Negative torque is treated as direction; stress and twist magnitudes should be interpreted with sign convention awareness.

Calculation steps

Resolve shaft geometrySolid circular shaft
Calculate torsional shear4.07437e+7 Pa
Calculate angle of twist2.03718e-5 rad
Resolve power relationship104,720 W at 104.72 rad/s

Circular shaft torsion

tau = T c / J; theta = T L / (J G); P = T omega

J = pi d^4 / 32 for solid shafts; tau = T c / J; theta = T L / (J G); P = T omega.

tau = 1,000 N*m x 0.025 m / 6.13592e-7 m^4

theta = 1,000 N*m x 0.001 m / (6.13592e-7 m^4 x 8.00000e+10 Pa)

Technical visual

Engineering interpretation

Use the stress and twist values as preliminary elastic checks. Transfer combined bending and torsion cases to the shaft design calculator.

What is the torsion calculator?

This calculator evaluates Saint-Venant elastic torsion for circular shafts. It reports shear stress, twist, polar second moment of area, torque-power-speed relationships, and simple diameter sizing for preliminary design review.

How to use the torsion calculator

Choose the torsion calculation mode that matches the hand calculation you want to perform.
Enter geometry, loads, material values, and units from a consistent source.
Review the primary result, secondary values, formula substitution, assumptions, and warnings.
Use the related-tool links only as a workflow aid; verify every transferred value before design use.

Input explanations

Shaft type controls whether J is computed from a solid diameter, hollow diameters, or a custom polar moment and outer radius.
Torque may be entered directly or derived exactly from power and angular speed.
Allowable stress and twist modes calculate the torque or diameter associated with the entered limit.
Shear modulus and length are required for angle-of-twist calculations.
Material properties vary with grade, condition, temperature, manufacturing process, loading direction, heat treatment, applicable standard, and data source. Verify all properties for the actual material.

Notation

Formula notation
Symbol	Meaning	Unit
T	Applied torque	N*m
J	Polar second moment of area	m^4
tau	Maximum torsional shear stress	Pa
theta	Angle of twist	rad
GJ	Torsional rigidity	N*m^2

Worked examples

A 50 mm solid steel shaft carrying 1000 N*m has J = 6.135923152e-7 m^4 and torsional shear stress about 81.487 MPa.

The same shaft with 1 m length and G = 80 GPa twists about 0.020372 rad, or 1.167 deg.

A 10 kW shaft at 1000 rpm transmits 95.493 N*m by the exact SI power relationship.

Cross-tool workflow

The torsion workflow can feed preliminary shaft design, factor-of-safety checks, stress conversion, and power-torque-speed review.

Use torque in Shaft Design Calculator Use stress in Factor of Safety Calculator

Common mistakes

Using outside diameter where radius is required in Tc/J hand checks.
Using the rounded 9550 shortcut as the internal calculation instead of exact SI power and angular speed.
Applying circular-shaft torsion formulas to open thin-walled sections.
Ignoring keyways, shoulders, stress concentrations, and fatigue.

Limitations

Circular elastic shafts only.
No warping, plastic torsion, keyway, notch, fatigue, or combined bending assessment.
Uniform material, prismatic geometry, and Saint-Venant torsion are assumed.

References

Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design.
Roark's Formulas for Stress and Strain: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.
Mechanics of Materials: Common mechanics-of-materials texts covering torsion, beam deflection, column buckling, and elastic stress relationships.
Machinery's Handbook: Machinery's Handbook reference material for machine-element formulas and terminology.

How is torsional stress calculated?

For elastic circular shafts, maximum shear stress is T c / J, where c is the outside radius.

What is polar moment?

The polar second moment of area J measures resistance to twisting. For a solid circular shaft, J = pi d^4 / 32.

How is angle of twist calculated?

Angle of twist is theta = T L / (J G) for a prismatic circular shaft in the elastic range.

How are power, speed, and torque related?

Power equals torque times angular speed. The calculator converts rpm to rad/s before calculating torque.

Can the tool handle hollow shafts?

Yes. It uses J = pi (D^4 - d^4) / 32 and checks that the outer diameter exceeds the inner diameter.

Does this include stress concentrations?

No. Keyways, shoulders, grooves, and surface defects must be evaluated separately.

Can I use this for fatigue?

No. Transfer the stress range to the fatigue life estimator when cyclic loading matters.

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Engineering disclaimer

This tool is intended for educational, estimation, and preliminary engineering use. It does not certify designs or replace applicable codes, standards, validated software, physical testing, or qualified professional engineering judgment.

Need Professional Support?

ScholarTool is powered by ScholarEase Consultancy Services LLP. Professional engineering, simulation, research, and technical documentation support is available through the connected services website.

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Last reviewed: 2026-06-30

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Torsion inputs

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.

Calculation mode

Display precision

Material properties

Material property mode

Material preset

Young's modulus

Young's modulus unit

Shear modulus

Shear modulus unit

Yield strength

Yield strength unit

Ultimate strength

Ultimate strength unit

Generic steel values. Use grade-, standard-, temperature-, and process-specific data for real design.

Shaft geometry and loading

Shaft type

Torque

Torque unit

Diameter

Diameter unit

Length

Length unit

Allowable shear stress

Allowable shear stress unit

Allowable angle of twist

Radians.

Hollow diameter ratio

Inner diameter divided by outer diameter for sizing mode.

Power

Power unit

Rotational speed

Rotational speed unit

Length output unit

Stress output unit

Result section

warningshear-stress

Maximum torsional shear stress

40.7437 MPa

Secondary results

Maximum torsional shear stress: 40.7437 MPa
Angle of twist: 2.03718e-5 rad
Angle of twist: 0.00116722 deg
Polar second moment of area: 6.13592e-7 m^4
Outer radius: 0.025 m
Torsional rigidity GJ: 49,087.4 N*m^2
Torque: 1,000 N·m
Power: 104.72 kW
Rotational speed: 1,000 rpm
Solid diameter for entered allowable shear: 46.7018 mm
Hollow outer diameter for entered ratio: 48.9133 mm

Unit conversion summary

Unit-normalization summary: 1,000 N*m; 6.13592e-7 m^4; 8.00000e+10 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Material properties vary with grade, condition, temperature, manufacturing process, loading direction, heat treatment, applicable standard, and data source. Verify all properties for the actual material.
These calculations are preliminary scalar mechanics checks and do not certify design adequacy.
No keyway, shoulder, notch, warping, plastic torsion, fatigue, or combined loading assessment is included.

Assumptions

Circular prismatic shaft, uniform material, elastic Saint-Venant torsion.
Negative torque is treated as direction; stress and twist magnitudes should be interpreted with sign convention awareness.

Calculation steps

Resolve shaft geometrySolid circular shaft
Calculate torsional shear4.07437e+7 Pa
Calculate angle of twist2.03718e-5 rad
Resolve power relationship104,720 W at 104.72 rad/s

Circular shaft torsion

tau = T c / J; theta = T L / (J G); P = T omega

J = pi d^4 / 32 for solid shafts; tau = T c / J; theta = T L / (J G); P = T omega.

tau = 1,000 N*m x 0.025 m / 6.13592e-7 m^4

theta = 1,000 N*m x 0.001 m / (6.13592e-7 m^4 x 8.00000e+10 Pa)

How to use the torsion calculator

Choose the torsion calculation mode that matches the hand calculation you want to perform.
Enter geometry, loads, material values, and units from a consistent source.
Review the primary result, secondary values, formula substitution, assumptions, and warnings.
Use the related-tool links only as a workflow aid; verify every transferred value before design use.

Input explanations

Shaft type controls whether J is computed from a solid diameter, hollow diameters, or a custom polar moment and outer radius.
Torque may be entered directly or derived exactly from power and angular speed.
Allowable stress and twist modes calculate the torque or diameter associated with the entered limit.
Shear modulus and length are required for angle-of-twist calculations.
Material properties vary with grade, condition, temperature, manufacturing process, loading direction, heat treatment, applicable standard, and data source. Verify all properties for the actual material.

Symbol

Meaning

Unit

Applied torque

N*m

Polar second moment of area

m^4

tau

Maximum torsional shear stress

theta

Angle of twist

rad

Torsional rigidity

N*m^2

Common mistakes

Using outside diameter where radius is required in Tc/J hand checks.

Using the rounded 9550 shortcut as the internal calculation instead of exact SI power and angular speed.

Applying circular-shaft torsion formulas to open thin-walled sections.

Ignoring keyways, shoulders, stress concentrations, and fatigue.

References

Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design.

Roark's Formulas for Stress and Strain: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.

Mechanics of Materials: Common mechanics-of-materials texts covering torsion, beam deflection, column buckling, and elastic stress relationships.

Machinery's Handbook: Machinery's Handbook reference material for machine-element formulas and terminology.

Torsion inputs

Calculation mode

Material properties

Shaft geometry and loading

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Circular shaft torsion

Technical visual

Engineering interpretation

What is the torsion calculator?

How to use the torsion calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?

Torsion inputs

Calculation mode

Material properties

Shaft geometry and loading

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Circular shaft torsion

Technical visual

Engineering interpretation

What is the torsion calculator?

How to use the torsion calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?