AtlasCalcsUtility Calculation Suite
2025-08-21 14 min

Torsion Calculator

Calculate torsional shear stress and angle of twist for circular shafts using tau = Tr/J and theta = TL/(GJ).

Calculator

Unit System
Applied Torque(N*m) Help
N*m

Torque about shaft axis

Outer Diameter(mm) Help
mm

Leave polar moment blank to auto-calc

Shaft Length(mm) Help
mm

Length under torsion

Shear Modulus (G)(GPa) Help
GPa

Steel ~80, Aluminum ~26

Polar Moment (J)(mm^4) Help
mm^4

Leave blank to compute from diameter

Decimal Precision
Results update live as you type

Results

Normal

Shear Stress (tau)

0.02MPa

Angle of Twist (theta)

0.00deg

Polar Moment (J)

613592.32mm^4

Visualizations

Results Overview

0MPaShear Stress
613,592MPaPolar Moment

Formula

Torsional shear stress equals applied torque times radial distance divided by polar moment of inertia. Angle of twist equals torque times length divided by shear modulus times polar moment.

tau = Tc / J and theta = TL / (GJ)

Alternative Form

For solid shaft: J = pi*d^4/32, c = d/2

Variables

SymbolNameUnitDescription
tauShear StressMPaMaximum torsional shear stress at the outer surface
TApplied TorqueN*mTorque applied about the shaft axis
cOuter RadiusmmDistance from center to outer fiber
JPolar Moment of Inertiamm^4Resistance to torsional deformation
thetaAngle of TwistdegreesAngular deformation over the shaft length
LShaft LengthmmLength of the shaft under torsion
GShear ModulusGPaMaterial rigidity (modulus of rigidity)

Explanation

When torque is applied to a circular shaft, shear stresses develop varying linearly from zero at the center to maximum at the outer surface. Angle of twist accumulates proportional to torque and inversely proportional to GJ.

Derivation

From equilibrium: d(theta)/dx = T/(GJ). Integrating 0 to L gives theta = TL/(GJ). From stress equilibrium: tau = Tr/J, maximized at r = c.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Determine applied torque T and shaft geometry (diameter, length)

Result: T and dimensions known
Step 2

Calculate polar moment J = pi*d^4/32 for solid shaft

Result: J computed
Step 3

Compute shear stress: tau = Tc/J where c = d/2

Result: tau computed
Step 4

Compute angle of twist: theta = TL/(GJ)

Result: theta computed in radians, convert to degrees
Step 5

Compare results against allowable shear stress for the material

Result: Design check complete
Final Answer
0.02

Worked Examples

See how the calculator works with different scenarios.

Scenario

A 50 mm diameter solid steel shaft (G = 80 GPa) transmits 500 N*m torque over 1.5 m.

Inputs

Torque:500 N*m
Outer Diameter:50 mm
Length:1500 mm
Shear Modulus:80 GPa
Polar Moment:0 (auto)

Calculation

J = pi*50^4/32 = 613,592 mm^4. tau = 500*1000*25/613592 = 20.4 MPa. theta = 500*1000*1500/(80000*613592) = 0.876 deg.

Result

tau = 20.4 MPa, theta = 0.876 degrees

Interpretation

Low shear stress, well within steel capacity. Small twist angle indicates adequate stiffness.

Understanding Your Result

Your Result: Shear Stress (tau): 0.02 MPa

Typical range: Steel shafts: tau_allow = 40-80 MPa. Angle limit: 1-3 deg/m for precision, up to 10 deg/m for general use.

What This Means

Torsional shear stress is the internal resistance that develops when a shaft transmits torque. Angle of twist measures how much one end rotates relative to the other.

Why It Matters

Excessive shear stress causes yielding and fatigue failure. Excessive twist degrades positional accuracy and causes vibration in drive systems.

Professional Interpretation

Compare calculated shear stress against 50-60% of material shear yield strength. Check twist per unit length against application requirements.

Recommendations

  • •Increase shaft diameter to reduce stress and twist
  • •Use higher shear modulus materials for stiffness-critical applications
  • •Consider hollow shafts for weight savings
  • •Apply stress concentration factors at keyways

Educational Content

Introduction

Torsion is one of the primary loading modes in mechanical engineering, alongside bending, axial, and shear loading.

Concept Explanation

In circular shafts, tau = Tc/J gives elastic shear stress distribution. Theta = TL/(GJ) quantifies angular deformation. Both depend on polar moment of inertia J.

Applications

▸Automotive drive shafts
▸Industrial transmission shafts
▸Aerospace control linkages
▸Wind turbine drivetrain shafts
▸Screw conveyor augers

Benefits

  • •Directly applicable to any circular cross-section
  • •Linear elastic range is well defined
  • •J can be computed for solid, hollow, or composite sections

Limitations

  • •Applies only to circular cross-sections in the elastic range
  • •Does not account for stress concentrations
  • •Assumes uniform torque along the length

Best Practices

  • •Include stress concentration factors at discontinuities
  • •Check both stress and deflection criteria
  • •Use consistent units throughout

Common Misconceptions

Myth: Polar moment equals area moment of inertia
Truth: For circular sections J = 2I, but they are distinct properties. J applies to torsion, I to bending.

Practical Tips

  • •For solid shafts J = pi*d^4/32, for hollow J = pi*(D^4-d^4)/32
  • •Shear modulus G is approximately E/(2*(1+nu))
  • •Steel G = 80 GPa, aluminum G = 26 GPa
  • •Angle of twist is proportional to length and torque

Common Mistakes

Using diameter instead of radius for c
c is the outer radius (d/2), not the diameter.
Confusing polar moment J with area moment I
J = 2I for circular sections. Use J for torsion, I for bending.
Forgetting unit conversions
Ensure torque is in N*mm when stress is in MPa (N/mm^2).

Assumptions

This calculator makes the following assumptions to provide estimates:

Linear Elastic Material: Shaft material behaves elastically with constant shear modulus G.
Circular Cross-Section: The torsion formula applies only to circular shafts.
Uniform Torque: Torque is constant along the shaft length.
Small Deformation: Angle of twist is small enough for small-angle approximations.

Limitations

Be aware of these limitations when using this calculator:

Non-Circular Sections: Rectangular or irregular sections require different methods.
Plastic Deformation: Beyond yield, elastic formulas are no longer valid.
Dynamic Loading: Fatigue and impact require additional analysis.

Frequently Asked Questions

Answers to common questions about this calculator.

What is torsion in mechanical engineering?
Torsion is the twisting of a structural member caused by an applied torque about its longitudinal axis. It produces shear stresses that vary from zero at the center to maximum at the outer surface of a circular shaft.
What is the torsion formula?
The primary formulas are tau = Tc/J for shear stress and theta = TL/(GJ) for angle of twist, where T is torque, c is outer radius, J is polar moment of inertia, L is length, and G is shear modulus.
What is the polar moment of inertia?
J measures a cross-section resistance to torsional deformation. For a solid shaft J = pi*d^4/32 and for a hollow shaft J = pi*(D^4-d^4)/32. It has units of length^4.
What is shear modulus?
Shear modulus G (modulus of rigidity) relates shear stress to shear strain. Steel: G ~ 80 GPa, Aluminum: G ~ 26 GPa. G = E/(2*(1+nu)).
How does torque affect shear stress?
Shear stress is directly proportional to torque. Doubling the torque doubles the shear stress in the elastic range.
What is the angle of twist?
Angle of twist theta is the angular deformation between two cross-sections under torsion. theta = TL/(GJ) radians or TL*180/(pi*GJ) degrees.
What is the difference between torsion and bending?
Torsion applies torque about the longitudinal axis producing shear stress. Bending applies a moment about a transverse axis producing normal stress.
How do I increase torsional strength?
Increase shaft diameter (strength scales with d^3, stiffness with d^4), use higher shear yield strength material, or use hollow shafts for weight efficiency.
What is the maximum torque a shaft can transmit?
T_max = tau_allow * J / c. For steel shafts, tau_allow is typically 40-80 MPa depending on grade and safety factor.
How does temperature affect torsion?
Temperature changes alter shear modulus G and introduce thermal stresses if constrained. At elevated temperatures G decreases, reducing torsional stiffness.
What is torsional rigidity?
Torsional rigidity is the product GJ, representing the shaft resistance to angular deformation. Higher GJ means less twist for a given torque.
What is the torsion constant for non-circular sections?
For non-circular sections, the torsion constant K replaces J and must be determined from tables or numerical analysis. Rectangular bars have significantly lower torsion constants than circular ones.
How do keyways affect torsion?
Keyways create stress concentrations that can increase local stress by 2-3 times. Use stress concentration factors Kt = 2.0-3.0 in the design calculation at keyway locations.
What materials are best for torsion shafts?
Alloy steels (4140, 4340) offer the best combination of shear strength and fatigue resistance. Stainless steels provide corrosion resistance. Aluminum alloys are used where weight is critical.
What is plastic torsion?
Plastic torsion occurs when shear stress exceeds the yield point. The elastic-plastic boundary moves inward as torque increases. Fully plastic torque is typically 4/3 times the elastic limit torque for solid shafts.
How is torsion different from pure shear?
Pure shear is a state of stress where shear acts on perpendicular planes equally. Torsion in circular shafts produces pure shear at the surface, but the stress state varies radially.
What is the Tresca yield criterion for torsion?
The Tresca criterion states that yielding occurs when the maximum shear stress equals half the tensile yield strength: tau_max = sigma_y/2. For torsion: tau = Tc/J <= sigma_y/2.
What is the von Mises criterion for torsion?
Von Mises predicts yielding when the distortion energy reaches a critical value. For pure torsion: tau <= sigma_y/sqrt(3) = 0.577*sigma_y, which is less conservative than Tresca.
How do you measure angle of twist experimentally?
Angle of twist is measured using optical encoders, LVDTs attached to rotating arms, or digital protractors mounted at two cross-sections along the shaft length.
What is torsional vibration?
Torsional vibration is oscillatory angular motion superimposed on steady rotation. It occurs in drive systems with varying torque and can cause fatigue failure if resonance occurs.
How do you prevent torsional resonance?
Torsional resonance is avoided by ensuring the excitation frequency does not match the natural frequency. Add torsional dampers, change shaft stiffness, or modify mass distribution of rotating elements.
What is a St Venant torsion problem?
St Venant torsion refers to the uniform torsion of prismatic bars where warping is free. The solution assumes no end constraints, giving the classic tau = Tc/J for circular sections.
What is warping in torsion?
Warping is the out-of-plane deformation of cross-sections during torsion. It occurs in non-circular sections and at constrained ends of circular sections. Warping restraint introduces additional normal and shear stresses.
What is a torsion spring?
A torsion spring stores elastic energy through twisting. Its rate is k = Ed^4/(64*D*N), where d is wire diameter, D is coil diameter, and N is active coils.
How do you calculate J for a hollow shaft?
For a hollow shaft with outer diameter D and inner diameter d: J = pi*(D^4 - d^4)/32. This is more efficient than solid shafts for the same weight.
What is the thin-walled tube approximation for torsion?
For thin-walled tubes (t << r), J approximates to 2*pi*r^3*t where r is mean radius and t is wall thickness. This simplifies calculations for tubing.
What is combined torsion and bending?
When a shaft carries both torque and bending moment simultaneously, use von Mises or Tresca equivalent stress: sigma_eq = sqrt(sigma_b^2 + 3*tau^2) and compare to yield strength.
How does fatigue affect torsional shafts?
Fatigue is the most common cause of torsion shaft failure. Cyclic torsion causes crack initiation at stress concentrations. Use endurance limit (typically 0.5*UTS) and Goodman or Soderberg criteria.
What is a torsion balance?
A torsion balance measures very small forces by the twist of a thin fiber. Cavendish used one to measure gravitational constant G. Sensitivity depends on fiber torsional rigidity and fiber length.
What are typical shaft twist limits?
Precision machinery: 0.25-0.5 deg/m. General drive shafts: 1-3 deg/m. Allowable twist depends on application requirements for positioning accuracy and vibration control.
How does diameter affect torsional performance?
Shear stress scales with d^-3 and angle of twist with d^-4. Doubling diameter reduces stress by 8x and twist by 16x. Diameter is the most effective parameter to increase torsional capacity.
What is torsional stiffness?
Torsional stiffness is the torque required per unit angle of twist: k_t = T/theta = GJ/L. Units are N*m/rad. Higher stiffness means less angular deflection under load.
How do you select shaft diameter for torsion?
Start from T_max = (pi/16)*tau_allow*d^3 for solid shafts. Solve for d. Apply safety factor of 1.5-3.0 and stress concentration factors. Verify the angle of twist is within limits.
What is the Bredt formula for thin-walled tubes?
Bredt formula gives shear stress in thin-walled closed sections: tau = T/(2*A_m*t), where A_m is the enclosed area and t is wall thickness. This is essential for structural tubing design.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Torsion
Twisting of a member due to applied torque about its longitudinal axis.
Polar Moment of Inertia
Second moment of area about the polar axis, J = integral(r^2 dA).
Shear Modulus
Material property relating shear stress to shear strain, G = E/(2*(1+nu)).
Angle of Twist
Angular deformation between two cross-sections under torsion.
Torsional Rigidity
Product GJ representing resistance to angular deformation.
Stress Concentration
Local increase in stress at geometric discontinuities like keyways.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

ASME B17.1ISO 6336DIN 743

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

License:

Free to use for educational and professional purposes.