Torsion

Learning Objectives

  • Relate applied torque to internal torsional shear stress in circular shafts.
  • Compute polar moment of inertia for solid and hollow circular sections.
  • Determine angle of twist using material and geometric stiffness.
  • Connect torque, rotational speed, and transmitted power.
  • Solve basic statically indeterminate torsion systems using equilibrium and compatibility.

Torque

Torque is a moment about a member's longitudinal axis that tends to twist the member.

Elastic Torsion Formula

Shear stress distribution in a circular shaft under Saint-Venant torsion.

TJ=τr=GθL\frac{T}{J}=\frac{\tau}{r}=\frac{G\theta}{L}

Solid Circular Polar Moment

Polar second moment of area of a solid circular shaft.

J=πd432J=\frac{\pi d^4}{32}

Hollow Circular Polar Moment

Polar second moment of area of a hollow circular shaft.

J=π(D4−d4)32J=\frac{\pi(D^4-d^4)}{32}
Solid and Hollow Shaft Sections

The connected cutaway comparison makes the solid core and hollow bore visible; exact polar-moment relationships remain in the formulas.

Text-free connected shaft assembly showing a solid circular section beside a hollow tubular section.

Reading Solid and Hollow Sections

Compare the material layout at the two coaxial sections: the left segment is solid while the right segment surrounds an open bore. This contextual raster carries no dimensions, polar-inertia value, or stress distribution; use the formulas for those relationships.

Shear-Stress Distribution

For a circular shaft in elastic torsion, shear stress varies linearly with radius: it is zero at the center and maximum at the outer surface. The model assumes circular sections and Saint-Venant torsion; open thin-walled sections require different treatment.

Angle of Twist

Elastic twist of a prismatic circular shaft.

θ=TLJG\theta=\frac{TL}{JG}
Qualitative Shaft Twist

The intact shaft's restrained helical surface cue helps make rotation along its length visible without assigning an angle or measured result.

Text-free intact circular shaft between bearing supports with a restrained helical surface cue.

Reading the Twist Cue

The surface cue gives a qualitative sense of how a marked line on an intact shaft can change orientation along its length. It is not an angular scale, pointer, measured result, or substitute for the angle-of-twist formula and simulator.

Interactive Exploration

Adjust torque, diameter, and shaft length. Compare how the maximum shear stress and angle of twist respond, noting the strong fourth-power influence of diameter through JJ.

Torsion of a Circular Shaft

Concept and model scope

Explore elastic torsional shear stress and angle of twist for a solid circular steel shaft with G = 80 GPa.

Controls

Torque

Applied twisting moment. Both maximum shear stress and elastic twist vary linearly with torque.

Range: 0.1–5.0 kN·m. Step: 0.1 kN·m.

1.5 kN·m

Shaft diameter

Solid circular-shaft diameter. Polar moment J varies with the fourth power of diameter.

Range: 20–100 mm. Step: 2 mm.

50 mm

Shaft length

Physical shaft length. Elastic angle of twist varies linearly with length for constant T, J, and G.

Range: 0.5–4.0 m. Step: 0.1 m.

2.0 m
L = 2.0 m · d = 50 mmT = 1.5 kN·m
Polar moment J
0.614 ×10⁶ mm⁴

J = πd⁴/32

Maximum shear stress
61.1 MPa

Occurs at the outer radius for elastic circular-shaft torsion.

Angle of twist
3.50°

0.0611 rad

Power and Torque

Mechanical power transmitted by a rotating shaft.

P=Tω=2πfTP=T\omega=2\pi fT
Driver–Shaft–Driven Machine

The compact machine train shows where a shaft and coupling connect a driver to driven equipment; use the power formula for exact torque and speed relationships.

Text-free compact drive showing an electric motor, coaxial shaft, flexible coupling, and driven pump housing on a common base.

Reading the Power-Transmission Context

Follow the connected shaft from the motor-side driver through the coupling to the driven housing. This is a static physical context rather than operating guidance; it provides no power, speed, torque, rating, or rotation instruction.

Power-Transmission Design

For fixed power, higher rotational speed requires lower torque. Shaft sizing still has to satisfy both strength and stiffness: a shaft may meet an allowable shear stress but exceed a permissible angle of twist.

Shaft-Line Torque Context

A continuous circular shaft and coupling provide physical context for torque transmission; use the adjacent formulas and simulator for stress, twist, speed, and power values.

Idle shaft line with a circular shaft, bearing supports, flanged coupling, motor housing, and load hub.

Reading the Shaft-Line Context

The motor-side hub, circular shaft, coupling, and load-side hub show where a twisting action is transmitted through a shaft line. This is a static educational context rather than a machine-safety or equipment-rating instruction; it gives no torque, speed, power, or twist value. Use the sign convention, P=TωP=T\omega, and the torsion simulator for exact calculations.

Compatible Shaft Coupling

The interface view shows both shaft ends seated on the same axis through a coupling; it gives no alignment tolerance or torque capacity.

Text-free close-up of two coaxial shaft ends seated in a flexible coupling with exposed hubs.

Reading the Coupling Interface

Observe the common shaft axis and the way each end is seated in its coupling hub. The close-up explains physical compatibility only; it supplies no alignment gauge, tolerance, torque value, or equipment-approval claim.

Statically Indeterminate Torsion

A torsional system is statically indeterminate when equilibrium alone cannot determine all reaction torques. Add compatibility of rotation. For a shaft fixed at both ends, the net relative rotation between the fixed supports is zero, so the signed segment twists must satisfy ∑θi=0\sum\theta_i=0.

Compatibility Requires Signed Twist

Use a consistent torque and rotation sign convention across all shaft segments. Magnitude-only twist equations can produce incorrect reaction torques in indeterminate systems.

Indeterminate Torsion: Equilibrium and Compatibility

For a fixed-end segmented circular shaft within the linear-elastic torsion model, use one consistent torque and rotation sign convention. For each segment, use its own length Lᵢ, polar moment Jᵢ, and shear modulus Gᵢ to compute signed twist θᵢ = TᵢLᵢ/(JᵢGᵢ); fixed-end compatibility requires Σθᵢ = 0. Solve this condition together with torque equilibrium.

Indeterminate Torsion: Equilibrium and CompatibilityFor a fixed-end segmented circular shaft within the linear-elastic torsion model, use one consistent torque and rotation sign convention. For each segment, use its own length Lᵢ, polar moment Jᵢ, and shear modulus Gᵢ to compute signed twist θᵢ = TᵢLᵢ/(JᵢGᵢ); fixed-end compatibility requires Σθᵢ = 0. Solve this condition together with torque equilibrium.. Fixed-end shaft; choose torque sign → Write torque equilibrium for reactions and segments; Write torque equilibrium for reactions and segments → Use signed θᵢ = TᵢLᵢ/(JᵢGᵢ); impose Σθᵢ = 0; Use signed θᵢ = TᵢLᵢ/(JᵢGᵢ); impose Σθᵢ = 0 → Solve jointly; verify torque balance and zero twist

Fixed-end shaft; choose torque sign → Write torque equilibrium for reactions and segments; Write torque equilibrium for reactions and segments → Use signed θᵢ = TᵢLᵢ/(JᵢGᵢ); impose Σθᵢ = 0; Use signed θᵢ = TᵢLᵢ/(JᵢGᵢ); impose Σθᵢ = 0 → Solve jointly; verify torque balance and zero twist

  • Fixed-end shaft; choose torque sign: terminator. Model the shaft as connected segments between fixed supports and use one consistent torque and rotation sign convention.
  • Write torque equilibrium for reactions and segments: process. Relate the support reaction torques and each segment's internal torque using equilibrium.
  • Use signed θᵢ = TᵢLᵢ/(JᵢGᵢ); impose Σθᵢ = 0: process. Use the linear-elastic twist relation for each segment with its own properties; algebraic segment twists sum to zero because both supports are fixed.
  • Solve jointly; verify torque balance and zero twist: terminator. Solve the equilibrium and compatibility equations together, then check both torque balance and zero net rotation.
Key Takeaways
  • Circular-shaft shear stress increases linearly with radius and is largest at the outer surface.
  • Torsional stiffness is JG/LJG/L; diameter strongly affects both stress and twist.
  • Power transmission links torque to angular speed through P=TωP=T\omega.
  • Strength and stiffness are separate design checks.
  • Indeterminate torsion requires both equilibrium and rotational compatibility.

References