Torsion (engineering)
Torsion is the twisting of a structural member caused by an applied torque (twisting moment) about its longitudinal axis. In circular shafts, torsion produces a shear stress distribution that varies linearly from zero at the neutral axis to a maximum at the outer surface. Torsion analysis is fundamental for the design of drive shafts, axles, springs, and any component that transmits rotary power.
Key Formula
tau_max = (T × r) / J
LaTeX: \tau_{max} = \frac{T \cdot r}{J}
| Symbol | Meaning | Unit |
|---|---|---|
| τ_max | Maximum shear stress (at outer surface) | Pa |
| T | Applied torque | N·m |
| r | Outer radius of the shaft | m |
| J | Polar moment of inertia of the cross-section | m⁴ |
Worked Example
Problem
A solid circular steel shaft of diameter 40 mm is subjected to a torque of 500 N·m. Calculate the maximum shear stress.
Solution
Step 1: Calculate polar moment of inertia for a solid circle. J = π × d⁴ / 32 = π × (0.040)⁴ / 32 = π × 2.56 × 10⁻⁶ / 32 = 2.513 × 10⁻⁷ m⁴ Step 2: Identify outer radius. r = d/2 = 0.040/2 = 0.020 m Step 3: Apply torsion formula. τ_max = T × r / J = 500 × 0.020 / 2.513 × 10⁻⁷ = 10 / 2.513 × 10⁻⁷ = 3.979 × 10⁷ Pa
Answer
Maximum shear stress τ_max ≈ 39.8 MPa
Polar Moments of Inertia for Common Cross-Sections
| Cross-Section | Polar Moment J | Notes | Application |
|---|---|---|---|
| Solid circle (dia d) | πd⁴ / 32 | Full material | Drive shafts |
| Hollow circle (do, di) | π(do⁴ − di⁴) / 32 | Saves weight | Automotive axles |
| Thin-walled tube (t, r) | 2πr³t | Approx. for thin walls | Aircraft fuselage |
| Square (side a) | 0.1406 a⁴ | Torsional flexibility varies | Machine keys |
Interactive Tools
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Related Terms
Shear Stress
Shear stress is the component of stress that acts parallel (tangential) to a cross-sectional surface, as opposed to normal stress which acts perpendicular to it. It arises when equal and opposite forces act along parallel planes in a material, causing layers to slide relative to one another. Shear stress is critical in the design of bolts, welds, shafts, beams, and adhesive joints, where failure along a plane is the governing mode.
Beam Deflection
Beam deflection is the displacement of a beam's neutral axis from its original position under transverse loading, measured perpendicular to the beam's length. It depends on the load magnitude, span length, cross-sectional geometry (moment of inertia), and material stiffness (Young's modulus). Limiting beam deflection is a key serviceability requirement in structural design to prevent cracking of finishes, damage to supported equipment, and discomfort to occupants.
Engineering Stress
Engineering stress is defined as the applied force divided by the original cross-sectional area of a specimen, regardless of any deformation that occurs during loading. It is the conventional measure used in materials testing and structural analysis because the original dimensions are easily measured before the test begins. Engineering stress is widely used in design calculations, material data sheets, and stress-strain curves to characterise material behaviour under uniaxial loading.
From Latin "torsio" (a twisting), derived from "torquere" (to twist). The mathematical treatment of torsion in circular shafts was developed by Augustin-Louis Cauchy and Claude-Louis Navier in the early 19th century, and later extended to non-circular sections by Saint-Venant (1853).