Torsion of Shafts

Why it matters

Torsion of shafts is a fundamental concept in mechanical engineering, particularly in the design and analysis of rotating machinery such as engines, turbines, and gearboxes. Understanding how shafts behave under torsional loads helps engineers ensure the reliability and safety of mechanical systems.

Key ideas

• Torsion: It refers to the twisting of an object due to an applied torque. In mechanical systems, shafts often experience torsion.
• Shear Stress: When a shaft is subjected to torsion, shear stress is induced across its cross-section.
• Angle of Twist: This is the angle through which one end of the shaft rotates relative to the other end under the action of torque.
• Polar Moment of Inertia (J): A measure of an object's ability to resist torsion, dependent on the shape and size of the cross-section.
• Torsional Rigidity: The product of the modulus of rigidity (G) and the polar moment of inertia (J), indicating the resistance of a shaft to twisting.

Scope of the circular-shaft equations

These equations apply to homogeneous, isotropic, linearly elastic circular shafts under Saint-Venant torsion, away from local load-introduction effects. J is a polar second moment of area, with units m⁴, not a mass moment of inertia. For noncircular sections, the torsion constant generally differs from the polar area moment, and τ = Tr/J is not a general stress formula.

For a hollow circular shaft, J = π(D⁴ − d⁴)/32, where D and d are outer and inner diameters. Torsional rigidity is GJ; the end-to-end torsional stiffness of a uniform shaft is GJ/L. When torque or section properties vary, θ = ∫[T(x)/(G(x)J(x))] dx.

Formulas

• τ = T·r / J
  • τ: Shear stress (Pa)
  • T: Torque applied (N·m)
  • r: Radial distance from the shaft axis (m); use the outer radius for maximum stress
  • J: Polar moment of inertia (m⁴)
• θ = T·L / (J·G)
  • θ: Angle of twist (radians)
  • L: Length of the shaft (m)
  • G: Modulus of rigidity (Pa)
• J = π·d⁴ / 32 for a solid circular shaft
  • d: Diameter of the shaft (m)

Worked example

Given: A solid circular shaft with a diameter of 0.05 m, length 2 m, subjected to a torque of 100 N·m. Modulus of rigidity, G = 80 GPa.

1. Calculate the polar moment of inertia (J):

   • Formula: J = π·d⁴ / 32
   • Calculation: J = π·(0.05)⁴ / 32 = 6.13592 × 10⁻⁷ m⁴

2. Determine the shear stress (τ):

   • Formula: τ = T·r / J
   • Calculation: τ = 100·0.025 / 6.13592 × 10⁻⁷ = 4.07437 × 10⁶ Pa

3. Find the angle of twist (θ):

   • Formula: θ = T·L / (J·G)
   • Calculation: θ = 100·2 / (6.13592 × 10⁻⁷·80 × 10⁹) = 4.07437 × 10⁻³ radians

Final Answer: The shear stress is 4.074 MPa and the angle of twist is 4.074 mrad.

Common mistakes

• Confusing the radius with the diameter when calculating the polar moment of inertia.
• Forgetting to convert units, especially when dealing with GPa and mm.
• Misapplying the formulas for hollow versus solid shafts.

For GATE ME

Questions often involve calculating shear stress, angle of twist, or designing shafts to withstand specific torsional loads. Practice problems involving both solid and hollow shafts, and ensure familiarity with unit conversions.

Quick check

1. What is the formula for shear stress in a shaft under torsion?
2. How does the polar moment of inertia affect a shaft's resistance to torsion?
3. What unit is used for the modulus of rigidity?

Answers: 1. τ = T·r / J; 2. Higher J means greater resistance; 3. Pascal (Pa).