Torsion of Circular Shafts

Why it matters

Torsion of circular shafts is a fundamental concept in mechanical engineering, especially in the design and analysis of components like drive shafts, axles, and other cylindrical structures that transmit torque. Understanding torsion helps ensure these components can withstand operational stresses without failure, which is critical for safety and performance.

Key ideas

• Torsion refers to the twisting of an object due to an applied torque.
• Circular Shafts are commonly used in mechanical systems to transmit power and motion.
• Shear Stress is induced in the material of the shaft due to torsion, and it varies linearly from zero at the center to a maximum at the outer surface.
• Angle of Twist is the angular displacement experienced by a shaft under torsion, which depends on the material properties, shaft dimensions, and applied torque.
• Polar Moment of Inertia (J) is a geometric property of the cross-section that affects the torsional resistance of the shaft.

Scope and hollow shafts

The formulas assume homogeneous isotropic linear-elastic circular shafts, small strains, and regions away from end effects. J is the polar second moment of area. For a hollow circular shaft, J = π(D⁴ − d⁴)/32. A noncircular section generally requires a different torsion constant and stress distribution. For variable torque or section, θ = ∫T/(GJ) dx; TL/(GJ) is the uniform case.

Formulas

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

Worked example

Given: A solid circular shaft with a diameter of 0.05 m, length of 2 m, subjected to a torque of 100 N·m. The modulus of rigidity (G) is 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. Calculate the shear stress (τ) at the outer surface:

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

3. Calculate the angle of twist (θ):

   • Formula: θ = T·L / (J·G)
   • Calculation: θ = 100·2 / (6.13592×10⁻⁷·80×10⁹) = 0.00407437 radians

Final Answer: The shear stress is 4.07437 MPa and the angle of twist is 0.00407437 radians.

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.
• Neglecting the linear variation of shear stress across the radius.

For GATE ME

Questions often involve calculating shear stress, angle of twist, or the required diameter for a given torque and material properties. Practice problems involving both solid and hollow shafts, and ensure familiarity with unit conversions and the use of the torsion formulas.

Quick check

1. What is the relationship between torque and shear stress in a circular shaft?
2. How does the angle of twist change with increasing shaft length?
3. What is the polar moment of inertia for a shaft with a diameter of 0.1 m?

Answers: 1. τ = T·r / J; 2. It increases; 3. J = π·(0.1)⁴ / 32 = 9.82×10⁻⁶ m⁴.