Torsion of Circular Shafts

Why it matters

Torsion of circular shafts is a fundamental concept in civil engineering, particularly in the design of mechanical components such as drive shafts, axles, and other cylindrical structures. Understanding how these components behave under twisting forces is essential for ensuring their structural integrity and functionality in real-world applications.

Key ideas

• Torsion refers to the twisting of an object due to an applied torque. In circular shafts, this results in shear stress and angular deformation.
• Polar Moment of Inertia (J) is a measure of an object's ability to resist torsion. It depends on the geometry of the shaft.
• Shear Stress (τ) in a shaft under torsion is directly proportional to the applied torque and the radial distance from the center of the shaft.
• Angle of Twist (θ) is the angular displacement experienced by a shaft under torsion, which depends on the material properties, length, and geometry of the shaft.
• Assumptions: The shaft is circular, homogeneous, and isotropic, and the material follows Hooke's Law.

Formulas

• τ = T·r / J
  • τ: Shear stress (Pa)
  • T: Torque applied (N·m)
  • r: Radial distance from the center (m)
  • 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: Torque applied (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 0.1 m, length of 2 m, subjected to a torque of 500 N·m. The modulus of rigidity (G) is 80 GPa.

1. Calculate the polar moment of inertia (J): J = π·d⁴ / 32 J = π·(0.1)⁴ / 32 = 9.82 × 10⁻⁶ m⁴

2. Calculate the shear stress (τ) at the outer surface: τ = T·r / J τ = 500·(0.1/2) / (9.82 × 10⁻⁶) = 2.55 × 10⁶ Pa

3. Calculate the angle of twist (θ): θ = T·L / (J·G) θ = 500·2 / (9.82 × 10⁻⁶·80 × 10⁹) = 0.00128 radians

Final Answer: The shear stress is 2.55 MPa and the angle of twist is 0.00128 radians.

Common mistakes

• Confusing the diameter with the radius when calculating the polar moment of inertia.
• Forgetting to convert units, especially when dealing with GPa and mm.
• Applying formulas for non-circular shafts, which require different considerations.

For GATE CE

• Questions often involve calculating shear stress, angle of twist, or polar moment of inertia for given torque and dimensions.
• Practice problems involving both solid and hollow shafts, as well as composite shafts.

Quick check

1. What is the polar moment of inertia for a shaft with a diameter of 0.2 m?
2. How does the angle of twist change if the length of the shaft is doubled?
3. What is the relationship between torque and shear stress in a circular shaft?

Answers: 1. J = π·(0.2)⁴ / 32, 2. It doubles, 3. Directly proportional.