Theory of Torsion in Mechanics

Theory of Torsion & Assumptions

• Torsion is a twisting force applied to an object, resulting in shearing stress and strain.
• Assumptions for theory of torsion:
  • The material is homogenous, isotropic, and linearly elastic.
  • The shaft is circular in cross-section.
  • The applied torque is constant along the shaft.
  • The deformation is small compared to the dimensions of the shaft.

Derivation of Torsion Equation

• The torsion equation relates the applied torque to the shear stress and the geometry of the shaft.
• The torsion equation is: T = (π/16) * τ * d^3, where:
  • T is the applied torque.
  • τ is the shear stress.
  • d is the diameter of the shaft.

Polar Modulus

• The polar modulus (J) represents the shaft's resistance to torsion.
• It is calculated as (π/32) * d^4 for a solid circular shaft and (π/32) * (D^4 - d^4) for hollow circular shaft, where D and d are the outer and inner diameters respectively.

Stresses in Solid and Hollow Circular Shafts

• The shear stress in a circular shaft is maximum at the surface and decreases linearly towards the center.
• For a solid shaft, the maximum shear stress is τ = (16 * T) / (π * d^3).
• For a hollow shaft, the maximum shear stress is τ = (16 * T) / (π * (D^4 - d^4)).

Power Transmitted by Shaft

• Power transmitted by a rotating shaft is calculated as P = (2π * N * T) / 60, where:
  • P is the power transmitted in watts.
  • N is the rotational speed in rpm.
  • T is the torque in Nm.

Closed Coiled Helical Spring with Axial Load

• A closed coiled helical spring stores energy when subjected to an axial load.
• The deflection of the spring is proportional to the applied load.
• The spring constant is determined by the material properties, coil diameter, and wire diameter.

Thin Cylinders Subjected to Internal Pressure

• Thin cylinders subjected to internal pressure experience hoop stress, which is the stress acting circumferentially.
• This stress is calculated as σh= (P * d) / (2 * t), where:
  • σh is the hoop stress
  • P is the internal pressure
  • d is the mean diameter of the cylinder
  • t is the wall thickness
• Thin cylinders also experience longitudinal stress, which acts along the length of the cylinder.
• The longitudinal stress is equal to half of the hoop stress: σl = (P * d) / (4 * t)

This quiz covers the fundamentals of torsion theory, including key assumptions, the derivation of the torsion equation, and the concept of the polar modulus. It is ideal for students seeking to understand the principles governing twisting forces in engineering applications.