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### Torsional Deflection of Shaft

Angular Deflection of Solid Cylinder or Shaft with Torsion Applied

The angular deflection of a solid cylinder with torsion loading applied:

θ = L T / (G J )

Where:

θ = angular shaft deflection (radians)
T = torque (N-mm, in-lb)
L = length of shaft (mm, in)
G = modulus of rigidity (Mpa, psi)
J = Polar moment of inertia (mm4, in4)

#2 Equation and Calcuator for Angular Deflection of Solid Cylinder or Shaft with Torsion Applied

θ = 32 L T / (G π D4)

The angle in degrees can be achieved by multiplying the angle θ in radians with 180/π

Solid shaft (π substituted)

θdegrees ≈ 584 L T / (G D4)

Where:

D = outside diameter (mm, in.)
T = torque (N-mm, in-lb)
L = length of shaft (mm, in)
G = modulus of rigidity (Mpa, psi)

Angular Deflection of Hollow Shaft or Cylinder Equation and Calculator

The angular deflection of hollow shaft

θ = 32 L T / (G π (D4- d4))

Where:

D = outside diameter (mm, in.)
d = inside diameter (mm, in.)
G = modulus of rigidity (Mpa, psi)
T = torque (N-mm, in-lb)
L = length of shaft (mm, in)

Hollow shaft (π substituted)

θdegrees ≈ 584 L T / (G (D4- d4)

Reference

Shigley, J. E., Machine Design, McGraw-Hill Publishing Company, New York, 1956.

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