**Related Resources: mechanics machines**

### Torsional Deflection of Shaft

**Strength Mechanics of Materials**

**Mechanics of Machine Design, Equations and Calculators**

**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 )

Open **Torsional Deflection of Shaft Calculator**

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 (mm ^{4}, in^{4)}*

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

θ = 32 L T / (G π D^{4})

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

Solid shaft (π substituted)

θ_{degrees} ≈ 584 L T / (G D^{4})

Open **Angular Deflection of a Torsion Solid Shaft Calculator**

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**

Open **Angular Deflection of Hollow Shaft Calculator **

The angular deflection of hollow shaft

θ = 32 L T / (G π (D^{4}- d^{4}))

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)

θ_{degrees} ≈ 584 L T / (G (D^{4}- d^{4})

Reference

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

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