Torsion of Shafts

Torsion of solid and hollow shafts - Polar Moment of Inertia of an Area

Hollow shaft

Shear Stress in the Shaft

When a shaft is subjected to a torque or twisting, a shearing stress is produced in the shaft. The shear stress varies from zero in the axis to a maximum at the outside surface of the shaft.

The shear stress in a solid circular shaft in a given position can be expressed as:

τ = T r / J     (1)

where

τ = shear stress (MPa, psi)

T = twisting moment (Nmm, in lb)

r = distance from center to stressed surface in the given position (mm, in)

J = Polar Moment of Inertia of an Area (mm4, in4)

Note

  • the "Polar Moment of Inertia of an Area" is a measure of a beam's ability to resist torsion. The "Polar Moment of Inertia" is defined with respect to an axis perpendicular to the area considered. It is analogous to the "Area Moment of Inertia" - which characterizes a beam's ability to resist bending - required to predict deflection and stress in a beam.

"Polar Moment of Inertia of an Area" is also called "Polar Moment of Inertia", "Second Moment of Area", "Area Moment of Inertia", "Polar Moment of Area" or "Second Area Moment".

Circular Shaft and Maximum Moment

Maximum moment in a circular shaft can be expressed as:

Tmax = τmax J / R         (2)

where

Tmax = maximum twisting moment (Nmm, in lb)

τmax = maximum shear stress (MPa, psi)

R = radius of shaft (mm, in)

Combining (2) and (3) for a solid shaft

Tmax = (π / 16) τmax D3         (2b)

Combining (2) and (3b) for a hollow shaft

Tmax = (π / 16) τmax (D4 - d4) / D         (2c)

Circular Shaft and Polar Moment of Inertia

Polar Moment of Inertia of a circular solid shaft can be expressed as

J = π R4 / 2

   = π (D / 2)4 / 2

   = π D4 / 32         (3)

where

D = shaft outside diameter (mm, in)

Polar Moment of Inertia of a circular hollow shaft can be expressed as

J = π (D4 - d4) / 32         (3b)

where

d = shaft inside diameter         (mm, in)

Diameter of a Solid Shaft

Diameter of a solid shaft can calculated by the formula

D = 1.72 (Tmax / τmax)1/3         (4)

Torsional Deflection of Shaft

The angular deflection of a torsion shaft can be expressed as

θ = L T / (J G)         (5)

where

θ = angular shaft deflection (radians)

L = length of shaft (mm, in)

G = modulus of rigidity (Mpa, psi)

The angular deflection of a torsion solid shaft can be expressed as

θ = 32 L T / (G π D4)         (5a) 

The angular deflection of a torsion hollow shaft can be expressed as

θ = 32 L T / (G π (D4- d4))         (5b)

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

Solid shaft (π replaced)

   θdegrees ≈ 584 L T / (G D4)         (6a) 

Hollow shaft (π replaced)

   θdegrees ≈ 584 L T / (G (D4- d4)         (6b) 

Torsion Resisting Moments of Shafts of Various Cross Sections

Shaft Cross Section Area Maximum Torsional
Resisting Moment
- Tmax -
(Nm, in lb)
Nomenclature
Solid Cylinder Shaft ( π / 16) σmax D3
Hollow Cylinder Shaft ( π / 16) σmax (D4 - d4) / D
Ellipse Shaft ( π / 16) σmax b2 h h = "height" of shaft
b = "width" of shaft
h > b
Rectangle Shaft (2 / 9) σmax b2 h h > b
Square Shaft (2 / 9) σmax b3
Triangle Shaft (1 / 20) σmax b3 b = length of triangle side
Hexagon Shaft 1.09 σmax b3 b = length of hexagon side

Example - Shear Stress and Angular Deflection in a Solid Cylinder

A moment of 1000 Nm is acting on a solid cylinder shaft with diameter 50 mm and length 1 m. The shaft is made in steel with modulus of rigidity 79 GPa (79 109 Pa)

Maximum shear stress can be calculated as

τmax = T r / J

  = T (D / 2) / (π D4 / 32)

  = (1000 Nm) ((0.05 m) / 2) / (π (0.05 m)4 / 32)

  = 40.8 MPa

The angular deflection of the shaft can be calculated as

θ = L T / (J G) 

  = L T / ((π D4 / 32) G)

  = (1 m) (1000 Nm) / ((π (0.05 m)4 / 32) (79 109 Pa))

  = 0.021 (radians)

  = 1.2 o

Example - Shear Stress and Angular Deflection in a Hollow Cylinder

A moment of 1000 Nm is acting on a hollow cylinder shaft with outer diameter 50 mm, inner diameter 30 mm and length 1 m. The shaft is made in steel with modulus of rigidity 79 GPa (79 109 Pa)

Maximum shear stress can be calculated as

τmax = T r / J

  = T (D / 2) / (π (D4 - d4) / 32)

  = (1000 Nm) ((0.05 m) / 2) / (π ((0.05 m)4 - (0.03 m)4) / 32)

  = 46.8 MPa

The angular deflection of the shaft can be calculated as

θ = L T / (J G) 

  = L T / ((π D4 / 32) G)

  = (1 m) (1000 Nm) / ((π ((0.05 m)4 - (0.03 m)4) / 32) (79 109 Pa))

  = 0.023 (radians)

  = 1.4 o

Polar Moment of Inertia vs. Area Moment of Inertia

  • "Polar Moment of Inertia" as a measure of a beam's ability to resist torsion - which is required to calculate the twist of a beam subjected to torque
  • "Area Moment of Inertia" is a property of shape that is used to predict deflection, bending and stress in beams

Related Topics

  • Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects
  • Statics - Loads - force and torque

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