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Shafts Torsion

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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 (Pa, lbf/ft2(psf))

T = twisting moment (Nm, lbf ft)

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

J = Polar Moment of Inertia of Area (m4, ft4)

Note

  • the " Polar Moment of Inertia of an Area " is a measure of a shaft'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.
  • 1 ft = 12 in
  • 1 ft4 = 20736 in4
  • 1 psf (lbf/ft2) = 1/144 psi (lbf/in2)

" 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 ".

Polar Moment of Inertia vs. Area Moment of Inertia

  • " Polar Moment of Inertia " - 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 " - a property of shape that is used to predict deflection, bending and stress in beams
.

Circular Shaft and Maximum Moment or Torque

Maximum moment in a circular shaft can be expressed as:

Tmax = τmax J / R                               (2)

where

Tmax = maximum twisting torque (Nm, lbf ft)

τmax = maximum shear stress (Pa, lbf /ft2)

R = radius of shaft (m, ft)

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 (m, 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 (m, 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

α = L T / (JG)                                  (5)

where

α = angular shaft deflection ( radians)

L = length of shaft (m, ft)

G = Shear Modulus of Rigidity - or Modulus of Rigidity (Pa, psf)

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

α = 32 L T / (G π D4)                             (5a)

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

α = 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.

S olid shaft (π replaced)

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

Hollow shaft (π replaced)

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

.

Torsion Resisting Moments from Shafts of Various Cross Sections

Shafts - Torsional Resisting Moments
Shaft Cross Section AreaMaximum Torsional
Resisting Moment
- Tmax -
(Nm, lbf ft)
Nomenclature
Solid Cylinder Shaft

(π / 16) τmax (2 r)3

= (π / 16) τmax D3

Hollow Cylinder Shaft

(π / 16) τmax ((2 R)4 - (2 r)4) / (2 R)

= (π / 16) τmax (D4 - d4) / D

Ellipse Shaft (π / 16) τmax b2h h = "height" of shaft
b = "width" of shaft
h > b
Rectangle Shaft (2 / 9) τmax b2h h > b
Square Shaft (2 / 9) τmax H3
Triangle Shaft (1 / 20) τmax b3 b = length of triangle side
Hexagon Shaft

0.123τmax D3

0.189 τmax b 3

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 (0.05 m) 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)

= 40764331 Pa

= 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)

= 1.2o

.

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 (0.05 m) , inner diameter 30 mm (0.03 m) 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 radian)

= 1.4o

Example - Required Shaft Diameter to Transmit Power

A 15 kW electric motor shall be used to transmit power through a connected solid shaft. The motor and the shaft rotates with 2000 rpm . The maximum allowable shear stress - τmax - in the shaft is 100 MPa .

The connection between power and torque can be expressed

P = 0.105 nrpm T                    (7)

where

P = power (W)

nrpm = speed of shaft (rpm)

Re-arranged and with values - the torque can be calculated

T = (15 103 W) / (0.105 (2000 rpm))

= 71 Nm

Minimum diameter of the shaft can be calculated with eq. 4

D = 1.72 ((71 Nm) / (100 106 Pa))1/3

= 0.0153 m

= 15.3 mm

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