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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:

= T r / I p (1)

where

= shear stress (MPa, psi)

T = twisting moment (Nmm, in lb)

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

I p = "polar moment of inertia" of cross section (mm 4 , in 4 )

The "polar moment of inertia" is a measure of an object's ability to resist torsion.

C ircular Shaft and Maximum Moment

M aximum moment in a circular shaft can be expressed as:

T max = max I p / R (2)

where

T max = maximum twisting moment (Nmm, in lb)

max = maximum shear stress (MPa, psi)

R = radius of shaft (mm, in)

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

T max = ( /16) max D3 (2b)

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

T max = ( /16) max (D4 - d 4 ) / D (2c)

C ircular Shaft and Polar Moment of Inertia

P olar moment of inertia of a circular solid shaft can be expressed as

I p = R 4 /2 = D 4 /32 (3)

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where

D = shaft outside diameter (mm, in)

P olar moment of inertia of a circular hollow shaft can be expressed as

I p = (D 4 - d 4 ) /32 (3b)

where

d = shaft inside diameter (mm, in)

D iameter of a Solid Shaft

D iameter of a solid shaft can calculated by the formula

D = 1.72 (T max / max )1/3 (4)

T orsional D eflection of Shaft

The angular deflection of a torsion shaft can be expressed as

= L T / I p G (5)

where

= 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

= 32 L T / (G D 4 ) (5a)

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

= 32 L T / (G (D 4- d 4 )) (5b)

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

S olid shaft ( replaced)

degrees 584 L T / (G D4 ) (6a)

Hollow shaft ( replaced)

degrees 584 L T / (G (D 4- d 4 ) (6b)

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T orsion Resisting Moments of Shafts of Various C ross Sections

S haft Cross S ectionArea

M aximum TorsionalResisting M oment

- T max - (Nm, in lb)

Nomenclature

S olid Cylinder S haft ( /16) max D3

Hollow Cylinder S haft ( /16) max (D4 - d 4 ) / D

Ellipse S haft ( /16) max b2 h

h = "height" of shaft b = "width" of shaft

h > b

Rectangle S haft (2/9) max b2 h h > b

S quare S haft (2/9) max b3

Triangle S haft (1/20) max b3 b = length of triangleside

Hexagon S haft (1/1.09) max b3