Determine the shear stress at point A on the surface of the

Determine the shear stress at point A on the surface of the shaft. Represent the state of stress on a volume element at this point. The shaft radius has a radius of 40 mm.

Solution

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 / J     (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)

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

Circular Shaft and Maximum Moment or Torque

Maximum moment in a circular shaft can be expressed as:

T_max = _max J / 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   

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

T_max = ( / 16) _max (D⁴ - d⁴) / D

Maximum shear stress can be calculated as

_max = T r / J