Elasticity/Torsion of circular cylinders

Torsion of Circular Cylinders

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Torsion of a cylinder with a circular cross section

About the problem:

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  • Circular Cylinder.
  • Centroidal axis thru the center of each cross section (c.s.)
  • Length  , Outer radius  .
  • Applied torque  .
  • Angle of twist  .

Assumptions:

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  • Each c.s. remains plane and undistorted.
  • Each c.s. rotates through the same angle.
  • No warping or change in shape.
  • Amount of displacement of each c.s. is proportional to distance from end.

Find:

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  • Shear strains in the cylinder ( ).
  • Shear stress in the cylinder ( ).
  • Relation between torque ( ) and angle of twist ( ).
  • Relation between torque ( ) and shear stress ( ).

Solution:

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If   is small, then

 

Therefore,

 

If the material is linearly elastic,

 

Therefore,

 

The torque on each c.s. is given by

 

where   is the polar moment of inertia of the c.s.

 

Therefore,

 

and