Elasticity/Torsion of rectangular sections

Sample homework problem

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Torsion of a bar with a rectangular cross section

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Given:

A cylinder with a rectangular cross sections   under torison.

 
A bar with a rectangular cross section

Show:

Why a function of the form

 

cannot be used as a Prandtl stress function for this cross section.

Solution

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For compatibility, we need

 

However, for the given  ,

 

which is not constant. Hence, the given function cannot be used as a Prandtl stress function.