Introduction to Elasticity/Torsion of rectangular sections

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Sample homework problem [edit]

Torsion of a bar with a rectangular cross section [edit]

Given:

A cylinder with a rectangular cross sections a\times b under torison.

A bar with a rectangular cross section

Show:

Why a function of the form


   \varphi = m\left(\frac{x_1^2}{a^2}-1\right)\left(\frac{x_2^2}{b^2}-1\right)

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

Solution [edit]

For compatibility, we need


   \nabla^2{\varphi} = C

However, for the given \varphi,


   \nabla^2{\varphi} = -\frac{2m}{a^2b^2}\left[a^2+b^2-(x_1^2+x_2^2)\right]

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