Elasticity/Torsion of rectangular sections

Sample homework problem

Torsion of a bar with a rectangular cross section

Given:

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

File:Torsion rectangle.png A bar with a rectangular cross section

Show:

Why a function of the form

$\phi =m(\frac{x_1^2}{a^2}-1)(\frac{x_2^2}{b^2}-1)$

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

Solution

For compatibility, we need

${\mathrm{\nabla }}^2\phi =C$

However, for the given $\phi$,

${\mathrm{\nabla }}^2\phi =-\frac{2m}{a^2{b}^2}[a^2+b^2-(x_1^2+x_2^2)]$

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