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.

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

For compatibility, we need

$\nabla ^{2}{\varphi }=C$

However, for the given $\varphi$,

$\nabla ^{2}{\varphi }=-{\frac {2m}{a^{2}b^{2}}}\left[a^{2}+b^{2}-(x_{1}^{2}+x_{2}^{2})\right]$

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

Elasticity