Introduction to 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 $\varphi$,

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

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