= Elliptical-bar torsion and first yield
{title2=$R=\frac{\pi\mu a^3b^3}{a^2+b^2},\quad |\Omega|_Y=\frac{(a^2+b^2)\tau_Y}{2\mu a^2b}\quad(a>b)$}
On $x^2/a^2+y^2/b^2<1$, the <torsion stress function for a simply connected bar> is $\phi=\mu\Omega a^2b^2(1-x^2/a^2-y^2/b^2)/(a^2+b^2)$. Its Laplacian is $-2\mu\Omega$ and its boundary value is zero, proving the formula by uniqueness of the <Poisson equation> with Dirichlet data. Integration gives $M=\pi\mu\Omega a^3b^3/(a^2+b^2)$, hence the displayed stiffness $R=M/\Omega$.
The three <principal stresses> are $0,\pm|\nabla\phi|$. With $\tau_Y$ defined as the shear yield stress, the <Tresca yield criterion> is $|\nabla\phi|\leq\tau_Y$. For $a>b$ its maximum is $2|C|/b$ at $(0,\pm b)$, where $C=\mu\Omega a^2b^2/(a^2+b^2)$. Equating that maximum to $\tau_Y$ gives the displayed first-yield twist. The ends of the minor axis yield first; geometric distance from the bar axis alone does not determine stress in a noncircular section.
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