= First yield at a braced circular tunnel
{title2=$P_{\rm crit}=p_\infty(1-\sin\phi)$}
For a circular tunnel of radius $a$ in a plane-strain elastic medium, isotropic far compression $p_\infty$ and wall bracing pressure $P$ give $\tau_{rr}=-p_\infty-(P-p_\infty)a^2/r^2$ and $\tau_{\theta\theta}=-p_\infty+(P-p_\infty)a^2/r^2$. The <Mohr circle> center is $-p_\infty$ and its radius is $|P-p_\infty|a^2/r^2$. Its maximum is at the wall, proving that first yield occurs there. Cohesionless frictional tangency requires this radius to equal $p_\infty\sin\phi$. Lowering $P$ from hydrostatic confinement yields the displayed threshold and stress ratio $\tau_{\theta\theta}/\tau_{rr}=(1+\sin\phi)/(1-\sin\phi)$. Increasing bracing instead reaches $P=p_\infty(1+\sin\phi)$ with the reciprocal ratio. The two cases must not be confused when identifying the first loading threshold.
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