First yield at a braced circular tunnel ($P_{\mathrm{crit}}=p_{\infty }(1-\sin \varphi )$)

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 $p_{\infty }\sin \varphi$. Lowering $P$ from hydrostatic confinement yields the displayed threshold and stress ratio ${\tau }_{\theta \theta }/{\tau }_{rr}=(1+\sin \varphi )/(1-\sin \varphi )$. Increasing bracing instead reaches $P=p_{\infty }(1+\sin \varphi )$ with the reciprocal ratio. The two cases must not be confused when identifying the first loading threshold.