Kirsch Equations Tunnel Stress Calculator
Underground geomechanics & tunneling: Calculate exact Kirsch (1898) elastic stresses ($\sigma_r, \sigma_\theta, \tau_{r\theta}$) around a circular tunnel under anisotropic in-situ stress ($K_0$).
Tunnel Geometry & In-Situ Stresses
Induced Elastic Stresses
Tangential Stress $\sigma_\theta$
--
MPa (hoop stress)
Radial Stress $\sigma_r$
--
MPa (confinement)
Shear Stress $\tau_{r\theta}$
--
MPa (cross-plane)
Crown Hoop Stress ($r = a$)
--
$\theta = 90^\circ$ boundary
Sidewall Hoop Stress ($r = a$)
--
$\theta = 0^\circ$ boundary
Concentration Factor $K_t$
--
$\sigma_\theta / \sigma_v$ at boundary
Polar Tangential Hoop Stress Distribution
Polar envelope represents boundary tangential stress $\sigma_\theta(\theta)$. For hydrostatic stress ($K_0 = 1$), hoop stress is uniform $2\sigma_v$. For $K_0 < 1/3$, crown stress becomes tensile!
The Kirsch Equations Formulation (1898)
The Kirsch analytical solution evaluates stresses in polar coordinates $(r, \theta)$ around a circular opening of radius $a$ in an isotropic, linearly elastic rock medium:
$$\sigma_r = \frac{\sigma_v}{2} \left[ (1+K_0)\left(1 - \frac{a^2}{r^2}\right) - (1-K_0)\left(1 - \frac{4a^2}{r^2} + \frac{3a^4}{r^4}\right)\cos 2\theta \right] + p_i \frac{a^2}{r^2}$$
$$\sigma_\theta = \frac{\sigma_v}{2} \left[ (1+K_0)\left(1 + \frac{a^2}{r^2}\right) + (1-K_0)\left(1 + \frac{3a^4}{r^4}\right)\cos 2\theta \right] - p_i \frac{a^2}{r^2}$$
$$\tau_{r\theta} = \frac{\sigma_v}{2} (1 - K_0) \left( 1 + \frac{2a^2}{r^2} - \frac{3a^4}{r^4} \right) \sin 2\theta$$
At the boundary ($r = a$), radial and shear stresses vanish ($\sigma_r = p_i$, $\tau_{r\theta} = 0$), concentrating all load into hoop stress: $\sigma_\theta = \sigma_v [ (1+K_0) + 2(1-K_0)\cos 2\theta ] - p_i$.
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