Stress state around circular openings¶
The calculation of the stress state around circular openings is relevant for tunnels and boreholes among others. Under certain assumptions analytical solutions are available for this problem. The most widely known is the solution introduced in 1898 by E.G. Kirsch [1], which is known as the Kirsch solution. The Kirsch solution is a special case of the Michell solution [2], which was introduced slightly later. The assumptions underlying the Kirsch solution are:
- The rock mass is elastic, uniform and isotropic.
- The initial stress state is uniform.
- The problem can be reduced to two dimensions.
- The stress state is coaxial to the tunnel or borehole.
In the specific case of tunnels and boreholes the third condition is almost automatically satisfied. As shown in the figure below, their diameter is much smaller than their length, so that they may be considered infinitely long. As a result assuming that no deformation can take place parallel to the tunnel or borehole axis is reasonable. The problem reduces to one of plane strain.
The assumption that the stress state is uniform means that gravity needs to be neglected. This in addition to the requirement of elasticity indicates that this solution is more suitable to deep tunnels or boreholes in competent rock mass.
Assuming that the stress state before excavation is the one shown in the figure above, the Kirsch solution yields the stress state after excavation in polar coordinates as
$ \sigma_{rr} = \dfrac{\sigma_{V}}{2}\left[ \left( 1+\lambda \right)\left( 1-a^2 \right)-\left( 1-\lambda \right)\left( 1-4a^2+3a^4 \right)\cos(2\theta) \right] $
$ \sigma_{\theta\theta} = \dfrac{\sigma_{V}}{2}\left[ \left( 1+\lambda \right)\left( 1+a^2 \right)+\left( 1-\lambda \right)\left( 1+3a^4 \right)\cos(2\theta) \right] $
$ \sigma_{r\theta} = \dfrac{\sigma_{V}}{2}\left[ \left( 1-\lambda \right)\left( 1+2a^2-3a^4 \right)\sin(2\theta) \right]$
where $a=\dfrac{R}{r}$, with $r$ being the radial coordinate and $R$ the tunnel radius.
$\lambda$ is called the lateral stress coefficient and is defined as
$\lambda = \dfrac{\sigma_H}{\sigma_V}$
Notes¶
- The case described above refers to a tunnel or a horizontal borehole. For a vertical borehole $\sigma_V$ can be substituted with the maximum horizontal stress and $\sigma_H$ with the minimum horizontal stress.
- The lateral stress coefficient $\lambda$ can also be larger than 1.
- When $\lambda$ = 1 the stress state is isotropic. The circumferential stress $\sigma_{\theta\theta}$ is the same in all directions and equal to $2\sigma_V$ at the tunnel wall.
- As $\lambda$ becomes smaller or larger than 1, at the tunnel wall the circumferential stress increases normal to the maximum in situ stress and decreases parallel to it.
- If $\lambda$ < 1/3 or $\lambda$ > 3, tensile circumferential stresses appear parallel to the maximum in situ stress.
