Convergence-Confinement method¶
The characteristic line method is a tool used to dimension the support of tunnels and to assess the pressure acting on it. It allows the graphical visualization of the convergence as a function of the support pressureand the distance of the installation of the support from the tunnel face. For the application of the method three characteristic curves are necessary:
- The ground reaction curve, expressing the convergence as a function of the internal pressure in the tunnel.
- The longitudinal displacement profile, expressing the convergence as a function of the distance from the tunnel face.
- The support reaction curve, expressing the convergence of the lining as a function of the pressure on the lining.
Each of these curves is discussed briefly below.
Ground reaction curve¶
The ground reaction curve expresses the convergence, meaning the displacement ofthe tunnel wall towards the symmetry axis of the tunnel, as a function of the internal pressure acting in the tunnel. It is customary to plot the internal pressure on the vertical axis and the convergence on the horizontal axis. If the internal pressure $p_i$ is equal to the far field stress $\sigma_0$ the convergence will be equal to zero, as no unloading takes place. In the following figure three examples of ground reaction curves are shown for different mechanical responces of the rock mass. On the ricght of the figure the corresponding mechanical response of the rock mass is shown. $R$ stands for the tunnel radius.
- Black curve: The rock mass is elastic. The convergence $u$ is a linear function of the internal pressure $p_i$.
- Blue curve: The rock mass is elastic-perfectly plastic. The convergence $u$ is a linear function of the internal pressure $p_i$ for values higher than the critical internal pressure $p_{cr}$. Beyond this point the rock mass starts to plastify forming a plastic annulus with increasing radius $R_{pl}$ around the tunnel and the convergence increases faster as the internal pressure decreases.
- Red curve: The rock mass shows softening, meaning that its bearing capacity decreases with increasing deformation. Beyond the elastic domain, the convergence becomes a nonlinear function of the internal pressure. The internal pressure exhibits a minimum value and then increases again, meaning that the rock mass is so weakened beyond this point, that increasing internal pressure is needed to stabilise it.
While the ground reaction curve can be measured, it is usually not known in the phase of the tunnel planning. The mechanical properties of the rock mass on the other hand may be known through site investigation. Several alternatives are available for the evaluation of the ground reaction curve as a function of the mechanical response of the rock mass. Most are analytical solutions, such as [1] for an elastic-perfectly plastic rock mass following the Mohr-Coulomb failure criterion and [2] for dilating rock mass following the Hoek-Brown failure criterion.
Longitudinal displacement profile¶
Beyond the internal pressure the convergence also depends on the distance from the tunnel face. Convergence starts taking place generally before the tunnel place and reaches its maximum value several radii behind the tunnel face. The figure below, after [3], illustrates how the convergence varies with distance from the tunnel face.
The curve describing the convergence as a function of the distance from the tunnel face is called longitudinal displacement profile. Its analytical derivation is much more complex than that of the ground reaction curve. Some analytical solutions exist, but numerical approaches are also common.
An example of a longitudinal displacement profile is shown in the figure below. The horizontal axis expresses the normalized distance from the tunnel face. Negative values indicate the area that has not yet been excavated. These are left of the vertical axis. Positive values indicate the distance from the tunnel face inside the tunnel. The vertical axis is the convergence in meters.
Support reaction curve¶
As the tunnel wall converges inwards load is tranferred from the rock mass to the support. Under the load the support begins to deform and converge inwards in its turn. The curve expressing the relationship between load on the support and convergence of support is called the support reaction curve. Examples are shown qualitatively in the figure below.
As can be seen, the support reaction curve does not generally start at zero convergence. This is so because the total convergence of the tunnel wall is used. As a result, the support reaction curve starts at the point where the support is installed.
- Black curve: Stiff, linearly elastic support. Segmental lining could be an example of this type of behavior.
- Blue curve: Hardening support installed at a later point or further from the tunnel face in comparison to the previous example. An example could be shotcrete.
- Red curve: Linearly elastic and perfectly plastic support.
Application of the method¶
The following figure shows the ground reaction curve in black, the longitudinal displacement profile in red and the support reaction curve in purple.
The procedure to evaluate the pressure on the support and the convergence is as follows:
- The distance from the tunnel face at which the support is to be installed is selected on the right vertical axis. The corresponding point is marked as A in the figure.
- The convergence that has taken place up to that point is determined by using the longitudinal displacement profile. From point A the point B is marked and the corresponding convergence is read at point C.
- At point C the support is installed in the form of the support reaction curve.
- The intersection of the support reaction curve and the ground reaction curve, point D, gives the pressure on the support and the final convergence.
Note¶
All three curves can be time dependent, for example when the rock mass tends to creep or when shotcrete is used for the lining, as it requires time to harden. Time dependence has not been considered here.
Example¶
For the sake of simplicity a circular tunnel under an isotropic far stress $\sigma_0$ is considered. It is assumed that the rock mass is elastic - perfectly plastic and obeys the Mohr-Coulomb failure criterion:
$\sigma_1 = \lambda \sigma_3 + \sigma_u$
where
$\lambda=\frac{1+\sin(\phi)}{1-\sin(\phi)} $ and $\sigma_u = \frac{2c\cdot \cos(\phi)}{1-\sin(\phi)}$
$c$ being the cohesion and $\phi$ the friction angle. For the ground reaction curve the analytical solution suggested by Salencon [4] is used, while for the longitudinal displacement profile the approximation by Vlachopoulos and Diedrichs [5] is used. It should be noted that this approximation is based on numerical simulations using the Hoek-Brown failure criterion. It was nonetheless selected here because it takes into account the plastic radius and is in general widely used.
The parameters that can be controlled by the user in this widget are:
$G/p$: Shear modulus of the rock mass $G$ normalised with the in situ stress $p$
$\sigma_u/p$: Uniaxial strength of the rock mass $\sigma_u$ normalised with the in situ stress
$\phi$: Friction angle of the rock mass
$N/p$: Stiffness of the support $N$ normalised with the in situ stress $p$
$u_i/R$: Convergence at support installation $u_i$ normalised with the tunnel radius
