**4. Analysis of tunnel behaviour during construction**

**Figure 4** shows the deformation vectors and the plastic zone in the tunnel propagation direction in weak rocks. **Figure 5** gives a summary of the deformations that occur in the tunnel. The elastic deformations that occur in the tunnel start at a distance of two tunnel diameters in front of the tunnel and reach the maximum level after two tunnel diameters behind the tunnel face [16]. The maximum displacement occurring in the tunnel face is one-third of the total displacements (**Figure 5**).

The unsupported deformation situation for the rock-soil interaction is given in **Figure 6**. If the uniaxial compressive strength of the rock mass is *σcm* > 2*po* (*pi* = 0), the displacements are elastic and continue linearly. If a failure occurs, the displacements are plastic and are curved in **Figure 6**.

Ground reaction curve or characteristic line depends on the convergence occurring in the tunnel with the internal support pressure.

Here, the tunnel radius is taken as *r*0, the hydrostatic pressure *p*<sup>0</sup> and the support pressure as *pi* (**Figure 7**).

#### **Figure 4.**

*Vertical section through a three-dimensional finite element model of the failure and deformation of the rock mass surrounding the face of an advancing circular tunnel [16].*

**Figure 5.** *Radial displacements around the tunnel [17].*

**Figure 6.**

*Graphical representation of relationships between support pressure and radial displacement of tunnel walls [16].*

If the support pressure *pi* is less than the critical pressure *pcr*, the rock mass will fail. If the pi pressure is greater than *pcr*, no failure occurs around the tunnel and the rock mass behaves elastically. The critical support pressure is given in Eq. (5):

$$Pcr = \frac{2p\,\mathrm{0} - \sigma cm}{\mathrm{1} + k} \tag{5}$$

Elastic displacement is given in Eq. (6):

$$
uie = \frac{r\mathbf{0}(\mathbf{1} + \theta)(p\mathbf{0} - pi)}{Em} \tag{6}$$

Here *Em* is the rock mass deformation modulus and ν is Poisson's ratio.

If the support pressure (*pi*) is less than the critical pressure *pcr*, failure occurs around the tunnel and a plastic zone is formed. In this case, plastic deformation and plastic zone radius are defined by Eq. (7). The resulting plastic deformation is given in Eq. (8):

$$rp = r o\left[\left(\frac{2(p0(k-1) + \sigma cm)}{(1+k)((k-1)pi + \sigma cm)}\right)^{\frac{1}{k-1}}\right] \tag{7}$$

$$uip = \left(\frac{ro(1+\theta)}{Em}\right) \left[2(1-\theta)(p0-pc)\left(\frac{rp}{r0}\right)^2 - (1-2\theta)(p0-pi)\right] \tag{8}$$

The graph of radial displacement with Pi to the support pressure drawn with the help of the given equations is given in **Figure 8**. Here,

if *pi* = *po*, no deformation occurs,

*pi* > *pc* elastic deformation occurs,

*pi* < *pcr*ise plastic deformation occurs.

After the support installation, the deformations continue elastically. Maximum elastic deformation is defined as *usm* and maximum support pressure is defined as *psm*.

Support interaction analysis depends on three main parameters. These are


*Response of support system to tunnel wall displacement resulting in the establishment of equilibrium [16].*


#### **Table 6.**

*Support capacity equations [18, 19].*

In the support reaction curve, elastic deformations occur in the tunnel after the tunnel excavation. The support reaction curve has reached equilibrium if the ground reaction curvature crosses the curve before deformations in the rock mass increase substantially. However, if the deformations that occur develop very quickly and the reinforcements are insufficient, failure occurs and balance cannot be achieved.

Equations for the stiffness and capacity of supports have been published by Hoek and Brown [18] and Brady and Brown [19]. The equations are given in **Table 6**.
