Analysis and Design of Tunnels

### Chapter 3

## Topics of Analytical and Computational Methods in Tunnel Engineering

Michael G. Sakellariou

### Abstract

In this chapter, a selection of tunneling topics is presented, following the evolution of methods and tools from analytical to computational era. After an introductory discussion of the importance of elasticity and plasticity in tunneling, some practical topics are presented as paradigms to show the successful application of them in achieving a solution. The circular and horseshoe tunnel sections served as the basis of the elastic analysis of deep tunnels. Practical aspects such as influence zone and elastic convergences in both cases are examined. In the case of circular tunnels, the estimation of plastic zone formation is discussed for a selection of strength criteria. After a detailed discussion of the influence of surface proximity, the elastic and plastic analysis of shallow tunnels is examined in some detail. The presentation is completed by a short presentation of computational methods. An overview of recent developments and a classification of the methods are presented, and then some problems for the case of anisotropic rocks have been presented using finite element method (FEM). The last topic is the application of artificial intelligence (AI) tools in interpreting data and in estimating the relative importance of parameters involved in the problem of tunneling-induced surface settlements. In the conclusions a short discussion of the main topics presented follows.

Keywords: elasticity, plasticity, deep tunnels, shallow tunnels, influence zone, plastic zone, circular tunnels, horseshoe tunnels, computational methods, ANN, surface settlements

### 1. Introduction

Underground work history goes back to prehistoric times [1]. The oldest known tunnel is the one underpassing the River Euphrates in Babylon constructed 4000 years ago. Hezekiah, King of Judea, built a tunnel 2700 years ago, whereas Eupalinos built the Eupalinion tunnel constructed in Samos Island, Greece, 2600 years ago, both for water supply purposes. For further information see [2–5]. Shelters, underground works for warfare purposes, mineral resource exploitation, traffic tunnels and conveyance tunnels like water supply tunnels, etc. are the main categories of underground structures [3].

Tunnel engineering could be characterized as an art, not fine art of course, with the meaning of know-how based on past experience, lessons learned from tunnel disasters [6], knowledge of geological conditions and behaviour of the ground,

innovations to overcome encountered difficulties, scientific knowledge from applied mechanics and advances in mechanized excavation tools and methods. In recent years, the experience from South Africa and North Europe rock engineering practice resulted in the introduction of comprehensive classification systems, namely, RMR [7, 8] and Q system [9]. In theory, Lamé in 1852 and Kirsch in 1898 [10] paved the road of scientific development of solutions to the problem of stresses and strains around underground holes [10]. Terzaghi [11] examined shallow tunneling through sands and cohesive soils. For a comprehensive presentation of the fundamentals of theoretical rock mechanics, see Jaeger et al. [12]. For a detailed coverage of underground excavations in rock, see Hoek and Brown [13]. Experimental methods such as photoelasticity [14, 15] and Moiré [16] and computational methods such as finite elements [17], finite difference, direct and indirect boundary elements [13, 18], distinct elements [19] and meshfree methods [20, 21] have been developed and extensively applied successfully. On the other hand, the progress regarding the material behaviour resulted in advanced constitutive laws [22] and in the more realistic modeling of continua and discontinua media [23]. Finally, the information systems and in particular the artificial intelligence offered new tools like artificial neural networks (ANN) and other machine learning methods.

### 2. Analytical methods: elastic problems

### 2.1 The case of deep tunnels

Deep tunnels can be considered as the limiting case of a cylinder with thick walls when the radius of the exterior wall tends to infinity. In that case the ground surface has no any influence on the stress field around the opening.

Jeffery [24] solved the problem of stress distribution inside the walls of a cylinder bounded by two non-concentric circles, with the distance of their centres denoted by d. In this way, Jeffery's approach solves the problems of an infinite plate containing a circular hole as the limit of a thick cylinder with concentric boundaries (d = 0) and the problem of a semi-infinite plate containing a circular hole as a particular case of two eccentric boundaries. So, he gave the general solution of deep and shallow tunnels, respectively, as particular cases of a general problem.

For a historical account of the mathematical solutions of stress distribution around underground openings, see Gerçek [25].

### 2.1.1 Cyclic section

Assuming a homogeneous isotropic linear elastic medium (HILE medium) and considering the geometry of the problem and the equations of equilibrium, we obtain the following set of equations (Eqs. (1)–(4)) (Figure 1):

$$
\sigma\_r = -\frac{2G}{1-2\nu} \left\{ \frac{A}{2} - (1-2\nu)\frac{B}{r^2} \right\}, \\
\Rightarrow \sigma\_r = C - \frac{D}{r^2}, \tag{1}
$$

$$
\sigma\_{\theta} = -\frac{2G}{1 - 2\nu} \left\{ \frac{A}{2} + (1 - 2\nu)\frac{B}{r^2} \right\},
\Rightarrow \sigma\_{\theta} = C + \frac{D}{r^2},\tag{2}
$$

$$
\pi\_{\theta} = \mathbf{0} \tag{3}
$$

$$u\_r = -\frac{1}{2G} \left\{ (1 - 2\nu)Gr + \frac{D}{r} \right\} \tag{4}$$

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

### Figure 1. The cylinder with thick walls.

The constants C and D are defined as:

$$\begin{aligned} C &= -\frac{GA}{1 - 2\nu}, \\ D &= -2GB \end{aligned}$$

Considering the boundary conditions: for r ¼ a, σ<sup>r</sup> ¼ pa. r ¼ b, σ<sup>r</sup> ¼ pb.

therefore:

$$\begin{aligned} p\_a &= C - \frac{D}{a^2}, \\ p\_b &= C - \frac{D}{b^2} \\ D &= \frac{(p\_b - p\_a)a^2b^2}{(b^2 - a^2)}, \\ C &= \frac{(b^2p\_b - a^2p\_a)}{(b^2 - a^2)} \end{aligned}$$

When b tends to infinity, we obtain the special case of a cyclic opening excavated in an infinite medium under hydrostatic stress field p, by superposing the initial stresses induced by the excavation:

$$\begin{aligned} \sigma\_r &= p \left( 1 - \frac{a^2}{r^2} \right), \\ \sigma\_\theta &= p \left( 1 + \frac{a^2}{r^2} \right), \\ \tau\_{r\theta} &= 0, \\ u\_r &= -\frac{1}{2G} \frac{a^2}{r} p \end{aligned} \tag{5}$$

For the general case of stress field, denoting the ratio of horizontal to vertical in situ principal stress with k, Kirsch [10] obtained the following equations:

$$\begin{aligned} \sigma\_r &= \frac{p}{2} \left[ 1 + k \left( 1 - \frac{u^2}{r^2} \right) - \left( 1 - k \left( 1 - 4 \cdot \frac{u^2}{r^2} + 3 \cdot \frac{u^4}{r^4} \right) \cos 2\theta \right), \\ \sigma\_\psi &= \frac{p}{2} \left[ 1 + k \left( 1 + \frac{a^2}{r^2} \right) + \left( 1 - k \right) \left( 1 + 3 \cdot \frac{a^4}{r^4} \right) \cos 2\theta \right], \\ \sigma\_{r^{\text{eff}}} &= \frac{p}{2} \left[ 1 - k \left( 1 + 2 \cdot \frac{a^2}{r^2} - 3 \cdot \frac{a^4}{r^4} \right) \sin 2\theta \right], \\ \sigma\_r &= -\frac{p\alpha^2}{4Gr} \left( 1 + k \right) - \left( 1 - k \right) \left[ 4(1 - \nu) + \frac{a^2}{r^4} \right] \cos 2\theta \end{aligned} , \tag{6}$$

$$\begin{aligned} u\_\psi &= -\frac{p\alpha^2}{4Gr} \left[ (1 - k) \left\{ 2(1 - 2\nu) + \frac{a^2}{r^4} \right\} \sin 2\theta \right], \end{aligned}$$

It is interesting to note that in the case of hydrostatic or isotropic stress field (k = 1), we can estimate an influence zone accepting a � 0.04% deviation from the initial stress field. Then by introducing this condition in the above equations, we obtain the radius of influence zone as 5α, where α is the radius of the excavation.

### 2.1.2 The case of horseshoe section

Let us examine now the case of horseshoe section (Figures 2 and 3). This problem is very useful in the case of excavations without using a tunnel boring machine (TBM). Figure 2 shows a typical metro section, whereas Figure 3 shows the simplified section used in the analysis.

First, the maximum principal stress σ<sup>1</sup> contours obtained by a numerical analysis are shown Figure 4. A slight influence of the asymmetric shape of the tunnel's section can be observed. Schürch and Anagnostou [26] examined the influence of rotational symmetry violation on the applicability of the ground response curve.

To obtain a solution to this problem, we adopted Muskhelishvili approach, using functions of complex variables and conformal mapping techniques [27].

Figure 2. Typical horseshoe tunnel section.

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

Figure 3. A simplified horseshoe tunnel section.

Adopting Gerçek solution [28], we introduce the following mapping function, which transforms the infinite region surrounding the opening onto the interior of the unit circle (Figure 5):

$$z = a(\zeta) = R\left(\frac{\mathbf{1}}{\zeta} + \sum\_{k=1}^{3} a\_k \cdot \zeta^k\right) \tag{7}$$

In Eq. (7) R is a real constant, and the complex coefficients α<sup>k</sup> are defined as:

$$a\_k \quad a\_k + b\_k \text{, for } k \quad 1, 2, 3 \tag{8}$$

Figure 5. Conformal mapping of an infinite region surrounding a hole onto the unit circle [28].

Figure 6. The final section that best fits the initial horseshoe section [29, 30].

In order to find the optimum values of the above coefficients to fit best the given horseshoe section, a (123x3) system of equations has been solved [29, 30]. The values of the coefficients are:

$$\begin{aligned} \mathbf{a}\_1 &= \mathbf{b}\_3 = \mathbf{0} \\ \mathbf{a}\_2 &= \mathbf{b}\_2 = \mathbf{0}.\mathbf{047}, \\ \mathbf{a}\_3 &= \mathbf{0}.\mathbf{029}, \\ \mathbf{b}\_1 &= \mathbf{0}.\mathbf{104}, \\ \mathbf{R} &= \mathbf{3.756} \end{aligned} \tag{9}$$

In [29, 30] an extended investigation of this problem is presented resulting in the calculation of the set of coefficients for 49 sections. In Figure 6, the final section, which is the best for our problem, is presented superimposed to the original horseshoe shape.

Now, we can calculate the stresses around the opening as well as the strains for initial stress fields with k = 0, 0.333, 1 and 3 [29, 30]. In Figures 7 and 8, the variation of σθ at the boundary of the excavation is shown for k = o.333 (Figure 7) and k = 1 (Figure 8).

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

Figure 7. Hoop stress σθ variation along the boundary of the excavation for k = 0.333 [29].

Figure 8. Hoop stress σθ variation along the boundary of the excavation for k = 1 [30].


### Table 1.

Convergences of circular and horseshoe sections for different stress fields [29, 30].

A further result of practical significance obtained by this method is the calculation of the convergence of the excavation boundaries and the extent of the zone of influence. In Tables 1 and 2, these values are presented in comparison with the corresponding values for the circular section. We can observe the greater deviation at the bottom of the horseshoe section because of the greater radius of this segment compared to the circle.

In all cases presented in this paragraph, we assumed the initial stress field p = 1 MPa, modulus of elasticity E = 1GPa and Poisson ratio ν = 0.3. Using these values we are able to calculate the corresponding values for any p value and every kind of rock. The only constraint is that the value of Poisson ratio is equal to 0.3. In the following paragraph, we shall discuss the influence of ν upon the stresses.


### Table 2.

Influence zone extends around circular and horseshoe sections for different stress fields.

Figure 9. Stress distribution around a circular deep tunnel (Z = 25R, k = 1) [32].

### 2.2 The case of shallow tunnels: the circular section

Assuming that the tunnel is close to the ground surface, we can observe an influence of the boundary as the vertical stresses depend on the depth. So, the initial stresses at the roof and the bottom levels of the tunnel section are not equal. Bray [31] presented an extensive study on this problem. Some practical rules are summarized below.

By assuming an acceptable deviation, as in the case of the influence zone estimation, we may distinct three cases.

### 2.2.1 Case I

The tunnel depth, measured from the section's centre, is Z > 25R, where R is the tunnel radius. In this case Kirsch equations apply. In Figure 9, the distribution of stresses around a circular deep tunnel is shown [32]. There is no influence of the boundaries. Note that only a window of the stress field is shown.

### 2.2.2 Case II

The tunnel depth is 7R < Z < 25R. In this case we must add a correction term to the Kirsch equations depending on the Poisson ratio. That term is given by Bray as [31]:

$$\sigma\_{\theta} = \gamma h[1 + k + 2(1 - k)\cos 2\theta] - \gamma a \sin \theta \left[ \frac{3 - 4\nu}{2(1 - \nu)} + 2(1 - k)\cos 2\theta \right] \tag{10}$$

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

Figure 10.

Stress distribution around a circular shallow tunnel (Z = 7R, k = 1) [32].

where α is the tunnel's radius. This expression agrees with an equation given by Savin [33].

In Figure 10, the distribution of stresses around a circular shallow tunnel is shown. The influence of the surface boundary starts to be noticeable [32].

### 2.2.3 Case III

The tunnel depth is Z < 7R. This case is more difficult because the influence of the boundary becomes greater. Then Mindlin's closed-form solutions apply [34, 35]. Mindlin obtained solutions to this problem for three cases of in situ stress fields at depth z (remote from the tunnel):

### 2.2.3.1 Case I

pz = wz. ph = wz.

i.e. isotropic, or hydrostatic, gravitational pressure and w being the unit weight of mass.

2.2.3.2 Case II

$$\begin{aligned} \mathbf{p}\_{\mathbf{z}} &= \mathbf{w}\mathbf{z}. \\ \mathbf{p}\_{\mathbf{h}} &= \{\nu/(1 \text{-} \nu)\}\text{wz.} \end{aligned}$$

This is the case where the lateral deformation is a constraint remote from the tunnel.

2.2.3.3 Case III

pz = wz. ph = 0.

This is the case of nonlateral constraint of the mass remote from the tunnel.

The solutions of Mindlin were in terms of bipolar coordinates α and β. The expression giving the stress on the circular boundary for the first case is [35]:

$$\begin{aligned} \left[\sigma\_{\beta}\right]\_{a=a\_1} &= \frac{2w\mathcal{A}(\cosh a\_1 - \cos \beta)}{\sinh a\_1} \left\{ \frac{1 - \cosh a\_1 \cos \beta}{\left(\cosh a\_1 - \cos \beta\right)^2} \right. \\\\ &- \coth a\_1 - \frac{\left(\Im - 8\nu\right)}{4(1 - \nu)} \frac{\cos \beta}{\sinh a\_1} + 2e^{-a\_1} \cos \beta \sum\_{n=2}^{\infty} R\_n \cos n\beta \right\} \end{aligned} \tag{11}$$

where α<sup>1</sup> is the value of α corresponding to the boundary of the tunnel:

$$\begin{aligned} \frac{Z}{R} &= \cosh a\_1\\ A &= Z \tanh a\_1 = R \sinh a\_1 \\ R\_n &= N\_n - n e^{-na\_1} \\ N\_n &= \frac{n e^{-na\_1} (\sinh n a\_1 \cosh n a\_1 - n \sinh a\_1 \cosh n a\_1)}{\sinh^2 n a\_1 - n^2 \sinh^2 a\_1} \end{aligned} \tag{12}$$

Poulos and Davis [35] gave in figures and tables values of σβ for 1 < (Z/R) < 4.

For Cases II and III above, the solution for σβ is obtained by adding in the solution given by Eq. (11) a further expression [33]. From the above analysis, the importance of taking into account the influence of the ground surface on the stress distribution on tunnel boundary is obvious. A further important notice is the influence of Poisson ratio on the expressions of stresses. From Eq. (10), it is clear that this influence is important when Z < 25R.

Malvern [36] in a rigorous analysis concluded that when the resultant force on each boundary is zero, then the stress distribution is independent of the elastic constants. Otherwise the stress distribution will depend on ν. When the boundary conditions include displacement constraints, then the stress distribution will, also, depend on ν. These conclusions are important to be known for the stress analysis of tunnels by using computational methods like finite elements or boundary elements. Frocht [14] examined the influence of stresses from ν by extensive photoelastic experimental study. In this short presentation of the elastic solutions for the shallow circular tunnel, we restricted to the stress field influence. For an extensive presentation of elastic solutions for the important problem of surface settlements induced by tunneling, which is of great practical significance, see [37–39]. Another important topic in the case of shallow tunnels is the problem of stress field due to seismic loading. Recently, Pelli and Sofianos [40] published a paper addressing the very important topic of the stress field around shallow tunnels under seismic loading of SW waves.

### 3. Analytical methods: plastic problems

### 3.1 The case of deep tunnels

We now turn our interest in plastic problems. This is a case of great theoretical and practical significance, as every underground excavation must be analysed against failure. For a detailed coverage, see, almost, every standard textbook of soil and rock mechanics. For a more detailed coverage, see [41–44]. Here, we examine

### Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

special topics related to tunnel engineering. In order to analyse the competence of an excavation in soil or rock, we must extend our analysis beyond elasticity to the plastic behaviour of the surrounding medium. To this end, we have to introduce to our analysis a failure criterion. The most common criterion is the Mohr-Coulomb [11]. Extensive research resulted in a remarkable advancement in understanding the material behaviour and the implementation of advanced models in computational methods. A milestone was the introduction of critical state theory in the 1960s [45]. Further research in rock mechanics resulted in the introduction of the Hoek-Brown criterion, which is widely used in rock engineering practice [13]. Yu [46], in an extensive review, presented a wide range of failure criteria applicable on, almost, every kind of materials under every possible type of loading. In this review Yu discussed as well the soil and rock materials.

Let us examine the problem of practical significance of the formation of plastic zone around a deep tunnel of circular section. Closed-form solutions can be found for every criterion of soil and rock literature in the case of isotropic or hydrostatic stress field (k = 1), because of the axisymmetric type of this case. Recently, Vrakas and Anagnostou presented an extension of the small strain analyzes to obtain finite strain solutions [47]. For the general case of stress fields (k 6¼ 1), closed-form solutions have been obtained so far for the Tresca criterion [48] and for the Mohr-Coulomb criterion [49]. Here, the elliptic paraboloid criterion developed by Theocaris [50, 51] is introduced to solve the problem of plastic zone around a circular deep tunnel in rock. First, the mathematical expression of the criterion is given, and a comparison of it with Griffith and Hoek-Brown criteria is presented.

This criterion is expressed through the three principal stresses. It is an energy criterion having as parameter the absolute value of the ratio R of uniaxial compressive strength over the uniaxial tensile strength. The elliptic paraboloid criterion is developed for applications in applied mechanics in general, so the tensile stresses are positive, and the compressive stresses are negative. Then:

$$\left(\sigma\_1 - \sigma\_2\right)^2 + \left(\sigma\_2 - \sigma\_3\right)^2 + \left(\sigma\_3 - \sigma\_1\right)^2 + 2(R - 1)\left(\sigma\_1 + \sigma\_2 + \sigma\_3\right)\sigma\_t = 2\operatorname{Re}\sigma\_t^2 \quad \text{(13)}$$
  $\text{where } \mathbf{R} = \frac{|\sigma\_t|}{\sigma\_{t,\cdot}}$ 

Eq. (13) is the addition of two components, which express two parts of the total elastic energy. Indeed, the first part is, up to a constant multiplicative factor, the distortion energy expressed through the deviatoric stresses. This part represents the energy of the elastic change of shape [52] given by Eq. (14):

$$\left(\sigma\_1-\sigma\_2\right)^2+\left(\sigma\_2-\sigma\_3\right)^2+\left(\sigma\_3-\sigma\_1\right)^2\tag{14}$$

The second part depends on the hydrostatic or spherical part of the stresses and represents the energy of elastic change in volume. In this way, the elliptic paraboloid criterion combines both the change of shape and volume. The latter is important for soil and rock materials as their strength depends on the confining pressure.

For the isotropic case of field stress, conditions of axisymmetry are valid, i.e. σ<sup>1</sup> = σ2. Then from Eq. (13), we obtain:

$$\left(\left(\sigma\_3-\sigma\_1\right)^2+\frac{R-1}{R}(2\sigma\_1+\sigma\_3)\sigma\_\varepsilon-\frac{\sigma\_\varepsilon^2}{R}=0\tag{15}$$

In the case of axisymmetric conditions, the criterion becomes paraboloid of revolution. In Figure 11, a comparison of Griffith criterion with Hoek-Brown criterion being shown. It is obvious that the deviation between their representation

### Figure 11.

Comparison of Griffith criterion with Hoek-Brown criterion for different m values [52].

for m value being equal to 7.88, close to the restriction of Griffith's corresponding value being 8 [52].

Now, we can proceed to a comparison of elliptic paraboloid criterion with Griffith. We assume an R value of 8 (Figure 12).

From the above figures, we may conclude that Griffith and elliptic paraboloid criteria agree, although their assumptions are completely different. Griffith assumed that failure initiates from pre-existed cracks because of the stress concentration at the tips of the cracks. This assumption is fundamental in fracture mechanics. On the other hand, the assumption of elliptic paraboloid criterion is an extension of von Mises criterion. Both criteria are identical in the case of equal strengths under compression and tension (R = 1). This assumption is close to the experimental results for metals. Theocaris applied this criterion in igneous and metamorphic rocks [45].

We, now, come to examine the problem of plastic zone formation around a deep tunnel of circular section under isotropic stress field (k = 1) [53] (Figure 13).

To solve the problem, we introduce the equation of equilibrium in polar coordinates (Eq. (16)) and the flow rule (Eq. (17)). Then, by introducing the expression of elliptic paraboloid criterion in Eq. (17), we obtain Eq. (18) [53]:

$$\frac{d\sigma\_r}{dr} + \frac{\sigma\_r - \sigma\_\theta}{r} = \mathbf{0} \tag{16}$$

$$\frac{\partial \mathbf{f}}{\partial \sigma\_2} = \mathbf{0} \tag{17}$$

$$\frac{\partial \left[ \left( \sigma\_1 - \sigma\_2 \right)^2 + \left( \sigma\_2 - \sigma\_3 \right)^2 + \left( \sigma\_3 - \sigma\_1 \right)^2 + 2 \cdot \left( R\_b - 1 \right) \cdot \left( \sigma\_1 + \sigma\_2 + \sigma\_3 \right) \cdot \sigma\_{tb} - 2 \cdot R\_b \cdot \sigma\_{tb}^2 \right]}{\partial \sigma\_2} = 0 \tag{18}$$

After complicated algebraic manipulations, we obtain the condition for plastic zone formation around the tunnel (Eq. (19)) [53]:

$$p\_o < \frac{(R\_i - 1) \cdot \sigma\_{ti}}{2} - p\_i - \sqrt{-(R\_i - 1) \cdot \sigma\_{ti} \cdot p\_i + \frac{1}{3} \cdot R\_i \cdot \sigma\_{ti}^2 + \frac{(R\_i - 1)^2 \cdot \sigma\_{ti}^2}{3}} \tag{19}$$

Finally, the radius of the plastic zone rc is obtained (Eq. (20)) [53]:

$$r\_{\varepsilon} = r\_i \cdot e^{\frac{L-K}{2\cdot (\mathbb{R}\_b - 1)\cdot \sigma\_{ib}} - \frac{1}{2} \ln \frac{K}{L}} \tag{20}$$

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

L and K above are constants given by Eq. (21), (22):

$$K = -(R\_b - \mathbf{1}) \cdot \sigma\_{tb} - \sqrt{\frac{4}{3} \cdot R\_b \cdot \sigma\_{tb}^2 - 4 \cdot (R\_b - \mathbf{1}) \cdot \sigma\_{tb} \cdot \sigma\_{rc} + \frac{4 \cdot (R\_b - \mathbf{1})^2 \cdot \sigma\_{tb}^2}{3}} \tag{21}$$

$$L = -(R\_b - \mathbf{1}) \cdot \sigma\_{tb} - \sqrt{\frac{4}{3} \cdot R\_b \cdot \sigma\_{tb}^2 - 4 \cdot (R\_b - \mathbf{1}) \cdot \sigma\_{tb} \cdot p\_i + \frac{4 \cdot (R\_b - \mathbf{1})^2 \cdot \sigma\_{tb}^2}{3}} \tag{22}$$

Figure 12. Comparison of elliptic paraboloid and Griffith criteria (R = 8) [52].

Figure 13. The geometry of the problem. The plastic zone around the circular tunnel is shown [53].

Hoek and Brown [9] obtained the radius of the plastic zone for Hoek-Brown criterion as follows [13]:

$$\mathbf{r\_e} = \mathbf{r\_i} \mathbf{e} \left\{ \mathbf{N} - \frac{\frac{2}{\mathbf{m}\_t \sigma\_\mathbf{c}} (\mathbf{m}\_t \sigma\_\mathbf{c} \mathbf{p}\_i + s\_\mathbf{r} \sigma\_\mathbf{c}^2)^{\frac{1}{2}}}{} \right\} \tag{23}$$

$$\mathbf{N} = \frac{2}{\mathbf{m}\_{\mathbf{r}}\sigma\_{\mathbf{c}}} \left( \mathbf{m}\_{\mathbf{r}}\sigma\_{\mathbf{c}}\mathbf{p}\_{0} + \mathbf{s}\_{\mathbf{r}}\sigma\_{\mathbf{c}}^{2} - \mathbf{m}\_{\mathbf{r}}\sigma\_{\mathbf{c}}^{2}\mathbf{M} \right) \tag{24}$$

$$\mathbf{M} = \frac{1}{2} \left( \left( \frac{\mathbf{m}}{4} \right)^2 + \mathbf{m} \frac{\mathbf{p}\_0}{\sigma\_\mathbf{c}} + \mathbf{s} \right)^{\frac{1}{2}} - \frac{\mathbf{m}}{8} \tag{25}$$

In Table 3 a comparison of the above criteria is shown. We can conclude that, except for low values of in situ stresses, both criteria are close in their predictions of radial stresses at the elastic–plastic interface. On the contrary, their predictions regarding the extent of the plastic zone differ with Hoek-Brown criterion being somehow more conservative.

### 3.2 The shallow tunnel case

The shallow tunnel case is complicated because of the influence of the proximity of ground surface on the stress field and the influence of gravity. As we notice for the case of deep tunnels, there are closed-form solutions for the plastic zone formation around circular tunnels for isotropic stress field and for the general case of Tresca [48] and Mohr-Coulomb [49] criteria. For the case of shallow tunnels, there were no closed-form solutions until 2009 when a solution was published for the Mohr-Coulomb solution under the assumption of isotropic stress field (k = 1) and no gravitational stresses, based on bipolar coordinates (Figure 14).


Table 3.

Comparison of elliptic paraboloid and Hoek-Brown criteria for σ<sup>c</sup> = 1 MPa, σ<sup>t</sup> = 0.125 MPa, R = 8.00, m = 8 and s = 1 [53].

Figure 14.

Bipolar coordinates used for the solution of shallow tunnel problem [54].

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

In the following the solution published in [54] and further applied [55] is presented omitting the detailed mathematical analysis:

$$\begin{split} P\_{cr} &= \frac{2\kappa^2}{2\left(d\_i^2 - r\_i^2\cos^2\beta\right) + \kappa^2(\lambda - 1)} \left[ P\_0 \left(\frac{d\_i^2 - r\_i^2\cos^2\beta}{\kappa^2}\right) - \frac{Y}{2} \right] \\ &= \frac{2\kappa^2}{2\left(\kappa^2 + r\_i^2\sin^2\beta\right) + \kappa^2(\lambda - 1)} \left[ P\_0 \left(\frac{\kappa^2 + r\_i^2\sin^2\beta}{\kappa^2}\right) - \frac{Y}{2} \right] \end{split} \tag{26}$$

where ri is the tunnel radius and di (= κcothαi) is the depth of the tunnel axis from the surface. Considering the conditions of the problem, following [54] the final form of the equation giving the shape of plastic zone is given below:

$$\left(\frac{r\_c}{r\_i}\frac{d\_i - r\_i\cos\beta}{d\_c - r\_c\cos\beta}\right)^{1-\lambda} = \frac{[2M\_0 + \kappa^2(\lambda - 1)][Y + P\_i(\lambda - 1)]}{2M\_0[Y + P\_0(\lambda - 1)]}\tag{27}$$

where <sup>M</sup><sup>0</sup> <sup>¼</sup> <sup>κ</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>c</sup> sin <sup>2</sup> β.

In Figure 15, a parametric study of the plastic zone shape is shown [54].

Eq. (27) can be solved, also, by using MATLAB [56]. In Figure 16, an example is shown.

The analytical solution of this problem apart from its theoretical interest could be used in conjunction with computational analysis in practical problems [57]. In [55] a very important case is presented where the task was for a new tunnel to

Plastic zone formation for different Pi/P0 values for P0 = 500 kPa, c = 100 kPa, φ = 250 , tunnel's depth = 20 m and r = 5 m [54].

### Figure 16.

Plastic zone around shallow tunnel with MATLAB. Example SMG1 [54]: D = 10 m, r = 5 m, γ = 25 kN/m<sup>3</sup> , P0 = 250 kPa, Pi = 50 kPa, c = 60 kPa, φ = 250 .

underpass the monumental Chandpole Gate in Jaipur, India. From the closed-form solution, the critical internal pressure was calculated to start with a further computational search for the optimum EPBM pressure in order to minimize the surface settlements induced by the excavation. For a more deep analysis of this problem, it seems that we have to proceed using semi-analytical methods. For the current status of this research, see [58, 59]. Important developments have been presented by Schofield [60] and Mair [61] in their Rankine Lectures of 1980 and 2008, respectively.

### 4. Applications of computational methods in tunnel engineering

In the last section of the chapter, a short, not complete, survey of computational methods in tunnel engineering is presented. It is a rather chronological account of methods and tools based on author's personal experience.

To start with, in Figure 17 a very useful and clear classification of methods of analysis is presented [62–64]. For an extensive review of numerical analysis, see Potts [65].

### 4.1 Level 1: basic numerical methods: 1:1 mapping

The available numerical methods (FEM, BEM, DEM) belong to category "C" making "one-to-one mapping". The meaning of this term is that they make a direct modeling of geometry and physical mechanisms [63]. In the 1970s finite element method (FEM) codes, based on differential equation formulation, were developed running in mainframe computers. Towards the end of the 1970s, the boundary element method (BEM) was developed, based on integral equation formulation. Bray and his co-workers introduced a 2D indirect formulation of BEM, and they developed a code included in [13].

Based on the example presented in Hoek and Brown ([13], p. 499), a twin cavern problem was analysed using the indirect BEM formulation [66]. In order to check the numerical analysis, a photoelastic model was, also, analysed. The BEM code was modified, to plot isochromes and isoclinics [14] obtained from

### Figure 17.

The four basic methods, in two levels, comprising eight different approaches to rock mechanic modeling [62–64].

Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

### Figure 18.

Plotting of isochromes of the twin cavern case. (a) Curves obtained from the numerical analysis and (b) curves obtained from the experiment [66].

photoelasticity, for comparison. In Figure 18, plotting of isochromes and those obtained from the experiment is shown. The agreement between the two is remarkable [66].

In a later stage (2002), with the advances in information systems and methods, the numerical methods to solve engineering problems advanced exponentially. A 3D FEM code was developed, and an extensive study was undertaken for the modeling of rock mass and underground excavations [52]. In that code several failure criteria were implemented. It may be the only 3D FEM code running the elliptic paraboloid criterion described in Section 3.1. In order to benefit from the 3D capabilities of the code, a more complicated problem was analysed. The cavern section included in [13] was modified to include, also, a tunnel of circular section cutting the cavern at a right angle (Figures 19 and 20). The rock mass is assumed to be anisotropic with two families of joints [51]. In formulating the problem, the following assumptions have been made: modulus of elasticity E = 7 GPa, ν = 0.25, γ = 25 kN/m3 , tunnel's depth Z = 400 m, vertical stress at infinity p = 10 MPa and in situ stress ratio k = 0.8. The model had 1441 nodes with 4323 degrees of freedom. About 236 isoparametric hexahedral elements with 20 nodes were used.

In the following figures, some characteristic results are presented. In Figure 19, the formation of plastic zones around the tunnels in the conjunction area is presented. Because of the anisotropy of the problem, the plastic zones are not symmetric. In Figure 20, the convergences of the excavation boundaries are shown.

Figure 20.

Convergences at the excavation boundaries [52].

### Figure 21.

Displacement contours (k = 0, β = 30°, r = 0.05, s = 0.013) and σ<sup>3</sup> stress contours (k = 0, β = 30°, r = 0.60, s = 0.91) [52].

Finally, in Figure 21, the displacement and stress contours are shown. The r and s coefficients are defined as r = E'/E and s = G'/G. All excavations are assumed to be unlined.

### 4.2 Level 2: system approaches: non-1:1 mapping

In the last paragraph of this chapter, the application of artificial intelligence (AI) methods and tools in tunnel engineering is shortly discussed. This category of methods belongs to Level 2 methods achieving a non-1:1 mapping (Figure 17). In this category of methods, the rock or soil mass is mapped indirectly by a network of nodes [67]. In the 1990s some applications of AI in geotechnical engineering were published [68–74]. From 2001 a great number of application of AI methods in geotechnical engineering have been published. There are two categories of approaches: the supervised learning and the unsupervised learning. Backpropagation, which was the first method applied in geotechnical engineering, belongs to the first category. Interconnected nodes corresponding to parameters involved in the problem represent the physical problem. The output of the training process is taken as a known target [67]. On the contrary, in unsupervised learning methods, the system has to extract knowledge from the data resulting in the underlying interconnections of the parameters of the problem [75]. Currently, a great number of publications are available using more advanced and sophisticated AI

### Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

methods. A critical factor is the quality of data and the engineering judgment of the users. There are cases where the overtraining of the system resulted in a "blind" learning of the data guiding to wrong conclusions. The ability to "include creative ability, perception and judgment" [67] is, still, not achieved.

Here, an attempt to train a backpropagation network system using surface settlements induced by underground excavations is presented [76]. The total amount of the available data is 90 records, coming from different types of ground profiles and referring to tunnels constructed with different methods of excavation [77]. The provided information includes tunnel size and depth (Zo), maximum settlement (w), settlement trough width (i), volume (Vs), ground description, geological properties and method of working. An indicative part of the available data is given in Table 4.

The range of values of the data, the type of geological profiles and the excavated methods are given in Table 5 [77].

In training the ANN, two approaches followed. In the first approach, all the available data were included. The tunnel depth and diameter have been used as input data and the surface settlement as output. Qualitative "data" as the geological conditions and the excavation method were not initially taken into consideration as parts of the training process. The training was not successful because we mixed the nonhomogeneous data. Therefore we proceeded to the second approach using all available data and information. To this end, we assigned a number to distinct different geological conditions and excavation method. In this training four inputs were used and the training was successful. In Figure 22, the relative importance of the parameters involved to the problem is shown. From this figure, we may conclude that the geometry of the tunnel, i.e. radius and depth, is the main factor with geological profile and excavation method having an important contribution too [76].


### Case history data on some tunnels and tunneling conditions [76, 77].

### Table 4.

Indicative examples of case history data [76, 77].


### Table 5.

Range of values of the available data and the categories of geological profiles and excavation methods [76, 77].

Figure 22. Relative importance of the parameters contributing to induced settlements due to tunneling [76].

Further tunneling applications using AI tools can be found in the literature. In Zaré and Lavasan [78] an objective system approach is adopted to quantify the interaction of parameters involved in the problem of tunnel face stability. The method is based on a backpropagation ANN approach. A different approach of objective system approach, which adopted the unsupervised type of learning based on self-organizing maps, is, also, published with applications in rock engineering problems [75, 79–85].

### 5. Conclusions

In this chapter, a rather subjective view of tunnel engineering is presented. Nevertheless, the progress made in analytical and computational methods was followed with an effort to be well documented. Starting from a well-known problem of the elastic stress field around a circular deep tunnel, we investigated the influence of tunnel's shape in stress and displacement fields around a tunnel of the horseshoe section, as well as the extent of the influence zone for several stress fields. To this end Muskhelishvili's complex variable formulations of stress functions were Topics of Analytical and Computational Methods in Tunnel Engineering DOI: http://dx.doi.org/10.5772/intechopen.90849

used. The next problem was the case of the shallow circular tunnel for which established elastic solutions are presented. Proceeding to the more difficult problem of the plastic analysis, we examined the case of deep tunnels, and then we presented a closed-form solution of the plastic zone formation for the shallow circular tunnel. This is a topic still needing further investigation because of its mathematical difficulty. Computational methods were the last part of the chapter. A classification of methods was presented followed by a problem of deep tunnels analyzed using 3D finite element analysis. The increasing exploitation of artificial intelligence tools in analyzing geotechnical problems was the last topic. The presentation was based on the tunneling-induced surface settlements as a paradigm. The cited references listed below are a basic and indicative selection of literature for further reading.

### Acknowledgements

The author acknowledges the contribution of his former PhD, MSc and MEng students at the National Technical University. Their theses are listed in the references, and they are referred in the text.

### Author details

Michael G. Sakellariou National Technical University of Athens, Athens, Greece

\*Address all correspondence to: mgsakel@mail.ntua.gr

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### **Chapter 4**

## Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures

*Prodromos Psarropoulos*

### **Abstract**

Depending on the circumstances, the design and construction of tunnels and underground spaces may be very challenging. In the case of an underground project located at a relatively shallow depth in an urban area, the design and construction will probably be more demanding since there is a potential interaction between the underground project and the overlying pre-existing structure(s) that are founded at the ground surface, such as buildings, bridges, etc. This interaction is generally related to the (usually differential) settlements at the ground surface due to the excavation and the consequent distress of the overlying structures. Nevertheless, in areas that are characterized by seismicity, this interaction may be more complicated, since, apart from the aforementioned static interaction, various phenomena of soil dynamics and dynamic interaction may take place, dominating thus the seismic excitation, response, and distress of the overlying structure(s). The current chapter deals with this interesting topic of geotechnical earthquake engineering. After a literature review, some indicative numerical analyses have been performed in order to determine the impact of the main parameters involved. Although the problem is generally complex and multi-parametrical, the numerical results are indicative of the dynamic interaction between the underground project, the ground, and the overlying structure(s).

**Keywords:** tunnel, underground space, seismic response, dynamic interaction

### **1. Introduction**

Undoubtedly, during the last decades, a significant progress has been made in the design and construction of underground projects worldwide. Apart from the construction of underground spaces for various purposes (e.g., environmental, military, etc.), the most important underground projects are long tunnels which usually comprise important elements of highways and railways. In urban areas and especially in big cities, the increase in population and the need for fast transportation means have led to the development of metropolitan railways (i.e., subways), and therefore, there has been a large increase in the number and size of underground structures (i.e., metro stations and tunnels). Depending on the local site conditions, in an urban area, one of the main issues during the construction of a tunnel at a relatively shallow depth is the potential (static) interaction between the tunnel and a pre-existing overlying structure at the ground surface, such as a building or a bridge. It is evident that the development of (usually differential) settlements at the ground surface will probably distress any pre-existing structure.

Nevertheless, in areas that are characterized by seismicity, the construction of a tunnel under a pre-existing structure may have an impact, not only on the seismic excitation of the structure but on its seismic (i.e., dynamic) response as well. In geotechnical (earthquake) engineering, the term "local site conditions" is usually used to describe the prevailing topographical and geotechnical conditions. In the case of the existence of an underground project at a relatively shallow depth, the local site conditions include the underground project as well. As shown in **Figure 1**, in the case of a structure founded at the ground surface and subjected to a seismic excitation, there exist four general cases of local site conditions: (a) a structure founded on rock without a tunnel, (b) a structure founded on rock with a tunnel underneath, (c) a structure founded on soil layers without a tunnel, and (d) a structure founded on soil layers with a tunnel underneath.

Additionally, it is generally acknowledged that underground structures suffer less from earthquakes than buildings on the ground surface. Nevertheless, earthquakes in Kobe (1995), Chi-Chi (1999), and Duzce (1999) (see [1–5]) have caused extensive failures in tunnels (and buried pipelines), reviving the interest in the associated analysis and design methods.

The current chapter attempts to shed some light on the seismic (i.e., dynamic) behavior of underground projects and mainly on their impact on the overlying structures. **Figure 2** shows a sketch of the problem under consideration. An underground project (i.e., a circular tunnel) is constructed within a soil layer. A structure is founded on the surface of the soil layer, at a relatively short distance from the underground project. For the sake of simplicity, the structure is considered to be a single-degree-of-freedom system (i.e., a concentrated mass on a single beam). The soil layer and the two structures are subjected to seismic loading, i.e., an acceleration time history applied at the base of the soil layer. Therefore, the dynamic

### **Figure 1.**

*Sketch showing the four potential cases of "local site conditions" of a structure founded at the ground surface and subjected to a seismic excitation.*

*Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

**Figure 2.**

*Sketch of the problem under consideration: a simple structure founded on a soil layer is overlying a tunnel. The soil layer and the two structures (i.e., tunnel and simple structure) are subjected to an acceleration time history being applied at the base of the soil layer.*

interaction between the underground project, the soil, and the overlying structure is investigated, while emphasis is given to the impact of the underground project on the dynamic response of the overlying structure.

The next sections are involved with various topics of the problem under examination. More specifically, a literature review on the impact of underground structures on the characteristics of the seismic motion at the ground surface is described in Section 2. The aim of the literature review is to identify the most important parameters of the problem. This section also includes some indicative numerical results related to the distribution of accelerations on the ground surface in the presence of an underground circular lined tunnel. Finally, Section 3 is devoted to the dynamic soil-structure interaction phenomena and the seismic response of structures overlying circular tunnels. Although the literature on this issue is relatively limited, various parameters are considered, and useful conclusions are drawn. In this section the numerical simulations also include the overlying structures. The results are indicative of the complexity of the dynamic interaction between the tunnel, the ground, and the structure.

### **2. Impact on the seismic motion at the ground surface**

One of the first studies on the impact of underground projects on the seismic motion was the study of Lee and Trifunac [6]. In their work, Lee and Trifunac analyzed the scattering and refraction of SH shear waves due to a circular tunnel in a homogeneous elastic half-space using an analytical solution.

The main parameters that determine the response at the ground surface are (a) the angle of incidence and the frequency content of the seismic waves, (b) the distance from the vertical axis of the tunnel, and (c) the tunnel depth.

During the last four decades, various researchers have studied similar problems. More specifically, many researchers have examined analytically the impact of a circular underground structure on the surface ground motion, while few researchers have examined the problem numerically using mainly the finite element method. In addition, some researchers have verified their analytical results with numerical simulations or vice versa. Finally, very few attempts have been made in order to simulate experimentally the seismic response of underground structures. For more details the reader may refer to [7–20].

The evaluation of the aforementioned publications shows that the impact of underground projects on the seismic motion at the ground surface consists of the following:


**Figure 3** shows the four numerical models that have initially been examined in order to indicatively demonstrate the potential impact of a tunnel on the seismic motion at the ground surface. Model 1 is actually a rigid rock, while Model 2 is a rigid rock with a lined tunnel. On the other hand, Model 3 consists of a soil layer


### **Figure 3.**

*The four examined numerical models showing the points of interest at the base and at the ground surface. Model 1 is a rigid rock, Model 2 is a rigid rock including a tunnel, and Model 3 is a soft soil layer on rock, while Model 4 is the same model with a tunnel. Point A is located at the base, while the points Bi, Ci, Di, and Ei (with i = 1 to 4) are at the ground surface.*

### *Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

with a height, *H*, of 25 m, overlying a rigid rock (i.e., one-dimensional model). The soil layer is characterized by a shear-wave velocity, *VS*, equal to 200 m/s. Model 4 is the same model including also the aforementioned lined tunnel. Two cases of tunnel radius, *R*, have been examined: *R1 =* 5 m and *R2* = 10 m.

All numerical analyses have been performed with PLAXIS2D which is a commercial finite element program capable to perform dynamic ground response analyses in the time domain. Special transmitting boundaries have been applied at the two vertical boundaries of both models in order to avoid unrealistic trapping of the seismic waves.

The four models have been horizontally excited by three acceleration time histories. As shown in **Figure 4**, the first is a sinusoidal motion with frequency *fo* = 2 Hz, the second is a simple Ricker wavelet of central frequency *fo* = 2 Hz (characterized by a wide range of frequencies up to 3*fo* = 6 Hz), while the third is a real accelerogram that has been recorded during the 1990 Upland earthquake, in California, with a peak ground acceleration (PGA) of the order of 0.15 g. All excitations have intentionally been scaled to low values of peak acceleration (i.e., 0.01 g) in order to keep the behavior of the geomaterials in the elastic range.

According to the wave propagation theory, the first eigenfrequency of Model 3 (i.e., a single soil layer) is equal to *f1* = *VS*/(4*H*) = 2 Hz, while its maximum theoretical response at resonance is 2/(π*ξs*), where *ξs* is the material damping. Therefore, the sinusoidal motion with frequency *fo* = 2 Hz has been used in order to verify the numerical simulations with the corresponding analytical solutions. If we suppose that the material damping of soil, *ξs*, is 5%, the amplification factor, *AF*, is 12.7.

**Figures 5–11** show some indicative numerical results.

More specifically, **Figure 5** shows the dynamic response of Model 3 in the case of sinusoidal excitation. As it was expected, resonance phenomena are evident at the ground surface. As aforementioned, the peak ground base acceleration is only 0.01 g, while the peak ground surface acceleration has been amplified almost 12 times. The discrepancy between the *AF* from the analytical solution and the corresponding *AF* from the numerical modeling is attributed mainly to some deficiencies of the numerical modeling, such as the Rayleigh-type material damping, the size of the finite elements, and/or the rather medium accuracy of the vertical transmitting boundaries.

**Figure 6** shows the dynamic response of Model 3 in the case of Ricker excitation and in the case of the record from Upland earthquake. In the case of Ricker excitation, the peak ground surface acceleration is almost 0.02 g, while the duration of the ground motion has been substantially increased from 1 second to almost 5 s. In the case of the recorded acceleration from the Upland earthquake, the ground acceleration has been amplified almost 2.5 times since the peak ground surface acceleration is almost 0.025 g.

As it was expected, in the case of Model 1, Model 2, and Model 3, there are no differences of the dynamic response at the ground surface. Model 1 and Model 2 are rigid, while Model 3 is characterized by one-dimensional conditions. **Figure 8** and **Figure 9** show the calculated time histories of horizontal acceleration at various locations at the ground surface in the case of Model 4 with the small tunnel (*R1* = 5 m) and in the case of Model 4 with the big tunnel (*R2* = 10 m), respectively. It is evident that the existence of the small tunnel has actually no impact on the variation of the response at the ground surface, a phenomenon that can be attributed to the fact that the size of the tunnel is relatively small compared to the examined wavelengths. On the contrary, in the case of the big tunnel, there is an impact, although it is rather minor. More specifically, the acceleration levels are lower right above the tunnel (i.e., a "shadow zone" has been created), while few meters away (at point C4) the acceleration is locally increased. Similar are the results shown in **Figure 10** where the seismic responses of the points at the ground surface are being compared in the case of Upland excitation.

### **Figure 4.**

*The three acceleration time histories that have been used as seismic excitations (all scaled to 0.01 g): (a) a sinusoidal excitation, (b) a Ricker pulse, and (c) the record from the Upland earthquake.*

As aforementioned, another phenomenon that may take place due to the presence of a tunnel is the development of parasitic vertical accelerations. Since the seismic excitation in all numerical analyses was only horizontal (i.e., S waves), no *Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

**Figure 5.** *The dynamic response of Model 3 subjected to the sinusoidal excitation.*

**Figure 6.**

*The dynamic response of Model 3 subjected to the Ricker excitation.*

vertical ground motion is expected. This is reasonable for Model 1 and Model 2 that are rigid and for Model 3 that is one-dimensional. Nevertheless, the points at the surface of Model 4 exhibit this vertical parasitic acceleration. **Figure 11** shows the vertical acceleration at various locations along the ground surface in the case of the big tunnel and the Upland excitation. It is noted that the maximum vertical acceleration is observed at point C4 where its value is almost 30% of the corresponding peak ground surface acceleration (i.e., 0.03 g) and in parallel comparable to the peak ground base acceleration (i.e., 0.01 g).

Judging from the numerical results of this rather limited parametric study, it becomes evident that the existence of the tunnel alters the acceleration pattern along the ground surface. The "shadow zone" right above the tunnel and the vertical parasitic seismic motion are rather obvious.

In any case, it has to be emphasized that the interaction between the soil, the structure, and the tunnel is a problem with several parameters, and therefore, in any other case the patterns of horizontal and vertical acceleration at the ground surface may be completely different.

**Figure 7.** *The dynamic response of Model 3 subjected to the Upland excitation.*

**Figure 8.**

*The dynamic response of Model 4 with the small tunnel (R1 = 5 m) subjected to the Ricker excitation.*

**Figure 9.**

*The dynamic response of Model 4 with the big tunnel (R2 = 10 m) subjected to the Ricker excitation.*

*Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

**Figure 10.** *The dynamic response of Model 4 with the big tunnel (R2 = 10 m) subjected to the Upland excitation.*

**Figure 11.**

*The vertical dynamic response of Model 4 with the big tunnel (R2 = 10 m) subjected to the Upland excitation.*

### **3. Dynamic interaction between tunnel, ground, and structure**

The seismic response of any structure founded at the ground surface is an issue that depends on various factors, such as the mechanical and geometrical properties of the structure and the characteristics of the seismic excitation.

When the structure is a single-degree-of-freedom (SDOF) structure with a fixed base, then it is characterized by its eigen-period *To,* which is given by the following simple expression: \_

$$T\_o = 2\pi r \sqrt{\frac{M}{K}} \tag{1}$$

where *M* is the concentrated mass of the structure and *K* is its stiffness.

Therefore, if the fundamental period of the seismic excitation is close to *T*o, resonance phenomena are expected, and therefore, the dynamic distress of the structure may have its maximum value.

Nevertheless, according to [21, 22], the potential existence of soft soil layers under the structure will lead to the following phenomena:

	- The soil compliance will reduce the stiffness of the structure, a fact that will certainly lead to an increase of the eigen-period of the structure.
	- The overall damping of the system will be increased since the existence of soil layers will introduce other means of energy dissipation apart from the material damping of the structure, such as the material damping of the soil and the radiation damping.

Although the increase of the damping is always beneficial, the reduction of the stiffness (and the subsequent increase of the eigen-period) may be either beneficial or detrimental for the distress of the structure, depending on the circumstances.

In this section all the previous numerical models have been modified in order to include four identical simple structures (4) at the ground surface. As shown in **Figure 12**, the four structures are above the tunnel, while the distance between them is the same (15 m). All of them are single-degree-of-freedom (SDOF) structures, and they are characterized by (a) a material damping of 5% and (b) an eigenperiod *To* = 0.5 s or eigenfrequency *fo* = 2 Hz (identical to the first eigenfrequency of the soil layer). Note that in Model 3 and Model 4, the actual eigenfrequency of the structures is smaller due to the soil compliance.

The following figures show some indicative numerical results. More specifically, **Figure 13** shows the horizontal acceleration time histories that have been developed on the top of the structures in the case of Model 4 with the big tunnel subjected to the Upland excitation. It is evident that the acceleration levels are relatively high (of the order of 0.1 g). This fact is attributed to the resonance phenomena between the soil and the structures (since they have comparable eigenfrequencies). The initial peak ground base acceleration (of 0.01 g) has been amplified up to 0.03 g (i.e., almost three times) at the ground surface, while the peak ground surface acceleration has been amplified again, reaching a value of the order of 0.1 g.

In parallel, minor differences exist between the structural responses of the four structures. As it was expected, the minimum response is observed in the case of the structure located at point B, while the maximum response is on the structure located at point C.

**Figure 14** shows the corresponding (parasitic) vertical accelerations that have been developed on the top of the structures. The maximum response is also observed in the case of the structure located at point C. Note that these accelerations are comparable to the acceleration levels at the ground surface (see **Figure 11**).

**Figure 12.**

*The modified Model 4 including four (4) equally spaced single-degree-of-freedom structures at the ground surface.*

*Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

**Figure 13.**

*The horizontal accelerations developed on the top of the four structures in the case of the modified Model 4 with the big tunnel subjected to the Upland excitation.*

### **Figure 14.**

*The vertical accelerations developed on the top of the four structures in the case of the modified Model 4 with the big tunnel subjected to the Upland excitation.*

This phenomenon was actually expected since the single-degree-of-freedom structures have no vertical response. Note that in a more realistic case with multidegree-of-freedom systems, the vertical component would have been amplified.

### **4. Conclusions**

In urban areas and especially in big cities, the increase in population and the need for fast transportation means will lead to the development of metropolitan railways (i.e., subways), and therefore, there will be a large increase in the number and size of underground structures (i.e., metro stations and tunnels).

In areas that are characterized by moderate or high seismicity, it is evident that the construction of an underground project (e.g., tunnel or underground space) under a pre-existing structure may alter more the seismic excitation of the structure, modify the soil-structure interaction pattern, and consequently have an impact on the structural response and distress.

The numerical results that have been presented in the previous sections have shown that the existence of a tunnel may alter the pattern of horizontal acceleration at the ground surface in the time domain (and in the frequency domain). This fact means that the construction of a tunnel under a pre-existing structure will complicate more the aforementioned dynamic soil-structure interaction phenomena.

Finally, it has to be emphasized that the anticipated vertical parasitic acceleration may have an impact on the structural response and distress of structures with many degrees of freedom, especially when the acceleration levels of the seismic excitation are high and a nonlinear behavior of the structure is expected.

Based on all aforementioned, when a new underground structure is constructed in urban areas, a special study should be performed in order to assess quantitatively the impact of the underground structure on the seismic response and distress of any pre-existing overlying structure.

### **Acknowledgements**

The author would like to thank George Christou, an undergraduate student at the School of Rural and Surveying Engineering of the National Technical University of Athens, for his contribution to some of the numerical analyses.

### **Author details**

Prodromos Psarropoulos

Laboratory of Structural Mechanics and Engineering Structures, School of Rural and Surveying Engineering, National Technical University of Athens, Greece

\*Address all correspondence to: prod@central.ntua.gr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Impact of Tunnels and Underground Spaces on the Seismic Response of Overlying Structures DOI: http://dx.doi.org/10.5772/intechopen.89338*

### **References**

[1] EQE Summary Report (1995): The January 17. Kobe Earthquake; 1995

[2] Sinozuka M. The Hanshin-Awaji earthquake of January 17, 1995: Performance of lifelines. NCEER-95-0015; 1995

[3] Chen WW, Shih B-J, Chen Y-C, Hung J-H, Hwang HH. Seismic response of natural gas and water pipelines in the Ji–Ji earthquake. Soil Dynamics & Earthquake Engineering. 2002;**22**:1209-1214

[4] Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TH. Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake. Soil Dynamics & Earthquake Engineering. 2002;**22**:73-96

[5] Uenishi K, Sakurai S, Uzarski SM-J, Arnold C. Chi-Chi Taiwan, earthquake of September 21, 1999: Reconnaissance report. Earthquake Spectra. 1999;**5**(19):153-173

[6] Lee VW, Trifunac MD. Response of tunnels to incident SH waves. Journal of Engineering Mechanics Division, ASCE. 1979;**105**:643-659

[7] St. John MC, Zahrah TF. Aseismic design of underground structures. Tunneling & Underground Space Technology. 1987;**2**(2):165-197

[8] Lee VW, Karl J. Diffraction of SV-waves by underground, circular, cylindrical cavities. Soil Dynamics and Earthquake Engineering. 1992;**11**:445-456

[9] Lee VW, Karl J. On the deformations near a circular underground cavity subjected to incident plane P waves. European Earthquake Engineering. 1993;**1**:29-39

[10] De Barros FCP, Luco JE. Diffraction of obliquely incident waves by a

cylindrical cavity embedded in a layered viscoelastic halfspace. Soil Dynamics & Earthquake Engineering. 1993;**12**:159-171

[11] Wang JN. Seismic Design of Tunnels: A State-of-the-Art Approach. New York, N.Y.: Parsons Brinckerhoff Quade & Douglas, Inc.; 1993, Monograph 7

[12] Luco JE, De Barros FCP. Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space. Earthquake Engineering & Structural Dynamics. 1994;**23**:321-340

[13] Manoogian ME, Lee VW. Diffraction of SH-waves by subsurface inclusions of arbitrary shape. Journal of Engineering Mechanics Division, ASCE. 1996;**122**:123-129

[14] Penzien J. Seismically induced racking of tunnels linings. Earthquake Engineering & Structural Dynamics. 2000;**29**:683-691

[15] Vanzi V. Elastic and inelastic response of tunnels under longitudinal earthquake excitation. Journal of Earthquake Engineering. 2000;**4**(2):161-182

[16] Davis CA, Lee VW, Bardet JP. Transverse response of underground cavities and pipes to incident SV waves. Earthquake Engineering & Structural Dynamics. 2001;**30**:383-410

[17] Kouretzis GP, Bouckovalas GD, Gantes CJ. 3-D shell analysis of cylindrical underground structures under seismic shear (S) wave action. Soil Dynamics and Earthquake Engineering. 2006;**26**:909-921

[18] Yiouta-Mitra P, Kouretzis G, Bouckovalas G, Sofianos A. Effect of underground structures in earthquake resistant design of surface structures. In: Geo-Denver 2007: New Peaks in Geotechnics. ASCE; 2007

[19] Park KH, Tantayopin K, Tontavanich B, Owatsiriwong A. Analytical solution for seismic-induced ovaling of circular tunnel lining under no-slip interface conditions: A revisit. Tunnelling and Underground Space Technology. 2009;**24**(2):231-235

[20] Chen S, Zhuang H, Quan D, Yuan J, Zhao K, Ruan B. Shaking table test on the seismic response of large-scale subway station in a loess site: A case study. Soil Dynamics and Earthquake Engineering. 2019;**123**:173-184

[21] Veletsos AS, Meek JW. Dynamic behavior of building-foundation systems. Journal of Earthquake Engineering and Structural Dynamics. 1974;**3**(2):121-138

[22] Kramer LS. Geotechnical Earthquake Engineering. Upper Saddle River, New Jersey: Prentice Hall; 1996

## **Chapter 5** Designing a Tunnel

*Spiros Massinas*

### **Abstract**

Designing a tunnel is always a challenge. For shallow tunnels under cities due to the presence of buildings, bridges, important avenues, antiquities, etc. at the surface and other infrastructures in the vicinity of underground tunnels, parameters like vibrations and ground settlements must be tightly controlled. Urban tunnels are often made in soils with very low values of overburden. Risks of collapse and large deformations at the surface are high; thus negative impact on old buildings are likely to occur if appropriate measures are not taken in advance, when designing and constructing the tunnel. For deep tunnels with high overburden and low rock mass properties, squeezing conditions and excessive loads around the excavation can jeopardize the stability of the tunnel, leading to extensive collapse. The aim of the chapter is to give details on advance computational modelling and analytical methodologies, which can be used in order to design shallow and deep tunnels and to present real case studies from around the world, from very shallow tunnels in India with only 4.5 m overburden to a deep tunnel in Venezuela with extreme squeezing conditions under 1300 m overburden.

**Keywords:** tunnel, shallow tunnelling, deep tunnels, surface settlements, tunnel squeezing, analytic solution, numerical analysis, lining stress controllers, sliding joints, monument underpassing, monitoring, plasticity, TBM, earth pressure balance shield, high overburden, high deformations, NATM, conventional tunnelling

### **1. Introduction**

The aim of the current chapter is to give details and guidance in designing underground tunnels, to be constructed with tunnel boring machines (TBM).

In the sequence of paragraphs to follow, the reader will get a grasp of tunnelling principles and theory related to mountainous and urban tunnelling—deep and shallow tunnels will be explained (paragraph 2).

In paragraph 3, details for designing mechanized constructed (TBM) Metro tunnels in urban environment will be given, and real cases from India will be presented. Special case for monument underpass with earth pressure balance machine (EPBM) under extreme low overburden will also be discussed, and the real case study from Chandpole Gate in Jaipur will be presented.

Mountainous and deep tunnels design will be presented in paragraph 4. Squeezing and non-squeezing conditions will be explained, and methods of designing will be given, along with examples from around the world.

The final paragraph will summarize the primary conclusions from the presented design methodologies. It is noted that all the examples presented herein are real cases of tunnels already constructed, with the personal involvement of the author in their designs elaboration.

### **2. Tunnelling principles**

Tunnelling is divided in two general categories: the deep tunnel case and the shallow tunnel case (**Figure 1**). The most common example of a shallow tunnel is the Metro Lines in a big city. For example, underground tunnels realize the connection between the underground stations. Since the stations are constructed, as underground structures, to serve the surface mass transit system, the depth of the train platform from the surface is limited to meters; in a normal typical station, the platform depth can vary from 15 to 25 m, while for other cases like flagship stations connecting different Metro Lines, platforms can be in different levels, and consequently their depth can reach and even exceed 40 m. Therefore, the tunnels connecting the stations can also vary in depth, and the typical overburden height (distance between the tunnel crown and the surface) can be 10–20 m, while for deeper sections can reach or even exceed 35 m. Of course along a Metro Line, there are always unique cases where the tunnel depth can be very limited, and thus the overburden height can be even less than a tunnel diameter (e.g., 5 m); for such a special case, the real case study from Chandpole Gate in Jaipur will be presented in the following paragraphs. Therefore, from a mathematical perspective, in shallow tunnelling, two boundaries are introduced, the tunnel geometry (circular or not) and the surface (**Figure 1**).

On the contrary, for a deep tunnel case, the problem can be described only by one boundary, the tunnel geometry (**Figure 1**). Furthermore, the influence of the variation of the in situ stress with the depth is more intense and critical in a shallow tunnel rather than in a deeper tunnel case. In the latter, the in situ stress difference between tunnel crown and invert is insignificant compared to the absolute value of the in situ stress at this depth. To understand the main difference between shallow and deep tunnel, a characteristic example of a motorway with multiple tunnels crossing a mountainous terrain can be used. In such a case where the tunnel pierces a mountain, the overburden height can be decades of meters (120, 150 m, etc.) or even hundreds of meters and even can exceed 1000 m; a real case study for a very deep tunnel in Venezuela will be presented later on.

The first to differentiate the shallow from the deep tunnel, covering, also, the intermediate zone between deep and shallow tunnels, was Bray [2]. In order to do so, he introduced the dimensionless ratio of the depth (from tunnel center—di) to the radius (ri) of a tunnel. For a ratio (di/ri) equal or greater to 25—e.g., a tunnel with radius 5 and 120 m overburden height—the deep tunnel case is described, while the shallow tunnel is defined by a ratio (di/ri) smaller or equal to 7—e.g., a

**Figure 1.** *Shallow and deep tunnels [1].*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

tunnel with radius 5 and 25 m overburden. For the intermediate values of the ratio, thus between 7 and 25, a transition zone is defined. The stress distribution around an underground cavity either shallow or deep is the key factor to design a tunnel.

In a similar fashion to the lines of flow in the current of a river, which are deviated by the pier of a bridge and increase in speed as they run around it, the flow lines of the stress field in a rock mass are deviated by the opening of a cavity (tunnel) and are channelled around it to create a zone of increased stress around the walls of the excavation. The channelling of the flow of stresses around the cavity introduces the arch effect (**Figure 2**).

Arch effect can occur, depending on the size of the stresses (overburden height) and the geomechanical properties of the rock mass (strength and deformation properties), (a) close to the profile of the tunnel, (b) far from the profile of the tunnel, and (c) not at all.

Case (a) occurs when the rock mass around the tunnel withstands the deviated stress flow, responding elastically in terms of strength and deformation. In case (b), due to the low properties of the rock mass, the ground around the excavation is not able to withstand the deviated stress flow and thus responds nonelastically, plasticizing and deforming in proportion to the volume of ground involved in the plasticization phenomenon. The latter, that often causes an increase in the volume of the ground affected, propagates radially and deviates the channelling of the stresses outwards into the rock mass until the triaxial stress state is compatible with the strength properties of the rock mass. In this situation, the arch effect is formed far from the line of the excavation and the ground around the tunnel which has been plasticized (plastic zone), contributing to the final tunnel stability with its own residual strength giving rise to deformations, which is often sufficient to compromise the safety of the excavation. With proper support measures, the ground can be "helped"; the plasticization phenomena can be limited, and thus the formation of the arch effect by natural means can be produced, and the tunnel stability can be ensured. In the third case (c), the ground around the cavity is completely unable to withstand the deviated stress flow and responds in the failure range producing the collapse of the tunnel. In such case the arch effect cannot be formed naturally, and thus pre-support measures must be used before the excavation, in order artificially to initiate the arch effect.

The reaction is the deformation response of the medium (ground) to the action of excavation (tunnelling). It is always generated ahead of the excavation face within the area that is disturbed, following the generation of greater stress in the medium around the cavity. The deformation response always depends on the

**Figure 2.** *Flow lines in the current of a river around a pier (left) and stress field flow lines around a tunnel [1].*

medium's properties and its stress state and is affected by the tunnel's face advance. As the face advances, the tunnel passes from a triaxial to a plane stress state. In case that the progressive decrease in the confinement pressure at the face (σ3 = 0) produces stress in the elastic range ahead of the face, then the excavation face remains stable with limited and absolutely negligible deformation. In this case the channelling of stresses around the cavity (arch effect) is produced by natural means close to the profile of the excavation, and no artificial support is required to secure the tunnel stability. If, on the other hand, the progressive decrease in the stress state at the face (σ3 = 0) produces stress in the elastoplastic range ahead of the face, then elastic-plastic deformation of the face will give rise to a condition of short-term stability. This means that in the absence of any intervention, plasticization is triggered, which, by propagating radially and longitudinally from the walls of the excavation, produces a shift of the "arch effect" away from the tunnel further into the rock mass. This shift from the theoretical profile of the tunnel can only be controlled by intervention to stabilize the ground.

Therefore, in respect to the in situ ground properties, the in situ stress field, and the applied support measures/pressure, the stress redistribution remains within the elastic domain, or a phenomenon of plasticization is triggered which may result to the formation of a plastic domain around the tunnel. Extent/width and shape of the plastic zone around the cavity are the main parameters for calculating/evaluating the stability conditions of an underground excavation. The impact of this mechanism is different in a shallow and a deep tunnel.

In a shallow tunnel, the overburden height is limited. Therefore, any underground deformation that may result from the soil plasticization will affect the development of the surface settlements. In case that the extent of the plastic zone gives rise to excessive surface settlements (**Figure 3**), damages on the surface structures can also be significant or even severe. For a shallow tunnel design, the key parameter is to minimize the magnitude of the developed surface settlements, and thus the redistribution of the stresses around the tunnel needs to be controlled accordingly by designing and applying proper support measures and consequently proper support pressure (Pi). However, for the deep tunnel case the redistribution of the stresses around the excavation can be the main challenge. The increased overburden height combined with low ground mass properties can give rise to excessive loads exerted around the excavation and thus can lead to tunnel collapse, if non-proper design of the support measures is elaborated. In such difficult cases, safe tunnel advance can only be achieved with controlled plasticization of the

**Figure 3.** *Impact of plastic zone formation around a shallow (left) and a deep (right) tunnel.*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

groundmass; intervening with application of proper yielding support measures if the groundmass is deformed plastically under controllable manner, and thus the exerted loads can significantly be reduced.

Three main different methods can be applied to solve tunnelling problems:


Analytic methods are related with the use of closed form solutions, while the finite element method (FEM) and the finite difference method (FDM) describe the computational procedures. Finally, combination of closed form solutions with FEM or FDM modelling can be more beneficial than the other two methods along; closed form solution can give the opportunity to the designer for quick and accurate calculations and thus to be properly "guided" in the elaboration of the final FEM/ FDM modelling. Indeed, empirical methods (e.g., Protodiakonov, Terzaghi, etc.) are also used in some cases for solving tunnelling problems.

### **3. TBM shallow tunnelling in urban environment**

The principle objective when designing a shallow tunnel in an urban environment is to minimize the induced surface settlements. Therefore, the face stability along with the tunnel's induced displacements are the key factors to control the extent of the plastic zone formation and consequently to secure the surface structures from undesirable settlements. In order to design the tunnel, the method of construction to be adopted for the work's execution needs to be defined. In the current paragraph, the principles for designing a Metro tunnel constructed with tunnel boring shield machine (earth pressure balance (EPB)-TBM) will be discussed, and the main design phases will be explained in detail.

After the alignment is fixed, the geological and geotechnical conditions along the alignment of a Metro project are always the main input required in order to further design the tunnels. Knowledge of the regional geology of the area along with geotechnical investigation campaign is always mandatory in order to determine the ground properties (**Figure 4**).

Boreholes with sampling, executed typically every 50–150 m, aim to investigate the ground conditions at least one to two diameters below the tunnel invert.

Mainly the complexity of the geological model of the area is a key factor to determine the required number of the boreholes along the alignment and if additional investigations such as geophysical surveys will be needed. For example, for the phase 3 of Delhi Metro, the general geology of the area (**Figure 4**) reveals the existence of polycyclic sequence of brown silty clay with kankar, fine- to medium-grained micaceous sand, c-silty/clay, and s-sandy facies. Those thick layers of alluvium deposits mainly are known as Delhi Silt with almost uniform properties along the alignment. Indeed, the presence of quartzite rock is also visible in certain areas of Delhi. Although the exposures are very less in number, structurally it occurs in the form of anticline and syncline with axial trend as N-S and SE-NW; their appearance within the tunnel excavation face (mainly consisted of soft soil—Delhi silt) can be detrimental for the tunnel construction with TBM. Therefore, geotechnical campaign with dense boreholes were executed (varying from 50 to 70 m distances) with the aim to investigate

### **Figure 4.**

*Geological map of Delhi (Source: Geological Survey of India) (Delhi Metro Phase 3 area marked with blue circle).*

the actual depth of the bedrock (**Figure 5**), to quantify the risk from having mixed face conditions (soft soil with hard rock), and finally to determine the arrangement of the EPB-TBM cutterhead (to be designed for mixed face conditions) (**Figure 6**).

For another case, where the stratigraphy is uniform along the entire alignment, the boreholes can be in greater distance. Let us investigate another example, again from India but from another area, the state of Rajasthan and the Jaipur Metro Phase 1B.

**Figure 6.** *EPB-TBM cutterhead designed by Herrenknecht for mixed face conditions. Cutter discs are visible (Delhi Metro Phase 3—CC23). Source: FEMC-Pratibha JV.*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

Quaternary sediment occupies the major northeastern part of Jaipur, but hillocks and ridge are present around the city (**Figure 7**). Granite gneisses occupy the southern part of the district. Other important lithological unites exposed in Jaipur are limestone, sandstone, etc. Soil alluvium is the main formation along the alignment of the project, in the form of sand to silty sand with gravels. The uniformity on the geological conditions dictated the determination of a geotechnical campaign with boreholes in greater distances varying from 100 to 150 m (**Figure 8**).

On the contrary to the Delhi Metro, the EPB-TBM for Jaipur is designed with an open spoke-type cutterhead with opening ratio of 60%, suitable for the sandy formations (**Figure 9**). The increased opening ratio of the cutterhead is more suitable in applying uniformly the earth pressure on the excavation face but is essential to maintain always the pressure in every stroke of the machine as it moves forward, in order to avoid face instabilities and excessive surface settlements.

For both cases, Delhi and Jaipur Metro, the water table is revealed below the tunnel invert.

### **3.1 Geotechnical assessment and interpretation**

Following the geotechnical campaign and the laboratory tests on the samples, which are taken from the executed boreholes, the geotechnical model will be determined along with the characteristic geotechnical parameters of the soil formations. The modulus of elasticity (E), the effective cohesion (c′), effective friction angle (φ′ or phi), and the unit weight (γ) of the soil formations are the main geomechanical properties required for designing the tunnel excavation. While the effective cohesion and friction angle are critical parameters that will govern the extent of the plastic zone shape, the modulus of elasticity controls the magnitude of the surface settlements, and the tunnel depth is the geometrical parameter that will affect the shape of the surface settlement trough.

When tunnelling with EPB shields, the surface settlement development is related with the machine operation in conjunction with the soil properties. The so-called volume loss, resulted from the face extrusion (face loss), the steering gap closure (radial loss on the shield), and the annular void between the segmental lining and the soil, are basic key parameters that will affect the magnitude of the soil's deformations. It is obvious that the soil properties are the governing factors for determining the operation parameters of the TBM, in order to keep the volume loss within acceptable limits, to control the plastic zone formation, and thus to minimize the surface settlements.

Since the tunnel excavation with EPB shields is a short-term procedure, the properties of the soil should be addressed in a careful manner in order to describe the actual geotechnical conditions. Continuous support on the tunnel face is applied by using the freshly excavated soil, which completely fills up the work chamber (muck). The supporting pressure is achieved through control of the incoming and outgoing materials in the chamber, i.e., through regulation of the screw conveyor rotation speed and of the excavation advance rate.

Underestimation of the mechanical characteristics of the soil medium and especially underestimation of the modulus of elasticity E can lead to the calculation of unreasonable and extremely high values of support pressure, with detrimental effect on the operation of the machine, as demonstrated below:

a.The muck exhibits a shear resistance that, for a given internal friction angle, increases with the support pressure. Since muck with shear resistance does not behave like a fluid, the stress field in the work chamber, and thus the distribution of the support pressure along the tunnel face, is not under control (**Figure 10a**).

*Geological map of greater area of Jaipur—Rajasthan (Source—Geological Survey of India) (Jaipur Metro Phase 1B area marked with magenta circle).*

**Figure 7.**

**Figure 8.** *Geological longitudinal section of shallow tunnel in Jaipur Metro Phase 1B, with boreholes [4]—Silty sand along the entire alignment marked with orange color.*

### **Figure 10.**

*Typical problems caused by high support pressure in EPB shields.*

This is a nonoptimal situation from a stability point of view. The problem of nonuniform distribution of support pressure becomes even worse in the case of a mixed tunnel face due to the widely differing stiffnesses of the rock and soil layers.


Conservatism in determining the geotechnical properties of the soil not only can result in overdesigning the permanent segmental lining of the tunnel but also can lead to unreasonably high support pressures with immediate detrimental effect on the operation of the machine. Furthermore, for very shallow tunnel cases, the blowout is also a case that needs to be carefully studied before determining the final EPB support pressure.

For soft soil formations, modulus of elasticity can be evaluated on the basis of the SPT results and the respective uncorrected N60 value, as per CIRIA report 143. Based on this report, the consistency of soils is determined by SPT N values as illustrated in **Table 1**.


**Table 1.**

*Consistency of soil formation based on SPT N values as per CIRIA, R143.*

**Figure 11.**

*Characteristic plots showing the variation on SPT N value with depth along the tunnel alignment. Delhi Metro (CC23), left plot; Jaipur Metro Phase 1B, right plot.*

In order to establish the representative SPT N60 values along the tunnel, the respective values should be plotted versus depth for the entire alignment considering the results from all the boreholes (**Figure 11**).

Further plots can also be prepared for different stretches, e.g., considering the boreholes results of the same stretch (between two stations). With this approach the entire tunnel stretch can be divided in sub-areas, and different fit lines can be determined as linear equations linking the representative N value with depth (e.g., N = 2.8z + 5, whereas z = depth below ground surface). Therefore, by considering the CIRIA report, the modulus of elasticity can be calculated versus the SPT N60 values. As an example, for silts, sandy silts, and silty sands, the equation E = 0.7–1.0 N60 can determine the modulus of elasticity. For the derivation of the cohesion and friction angle, consolidated undrained triaxial strength tests (CU) when possible and direct shear strength tests can be used.

As already mentioned, underestimation of the soil properties when calculating the EPB pressure can lead to excessive problems during construction of the tunnel. Therefore, it is essential to study in detail the soil behavior not only through the results of the laboratory tests but also by investigating physical exposed open cuts in order to understand the "stand-up time" of the formations. "Stand-up time" is the short-term stability of the soil without the application of support measures and mainly reveals the existence of increased shear strength on the formation which sometimes is difficult to be determined by laboratory tests.

As a conclusive remark, it is worth mentioning that when designing the excavation of a tunnel with shield machine, it is proper to use the upper values of the soil parameters, while when designing the permanent lining, lower values can be used with caution to avoid unnecessary overdesign.

### **3.2 EPB-TBM principles**

The tunnel boring machines that provide immediate peripheral and frontal support simultaneously belong to the closed-face group. They excavate and support both the tunnel walls and the face at the same time. Except for mechanical support machines, they all have the, so-called, cutterhead chamber at the front, separated by the remaining part of the machine by a bulkhead, where a confinement pressure is maintained in order to actively support the excavation and/or balance the hydrostatic pressure of the groundwater. The TBM is moving forward through hydraulic cylinders that are pushing the already erected segmental lining (**Figure 12**). Respective video for the operation of an EPB can be found in the following link from Herrenknecht: https://www.herrenknecht.com/en/products/productdetail/epb-shield/.

**Figure 12.**

*Typical view of earth pressure balance (EPB) shield (Source: Herrenknecht).*

With earth pressure balance shields, the cohesive soil loosened by the cutting wheel serves to support the tunnel face, unlike other shields which are dependent on a secondary support medium (e.g., slurry shields). The area of the shield in which the cutting wheel rotates is known as the excavation chamber and is separated from the section of the shield under atmospheric pressure by the pressure bulkhead. The soil is loosened by the cutters on the cutting wheel, falls through the openings of the cutting wheel into the excavation chamber, and mixes with the plastic soil already there. Uncontrolled penetration of the soil from the tunnel face into the excavation chamber is prevented because the force of the thrust cylinders is transmitted from the pressure bulkhead onto the soil. A state of equilibrium is reached when the soil in the excavation chamber cannot be compacted any further by the native earth and water pressure. The excavated material is removed from the excavation chamber by an auger conveyor (screw of Archimedes). The amount of material removed is controlled by the speed of the auger and the cross section of the opening of the upper auger conveyor driver. The auger conveyor conveys the excavated material to the first of a series of conveyor belts. The excavated material is conveyed on these belts to the so-called reversible conveyor from which the transportation gantries in the backup areas are loaded when the conveyor belt is put into reverse. The tunnels are normally lined with steel- or fiber-reinforced lining segments, which are positioned under atmospheric pressure conditions by means of erectors in the area of the shield behind the pressure bulkhead and then temporarily bolted in place. Mortar is continuously forced into the remaining gap between the segments' outer side and the rock through injection openings in the tailskin or openings directly in the segments.

The principle of EPB-TBM operation is that pressurizing the spoil held in the cutterhead chamber to balance the earth pressure exerted holds up the excavation. If necessary, the bulkhead spoil can be made more plastic by injecting additives from the openings in the cutterhead chamber, the pressure bulkhead, and the muckextraction screw conveyor. By reducing friction, the additives reduce the torque required to churn the spoil, thus liberating more torque to work on the face. They also help maintain a constant confinement pressure at the face. The hydrostatic pressure is withstanding by forming a plug of confined earth in the chamber and screw conveyor; the pressure gradient between the face and the spoil discharge point is balanced by pressure losses in the extraction and pressure relief device (**Figure 13**).

Face support is uniform. It is obtained by means of the excavated spoil and additives. Injecting products through the shield can enhance additional peripheral support. For manual work to proceed in the cutterhead chamber, it may be necessary to create a sealing cake at the face through controlled substitution (without loss of confinement pressure) of the spoil in the chamber with bentonite slurry. The architecture of this type of TBM allows for rapid changeover from closed mode to open mode operation and vice versa. The tunnel lining is erected inside the TBM tail skin, with a tail skin seal, ensuring there are no leaks. Back grout is injected behind the lining as the TBM advances.

### **Figure 13.**

Following the EPB operation principles, it is obvious that the tunnelling-induced soil movements can occur at the longitudinal and radial direction. The face extrusion can give rise to longitudinal displacements, while the gap around the shield and the tail of the machine will introduce radial convergence. The volume of soil that intrudes into the tunnel owing to the pressure release at the excavated face will be excavated eventually. This movement of soil is defined as the volume of material that has been excavated in excess of the theoretical design volume of excavation and is called "ground loss" or "volume loss." Therefore the volume loss for a tunnel excavated with TBM occurs in three stages:


Since the above ground losses are related with the cavity displacements, consequently this is the mechanism that will give rise to the development of the surface settlements. Designing a TBM tunnel means control of the volume loss, and this can be achieved with:


**Figure 14.** *Overview of a TBM tunnel interstation design.*

### **3.3 TBM tunnel interstation design (TID)**

Following the finalization of the ground design parameters, documented in the Geotechnical Interpretation Report (GIR), the design for the excavation and support of the tunnel with the TBM will follow and is called TBM tunnel interstation design. The scope of the TBM TID is the complete tunnel boring design in order to ensure, the safety of the tunnel structure itself, the surface and subsurface structures adjacent to the project and, finally, the confinement of the ground deformations within permissible limits. A TBM TID (**Figure 14**) is divided in the following parts:


According to the analysis methodology that is related with the control of the induced surface settlements, the following basic phases are considered in a TBM TID:

	- Category 1: "Stable," elastic behavior
	- Category 2: "Stable for limited time," elastic-plastic behavior with limited plastic zone width
	- Category 3: "Unstable," elastic-plastic behavior with extended plastic zone width

Two categories to describe the deformation phenomena:


And finally two categories to describe the cavity behavior:


### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

iii. *Therapy phase*: According to the results derived from the previous phase (II), the required parameters (e.g., support pressure, TBM operation parameters) are determined in order to secure the face and cavity stability and thus to minimize the induced surface settlements.

After studying the tunnel alignment, the designer will define the critical sections of the project, which is a combination of ground properties, water table height, overburden height, axial distance between the two tunnel bores (in cases of twin bore tunnels), locations of the existing buildings, and important structures. The outcome of the study will determine the tunnel sections of the TID. In detail the characteristic tunnel sections that need to be used in a TID will include mainly the following cases:


During the tunnel excavation, the development of the surface settlements will be recorded through an established geomechanical monitoring program, in order to check the predictions of the TID and to calibrate if required the operation parameters of the EPB machines (e.g., support pressure).

### *3.3.1 Part A: face pressure calculation*

The tunnel face and cavity reaction is always related with the following main key parameters:



### **Table 2.**

*Critical sections along CC23 of Delhi Metro Phase 3.*


### **Table 3.**

*Critical sections along Jaipur Metro Phase 1b.*

For the determination of the required support pressure, different calculation sections are required to be defined as explained in the previous paragraph. To present the design methodology in more detail, the Delhi Metro Phase 3 and Jaipur Metro Phase 1b will be used as case studies hereafter.

Considering the geological longitudinal section of the contract CC23 of Delhi Metro Phase 3, snapshot of it presented in **Figure 5**, the critical sections defined along the entire alignment are given in **Table 2**.

Considering now the geological longitudinal section of the Jaipur Metro Phase 1b, snapshot of it presented in **Figure 8**, the critical sections defined along the entire alignment are given in **Table 3**.

For all sections of **Tables 2** and **3**, where the tunnel alignment is below roadway and buildings with less or equal to 5 storeys, a uniform surface load of 50 kPa is considered. This approach is common when designing shallow tunnels in urban environment, since a minimum surface load is always required in order to consider any unforeseen load case or any low height (<5 storeys) future structure. Sections U1, U2, and CH are the most critical cases, since the existing structures are in very close proximity with the tunnel crown; thus special considerations will be presented hereafter.

The geomechanical parameters of the soils for the abovementioned Metro projects (Delhi and Jaipur) are also given in **Table 4**.

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*


### **Table 4.**

*Geotechnical design parameters.*

### **Figure 15.**

*CC23 Delhi Metro. Calculation of plastic zone width according to Massinas and Sakellariou solution [6] for different tunnel depths (as per sections of* **Table 2***). Proposed range of support pressure also shaded with orange color.*

According to the international literature and case studies from various projects around the world, the applied support pressure for the tunnel construction with TBM shields is considered according to the earth pressure at rest and active earth pressure or according to Anagnostou and Kovari [5]. Recently, the Massinas and Sakellariou [6] analytic solution is also used to determine the required support pressure for EPB

### **Figure 16.**

*CC23 Delhi Metro. Calculated EPB support pressure for different overburden heights as per the critical sections F1–F8 of* **Table 2***. Pressures Pr and Pa are calculated based on earth pressure at rest and active earth pressure, respectively. Psm (min) and (max) give the range of calculated support pressure based on Massinas and Sakellariou closed form solution [6].*

shields [7, 8]. Considering the geotechnical parameters of the soil, the Massinas and Sakellariou solution [6] is used to calculate the min and max value of the support pressure (**Figure 15**). The proposed range of pressure, given by Massinas and Sakellariou, coincides with minimum plastic zone width (less than 0.5–1 m). Since the aim is to minimize the induced surface settlements, controlling the plastic zone formation to minimum width can determine the required value of support pressure.

Considering also earth pressure at rest (Ko) and active (Ka), the EPB support pressure is calculated for all the different sections F1–F8 of **Table 2**, for CC23 Delhi Metro. All the calculation results are plotted in the diagram of **Figure 16**.

It is obvious that the calculation of Pr by considering the earth pressure at rest leads to non-pragmatic values, not feasible for the tunnel construction. On the other hand, the active earth pressure seems to give more realistic results since Pa is almost coincide with the upper values of the pressures range, as calculated according to Massinas and Sakellariou solution. Most realistic results can be derived by Massinas and Sakellariou analytic solution; since the soil-tunnel-surface interaction is considered and the plastic zone width is derived, therefore the proposed support pressure to be applied by the EPB is concluded, and the input for part B is defined. More details for the application of Massinas and Sakellariou method can be found in Refs. [6–8].

For the special case U1, the bored tunnels of CC23 (Line-8) underpass the existing tunnels of Line-2 as shown in **Figure 17**. The objective of securing the safe operation of the existing Metro Line-2 is related with the reduction of the soil deformations and thus minimizing the displacement of the existing segmental lining. In order to examine the soil-structure interaction by calculating the plastic zone formation around the tunnel and calculating the EPB support pressure (Psm) range, again the method of Massinas and Sakellariou needs to be applied, with the assumptions that follow.

The presence of the existing Line-2 is taken into consideration by assuming as an upper boundary of the semi-infinite space its invert foundation level; thus a total overburden of 4 m (conservatively instead of 4.5 m) is considered, to examine the interaction between the new and the existing tunnel (**Figure 18**—left); the total earth pressure (due to gravity) at the real depth of the tunnel is taken into account, by applying a uniform load (Po) at the upper boundary of the half-space.

**Figure 17.** *CC23 Delhi Metro. Key plan and typical cross section of Line-8 bored tunnels under existing Metro tunnels of Line-2 [6, 7].*

**Figure 18.**

*CC23 Delhi Metro. Physical and analytical model for calculating the required support pressure of Line-8 TBM under the existing Metro tunnels of Line-2 (left). Analytic calculation of plastic zone width (right) [6, 7].*

By using the analytic solution formula [6], the calculation of the plastic zone shape around the tunnel is performed for different values of the support pressure (**Figure 18**—right). Minimum plastic zone of less than 0.5 m is derived for support pressure range of 1.8–2.2 bar, while for 2.4 bar, the soil around the cavity remains within the elastic domain. Therefore by considering a support pressure within the range of 1.8–2.2 bar, the stress redistribution remains within the elastic domain, and thus minimum displacements are expected to be developed. The above-derived conclusion is the initial observation for the tunnel interactions and is further analyzed (part B of TID) in full detail through 3D elastic-plastic FDM multistaged simulations with powerful software Fast Lagrangian Analysis of Continua in 3 Dimensions (FLAC3D) [9].

For the second special case U2 of CC23 Line-8, the TBM working on the up line underpass the existing Nala, while for the construction of the down line, the TBM will underpass both the abutments of the existing bridge as well as the Nala. Therefore, the critical area is approximately 50 m, starting from the neutral zone (cut and cover structure). Typical plan view of the horizontal alignment and the longitudinal sections of the down line tunnel is given in **Figure 19**.

Special survey on the bridge's foundation (trial pit) revealed pad foundation with thickness of approximately 2.33 m. The bridge is new, and the span is approx. 26 m with general dimensions in plan 35 × 14 m (length × width). The superstructure is made of prestressed reinforced concrete box and lay on the abutments through bearings, as presented in **Figure 20**.

As in the previous case U1, also for this critical section U2, the presence of the existing bridge is considered by assuming as an upper boundary of the semi-infinite space its foundation level; thus a total overburden of 4 m is considered, to examine the interaction between the bridge and the tunnel. Furthermore, the total earth pressure (due to gravity) at the real depth of the tunnel is also taken into account, and a total uniform load (Po) at the upper boundary of the half-space is applied (**Figure 20**—right). Considering the longitudinal section of the tunnel, different overburden heights are considered for the analytic calculations in order to determine the required support pressure. **Table 5** summarizes the calculated support pressure as per TID—part A for different overburden heights along the bridge and Nala area.

For the calculation of Psm pressures, the same principle, as in previous case U1, is used. Thus, the required pressure is calculated in order to keep the plastic zone width around the tunnel below 0.5 m. As per part B of TID, further analysis of the

*CC23 Delhi Metro Line-8. Horizontal alignment (left) of up and down lines in Nala (water canal) area. Longitudinal section (right) of down line shows tunnel under the bridge abutments and the Nala [10].*

### *Tunnel Engineering - Selected Topics*

### **Figure 20.**

*CC23 Delhi Metro Line-8. Cross section of Line-8 tunnels under existing bridge (left). Physical and analytical model for calculating the required support pressure of Line-8 TBM under the existing bridge (right) [10].*


### **Table 5.**

*Support pressure along tunnel alignment, for different overburden heights.*

settlement development is required to be elaborated through 3D calculations with FLAC3D [9], considering the different support pressures given in **Table 5**.

For the case of Jaipur Metro Phase 1b, the critical section is more complex than in previous examples from Delhi Metro. The importance of the monument of Chandpole Gate and the extremely shallow depth of the tunnels (**Figure 21**) required further study to determine the most appropriate method for underpassing with the TBM.

For the determination of the structure's foundation system, dimensions and condition, special survey was carried out, consisting of nine trial pits that were executed in certain locations near the Gate. According to the findings from the trial pits on both sides of the structure, the Gate rests on stone masonry foundation with depth of approx. 2.0–2.4 m, whereas below the main arch, a boulder layer was found with depth ~1.5 m near the Gate's walls and ~1.0 m below the mid part. Water pipes and other utilities were found below the two passageways and the main arch. A typical section of the Gate with its foundations and the underpassing TBM tunnels is given in **Figure 21** (right), indicating the soil cover of approx. 4.5 m between the bottom of the Gate's foundation and the tunnels' crown.

Examination and co-evaluation of the geotechnical investigation and Gate foundation survey results was jointly performed by the designer and the contractor, aiming to decide on the necessity of soil strengthening through grouting below and around the Gate's foundation. The grain size distribution of the in situ soil practically excluded the successful and efficient application of low-pressure cement grouting, even with use of microfine cement. Other grouting methods that would be more appropriate for this soil type, e.g., jet grouting and compensation grouting, were excluded, as the risk of soil disturbance, instability, temporary liquefaction, and settlement below the Gate foundation was considered too high for

**Figure 21.** *Jaipur Metro Phase 1b. Horizontal alignment and typical cross section of TBM tunnels under Chandpole Gate [8].*

### **Figure 22.**

*Jaipur Metro Phase 1b. Physical and analytical model for TBM support pressure calculation under Chandpole Gate (left). Analytic calculation of plastic zone width (right) [8].*


### **Table 6.**

*Critical sections along CC23 of Delhi Metro Phase 3 with proposed support pressure for settlement analysis.*

the significance and vulnerability of the structure. Additionally, the loading of the structure itself and the long period of its application was naturally assumed to have made the soil below the Gate compact. This assumption could be partly verified by the soil inspection in the trial pits. In the light of the above considerations, the designer and the contractor decided to underpass the structure without any pretreatment of the soil, relying on the appropriate TBM operation, mainly in terms of applied face pressure and annular gap grouting application.

Following the same procedure as for the Delhi Metro cases, the presence of the existing Gate was considered by assuming its foundation level as an upper boundary of the semi-infinite space. Thus a total overburden of 4.5 m has been considered to examine the interaction between the Gate and the tunnel, as presented in **Figure 22**. Furthermore, the total earth pressure (due to gravity) at the real depth


**Table 7.**

*Critical sections along Jaipur Metro Phase 1b with proposed support pressure for settlement analysis.*

of the tunnel is also considered, and a total uniform load (Po) at the upper boundary of the half-space is applied. The comparison between the physical model of the problem and its equivalent model for the analytical calculations is illustrated in **Figure 22** along with the results derived by the application of Massinas and Sakellariou analytic solution.

A minimum plastic zone width (<0.5 m) is calculated for a mean support pressure within the range of 1.4–1.5 bar, while for 1.6 bar, the soil around the excavation remains within the elastic domain. The value of 1.5 bar was selected for the execution of 3D numerical analyses for settlement prediction.

To summarize the results from part A of TID, **Tables 6** and **7** give the final proposed TBM support pressures used for the settlement analysis in part B of TID.

For all the above cases as summarized in **Tables 6** and **7**, the adopted support pressure for the settlements analysis is based on the calculation results as per Massinas and Sakellariou solution [6]. Comparing the two tables, it is obvious that for the case of Jaipur Metro Phase 1b, higher pressures are considered, which resulted from the requirement to keep the plastic zone width around the tunnel less than 0.5 m since the entire tunnel alignment is very shallow (7–12 m) compared to CC23 Delhi Metro.

Furthermore, even along the Jaipur Metro Phase 1b, the tunnel alignment is below the busy roadway; in close vicinity with the tunnel's sides, the very old buildings of the "Pink City" and other important monuments exist (**Figure 23**), and thus the settlement needs to be kept to absolute minimum values to avoid any damages.

### *3.3.2 Part B: surface settlements*

Following the derivation of the support pressures in part A, settlement analysis is required in part B of TID in order to numerically calibrate the range of the derived support pressures. The outcome form this step of the design will finalize the set of values to be used during construction that will result to minimum volume loss and consequently will reduce the magnitude of the surface settlements. Since the

**Figure 23.** *Jaipur Metro Phase 1b. Important monuments of the "Pink City" along the tunnel alignment [12].*

"geometric constraints" of the problem is bounded with the construction sequence of the tunnel (excavation-segment installation-grouting) that takes place near the face, the most effective type of analysis in order to investigate the face behavior and the cavity convergence around the shield is to perform 3D calculations.

Assuming stress-induced failure mode of ground mass, the computational three-dimensional analysis is performed with a continuum model approach, using the finite difference software FLAC3D [9]. In order to ensure the accuracy of the results and to avoid any "boundary effects," certain parametric 3D analyses are performed (benchmark tests) with different 3D meshes in dimension and density. In all vertical boundaries, the horizontal movements normal to the boundary were restricted. On the bottom boundary, soil movements were restricted in all directions. In order to avoid any effects on the surface settlements shape and magnitude, due to boundary positions, the proper grid dimensions were implemented in the models. Thus, for the vertical boundaries on each side of the tunnels, a distance of 10 x diameters is selected. For the extension of the grid below the tunnel spring line, a distance of 2 x diameters is considered. The distance between the vertical boundary normal to the tunnel direction in front of tunnel's face and the final excavation face is selected equal to 10 x diameters [13]. Typical views of the models are presented in **Figure 24**.

For the special cases U1, U2, and CH, different models were constructed following the same boundary conditions and dimensions as presented before. For the underpass of Yellow Line (U1), both the existing Line-2 and the Hauz Khas station (in close vicinity with the tunnels) were simulated (**Figure 25**).

**Figure 24.**

*Typical FDM models used for settlement analysis. CC23 Delhi Metro on the left and Jaipur Metro Phase 1b on the right [11, 12].*

**Figure 26.**

*Section U2: underpass of bridge and Nala. Typical FDM model used for the settlement analysis (left). Additional support measures for soil strengthening under the bridge's abutment (right). CC23 Delhi Metro [10].*

*Section CH: underpass of Chandpole Gate. Typical FDM model used for the settlement analysis. Jaipur Metro Phase 1b [15].*

For the settlement analysis at bridge and Nala area (U2), the 3D model considered the cut and cover structure of the neutral station the bridge's foundation. The anaglyph of the Nala with the physical slopes is also detailed simulated (**Figure 26**). Due to the fact that the tunnel excavation below the Nala area and the bridge would start from neutral station, additional support measures are required for strengthening the soil (in front of the launching area) during the TBMs entrance in the ground, since at this stage there will not be adequate support pressure. Therefore, fully grouted self-drilling bolts 14 m in length 32 mm in diameter are foreseen to be installed perpendicular to the face wall of the neutral station, around the tunnel excavation area (**Figure 26**). The respective support measures have been also simulated in the respective 3D analysis as illustrated in **Figure 26**.

The last simulation (CH) is related with the settlement analysis of Chandpole Gate during the TBM tunnel boring below the monument. Detailed 3D model prepared simulating the Gate's foundations and the sequential construction of the tunnels with the two TBM shields (**Figure 27**).

In all the 3D analyses, the exact geometry of the segments and the annular gap of the tail shield grouting were simulated as presented in **Figure 28**. The excavation simulation is sequential, and in each step certain actions are considered. The advance of the TBM can be simulated either by considering the total length of the shield (e.g., 9 m in length) as one excavation step or slower advance of the shield equal to one segment can also be considered in the simulation. In the first case, the total load cases can be reduced, and thus the calculation time can be significantly

### **Figure 28.**

*Typical 3D view (left) of segmental lining and grout for backfilling the annular gap simulation. Typical 3D view (right) of segmental lining (blue) and grout (red) applied on soil (white) [11, 12].*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

minimized. Both methods of simulating the TBM advancement will give reliable results. It is preferable in cases of critical underpasses (e.g., U1, U2, and CH) for the excavation step to follow each segment length. Therefore, in such cases the analysis can commence with an initial excavation step equal to the shield's length, and as the TBM simulation reaches the important structure, the excavation step is to be reduced and should follow the segment length.

In each excavation step simulation, the EPB mean support pressure (as calculated in part A of TID) is applied at the tunnel face. Around the shield the same pressure with the face can be applied either with a triangular distribution reaching zero value on the tail of the shield or uniformly up to the half-length of the shield. Both methods are feasible, and the designer can decide based on the abilities of the software that he uses. In the previous step (behind the excavation), green grout (a modulus of elasticity equal to 1 GPa or less can be used) to simulate the backfill on the annular gap is applied along with the segmental lining. Two steps behind, the grout mature is simulated by applying its final properties (modulus of elasticity equal to 10 GPa). There are also other methods to simulate the TBM excavation, for example, interface elements can be used in order to simulate the behavior of the shield gap or the annular gap around the segments. Those methods need advance modelling and increased computational time, which are more appropriate for academic research and are not common practice in the design industry, since there are many unknown parameters that need to be defined in order to assign the correct properties to the interface elements.

In order for the calculated surface settlements to reach equilibrium behind the shield, a minimum excavation simulation of total tunnel length equal to 5 x diameters is mandatory. For special cases additional excavation length may be required to be simulated. In **Table 8** the total simulated length of tunnel excavation is given for all the performed 3D analyses along with geometrical required information.

In all the 3D analyses performed (**Table 8**), the plastic zone width was calculated below 0.5–1.0 m (**Figure 29**). It is obvious that this value was expected as it was initially derived based on Massinas and Sakellariou solution [6].

As it is evident from the diagram in **Figure 30**, the simulated support pressure (Pfdm, black dotted line) in the FDM 3D analysis is within the lower range of the Massinas and Sakellariou support pressure. The higher the overburden, the higher the pressure required to support the tunnel excavation, following second-order polynomial curve (e.g., the equation of trendline is y = 8.89x<sup>2</sup> –0.4x + 5.67). On the contrary, the calculation of the support pressure, based on the earth pressure at rest, follows, as expected, linear line that results to unrealistic values, the application of them can jeopardize the effective operation of the TBM.


### **Table 8.**

*3D models simulation properties.*

### **Figure 29.**

*Calculated plastic zone width (in red) as per TID of CC23 Delhi Metro (left) and as per TID of Jaipur Metro Phase 1b (right) [11, 12].*

**Figure 30.**

*CC23 Delhi Metro. Calculated EPB support pressure for different overburden height as per critical sections F1–F8 of* **Table 2** *and including the results from the 3D FDM analysis (Pfdm).*

Therefore, it is of greatest importance to avoid unnecessary conservatism when calculating the required TBM support pressure. The knowledge of the geotechnical conditions of the groundmass and the groundwater through proper established geotechnical campaign, in conjunction with the knowledge of the TBM operation principles, are the key factors for determining the required support pressure.

Following the establishment of the required support pressure in order to restrict the plastic zone formation width, the next step as per the TID is to evaluate the calculated induced surface settlements, in respect with the vulnerability of the existing surface and subsurface structures.

Following as example the CC23 Delhi Metro, the adopted support pressures in the 3D analyses resulted in minimum plastic zone widths in the order of 0.5–1.0 m and maximum surface settlements that are not exceeding 12 mm. After the total excavation of the two bores, the maximum calculated figures (independent of the overburden height) are bounded between 11 and 12 mm, as clearly illustrated in **Figure 31**.

It is evident that the range of maximum calculated surface settlements (7–11 mm) after the first bore excavation simulation is wider than the range calculated after both bore excavations. The overburden height is the key factor for this, and it is evident from the analyses results that the lower the overburden height, the greater the % of the developed surface settlements during the excavation of the first bore. As presented in the diagram of **Figure 32**, almost 90% of the total developed settlements are calculated during the first bore excavation for the shallowest case

### **Figure 31.**

*CC23 Delhi Metro. Maximum calculated surface settlements after first (left) and second (right) bore excavation simulations for sections F1–F8 of* **Table 2** *(horizontal axis in m and vertical axis in mm).*

**Figure 32.**

*CC23 Delhi Metro. Percentage of maximum developed surface settlements after first (left) bore excavation simulation for sections F1–F8 of* **Table 2***. Diagrammatic comparison of maximum calculated surface settlements after the first and second bore excavations (right).*

(11 m overburden), while for the deepest part of the alignment (25 m overburden), 60% of the total settlements are calculated (**Figure 32**). Of course the two bores' axial distance is another parameter which affects the surface settlements development range, but for shallow tunnelling in Metro projects, optimum value for the dimensionless ratio of the tunnels' axial distance to the diameter of the tunnel varies between 2.2 and 2.5. Even greater values than 2.5 can be applied, but it is economically preferable to keep the above range for the following reasons:


As it is evident from **Figure 32** (right), the projected maximum calculated surface settlements from first bore excavation, in respect to the different overburden heights, follow an exponential fit curve bounded by extreme values of 7 and 11 mm. With the proposed and simulated range of support pressure (0.8–1.5 bar), the totalv maximum calculated settlements after the excavation of the second bore follows a much steeper exponential fit curve bounded between 11 and 12 mm.

Further analysis of the curves of **Figure 32** (right), using the Gaussian formulae

$$\text{Further analysis of the curves of Figure 32 (right), using the Gaussian formula}$$

$$S = S\_{\text{max}} \exp\left(\frac{-\mathbf{y}^2}{2i^2}\right) = \frac{V\_L}{i\sqrt{2\pi}} \exp\left(\frac{-\mathbf{y}^2}{2i^2}\right) \tag{1}$$

to calculate the surface settlements through the volume loss and the constant K, give results that are in good agreement with the 3D FDM, considering volume loss per bore equal to 0.6% for all the different overburden heights. Thus, proper calculation of the support pressure range controls the surface settlements and can lead to the same volume loss across the alignment independent of the overburden height.

The analysis of section CH (Chandpole Gate underpass) is another example to discuss the surface settlement development. Following the simulation of the support pressure as derived by Massinas and Sakellariou solution [6], the maximum 3D FDM calculated surface settlements are given in **Figure 33**.

### **Figure 33.**

*Jaipur Metro Phase 1b. 3D FDM calculated surface settlements after the first and second bore excavation simulations versus actual final monitored surface settlements [8].*

After the first bore excavation simulation, the maximum surface settlement is calculated as 5 mm, while the second bore excavation only increases the surface settlement to 4 mm above the second bore without giving rise to the total maximum settlements. As already discussed in **Figure 32** (left), the shallow depth can give rise to the absolute maximum value of the surface settlements after the first bore excavation. The section CH, which is a case with extreme low overburden height, shows that even 100% of the final value of the surface settlements can be developed only during the first bore excavation, while the second tunnel construction can extend only the shape of the settlement trough without increasing the absolute maximum value. This is also visible from the actual monitored results; thus after the first bore construction (**Figure 34**), the maximum measured settlements at the Gate area are 6 mm, while after the second TBM underpass (**Figure 33**), only extent of the trough above its axis is measured, with absolute maximum value of the settlements equal to 2 mm.

Indeed, the application of the design support pressure (**Figure 35**—left) during the construction is the key factor along with the adequate tail shield grouting, in order to control the soil's settlements; the secret is to keep the chamber pressure constant in each stroke of the TBM, thus in each excavation step as the shield is moving forward (**Figure 35**—right).

To understand how the earth pressure is properly applied in the excavation chamber, the respective plots during the calibration of the TBM 1 are given in **Figure 36**. As it can be seen from the left diagram of **Figure 36**, during the calibration stage, the pressure drops at the beginning of the excavation for the construction of ring 78. This means that the chamber pressure during the last stroke of the previous ring 77 is not kept constant and as per the design requirements.

Therefore, at the beginning of ring 78 excavation, the support pressure inside the chamber is zero, and only after the excavation advanced 250 mm the pressure starts to rise. After the calibration stage, the support pressure inside the excavation chamber (**Figure 35**, right; and **Figure 36**, right) is kept constant along the entire excavation length. Along the calibration area, the maximum measured surface settlements were in the order of 10–14 mm. During the excavation simulation of the first tunnel, the maximum calculated deflection of the Gate's foundation (above the tunnel) is 1/1300, while the maximum differential settlement between the two foundations of the Gate is calculated 5 mm with an angular distortion of 1/1200.

### **Figure 34.**

*Jaipur Metro Phase 1b. Variation of measured surface settlements below the Gate for various relative positions of the first TBM (1) versus the final surface settlement along the Gate (left) and snapshot of recorded settlement values for cutterhead at ring position 110 (approx. 20 m, i.e., 3D ahead of the Gate) [8].*

**Figure 35.**

*Jaipur Metro Phase 1b. Variation of average face pressure per ring excavation versus the design range (as per section CH analysis), for TBM 1 (up line) and TBM 2 (down line) (left) [8].*

### **Figure 36.**

*Jaipur Metro Phase 1b. Variation of face pressure of middle sensors along excavation of last ring of calibration drive (left) and first ring under the Gate (right) vs. the machine stroke [8].*

Along the alignment of Jaipur Metro (3D FDM analysis J1–J3 as per **Table 7**), the maximum calculated surface settlements are in the order of 9–10 mm (**Figure 37**). As it is evident from **Figures 23** and **37**, all the buildings along the alignment are at 8 m distance from the tunnel bore axis. No structures are aligned above the tunnel centerline (except Chandpole Gate). Therefore, at the building location, the maximum calculated surface settlements are 4–5 mm. Maximum differential settlements are calculated 2 mm with an angular distortion of 1/2000.

For the case U1 of CC23 Delhi Metro, two analyses were performed. The first (as presented) includes the station structure, while the second is also performed without simulating the station. The purpose of the second analysis is to examine the effect of the station's stiffness on the development of the surface settlements. The presence of the stiffness of the existing tunnels and the Hauz Khas station affects the magnitude and the shape of the calculated surface settlements. In the first case, where the station is not simulated, the stiffness of the existing tunnels (Yellow Line) contributes to the development of the surface settlements by reducing their magnitude of approx. 10–15%.

This is evident in the diagram of **Figure 38** (left), where the maximum calculated surface settlements which are 30 m before the Yellow Metro Line is approx. 5 and 9 mm after the first and second bore excavations, respectively, while at a section just above and parallel with the up line axis of the Yellow Line, the surface settlements are calculated 4.5 and 8 mm, respectively. Therefore, a vertical shift on the settlement trough is observed, due to the stiffness of the existing tunnels, without affecting the overall shape of the trough but only the maximum value on

### **Figure 37.**

*Jaipur Metro Phase 1b. 3D FDM calculated surface settlements after total excavation of both bores (values in meters) (left). Projected maximum calculated surface settlements along tunnel alignment (right).*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

the settlement curve. On the other examined case, the simulation of the station has a more clear influence on both the shape and the magnitude of the surface settlements. As it is evident from **Figure 38** (right), both vertical and horizontal shift of the settlements curve is observed. The vertical shift is affecting the magnitude of the maximum calculated settlements which is in the order of approx. 3 and 6 mm after the first and second bore excavations, respectively. Thus, a reduction of approx. 30–40% is observed from the maximum values (5 and 9 mm) of the surface settlements. After the first bore underpass simulation, vertical displacements are calculated at the crown of the existing tunnels, with maximum values within the range of 3 and 5 mm (for both examined cases). After the second bore excavation, a slight increase on the crown vertical displacements is calculated, and the final maximum values are in the order of 7 and 8.5 mm (for the cases with and without station simulation, respectively). Again, the stiffness of the simulated station reduces the actual values of the crown displacements by producing vertical and horizontal eccentricity in the calculated curves, almost in the same degree as in the surface settlements which are presented before.

The maximum vertical displacements at the invert of the existing tunnels are calculated within the range of 3.5 mm (case with no station) and 6 mm (case with station) after the first bore excavation, with a maximum angular distortion in the order of 0.25‰ (no station simulation) having a peak value of 0.3‰ (station simulated) at the area which is in close vicinity with the station.

The advance of the second bore below the existing tunnels increases the invert displacements to 7 mm for the case where the station has been simulated and to 8.5 mm at the second case without the station been simulated, as it is evident in **Figure 39**. An increase on the differential displacements is also calculated with a maximum slope value reaching 0.4‰ (**Figure 40**). To sum up, for the case of U1 section, the maximum vertical displacements at the crown and the invert of the existing tunnels are calculated within the range of 3.5–6 mm with a maximum angular distortion of 0.3–0.35‰, after the first bore excavation simulation. The completion of the second bore excavation simulation gives rise to maximum vertical displacements at crown and invert equal to 7 and 8.5 mm, respectively, with a maximum angular distortion reaching the value of 0.4‰. The maximum horizontal displacements are calculated lower than 1 mm. The most critical aspect in U1 underpass is the execution of works with the existing tunnels under operation. Therefore minimum required displacements and differential settlements are acceptable at the track slab of the under operation tunnels. Details will be presented in the following chapter.

The case U2 of CC23 Delhi Metro was another critical underpass since in close vicinity with the tunnel launching shaft was the existing bridge and the water canal. The plastic zone width as it is evident in **Figure 41** is calculated less than 0.5 m width.

### **Figure 38.**

*Calculated surface settlements (m) at sections along the existing yellow line and 30 m before—case without station simulation in the left diagram and case with station simulation in the right diagram.*

### **Figure 39.**

*Calculated max vertical displacements (m) at the invert of the existing up line of yellow line for both examined cases—with (left above) and without (right above) station simulation.*

**Figure 40.**

*Calculated max differential displacements at the invert of the existing up line of yellow line for both examined cases.*

The maximum calculated surface settlements are in the order of 8–9 mm with a maximum deflection of 1/1000. At the area of the existing bridge, the surface settlements and angular distortion are calculated 2–8 mm and 1/1000, respectively. At the foundation level of the bridge (~4 m above tunnels crown), the maximum calculated vertical subsurface displacements are 10 mm with maximum angular distortion of 1/700 (**Figure 42**).

### *3.3.3 Part C: building risk assessment*

The permissible settlements and deflections in structures are commonly assigned according to their vulnerability which is the output from the precondition survey.

The relation between the categories of damage and the vulnerability index is given by Chiriotti et al. [16] in **Table 9**. Considering the results from part B of TID, the conclusive **Table 10** is prepared in order to be used in correlation with **Table 9**.

Following **Tables 9** and **10**, for all sections F1–F8 of CC23 Delhi Metro, the buildings within the tunnel influence area are expected with "Negligible" damage. Furthermore, for Jaipur Metro Phase 1b, all the buildings are not aligned above the tunnel centerline (except Chandpole Gate). For those cases, the derived results show also "Negligible" damage category even if the existing old buildings of "Pink City" will be categorized as "Highly" vulnerable. Considering that Chandpole Gate vulnerability is in category "Slight," according to the precondition survey, "Negligible" damage is expected for settlements <6.7 mm and angular distortion <1/750. As already derived from the 3D analysis, the maximum calculated settlements and angular distortion are 5 mm and 1/1200, respectively, values which are

**Figure 41.** *Calculated plastic zone width (left) and surface settlements (right) for case U2 of CC23 Delhi Metro [10].*

*Calculated surface settlements (left) and vertical displacements at the bridge's foundation depth (right), for case U2.*


### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

**Table 9.**

*Relation between buildings' categories of damage and vulnerability index.*

related with "Negligible" damage even in the case of "Highly" vulnerable structures. For the case of the bridge (U2), "Negligible" damage category is expected as per the calculated results presented synoptically in **Table 10**.

Following the damage assessment of the buildings within the influence zone of the tunnels, the trigger and alarm levels are mandatory to be established in order to monitor the actual the surface settlements and compare them with the calculated ones. Examples from real established trigger and alarm levels are given in **Table 11** for the critical sections U1, U2, and CH.

Concluding, the derived support pressures as per part A of TID resulted in acceptable surface settlements (calculated in part B), and consequently the building risk assessment (part C) proved that no additional measures are required to be


### **Table 10.**

*Synoptic presentation of calculated surface settlements.*


### **Table 11.**

*Established trigger and alarm levels for sections U1, U2, and CH.*

taken, in order to protect the existing surface and subsurface structures which are within the influence zone of the Metro tunnels.

### **4. Deep tunnelling in squeezing conditions**

As presented in the previous paragraphs, shallow tunnel design is mainly related with the control of the induced surface settlement, in order to protect the existing surface and subsurface structures (monuments, important buildings, tunnels, bridges, etc.), and this can be achieved by minimizing the plastic zone width around the cavity. On the other hand, for the case of deep tunnels, excavated in difficult geological-geotechnical conditions, the aim is to design the tunnel excavation and primary support by allowing the ground mass to get plasticized and redistribute the stresses around the cavity but at the same time to control the extent of the plastic zone in order to achieve equilibrium in each excavation stage. The excavation of the tunnel is a dynamic phenomenon, since the development of the ground displacement is taking place ahead of the excavation face (~ half to one tunnel diameter) and reaches its maximum value about one- and one-half tunnel diameters behind the excavation face. Whether or not the above deformations induce stability problems in the tunnel depends upon the ratio of ground mass strength (σcm) to the in situ stress (Po) level (**Figure 43**).

### **Figure 43.**

*Approximate relationship between strain and the degree of difficulty associated with tunnelling through squeezing rock, for unsupported tunnels [17].*

**Figure 44.** *Comparison between rigid (red line) and yielding (blue line) support [18].*

As it is evident from the curve of **Figure 43**, the smaller the ratio σcm/Po, the higher the strain, and thus critical in the tunnel design is the control of the rock mass deformations around the cavity. Therefore, in case of deep tunnels with high overburden and low rock mass properties, controlled plasticization is almost mandatory in order to reduce the exerted loads around the excavation. By intervening with application of proper yielding support measures, the rock mass is deformed plastically under controllable manner. This solution involves the introduction of deformable elements into the lining. These elements are allowed to deform by a predetermined amount, and when this limit is reached, the support system becomes rigid and starts to carry the full support load. This process allows progressive failure to occur, and a plastic zone is formed in the rock mass immediately surrounding the tunnel. This progressive failure results in a redistribution of the stresses in the rock, surrounding the tunnel, and in a significant reduction in the capacity of the support system required to stabilize the tunnel.

The concept is illustrated in **Figure 44**. Different behavior patterns of rigid (red line) and yielding (blue line) supports are compared. As the rigid support system is installed at a roof displacement of 100 mm, the rock mass loads are that high that the system fails. On the other hand, when a flexible yield support is installed at the same initial crown displacement (100 mm), the controllable deformation of the rock (to 300 mm) through the yielding joints reduces the loads transferred to the support after "locking" of the sliding joints (**Figure 44**, right), without failure of the system.

In the following paragraphs the case study from one of the most difficult tunnels around the world will be analyzed, and details from the design of critical sections will be presented.

### **4.1 Yacambu-Quibor tunnel overview**

A characteristic example of a deep tunnel case, where excessive loads and deformations are recorded around the excavation, is the Yacambu-Quibor tunnel in Venezuela. The 26.4-km-long tunnel (**Figure 45**) was originally conceived as an unlined TBM-driven tunnel to convey water from the Yacambu dam in the wet Orinoco River basin to the dry agricultural region around Quibor. Construction commenced in 1976 in anticipation that the majority of the rock mass through which the tunnel would be driven would be silicified phyllite. This rock forms very steep and stable cliffs in the area of the dam site, and numerous unsupported exploration and drainage tunnels have remained stable for many years in the dam abutments. Unfortunately, it soon became apparent that the rock mass through which the tunnel had to be driven is composed largely of graphitic phyllite which, in contrast to the silicified phyllite of the dam site, is a very weak and tectonically disturbed material. Unsupported tunnels cannot be driven in this rock, particularly at the cover depths of up to 1200 m that occur in the center of the tunnel. In addition, the Bocono fault and the Turbio fault had to be crossed by the tunnel, and it was anticipated that these would cause significant stability problems. Floor heave in the original TBM-driven tunnel resulted in abandoning the tunnel in 1979, and all subsequent tunnel driving decided to proceed with conventional drill and blast and mechanical excavation. Steel sets embedded in a shotcrete lining provided the main support system with the use of forepoling, spiling, and rockbolting where required. Spalling of the shotcrete lining and floor heave has continued as problems whenever the rock mass deformation has been more severe than anticipated.

During the driving of the Ventana Inclinada (**Figure 45**), an intermediate access tunnel designed to provide early access to the Bocono Fault, severe squeezing was encountered. This is a phenomenon characterized by closure of the tunnel, and it

**Figure 45.** *Yacambu-Quibor tunnel longitudinal section [18].* occurs when the rock mass surrounding the tunnel is overstressed and a "plastic zone" develops, as described previously. Unless adequate support measures are introduced, this squeezing can develop into uncontrolled closure and eventually collapse of the tunnel. The primary cause of squeezing is, as already described, a combination of a weak rock mass subjected to high in situ stresses due to a high overburden cover. The process is exacerbated by the presence of water and by gradual deterioration or "creep" of the rock mass.

### **4.2 Graphitic phyllite**

In the case of the graphitic phyllite, shown at the Salida heading in **Figure 46**, the tectonically disturbed rock mass makes it very difficult to collect samples for laboratory testing, and, consequently, very little reliable rock mass strength data is available.

The choice of an appropriate intact strength from results such as those presented in **Figure 46** is a highly subjective process. Anisotropic and foliated rocks such as slates, schists, and phyllites, whose behavior is dominated by closely spaced planes of weakness, cleavage, or schistosity, present particular difficulties in the determination of the uniaxial compressive strengths. Salcedo has reported the results of a set of directional uniaxial compressive tests on a graphitic phyllite from Venezuela. It will be noted that the uniaxial compressive strength of this material varies by a factor of about 5, depending upon the direction of loading. Evidence of the behavior of this graphitic phyllite in the field suggests that the rock mass properties are dependent upon the strength parallel to schistosity rather than that normal to it. In the case of Yacambu-Quibor, the many years of experience of the behavior of the tunnel gives adequate information to calibrate the rock mass strength to a certain degree. It is recommended that the parameter that should be varied in this calibration is the intact rock strength σci since this is both logical from a mechanics point of view and it has a very significant numerical influence on the estimated rock mass strength. For the good siliceous phyllites, the intact rock strength σci was determined as 40–50 MPa, while for the more carbonaceous and foliated phyllites (graphitic phyllite), the intact rock strength varies from 15 to 30 MPa.

Approximately 5 km of tunnel remained to be excavated, and it is anticipated that a significant portion of this length will be in graphitic phyllite and that this will be under the high overburden cover (up to 1200 m). The Turbio fault, which is clearly defined on surface, may extend to tunnel depth, and provision had to be made for

### **Figure 46.**

*Tectonically deformed graphitic phyllite exposed in excavated face of Salida heading of Yacambu-Quibor tunnel on 24 November 2003 (left) and influence of loading direction on strength of graphitic phyllite (right) [18].*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

excavating through and stabilizing the tunnel in this fault. It is also possible that high pressure water and methane gas could be encountered in excavating through the Turbio fault. Therefore, for designing the remaining un-excavated part (approx. 5 km), the information from two critical stations at KP 6+540 and KP 11+700 along the excavated tunnel (**Figure 47**) were used in order to design the most adequate support measures and excavation profile as well as the final lining of the tunnel.

### **4.3 Support types**

The support types along the tunnel were divided into two main categories. Three full face horseshoe types S1, S2 and S3 for their application in silicified phyllite with GSI values greater than 50 (Rock mass types A, B and C) were foreseen, for fault zones and areas with large intrusions of graphitic phyllite at the excavation face (Rock mass types D1 and D2) with GSI values even low to 25 (**Figure 48**).

The flexible support types (S4 and S5) consist of shotcrete with total final thickness up to 60 cm (to act also as final lining), steel sets with sliding joints embedded in the shotcrete lining placed in axial distances of 0.6–1.0 m, and 10–12 fully grouted rock bolts installed every round length. Two sliding joints were foreseen for class S4 at the vaulted area (**Figure 49**), while three flexible joints were equally spaced in category S5. In both flexible support classes, the sliding joints gap was 30 cm as shown in the detail of **Figure 49**.

### **4.4 Tunnel station at KP 11+700**

In order to investigate the behavior of the tunnel and the reaction of the flexible support class, to be used in the Turbio fault area, the particular case of the support near station 11+700 in the Salida heading was considered, and detailed 3D FEM analysis is elaborated. Near the station the conditions were as follows:

Rock mass classification: GSI 35 Tunnel cover: 1100 m Horizontal-vertical stress ratio (Ko): 1.15 Assumed radial deformation before final closing: 300–350 mm

An intact rock strength of 20 MPa is considered, and the following rock mass properties are considered in the simulation:


It is well noticed that the ratio σcm/Po (<0.1) for the above case reveals extreme squeezing conditions. Following the above, support type S4 has been examined. For the simulation, SOFiSTiK [19] software was used. Large model is constructed

**Figure 47.** *Geological-geotechnical section of Yacambu-Quibor tunnel [18].*

### **Figure 50.**

*3D finite element model of Yacambu-Quibor tunnel temporary support with sliding joints and roof displacement (ratio) versus tunnel advance [18].*

consisted of octahedral brick elements. An excavation diameter of 5.2 m is simulated with application of shotcrete shell, simulated with quadrilateral 4-noded elements. For the simulation of sliding joints, spring elements were used with closure gap of 300 mm. Excavation length is considered 1.5 m. Typical view of the 3D model is presented in **Figure 50** (left).

The 3D analysis results (**Figure 50** right) showed that the crown vertical displacement of the rock begins approximately one diameter ahead of the tunnel face and reaches its maximum value about one- and one-half tunnel diameters behind the excavation face.

The total crown vertical displacement is calculated in the order of 200 mm. It is noted that the 200 mm gap closure gives a crown vertical displacement equal to ~60 mm for a tunnel radius equal to 2.6 m, while for the same radius but for a total closure of 300 mm, the crown displacement is calculated 100 mm.

Since the 3D modelling requires enormous computational time, due to large strain conditions, 2D FEM models were also constructed, and further parametric analysis was executed in order to confirm the findings from the previous 3D analysis. Using the convergence-confinement method (Panet 1995), for an unsupported deep tunnel, to assess the rock mass displacement ahead of the excavation face, two cases were investigated. In the first FEM model (S4–1), detailed simulation technique of the sliding joint closure is considered using spring elements. In the alternative FEM model (S4–4), a more simplified approach was used. The principle of the flexible support is that during the movement of the sliding joints, the stresses are released from the shotcrete shell, and only after the total closure of the joints, the temporary shell starts to carry the rock mass loads. Considering this principle, the support system (shotcrete shell) is activated when the crown displacement reaches the value which includes also the locking of the joints, thus rock mass displacement ahead of the face plus additional vertical movement due to joint gap closure. In **Figure 51**, the simulation steps for the two models are presented.

Considering the convergence-confinement method, a crown displacement equal to 150–200 mm is calculated 2–3 m ahead of the excavation face. In model S4–1 the resulted relaxation as per the convergence-confinement method is considered in LC1, and the respective relaxation value is applied on the face core. As per the analysis results, this led to a crown displacement of 211 mm, thus is matching the convergence-confinement calculations. Following the simulation of the sliding joints, at the end of LC2, the spring displacement (gap closure) is calculated 320 mm and the crown displacement equal to 305 mm. After the final excavation of the tunnel, simulated in LC3, the total crown displacement is calculated 315 mm. For the case of model S4–4, a total relaxation is considered in order to simulate the rock mass relaxation ahead of the tunnel face plus the


**Figure 51.**

*Simulation stages for S4–1 model (left) and S4–4 model (right) [20].*


### **Table 12.**

*Simulation results from 2D models S4–1 and S4–4.*

closure of the sliding joints. Therefore, a value of 300 mm is considered in LC1 at the crown area, followed by a value of 308 and 309 mm at the end of LC2 and LC3, respectively. In both simulations the maximum axial force on the shotcrete shell is calculated after the total closure of the sliding joints, followed by the total excavation of the tunnel; 8500–9000 kN is calculated in model S4–1, while in S4–4 a maximum value of 8000–8500 kN is derived. In **Table 12** the results from both analyses are presented.

For tunnelling in extreme squeezing conditions, it is preferable for the parametric design to be performed with 2D FEM or FDM computational analysis models. Parametric analysis is mandatory in such cases, to examine the behavior of the tunnel for a range of the rock mass properties, since a small change in the intact rock strength or the overburden height or the GSI value can lead to different results. 2D FEM parametric modelling is quicker and, with the guidance provided herein, can lead to proper dimensioning of the required support measures. Of course, 3D FEM analysis can be used as additional to the 2D calculations in order to examine the longitudinal development of the displacements.

Following the results presented above, the S4 support class with 60 cm of shotcrete shell (which will be used as the permanent lining of the tunnel) and two sliding joints with 300 mm gap closure (**Figure 52**) proved adequate to be used for the un-excavated part of the tunnel. Further analysis of the results shows that possible reduction also on the final shotcrete shell thickness to 50 cm was also adequate, while 40 cm can also be utilized in special cases where the closure of the sliding joints was not fully mobilized (evidencing better rock conditions).

### **4.5 Tunnel station at KP 6+540**

Another also critical case that was mandatory to be examined in order to conclude on the support measures to be used for the un-excavated part was the findings near station KP 6+540 in the Entrada heading where a failure event was occurred.

After station 6+487, tunnel conditions were of GSI 50, 1225 m cover, and almost no ground deformation. For this situation a support type S3 (horseshoe) with 25 cm of shotcrete thickness was being used. In station 6+540 a tunnel slide occurred and was evaluated as a fault zone. While a geological, geotechnical, and support evaluation was in progress toward a heavier support, the contractor proceeded to place four steel ribs without sliding joints in S3 horseshoe geometry with 4.60 m excavation diameter. The measured deformations in sections before and after the main slide were of the order of 45 cm.

*S4 support class with 60 cm shotcrete thickness for station KP 11+700 [18].*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

The recommended excavation and support classes, according to Yacambu tunnel experience were:


The type S4 (horseshoe and circular geometry) has been examined in detail in this paragraph by 2D finite element analysis in order to investigate the implementation of type S4 section (horseshoe and circular geometry) during the long-term operation of the project as well as to use the findings for the un-excavated part of the tunnel.

Considering the event near the station KP 6+540, the conditions were considered as follows:

Rock mass classification: GSI 25 Tunnel cover: 1200 m Horizontal-vertical stress ratio (Ko): 1.0

An intact rock strength of 15 MPa is considered, and the following rock mass properties are adopted in the simulations:


### *4.5.1 Simulation stages in the area of the fault zone*

Using the convergence-confinement method (i.e., Panet) and elastoplastic response with Mohr-Coulomb law, the displacements 3–4 m ahead of the tunnel face are estimated 50–30 cm (see **Figure 53**).

Thus, before the tunnel sliding, the radial displacement is estimated about 30–50 cm. After the event, which happened in station 6+540 (tunnel slide) and was evaluated as a fault zone, additional displacements (45 cm), in sections before and after the main slide, were measured.

Considering the additional displacements (45 cm) due to the tunnel sliding as well as the radial displacements (30–35 cm) before the event, the total radial displacements in the area before and after the fault zone are estimated at about 75–80 cm. In the area of the fault zone, where the tunnel slid, the total radial deformations are estimated higher, about 80–100 cm. Following the above, four (4) 2D models are prepared using software SOFiSTiK [19] in order to check the adequacy of the shotcrete shell, as follows (**Table 13**).

### **Figure 53.**

*Inward radial displacement as per convergence-confinement method for unsupported tunnel [21].*


### **Table 13.** *2D FEM analysis for tunnel station KP 6+540.*

**Figure 54.** *Modulus of elasticity reduction as per convergence-confinement method [21].*

The initial rock deformations 45 cm (S4a–c), 80 cm (S4b–c, S4b–h), and 100 cm (S4c–h) are investigated in Load Case LC1, by considering an initial reduction in elasticity modulus of the core of the order of 98% (45 cm), 99.3% (80 cm), and 99.5% (100 cm), respectively, as per Panet curves. In the next step, the installation of a temporary support shell with minimum stiffness is simulated

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

(10 cm shotcrete in tunnel periphery) with further reduction in elasticity modulus of the core (as it is presented in **Figure 54**) (Load Case LC2).

According to the tunnel construction sequence, fully cemented rock bolts (with diam. 25 mm up to 35 mm) are implemented in the previously erected steel set. Additional thickness of temporary support shell (5 cm) is simulated with further reduction in elasticity modulus of the core. Shotcrete thickness of 15 cm has reached its final strength (Load Case LC21). The simulation of excavation is fulfilled in Load Case LC3 by considering that the support measures will be mobilized 100% (full stiffening of the 50 cm shotcrete shell for S4a–c and S4b–c and of the 40 cm shotcrete shell in the crown and 60 cm in the invert for S4b–h and S4c–h). The produced stresses in the lining (by considering normal (N) and shear (V) forces, as well as bending moments (M)) are considered, and the appropriate radial and shear reinforcement is then calculated.

In a second step, the long-term behavior of the lining is investigated. A reduction of 15–20% of the rock mass properties is considered, and strain-softening material is used, as follows:

Young modulus, E = 918 MPa Cohesion, c = 963 MPa Friction angle, φ = 16.12° Poisson ratio, ν = 0.3 Plastic ultimate strain, εu = 9.88‰ Ultimate friction angle, φu = 15.30° Ultimate cohesion, cu = 0.871 MPa

### **Figure 55.**

*Model S4a–c and S4b–c, calculated plastic zone (top-left and bottom-left, respectively) and axial force (12,500 and 8500 kN, respectively) distribution (top-right and bottom-right, respectively) within the final shotcrete shell [21].*

(Young modulus and ultimate shear strength are derived by the taken GSI = 25, mi = 10, and σci = 15 MPa).

Load Case LC4 is used as a long-term behavior calculation step. The thickness of the final lining is reduced to 40 cm for S4a–c and S4b–c and to 35 cm for S4b-h and S4c-h due to the risk of weathering of the external shotcrete layer. Rockbolts are considered as not functional.

### *4.5.2 Result presentation*

Four (4) different models have been elaborated in order to investigate the sensitivity of the reinforcement results (for the final lining of S4, in either a circular and a horseshoe geometry), in comparison with the initial rock mass deformation (45, 80, 100 cm) due to the tunnel sliding that occurred near station at KP 6+540. In all models the detailed construction sequence has been analyzed along with the respective initial rock mass deformation. In the first model S4a–c the initial rock mass convergence that is simulated, before the installation of the temporary support, is 45 cm.

At the second model (S4b–c), it is assumed that the reported deformation of 45 cm (ING-03-52) does not include initial deformation occurred before the slide (estimated of the order of 30–35 cm), and therefore the total deformation is of the order of 80 cm. The resultant axial forces, regarding the first approach, are higher than the axial forces that are calculated with the second model (**Figure 55**).

The second approach as mentioned above considers both convergences that are produced by (a) relaxation of the rock mass (30–35 cm) which is supposed to be occurred before the excavation and (b) additional measured displacement of the rock mass (before and after the fault zone) due to tunnel failure. The corresponding values of the axial and shear forces along with the moments that are developed in the flexible temporary lining is a result of the shell's displacement. Thus, the lower the displacement of the shell, the lower the axial and shear forces and moments. In a long-term basis, the stresses that were developed in the temporary lining are redistributed within the total thickness of the final lining. Considering the behavior of the tunnel near station 6+540, where no important shell displacements were occurred after the installation of the first 15 cm of shotcrete, it is assumed that the initial rock mass convergence before the installation of the shell was about 80 cm.

For models S4b–h and S4c–h, the radial initial displacement of the rock mass is estimated at about 80–100 cm, due to the fact that the area, which is investigated, is the fault zone, where the tunnel failed. The results for these analyses are unfavorable. In both approaches the bending moments and shear forces, in the connection area of the vault with the invert, are extremely higher than the previous model S4b–c (**Figure 56**).

### **Figure 56.**

*Model S4b–c, calculated plastic zone (left), axial force (9000 kN) distribution (middle), and bending moments (max 1300 kN m—right) within the final shotcrete shell [21].*

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

The reason for the unfavorable results is the "bad" geometry of the horseshoe support class (see **Figure 48**). Even in cases that further failure has not been observed in the areas of adjustment to KP 6+540, where the horseshoe geometry was applied, a decision had to be made in order to secure the tunnel lining on a long-term basis.

### **Figure 57.**

*Calculated required reinforcement for long-term tunnel operation. Model S4b–c (left) with max 14 cm2 and model S4b–h (right) with 43 cm<sup>2</sup> at crown and max 140 cm<sup>2</sup> [21].*

*Conversion of horseshoe tunnel excavation profile to equivalent circular final lining for long-term tunnel operation [21].*


**Table 14.**

*Support classes and application criteria for GSI 25–35.*

From the detailed FEM analysis results, it was clear that a circular geometry for the final lining during the long-term operation of the tunnel was able to withstand the rock load with a minimum required reinforcement of 14 cm<sup>2</sup> (**Figure 57**), while the horseshoe geometry was unfavorably affecting the capacity of the final lining. Therefore, special reinforcement arrangement was applied in order to convert the tunnel excavation profile to an equivalent circular final lining (**Figure 58**).

### **4.6 Conclusive remarks for extreme squeezing conditions**

The detailed analysis and back-analysis of the tunnel in the two critical stations at KP 11+700 and KP 6+540 describe the behavior of the excavation and consequently the support classes, under the worst geotechnical conditions which are governed by the presence of very weak carbonaceous and foliated phyllites (graphitic phyllite) with GSI <35 and intact rock strength from 15 to 30 MPa.

The tunnel excavation under maximum overburden height and in the presence of rock with GSI value of 35 and intact rock strength of 20 MPa can lead to a total radial closure of 300–350 mm. For the extreme case of GSI = 25 and σci = 15 MPa, the total radial closure can be in the order of 650–750 mm. For both cases, the circular excavation profile of S4 and S5 support classes is suitable and needs to be implemented in order to secure the long-term operation of the final lining. **Table 14** summarizes the findings from the parametric analysis.

As it is shown in detail in the respective chapters, the tunnel excavation under extreme squeezing conditions is only feasible by applying a flexible support category with sliding joints in a circular excavation profile.

### **5. Conclusions**

In the present chapter of the book, the shallow and deep tunnel cases are explained in detail, and methods of designing are presented. For the reader and the tunnel designer, the following conclusions are summarized for the shallow and deep tunnel problems.

### **5.1 Shallow tunnelling in urban environment**

Knowledge of the general macro-geology of the project area is always beneficial for the designer to establish a proper geotechnical campaign.

### *Designing a Tunnel DOI: http://dx.doi.org/10.5772/intechopen.90182*

The general geological formations of the under-study area will be the governing parameter for establishing the density of the investigation boreholes and the density and type of laboratory tests.

Knowledge of the geological-geotechnical conditions will determine the type of the TBM shield to be used as well as the cutterhead design.

Combination of analytic solutions and computational advance modelling (FEM or FDM) is the most adequate method to be used in the design, since it reduces the time required for executing the calculations.

TBM tunnel interstation design is mandatory to be followed in Metro projects, due to the existence of surface and subsurface structures. The aim of the design to control the surface settlements and thus the operation of the TBM shields need to be determined by the designer.

Conservatism in determining the geotechnical properties of the soil must be avoided since it will result in overdesigning the permanent segmental lining of the tunnel and will also lead to unreasonably high support pressures with immediate detrimental effect on the operation of the machine.

The discrete parts A, B, and C determined the TBM TID and as such need to be strictly followed.

The application of the design support pressure (as derived from the TID) is the key factor along with the adequate tail shield grouting, in order to control the soil's settlements. The adequate application is to keep the chamber pressure constant in each stroke of the TBM, thus in each excavation step, as the shield is moving forward. Grouting from the tail shield of the TBM must be performed, and the amount and pressure of grout injected into the annular gap must be controlled by the pilot of the machine, and the consumed volume must always be monitored per ring along with the applied grouting pressure. Grouting amount in general should be 1.1–1.2 times of the interspace cubage. Secondary grouting from the segments should be performed based always on the live monitoring data.

### **5.2 Deep tunnelling in extreme squeezing conditions**

Anisotropic and foliated rocks such as graphitic phyllites, when met under high in situ pressure, can always lead to extreme squeezing problems.

For extreme squeezing conditions only, flexible supports can secure the stability of the underground excavation. Any other type of stiff support will result in failure.

Long-term behavior of weak rocks should always be investigated when permanent lining is designed.

The most adequate excavation profile is the circular geometry. Horseshoe geometries should be avoided since the resultant bending moments will lead to undesignable conditions. Larger tunnels (>5 m diameter) should be designed with a geometry as close as possible to a circle.

For larger tunnel diameters to be constructed in extreme squeezing conditions with increased number of sliding joints (or lining stress controllers (LSC)), 3D FEM or FDM computational analysis will be required with advance modelling of the sliding interface in order to investigate the closure sequence of the joints in relation to the advance of the tunnel face. Using the results from the 3D analysis, relaxation factors can be used in 2D modelling, and the temporary support measures dimensioning can be achieved.

### **Acknowledgements**

I would like to express my gratitude to Omikron Kappa SA for the more than 17 years of continuous cooperation that we have in performing designs for demanding and difficult projects around the globe. Special thanks also to Dr. Evert Hoek for his valuable guidance in my early years as a designer in the Yacambu-Quibor tunnel project.

To download the chapter with high resolution images, please use the following link: https://cdn.intechopen.com/public/docs/70605.pdf

### **Author details**

Spiros Massinas1,2,3,4,5

1 Engineering Manager of ALYSJ JV (Aktor - Larsen and Toubro - Yapi Merkezi - STFA - Al Jaber Engineering JV), Gold Line Metro, Doha, Qatar

2 Middle East and India Regional Manager of Omikron Kappa Consulting, Doha, Qatar

3 Project Manager/Director of Omikron Kappa – Indus Consultrans JV, Gurgaon, India

4 Civil Engineer, PhD (NTUA), MSc (NTUA), CEng (UK), MICE UK

5 Associate Researcher, Laboratory of Structural Mechanics, Department of Infrastructure and Rural Development, National Technical University of Athens (NTUA), Greece

\*Address all correspondence to: spirosmas@yahoo.gr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

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[3] Omikron Kappa—Indus Consultrans JV. Geotechnical interpretation report for tunnels. CC23 Delhi Metro; 2013

[4] Omikron Kappa—Indus Consultrans JV. Geotechnical interpretation report for tunnels, stations and shafts. Jaipur Metro Phase 1B; 2014

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EPB machines under low overburden: The case of Chandpole Gate in Jaipur Metro, India. Geotechnical & Geological Engineering. 2018. DOI: 10.1007/ s10706-018-0565-0

[9] FLAC3D, Fast Lagrangian Analysis of Continua in 3 Dimensions, Itasca Consulting Group, Inc. US Minneapolis. Available from: www.itascacg.com

[10] Omikron Kappa—Indus Consultrans JV. Design of under passing existing bridge at Nala area. CC23 Delhi Metro; 2013

[11] Omikron Kappa—Indus Consultrans JV. TBM tunnel interstation design—Assessment and protection of existing buildings. CC23 Delhi Metro; 2013

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[14] Omikron Kappa—Indus Consultrans JV. Design of under passing existing mMetro line (yellow) near Hauz Khas—Stage 3 analysis. CC23 Delhi Metro; 2013

[15] Omikron Kappa—Indus Consultrans JV. Under passing scheme for Chandpole gGate. Jaipur Metro Phase 1B; 2014

[16] Chiriotti E, Marchionni V, Grasso P. Porto Light Metro System, Lines C, S and J. Compendium to the Methodology Report on Building Risk Assessment Related to Tunnel Construction. Normetro – Transmetro. Italian, Portuguese: Internal Technical Report; 2000

[17] Hoek E, Marinos P. Predicting tunnel squeezing problems in weak heterogeneous rock masses. Tunnels and Tunnelling International. 2000;**32**(11):45-51

[18] Hoek E—Omikron Kappa Consulting Ltd. Yacambu-Quibor tunnel lining; 2003

[19] SOFiSTiK, Finite Element Method Software, Sofistik AG. Available from: www.sofistik.com

[20] Omikron Kappa Consulting Ltd. Excavation, temporary and final lining design for Yacambu-Quibor tunnel near station in KP 11+700; 2003

[21] Omikron Kappa Consulting Ltd. Excavation, temporary and final lining design for Yacambu-Quibor tunnel near station in KP 6+540; 2003

Section 3
