1 Introduction

In most designs of concrete-lined hydropower tunnels, the rock mass supporting the tunnel is often assumed as an isotropic material (Schleiss 1986; Seeber 1985a, b; Simanjuntak et al. 2012a; Wannenmacher et al. 2012). This assumption is usually acceptable given that the rock mass exhibits non-significant anisotropy in strength and deformability. It has contributed not only to the knowledge of the mechanical and hydraulic interaction between the concrete lining and the rock mass, but also to the understanding of the behaviour of prestressed concrete-lined pressure tunnels embedded in an elasto-plastic isotropic rock mass subjected to either uniform (Simanjuntak et al. 2012b) or non-uniform (Simanjuntak et al. 2014b) in situ stresses.

Hydropower tunnels, nevertheless, may be constructed in inherently anisotropic rocks, such as sedimentary and metamorphic foliated rocks. These types of rocks, which are composed of laminations of intact rocks, can take the form of cross anisotropy or transverse isotropy commonly configured by one direction of stratifications perpendicular to the direction of deposition (Gao et al. 2010). Here, the rock supporting the tunnel can exhibit significant distinctive strength and deformability in the direction parallel and perpendicular to the stratification planes; rendering the behaviour of pressure tunnels embedded in such rock formations differs from that under the assumption of isotropic rocks.

As pointed out by Bobet (2011), aside from the anisotropic properties of the rock, the magnitudes of the in situ stress and the orientation of stratification planes in the rock mass are important aspects in the design of tunnel linings. So far, anisotropic rock behaviour is rarely taken into account for the design of concrete-lined pressure tunnels. Tunnel designers rather use a conservative model considering unfavourable isotropic rock behaviour when assessing the maximum stresses in the lining (Pachoud and Schleiss 2015). Particularly for the design of passively prestressed concrete-lined pressure tunnels, the aforementioned aspects cannot be overlooked since the load sharing between the lining and the rock and thus the bearing capacity of such tunnels will thoroughly rely on the support from the rock mass.

By employing a commercial finite element code DIANA, the behaviour of concrete-lined pressure tunnels embedded in transversely isotropic rock, whose in situ stresses are unequal in the vertical and horizontal direction is studied. To ensure the long-term bearing capacity of the tunnel, the concrete lining is prestressed by grouting according to the principles of Seeber (1999). To allow for the use of a two-dimensional model, the tunnel being considered is assumed to be driven in the direction parallel to the strike of the planes of transverse isotropy. The interplay between the in situ stress ratio and the transverse isotropy on the tunnel lining performance is explored based on the concept that there is no slip allowed to occur along the stratification planes. In accordance with the authors’ previous work, the study is focussed on a deep, straight ahead circular tunnel situated above the groundwater level.

This paper presents a series of cases that are relevant to tunnelling in transversely isotropic or cross-anisotropic rock. It begins with the response of the rock mass to circular excavation and continues to the prediction of stresses and deformations along the tunnel perimeter as a result of support (shotcrete) installation. A concrete final lining is installed behind the shotcrete and prestressed by injecting cement-based grout into the circumferential gap between the final lining and the shotcrete at high pressure. The maximum internal water pressure is assessed by offsetting seepage-induced hoop strains against prestress-induced hoop strains at the final lining intrados. Finally, locations where longitudinal cracks can occur in the final lining are identified, which are useful when taking measures in view of tunnel safety.

2 Tunnel Excavation in Transversely Isotropic Rocks

As long as discontinuities in the rock mass are more or less parallel and regularly spaced, the rock mass can at first be approximated as an elastic transversely isotropic or cross-anisotropic material. The effect of stratifications on the behaviour of the rock mass can be investigated by incorporating different deformability properties at directions parallel and perpendicular to the surface of dominant discontinuities (Fortsakis et al. 2012; Kolymbas et al. 2012). Figure 1 shows a representation of problems of a circular tunnel excavated in transversely isotropic rock, whose stratification planes are horizontal, i.e. α = 0°. While the z-axis is the tunnel axis, the x- and z-axes are the plane of transverse isotropy (Fig. 1a).

Fig. 1
figure 1

Circular tunnel in transversely isotropic rock with horizontal stratification planes

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In most cases, the in situ stresses in a rock mass are non-uniform or non-hydrostatic (Fig. 1b). The in situ horizontal stress, σ h, can be expressed in the product of the in situ vertical stress, σ v, and a coefficient of earth pressure at rest, k.

$$\sigma_{\text{h}} = k\sigma_{\text{v}}$$
(1)

The mean in situ stress, σ o, can be determined as (Carranza-Torres and Fairhurst 2000a, b):

$$\sigma_{\text{o}} = \frac{{\sigma_{\text{h}} + \sigma_{\text{v}} }}{2} = \frac{{(k + 1)\sigma_{\text{v}} }}{2}$$
(2)

in which the in situ stresses are non-uniform or non-hydrostatic, if k ≠ 1.

As long as the plane of transverse isotropy strikes parallel to the tunnel axis, plane strain conditions apply along the tunnel axis and the components ε z , ε yz , and ε xz vanish everywhere. The constitutive model in the plane strain conditions are given as:

$$\left( {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\varepsilon_{xy} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & 0 \\ {S_{21} } & {S_{22} } & 0 \\ 0 & 0 & {S_{33} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\tau_{xy} } \\ \end{array} } \right)$$
(3)

where σ x and σ y are the total stress along the x- and y-axes, respectively, τ xy is the shear stress, S 11, S 21, S 12, S 22, and S 33 are the compliance coefficients and related to the material parameters as follows:

$$\begin{aligned} S_{11} & = \frac{{1 - \nu_{\text{h}}^{2} }}{{E_{\text{h}} }} \\ S_{22} & = \frac{{1 - \nu_{\text{hv}} \nu_{\text{vh}} }}{{E_{\text{v}} }} \\ S_{12} & = S_{21} = - \frac{{\nu_{\text{vh}} (1 + \nu_{\text{h}} )}}{{E_{\text{v}} }} \\ S_{33} & = \frac{1}{{G_{\text{vh}} }} \\ \end{aligned}$$
(4)

in which E h and E v are the Young’s modulus in the plane of isotropy and in the direction normal to the plane of isotropy, respectively, ν h is the Poisson’s ratio in the plane of isotropy, ν hv is the Poisson’s ratio for the effect of stress in the plane of isotropy on the strain in the direction normal to the plane of isotropy, ν vh is the Poisson’s ratio for the effect of stress normal to the plane of isotropy on the strain in the plane of isotropy, and G vh is the shear modulus normal to the plane of isotropy.

A full mathematical treatise to determine the excavation-induced hoop stresses and radial deformations along the perimeter of a circular tunnel embedded in elastic transversely isotropic rock when the stratification planes are horizontal can be found in Hefny and Lo (1999). For completeness, the closed-form solution is included herein and can be written as:

Hoop stresses:

$$\begin{aligned} \sigma_{\theta } & = \frac{{2 + 2(\gamma_{1} + \gamma_{2} )^{2} - 2\gamma_{1}^{2} \gamma_{2}^{2} - 4(\gamma_{1} + \gamma_{2} )\cos 2\theta }}{{(1 + \gamma_{1}^{2} - 2\gamma_{1} \cos 2\theta )(1 + \gamma_{2}^{2} - 2\gamma_{2} \cos 2\theta )}}\left( {\frac{{(k + 1)\sigma_{\text{v}} }}{2}} \right) \\ & \quad + \frac{{4(\gamma_{1} + \gamma_{2} ) - 4(1 + \gamma_{1} \gamma_{2} )\cos 2\theta }}{{(1 + \gamma_{1}^{2} - 2\gamma_{1} \cos 2\theta )(1 + \gamma_{2}^{2} - 2\gamma_{2} \cos 2\theta )}}\left( {\frac{{(k - 1)\sigma_{\text{v}} }}{2}} \right) \\ \end{aligned}$$
(5)

with

$$\begin{aligned} \gamma_{1} & = \frac{{\mu_{1} - 1}}{{\mu_{1} + 1}};\quad \left| {\gamma_{1} } \right| < 1 \\ \gamma_{2} & = \frac{{\mu_{2} - 1}}{{\mu_{2} + 1}};\quad \left| {\gamma_{2} } \right| < 1 \\ \mu_{1}^{2} \mu_{2}^{2} & = \frac{{S_{11} }}{{S_{22} }} \\ \mu_{1}^{2} + \mu_{2}^{2} & = \frac{{2S_{12} + S_{33} }}{{S_{22} }} \\ \end{aligned}$$
(6)

Radial deformations:

$$u_{\text{r}} = \frac{R}{{2(\gamma_{1} - \gamma_{2} )}}\left\{ \begin{array}{ll} \left( {\frac{{(k + 1)\sigma_{\text{v}} }}{2}} \right)(\gamma_{2} \rho_{1} - \gamma_{1} \rho_{2} ) + \left( {\frac{{(k - 1)\sigma_{\text{v}} }}{2}} \right)(\rho_{1} - \rho_{2} ) \hfill \\ + \left[ {\left( {\frac{{(k + 1)\sigma_{\text{v}} }}{2}} \right)(\gamma_{2} \delta_{1} - \gamma_{1} \delta_{2} ) + \left( {\frac{{(k - 1)\sigma_{\text{v}} }}{2}} \right)(\delta_{1} - \delta_{2} )} \right]\cos 2\theta \hfill \\ \end{array} \right\}$$
(7)

with

$$\begin{aligned} \delta_{1} & = (1 + \gamma_{1} )\beta_{2} - (1 - \gamma_{1} )\beta_{1} \\ \delta_{2} & = (1 + \gamma_{2} )\beta_{1} - (1 - \gamma_{2} )\beta_{2} \\ \rho_{1} & = (1 + \gamma_{1} )\beta_{2} + (1 - \gamma_{1} )\beta_{1} \\ \rho_{2} & = (1 + \gamma_{2} )\beta_{1} + (1 - \gamma_{2} )\beta_{2} \\ \end{aligned}$$
(8)

and

$$\begin{aligned} \beta_{1} & = S_{12} - S_{22} \mu_{1}^{2} \\ \beta_{2} & = S_{12} - S_{22} \mu_{2}^{2} \\ \end{aligned}$$
(9)

As discussed by Hefny and Lo (1999) and Manh et al. (2015), two cases may arise, which are either all parameters δ 1, δ 2, ρ 1, and ρ 2 are real if γ 1 and γ 2 are real, or parameters δ 1 and δ 2 as well as ρ 1 and ρ 2 are complex conjugates if γ 1 and γ 2 are complex conjugates.

Figure 2 represents the general problem of a circular tunnel excavated in transversely isotropic rock, where the stratification planes make an angle α to the x-axis. Consequently, all deformation components including those appearing in Eq. (7) depend also on the orientation of stratification planes, α. The excavation-induced radial deformations, u r, can be written in dimensionless form as:

$$\frac{{u_{\text{r}} }}{R} = f\left( {\frac{{E_{\text{h}} }}{{E_{\text{v}} }},\frac{{G_{\text{vh}} }}{{E_{\text{h}} }},\frac{{\nu_{\text{h}} }}{{\nu_{\text{vh}} }},k,\alpha } \right)$$
(10)

Provided that there is no slip between the planes of transverse isotropy, the rock mass as a whole can be idealised as a continuous linear elastic and homogenous medium. To reveal the distribution of stresses and deformations around the tunnel, the elasto-plastic Jointed Rock model available in the commercial finite element code DIANA (2012) was used. The elastic behaviour of the rock mass is ensured by providing an adequate cohesion along the sliding planes (Simanjuntak et al. 2014a; Tonon 2004; Tonon and Amadei 2003; Wittke 1990).

Fig. 2
figure 2

Circular tunnel in transversely isotropic rock with non-horizontal stratification planes

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As an example, a circular excavation with a radius of 2 m is executed to the rock mass with transversely isotropic formations. The rock mass properties are given in Table 1 with reference to Hefny and Lo (1999). The orientations of the stratification planes being considered are 0°, 45°, 90°, and 135°, with 0° indicating the horizontal stratification planes and 90° the vertical.

Table 1 Rock mass properties (Hefny and Lo 1999)
Full size table

Regarding the in situ stresses in the rock mass, two cases are distinguished based on whether the in situ vertical stress is greater than the horizontal (k < 1), or the in situ horizontal stress is greater than the vertical (k > 1). The gravitational force can be assumed negligible since the pressure tunnel stands deep in a rock mass (Detournay and Fairhurst 1987).

In DIANA, the model domain was made free to move in the radial direction, but not in the longitudinal direction. To simulate the tunnel excavation, two calculation phases are considered. In the first phase, the corresponding magnitudes of the in situ stress were assigned to the elements representing the unperforated rock mass. Here, the deformations anywhere in the model are zero. In the second phase, the elements representing the cavity were deactivated. As a consequence, the first deformations occur in the rock mass.

Figure 3 illustrates the numerical results of excavation-induced radial deformations in the space surrounding the tunnel subjected to non-uniform in situ stresses. While Fig. 3a–c presents the predicted excavation-induced radial deformations for the case where the horizontal-to-vertical stress coefficient k = 0.80, Fig. 3d–f depicts the results for the case where k = 1.25.

Fig. 3
figure 3

Radial deformations after the tunnel excavation

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Corresponding to that has been observed by Hefny and Lo (1999), the distribution of excavation-induced radial deformations exhibits a symmetrical pattern if the stratification planes are horizontal. The distribution of radial deformations when the stratification planes are vertical is also presented in this paper (Fig. 3b, e). Notably, the distribution of radial deformations demonstrates an unsymmetrical pattern if the stratification planes are inclined (Fig. 3a, c, d, f). Comparing Fig. 3a with f or c with d, it can be seen that the distribution of excavation-induced radial deformations for a specific value α with coefficient k is identical to that for α + 90° with coefficient 1/k, by rotating the tunnel axis by 90°.

Polar representations of excavation-induced radial deformations along the tunnel perimeter for various k and α are depicted in Fig. 4. In view of model validation, the numerical results for cases where the stratification planes are horizontal, i.e. α = 0° are compared with those calculated by using the closed-form solution. It is seen that the numerical results for both k = 0.80 (Fig. 4a) and k = 1.25 (Fig. 4b) fit the analytical results with great accuracy, which implies that the numerical approach proposed herein is methodologically correct and thus can be applied to obtain the results for α = 45°, 90° and 135°.

Fig. 4
figure 4

Distributions of excavation-induced radial deformations along the tunnel perimeter

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In the following, the numerical results of excavation-induced radial deformations for cases where the in situ vertical stress is greater than the horizontal (k < 1) are analysed. For a ratio k = 0.80 (Fig. 4a) and α = 0°, the radial deformation, u r, at the tunnel roof and invert was found as 0.60%R, whereas at the sidewalls, it was 0.34%R. If the stratification planes are inclined at 45° or α = 45°, the maximum radial deformation equals to 0.56%R was found at the tunnel arcs, specifically at θ = 105° and 285° counted counterclockwise from the x-axis, while the minimum radial deformation equals to 0.36%R was located at θ = 15° and 195°. If the stratification planes are vertical or α = 90°, the maximum radial deformation equals to 0.48%R was observed at the tunnel roof and invert, whereas the minimum radial deformation equals to 0.44%R was found at the sidewalls. When the stratification planes are inclined at 135° or at α = 135°, the maximum radial deformation equals to 0.56%R was found at the tunnel arcs, specifically at θ = 75° and 255°, whereas the minimum radial deformation equals to 0.36%R was located at θ = 165° and 345°. These results suggest that if the in situ vertical stress is greater than the horizontal (k < 1), the maximum radial deformation will take place at the tunnel roof and invert when the direction of stratification planes coincides with the x-axis, i.e. horizontal bedding. The shape of the tunnel is oval with its major axis parallel to the x-axis. The ovalisation is augmented by two aspects: first, the loading in the vertical direction which is greater than that in the horizontal direction; second, the stiffness in the vertical direction which is lower than that in the horizontal direction. In such cases, the degree of ovalisation of the tunnel decreases if the stratification planes in the rock mass make an angle α to the x-axis.

The response of the rock mass to a circular tunnel excavation was also investigated for cases where the in situ horizontal stress is greater than the in situ vertical stress (k > 1). For a ratio k = 1.25 (Fig. 4b) and α = 0°, the maximum radial deformation equals to 0.48%R was found at the sidewalls, while the minimum radial deformation equals to 0.44%R was located at the roof and invert of the tunnel. If α = 45°, the maximum radial deformation equals to 0.56%R was found at θ = 165° and 345°, whereas the minimum radial deformation equals to 0.36%R was situated at θ = 75° and 255°. If the stratification planes are vertical, the maximum radial deformation equals to 0.60%R was found at the roof and invert, while the minimum radial deformation equals to 0.34%R was obtained at the sidewalls. If α = 135°, the maximum radial deformation equals to 0.56%R was observed at θ = 15° and 195°, while the minimum radial deformation equals to 0.36%R was situated at θ = 105° and 285°. These results indicate that if the in situ horizontal stress is greater than the vertical (k > 1), the maximum radial deformation occurs at the tunnel sidewalls if the direction of stratification planes coincides with y-axis, i.e. vertical bedding. The shape of the tunnel is still oval but its major axis is parallel to the y-axis. There are two reasons for this: one, the loading in the horizontal direction is greater than that in the vertical direction; two, the stiffness in the horizontal direction is lower than that in the vertical direction. Here, the degree of ovalisation of the tunnel increases if the stratification planes in the rock mass make an angle α to the x-axis.

With negative sign indicating a compressive state of stress, the numerical results of excavation-induced hoop stresses in the space surrounding the tunnel are illustrated in Fig. 5. To validate the numerical results, the analytical results of excavation-induced hoop stresses along the tunnel perimeter when the stratification planes are horizontal, are presented in Fig. 6. It is seen that the numerical results reproduce very well the analytical results with good accuracy.

Fig. 5
figure 5

Hoop stresses after the tunnel excavation

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Fig. 6
figure 6

Distributions of excavation-induced hoop stresses along the tunnel perimeter

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In accordance with that has been presented in Hefny and Lo (1999), the distribution of excavation-induced hoop stresses surrounding the tunnel demonstrates a symmetrical pattern when the stratification planes are horizontal. Herein, the distribution of hoop stresses when the stratification planes are vertical is depicted in Fig. 6a, b. For cases when the in situ vertical stress is greater than the horizontal (k < 1), the maximum hoop stress is located at the tunnel sidewalls, whereas the minimum hoop stress is observed at the tunnel roof and invert. Conversely, for cases when the in situ horizontal stress is greater than the vertical (k > 1), the maximum hoop stress is located at the tunnel roof and invert, while the minimum hoop stress is found at the tunnel sidewalls.

For k = 0.80 (Fig. 6a) and α = 0°, the excavation-induced hoop stress, σ θ , at the tunnel sidewalls was found as 2.55σ o, while at the tunnel roof and invert it was 1.65σ o. If α = 90°, the excavation-induced hoop stress at the tunnel sidewalls was obtained as 2.66σ o, whereas at the tunnel roof and invert it was 1.70σ o. For k = 1.25 (Fig. 6b) and α = 0°, the excavation-induced hoop stress at the tunnel roof and invert was obtained as 2.66σ o, while at the tunnel sidewalls it was 1.70σ o. When α = 90°, the excavation-induced hoop stress at the tunnel roof and invert was found as 2.55σ o, while at the tunnel sidewalls it was 1.65σ o.

Also shown in Fig. 6a, b, the distribution of hoop stresses is unsymmetric if the stratification planes in the rock mass make an angle α to the x-axis. For k = 0.8 (Fig. 6a) and if α = 45°, the maximum excavation-induced hoop stress equals to 2.34σ o was situated at the tunnel arcs, specifically at θ = 155° and 335°, whereas the minimum hoop stress equals to 1.43σ o was observed at θ = 95° and 275°. Under the same loading, the maximum hoop stress equals to 2.34σ o was found at θ = 25° and 205°, while the minimum hoop stress equals to 1.43σ o was situated at θ = 85° and 265° if α = 135°. For k = 1.25 (Fig. 6b) and when α = 45°, the maximum excavation-induced hoop stress equals to 2.34σ o was observed at θ = 115° and 295°, whereas the minimum hoop stress equals to 1.43σ o was found at θ = 175° and 355°. When α = 135°, the maximum hoop stress equals to 2.34σ o was observed at θ = 65° and 245°, while the minimum hoop stress equals to 1.43σ o was found at θ = 5° and 185°. These results again imply that the distribution of excavation-induced hoop stresses for a specific value α with coefficient k is identical to that for α + 90° with coefficient 1/k, by rotating the tunnel axis by 90°.

It has been recognised that squeezing problems can take place at the over-stressed regions in a rock mass where high compressive hoop stress is dominating. This problem can occur during and after tunnel excavation and may lead to tremendous operational difficulties (Panet 1996), and to a collapse of support lining. It is usually associated with poor rock mass deformability and strength properties as summarised in the review paper by Barla (2001). If the particular combination of induced stresses and material properties pushes some zones around the tunnel beyond the limiting shear stress at which creep starts, squeezing may be found along bedding surfaces in the rock mass. On the contrary, if the minimum hoop stress is too low, it can also cause a problem since the rock mass can resist only a very small tensile stress. If the hoop stresses around the tunnel fall in a tensile state, radial fracturing or separation of stratification planes may occur, which can endanger the tunnel stability. Using the modelling approach presented herein and the analysis example as shown in Fig. 6a, b, one can easily identify the extent of the over-stressed regions or the locations where stress relaxation may occur around the tunnel; even before the tunnel excavation is put on practice.

3 Radial Stresses and Deformations Transmitted to a Support System

If the stability of a pressure tunnel against squeezing can be guaranteed, a minimum support system, such as shotcrete lining, is still needed not only for stability but also for hydraulic reasons. Particularly for unreinforced concrete-lined pressure tunnels prestressed by grouting, a shotcrete lining is commonly placed between the final lining and the rock mass so as to achieve a smooth contact surface during contact grouting as well as prestressing processes.

Due to its capability to sustain a strain up to 1 %, shotcrete lining with a thickness of between 5 and 10 cm has commonly been used as a support system for hydropower tunnels. To prevent large tunnel deformations, stabilisation measures such as those recommended by Barla et al. (2011) and Hoek and Marinos (2000) can be used.

In this study, a shotcrete lining whose properties according to the concrete type C20/25 (Table 2), with a virtual thickness of 10 cm was used to support the tunnel excavation. In terms of radial stresses and radial deformations at the rock–shotcrete interface, the same setting as proposed by Bobet (2011) was adopted, i.e. elastic response of rock mass and shotcrete, continuous contact between rock mass and shotcrete, two-dimensional plane strain conditions along the tunnel axis, and simultaneous excavation and shotcrete installation. To reveal the radial stresses along the thickness of the shotcrete lining, the shotcrete was modelled by using continuum elements (Bonini et al. 2013; Simanjuntak et al. 2012b, 2014b). Here, the simultaneous tunnel excavation and support installation in DIANA were simulated as follows. First, the in situ stresses in rock mass are generated following the same procedure as that for the tunnel excavation without support. Then, the elements representing the cavity were deactivated and at the same time the shotcrete elements are activated.

Table 2 Concrete properties (ÖNORM 2001)
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Because the shotcrete is placed at the same time when the tunnel is excavated, it has to be acknowledged that this approach can result in stresses that are usually too large in the shotcrete and too small in the rock mass when compared to cases when the rock mass experiences deformations prior to shotcrete installation (Bobet and Nam 2007). This limitation can still be overcome if the deformations in the rock mass due to three-dimensional tunnel advance and prerelaxation ahead of the tunnel face are known. However, if the plane of transverse isotropy does not strike parallel to the tunnel axis, the plane strain conditions along the tunnel axis are violated and therefore three-dimensional models should be used.

Figure 7 depicts the numerical results of radial deformations along the rock–shotcrete interface for both k = 0.80 and k = 1.25. For k = 0.80 (Fig. 7a) and if α = 0°, the radial deformation at the tunnel roof and invert was reduced to 11.70 mm, while at the tunnel sidewalls it was 6.50 mm. If α = 45°, the radial deformation at the tunnel arcs located at θ = 105° and 285° was reduced to 10.87 mm, while at θ = 15° and 195° it was 7.11 mm. If α = 90°, the radial deformation at the tunnel roof and invert was found as 9.43 mm, whereas at the tunnel sidewalls it was reduced to 8.30 mm. If α = 135°, the radial deformation at θ = 75° and 255° was 10.87 mm, while at θ = 165° and 345° it became 7.11 mm.

Fig. 7
figure 7

Distributions of radial deformations along the rock–shotcrete interface

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For k = 1.25 (Fig. 7b) and when α = 0°, the radial deformation at the tunnel roof and invert was found as 8.30 mm, while at the tunnel sidewalls it was 9.43 mm. When α = 45°, the radial deformation at θ = 165° and 345° was reduced to 10.87 mm, whereas at θ = 75° and 255° it was 7.11 mm. When α = 90°, the radial deformation at the tunnel sidewalls became 11.70 mm, whereas at the tunnel roof or invert it was 6.50 mm. When α = 135°, the radial deformation located at θ = 15° and 195° was reduced to 10.87 mm, while at θ = 105° and 285° it became 7.11 mm.

The reduction in radial deformations along the perimeter of the rock–shotcrete interface is summarised in Table 3. For α = 0°, i.e. horizontal bedding, it is seen that the shotcrete takes more load at θ = 90° and 270°. For α = 90°, i.e. vertical bedding, the shotcrete takes more load at θ = 0° and 180°. The other cases, i.e. α = 45° and 135°, can be considered as intermediate between the two. These results suggest that the higher rock stiffness in the direction parallel to the stratification planes allows the rock to deform less in the direction parallel to the stratification planes; but requires more support in the direction perpendicular to the stratification planes.

Table 3 Reduction in radial deformation along the rock–shotcrete interface
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Figure 8 shows the numerical results of radial stresses in the space surrounding the tunnel after the shotcrete installation. It is again seen that the distribution of radial stresses for a specific value α with coefficient k is the same as that for α + 90° with coefficient 1/k, by rotating the tunnel axis by 90°.

Fig. 8
figure 8

Radial stresses after the shotcrete installation

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The distribution of radial stresses along the rock–shotcrete interface and therefore the bending of the shotcrete lining are depicted in Fig. 9. For k = 0.80 (Fig. 9a) and if α = 0°, the maximum scaled radial stress, σ r/σ o, equals to 3.08 % was found at the tunnel sidewalls, whereas the minimum scaled radial stress equals to 2.75 % was observed at the tunnel roof and invert. If α = 90°, the maximum scaled radial stress equals to 3.05 % was located at the tunnel sidewalls, whereas the minimum scaled radial stress equals to 2.97 % was found at the tunnel roof and invert. When α = 45°, the maximum scaled radial stress equals to 3.02 % was obtained at θ = 15° and 195°, whereas the minimum scaled radial stress equals to 2.80 % was situated at θ = 105° and 285°. If α = 135°, the maximum scaled radial stress equals to 3.02 % was found at θ = 165° and 345°, whereas the minimum scaled radial stress equals to 2.80 % was observed at θ = 75° and 255°.

Fig. 9
figure 9

Distributions of radial stresses along the rock–shotcrete interface

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For k = 1.25 (Fig. 9b) and when α = 0°, the maximum scaled radial stress equals to 3.05 % was found at the tunnel roof and invert, while the minimum scaled radial stress equals to 2.97 % was observed at the tunnel sidewalls. When α = 90°, the maximum scaled radial stress equals to 3.08 % was obtained at the tunnel roof and invert, while the minimum scaled radial stress equals to 2.78 % was located at the tunnel sidewalls. If α = 45°, the maximum scaled radial stress equals to 3.02 % was found at θ = 75° and 255°, whereas the minimum scaled radial stress equals to 2.80 % was found at θ = 165° and 345°. When α = 135°, the maximum radial stress equals to 3.02 % was found at θ = 105° and 285°, while the minimum scaled radial stress equals to 2.80 % was obtained at θ = 15° and 195°.

The numerical results of scaled radial stresses along the rock–shotcrete interface for both k = 0.80 and k = 1.25 are summarised in Table 4. Since the radial stresses remain in a compressive state, it can be concluded that there is no detachment of the shotcrete from the rock mass. The radial stresses are still below the compressive strength of concrete, f ck, type C20/25; meaning that the shotcrete with a thickness of 10 cm is adequate to stabilise the tunnel.

Table 4 Radial stresses along the rock–shotcrete interface
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4 Prestressed Concrete Lining

If the rock overburden is sufficient and the rock itself is of good quality, the long-term stability of concrete-lined pressure tunnels can be ensured by injecting the circumferential gap between the final lining and the shotcrete with cement-based grout at high pressure. As the grout is forced under high pressure, the gap is opened up and filled with densely compacted cement. In practice, axial pipes are embedded along the tunnel walls to allow precise injection in the gap (Fig. 10). The prestressing of the final lining aims not only to create adequate compressive hoop stresses in the final lining despite any pressure losses so that tensile stresses due to radial expansion are avoided during tunnel operation, but also to preserve the coupling of the lining and the rock mass. Due to the fact that the prestress in the final lining is produced by the support from the rock mass, this technique is called the passive prestressing technique.

Fig. 10
figure 10

Gap grouting procedure

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Principally, the level of the injection pressure applied into the gap depends on the properties of the rock mass, the concrete, and the thickness of the final lining. Taking into account a certain factor of safety, the grouting pressure must be maintained to a level below the smallest principle stress in the rock mass in order to avoid hydraulic jacking or fracturing of the rock mass.

It also has to be emphasised that consolidation grouting is prerequisite prior to prestressing the final lining. This is necessary to reinstate the mechanical properties of the rock mass and to provide stability to the loosened rock zone as a result of excavation (Schleiss 1986; Schleiss and Manso 2012). As the grout fills discontinuities in the rock mass and hardens, the permeability of the rock mass can also be reduced, which is favourable in view of limiting seepage into the rock mass. Therefore, if well grouted, consolidation grouting can improve the stress transfer from the lining to the rock, and vice versa.

The final lining herein is regarded as a pervious quasi-brittle material with low tensile strength. Its mechanical properties according to the concrete type C25/30 are given in Table 2. In view of the high compressive strength of concrete and the smallest principal stress in the rock mass, as high as 30 bar (3 MPa) of grouting pressure may be applied to prestress the final lining. However, considering the shrinkage during thermal cooling and creep, it was assumed that only 20 bar (2 MPa) of prestress, p p, remains effective at the shotcrete-final lining interface, according to the principle of Seeber (1999).

As illustrated in Fig. 10, there are five zones distinguished in this study: (1) the final concrete lining; (2) the shotcrete; (3) the circumferential gap between the concrete lining and the shotcrete; (4) the loosened zone of the rock mass (grouted zone); and (5) the non-disturbed rock mass. In order to reveal the prestress-induced hoop strains in the final lining, these five zones were modelled using continuum elements following the same approach as presented in Simanjuntak et al. (2012b, 2014b). The load sharing between the final lining and the rock mass was obtained according to the concept of compatibility conditions provided that the final concrete lining, the shotcrete, the circumferential gap, the grouted zone and the rock mass are continuous and tied with no slip conditions at their contact face.

In practice, the circumferential gap between the final lining and the shotcrete are opened up during grouting and sealed by the hardened grout. In DIANA, this mechanism was mimicked by applying thin continuum interface elements between the final lining and the shotcrete. The modulus of elasticity, E, of the grout and Poisson’s ratio, ν, are taken as 20 GPa and 0.20, respectively. The prestressing of the final lining was simulated in three steps and is explained below.

First, the final lining and the thin continuum elements representing the circumferential gap are activated as soon as the equilibrium condition after the support installation has been reached. Second, in order to take into account the consolidating effect of the grouting on the loosened rock zone, the material properties of the rock representing the loosened rock zone is replaced with the properties of the grouted rock mass. In order to ensure that the deformations calculated from the previous calculations will not affect the subsequent calculation steps, the deformations are reset to zero. Third, the grouting pressure is performed by assigning a positive value of volumetric strains to the elements representing the gap. Here, a positive value of the strain component represents an expansion. The grouting pressure is incremented such that the radial stress of 2 MPa in a compressive state along the extrados of the final lining is reached. To investigate the prestress-induced hoop strains in the final lining, the combined Rankine–Von Mises concrete model according to Feenstra (1993) was adopted. While the former bounds the tensile stresses, the latter is applicable in the compressive region.

It is important to note that if the grout completely fills the gap, the full contact between the final lining and the shotcrete is achieved. This process is called contact grouting, and the only load acting on the final lining during contact grouting is the grout pressure, which is constant along the perimeter of the final lining. As soon as the contact grouting has been completed, increasing volumetric strains of the interface elements will pull the final lining towards the tunnel axis on one side, and push the shotcrete and the surrounding rock mass away from the tunnel axis on the other side. In such circumstances, the rock mass also takes part of the load imposed during prestressing, especially when the final lining is pushed back against the shotcrete and the rock mass. Here, the prevailing hoop strains in the final lining depend not only on the relative stiffness of the different materials, i.e. the final lining, the shotcrete, the grouted zone and the rock mass, but also on the support from the rock mass characterised by the in situ stresses with respect to the planes of anisotropy.

In order to clarify whether the support from the rock affects the load sharing between the final lining and the rock mass during prestressing, the numerical results of prestress-induced hoop strains when the in situ stresses are uniform (k = 1) are included herein (Fig. 11) and will be compared with those when the in situ stresses are non-uniform, which is either when k = 0.80 (Figs. 12a, 13a) or when k = 1.25 (Figs. 12b, 13b). While Fig. 11a presents the distribution of prestress-induced hoop strains along the extrados of the final lining when the in situ stresses are uniform, Fig. 11b shows the results along the intrados of the final lining.

Fig. 11
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Distributions of prestress-induced hoop strains along the a extrados, and b intrados of the final lining (uniform in situ stresses)

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Fig. 12
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Distributions of prestress-induced hoop strains along the extrados of the final lining (non-uniform in situ stresses)

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Fig. 13
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Distributions of prestress-induced hoop strains along the intrados of the final lining (non-uniform in situ stresses)

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As an example, consider the case when the stratification planes are horizontal and the in situ stresses are uniform (α = 0° and k = 1). The increase in the volumetric strain in the interface elements (expansion) results in the final lining being pushed towards the tunnel axis harder in the horizontal direction than in the vertical direction. This is due to the fact that the stiffness of the rock mass in the horizontal direction is greater than that in the vertical direction. As the final lining pushed back harder in the horizontal direction, the compressive hoop strain at the roof (θ = 90°) and the invert (θ = 270°) of the final lining intrados will be greater than that at the sidewalls (θ = 0° and 180°) (Fig. 11b). This implies that as long as the in situ stresses are uniform (hydrostatic), the final lining will be under axisymmetric loading. The distribution of prestress-induced hoop strains at the final lining intrados is thus governed solely by the dip angle, α, since this is only the problem of rotational symmetry.

However, this not the case when the in situ stresses are non-uniform. For example, consider again the case when the stratification planes in the rock mass are horizontal, but the in situ vertical stress is greater than the horizontal stress illustrated in Fig. 13a (α = 0°, k = 0.80). Comparing Fig. 13a with Fig. 11b, it is seen that the compressive hoop strain at the roof (θ = 90°) and invert (θ = 270°) of the final lining intrados for the case when the in situ vertical stress is greater than the horizontal (α = 0°, k = 0.80) is higher than that of the case when the in situ stresses are uniform (α = 0°, k = 1). This is due to the fact that the support from the rock in the vertical direction is greater than that in the horizontal direction. Conversely, the compressive hoop strain at the sidewalls (θ = 0° and 180°) of the final lining intrados for the case when the in situ vertical stress is greater than the horizontal (α = 0°, k = 0.8) is lower than that of the case when the in situ stresses are uniform (α = 0°, k = 1). This is expected to occur since the support from the rock in the horizontal direction is lower than that in the vertical direction. Clearly, this implies that when the in situ stresses are non-uniform (non-hydrostatic), the final lining will not be under axisymmetric loading anymore. In such conditions, the distribution of prestress-induced hoop strain at the final lining intrados is governed not only by the dip angle, α, but also by the in situ stress ratio, k.

The numerical results of prestress-induced hoop strains along the extrados and intrados of the final lining when the in situ stresses are non-uniform are summarised in Tables 5 and 6, respectively. It is seen that a slight degree of compressive hoop strains was induced throughout the final lining as a result of prestress grouting. In the absence of internal pressure, the compressive hoop strains along the intrados of the final lining are greater than those along the extrados of the final lining (Timoshenko et al. 1970).

Table 5 Prestress-induced hoop strains along the extrados of the final lining
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Table 6 Prestress-induced hoop strains along the intrados of the final lining
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As mentioned previously, due to anisotropic properties of the rock and non-uniform in situ stresses, the final concrete lining is no longer under axisymmetric loading during prestress grouting and thus exhibits non-axisymmetrical deformations. While Fig. 12 shows the results of prestress-induced hoop strains along the extrados of the final lining, Fig. 13 displays the results along the intrados. Particular attention has to be paid to the smallest prestress-induced hoop strain at the intrados of the final lining since it will determine the maximum value of the internal water pressure that results in no tensile stresses in the final lining.

In the following, the results of prestress-induced hoop strains along the intrados of the final lining for cases when the in situ stresses are non-uniform as shown in Fig. 13 are analysed. Whereas Fig. 13a illustrates the results for k = 0.80, Fig. 13b depicts the results for k = 1.25. For k = 0.80 (Fig. 13a) and if the stratification planes are horizontal, the prestress-induced hoop strain, \(\varepsilon_{{\theta ,p_{\text{p}} }}^{i}\), at the roof and invert of the final lining intrados was found as 0.441‰, whereas at the sidewalls it was 0.440‰. If the stratification planes are vertical, the prestress-induced hoop strains along the final lining intrados are nearly uniform and were found as 0.440‰ with the hoop strain at the roof and invert slightly higher than that at the sidewalls. When α = 45°, the maximum prestress-induced hoop strain equals to 0.441‰ was found at the arcs of the final lining intrados, specifically at θ = 105° and 285°, while the minimum hoop strain equals to 0.440‰ was observed at θ = 15° and 195°. If α = 135°, the maximum prestress-induced hoop strain equals to 0.441‰ was found at θ = 75° and 255°, while the minimum hoop strain equals to 0.440‰ was observed at θ = 165° and 345°.

For k = 1.25 (Fig. 13b) and if α = 0°, the prestress-induced hoop strains along the final lining intrados are nearly uniform and were obtained as 0.440‰ with the hoop strain at the roof and invert slightly lower than that at the sidewalls. When α = 90°, the prestress-induced hoop strain at the roof and invert of the final lining intrados was found as 0.440‰, while at the sidewalls it was 0.441‰. When α = 45°, the maximum prestress-induced hoop strain equals to 0.441‰ was found at θ = 165° and 345°, whereas the minimum hoop strain equals to 0.440‰ was obtained at θ = 75° and 255°. When α = 135°, the maximum prestress-induced hoop strain equals to 0.441‰ was found at θ = 15° and 195°, whereas the minimum hoop strain equals to 0.440‰ was observed at θ = 105° and 285°. Once again, these results imply that the distribution of prestress-induced hoop strains in the final lining obtained for a specific value α with coefficient k will be the same as that for α + 90° with coefficient 1/k, if the tunnel axis is rotated by 90°.

Comparing Fig. 13a, b, it is obvious that higher compressive hoop strains at the intrados of final lining can be expected at the locations where higher in situ stresses in the rock mass are dominating. For cases where the in situ vertical stress is greater than the horizontal (k < 1), the maximum prestress-induced hoop strains are located around the roof and invert of the final lining intrados (Fig. 13a). When the in situ horizontal stress is greater than the vertical (k > 1), the maximum prestress-induced hoop strains are situated around the sidewalls of the final lining intrados (Fig. 13b).

It is essential to realise that the final concrete lining will be subjected to internal water pressure and exhibit radial expansion during tunnel operation. Once the tensile strength of concrete is exceeded, longitudinal cracks can occur in the final lining. Since the longitudinal cracks normally start to develop from the intrados of the final lining, the maximum value of the internal water pressure in which a passive prestressed concrete-lined pressure tunnel can convey, needs to be determined based on the smallest value of prestress-induced hoop strains at the intrados of the final lining.

5 Bearing Capacity of Prestressed Concrete-Lined Pressure Tunnels

Since much of the tensile strength of concrete has already been used in the thermal cooling, the final lining cannot transmit tensile stresses to the rock mass. Moreover, the low tensile strength of concrete cannot entirely be preserved throughout the lining due to inevitable construction joints.

To ensure that the hoop stresses in the final lining remain in a compressive state during tunnel operation, the following criterion has to be satisfied (Simanjuntak et al. 2014b):

$$\varepsilon_{{\theta ,p_{p} }}^{i} + \varepsilon_{{\theta ,p_{i} }}^{i} \le 0$$
(11)

where \(\varepsilon_{{\theta ,p_{\text{p}} }}^{i}\) and \(\varepsilon_{{\theta ,p_{\text{i}} }}^{i}\) are the prestress- and the seepage-induced hoop strain at the final lining intrados, respectively. As soon as the maximum internal water pressure is obtained using Eq. (11), a certain safety factor must be applied before applying the predicted internal water pressure into practice.

As an example, the rock mass permeability coefficient in the plane of isotropy, k rh, is taken as 10−5 m/s, while in the direction normal to the plane of isotropy, k rv, is 10−6 m/s. Considering the rock improvement due to the consolidating effect of the grouting, the permeability of the grouted rock mass becomes isotropic up to a radius of 3 m measured from the tunnel centre. The permeability of the hardened grout in the gap between the final and the shotcrete can be taken the same as that of the shotcrete lining. The data regarding the permeability coefficient of the grouted rock mass, k g, the shotcrete, k s, and the final lining, k c, are listed in Table 7.

Table 7 Permeability coefficient for grouted rock mass, shotcrete and final lining
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To determine the maximum internal water pressure, two hydraulic boundary conditions were introduced to the model, following the same approach as that has been introduced in Simanjuntak et al. (2012b, 2014b). The first boundary condition characterising the hydrostatic head imposed by the internal water pressure was applied to the perimeter of the final lining intrados, while the second boundary condition representing the groundwater level in the rock mass was assigned to the far-field boundary of the model domain. Since the pressure tunnel considered herein is situated above the groundwater level, the flow conditions at the far-field boundary of the model domain were set to open, except the one at the bottom of the model domain was set to close.

In DIANA, a uniform value of static water head can be assigned directly to the nodes along the intrados of the final lining. The fully coupled stress flow analysis was chosen so as to analyse combined flow and deformations around the pressure tunnel. In the simulation, the initial static water head was increased incrementally to a level yielding zero total hoop strain at the final lining intrados, according to the criterion given by Eq. (11).

The numerical analyses suggest that as soon as the static internal water pressure is increased up to 234 m, the total hoop strain at the final lining intrados became zero. The distributions of pore pressure head around the tunnel are presented in Fig. 14. Seepage, q, in the order of 0.35 l/s/bar per km length of the tunnel can be expected around the tunnel, which is below the acceptable value, i.e. 1 l/s/bar per km according to Schleiss (1988, 2013). A slight degree of hoop strains in a compressive state along the perimeter of the final lining extrados is presented in Fig. 15.

Fig. 14
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Distributions of pore pressure head around the pressure tunnel when the orientation of stratification planes in the rock mass is a horizontal, and b vertical

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Fig. 15
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Distributions of residual hoop strains along the extrados of the final lining

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In view of pervious concrete lining, seepage develops in the rock mass resulting in a bell-shaped saturated zone (Fig. 14), which corresponds to that was presumed by Schleiss (1997). Comparing Fig. 14a, b, it is interesting to note that the extent of the saturated zone around the tunnel depends on the direction-dependent permeability in the rock mass. While Fig. 14a shows the shape of saturated zone around the tunnel when the stratification planes in the rock mass are horizontal, Fig. 14b illustrates the shape of saturated zone around the tunnel when the stratification planes are vertical. Because the rock mass is very permeable compared to the final lining, the shotcrete, and the grouted zone, the vertical reach of the seepage flow is relatively small and may not negatively influence the hydrogeological conditions, such as the yield springs.

The distribution of residual hoop strains along the extrados of the final lining, \(\varepsilon_{{\theta ,{\text{res}}}}^{a}\), for k = 0.80 and k = 1.25 are depicted in Fig. 15. These results are summarised in Table 8. This once again implies that the redistribution of hoop strains along the extrados of the final lining as a result of the loading of internal water pressure for a specific value α with coefficient k will correspond to that for α + 90° with coefficient 1/k, by rotating the tunnel axis by 90°.

Table 8 Residual hoop strains along the extrados of the final lining
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Theoretically, as long as the static water head is not higher than 234 m or the internal water pressure applied is not greater than 23 bar (2.3 MPa), the final lining will be free from tensile stresses. However, a factor of safety between 1.35 and 1.50 with reference to ÖNORM (2001) can be applied to account for uncertainties in practice.

Utilising the information provided in Fig. 13, one can identify the areas in the final lining, which are vulnerable to cracking. Due to symmetry, a minimum of two longitudinal cracks can be expected to occur in a concrete lining. Cracks in a final lining can lead to either hydraulic jacking or fracturing of the rock mass and therefore has to be avoided. Whereas the hydraulic jacking is the opening of existing joints in the rock mass due to high internal water pressure, the hydraulic fracturing is the event that produces factures in a sound rock.

If the in situ vertical stress is greater than the horizontal (k < 1) (Fig. 13a), pressure tunnels embedded in transversely isotropic rock with horizontal stratification planes will be the most critical situation with regard to tunnel stability. If the in situ horizontal stress is greater than the vertical (k > 1) (Fig. 13b), pressure tunnels embedded in transversely isotropic rock with vertical stratification planes will be the most unfavourable situation in respect of tunnel stability. For the other situations where the stratification planes make an angle α to the x-axis, the location of longitudinal cracks in the final lining can be considered as intermediate between the two. The potential crack locations in the final lining are summarised in Table 9.

Table 9 Predicted crack locations in the final lining
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Once longitudinal cracks occur in a final lining, high local seepage can take place around the crack opening. Using an overall rock mass permeability equals to the highest permeability that the rock may have, one can quickly estimate the amount of seepage into the rock mass using the analytical approach proposed by Simanjuntak et al. (2013).

6 Conclusions

As long as the rock mass is not too pervious and the smallest principal stress in the rock mass is higher than the internal water pressure, employing concrete-lined pressure tunnels instead of steel linings to convey water from reservoirs to hydroelectric power plants may be economically attractive. The long-term bearing capacity of such tunnels can be ensured by using the passive presstressing technique. Adequate compressive hoop strains in the lining are induced by grouting the circumferential gap between the final lining and the support lining at high pressure, so that the lining will be free from tension during tunnel operation. Passively prestressed concrete-lined pressure tunnels are axisymmetrical multilayer structures composed of a final concrete lining, a support lining (shotcrete), a grouted rock zone and a sound far-field rock zone.

For the design of passively prestressed concrete-lined pressure tunnels, anisotropic behaviour of the rock mass is rarely considered. Tunnel designers rather assume isotropic rock behaviour, considering the most unfavourable elastic modulus of the rock mass measured in situ. This assumption is usually conservative in terms of maximum stresses in the final lining. Consequently, the load sharing between the lining and the rock when the pressure tunnels are embedded in anisotropic rock formations is not yet fully understood.

In this study, the interplay between the in situ stress ratio and the anisotropic behaviour of the rock mass on the lining performance are explored by means of a two-dimensional finite element model. While the rock mass supporting the pressure tunnel is considered as transversely isotropic or cross-anisotropic linear elastic material, the final lining and the shotcrete are pervious and elastic. Furthermore, tied contact is assumed between the layers and plane strain conditions apply along the tunnel axis.

A series of numerical analyses relevant in tunnelling, i.e. excavation, support installation, lining prestressing, and loading of internal water pressure is also presented herein. The pressure tunnel being considered is situated above the groundwater level and the orientations of stratification planes in the rock mass α are taken as 0°, 45°, 90°, and 135°, with 0° indicating the horizontal bedding and 90° the vertical. Two distinctive cases were examined based on whether the in situ vertical stress is greater than the horizontal (k < 1) or the in situ horizontal stress is greater than the vertical (k > 1). With regard to the mechanical behaviour of pressure tunnels, the study suggests that the distribution of stresses and deformations for a specific value α with coefficient k is identical to that for α + 90° with coefficient 1/k by rotating the tunnel axis by 90°.

Putting the prestress-induced hoop strains at the final lining intrados equals to the seepage-induced hoop strains, the maximum internal water pressure can be determined. In view of pervious concrete lining, seepage occurs in the rock mass, developing a saturated zone around the tunnel. It is interesting to note that the saturated zone is exclusively governed by the direction-dependent permeability of the rock mass. In addition to assessing the maximum internal water pressure, the proposed approach is applicable to identifying potential locations where longitudinal cracks can occur in the final lining.

When using the passive prestressing technique, longitudinal cracks in the lining must be avoided since it can cause hydraulic jacking or fracturing, which can endanger the overall tunnel stability. The study reveals that when the in situ vertical stress is greater than the horizontal (k < 1), the stability is unfavourable if the tunnel is embedded in transversely isotropic rock with horizontal bedding. Conversely, when the in situ horizontal stress is greater than the vertical (k > 1), the stability is critical if the tunnel is embedded in transversely isotropic rock with vertical bedding.

Looking at the large scale in the vicinity of pressure tunnels, it is essential to consider that cracks will always exist in any rock mass. Only if the stability of passively prestressed concrete-lined pressure tunnels against hydraulic jacking or fracturing can be guaranteed by adequate rock strength or overburden, seepage into the rock mass can be tolerated. Otherwise, no longitudinal cracks are allowed to occur in the final lining since the crack openings are difficult to control with the passive prestressing technique. The design criteria for passively prestressed concrete-lined pressure tunnels are therefore: avoiding cracks in the final lining, limiting seepage into the rock mass, and ensuring the bearing capacity of the rock mass supporting the tunnel.

It is however worth mentioning that this study was carried out based on the main assumption that the rock mass was considered as an elastic transversely isotropic material configured by one direction with no slip conditions along the bedding planes. If the behaviour of the rock mass supporting a pressure tunnel is controlled by persistent discontinuities or if the transverse isotropy does not strike parallel to the tunnel axis, the load sharing between the lining and the rock needs to be investigated using an approach going beyond the ones introduced herein so as to acquire more accurate results.