Design of Quasi-Rectangular Tunnel Built in the Rock Masses Following Hoek–Brown Failure Criterion

Abstract

Although quasi-rectangular tunnels have been widely used in underground projects, there is no related research work that was carried out to design this kind of tunnel excavated in the rock masses that followed the Hoek–Brown (HB) criterion. By transforming the strength parameters of the rocks that obey the HB failure criterion into equivalent strength parameters of the Mohr–Coulomb (MC) criterion, the lining internal forces of a quasi-rectangular tunnel are investigated in this manuscript in light of bending moments and normal forces using the Hyperstatic Reaction Method (HRM). After comparing with the values of a model test and verifying the present method, the influence of different tunnel depths, H, different rock unconfined compressive strengths, ${\sigma }_{ci}$, and the geological strength parameter GSI on the lining internal forces of the quasi-rectangular tunnel is then investigated. The obtained results indicate that the parameters H, ${\sigma }_{ci}$, and GSI have a significant effect on the lining internal forces of the quasi-rectangular tunnel. The work presented in this paper provides a theoretical reference for designing the quasi-rectangular tunnel support structure built in the rock masses following the Hoek–Brown criterion.

1. Introduction

The tunnel cross-section undergoes a development process from a rectangular shape to a circular shape and then to a non-circular shape. The trend of its development is to continuously improve the utilization rate of the cross-section and to meet the requirements of various actual projects [1], which has led to an increasing number of tunnels of different diameters and different cross-sections being applied to actual tunneling projects accompanied with the progress of tunnel construction technology. Since the quasi-rectangular tunnels have a higher utilization ratio of the cross-section and a lower probability of rolling than the circular ones, they are becoming increasingly common in underground engineering projects [1], for instance, the Jing’an Temple Station of the Metro Line 14 in Shanghai, the interval tunnel project between Luxiang Road and Qilianshan Road in Shanghai, etc.

The quasi-rectangular tunnels have greater bearing capacity compared to rectangular tunnels under the same overburden condition [1]. However, their stress distribution is greatly different from either rectangular tunnels or circular tunnels. The distribution of the lining internal forces caused by external loads changes depending on the form of a tunnel cross-section. Nakamura et al. [2] performed the full-scale loading tests on lining segments to confirm the design adequacy of the rectangular-shaped tunnel. By conducting a full-scale ring test, Liu et al. [3] investigated the mechanical behavior of quasi-rectangular segmental tunnels. Zhang et al. [4] performed a full-scale loading test for ‘standing’ special-shaped tunnel linings to estimate the impact of lining self-weight on the internal forces of lining under shallowly buried conditions. They developed a platform used to acquire data and the mechanical behaviors of the tunnel lining were evaluated. In particular, the internal force distribution pattern caused by the self-weight loading was different from that subjected to earth and water pressure. Du et al. [5] presented an optimization procedure of a quasi-rectangular tunnel cross-section by means of the HRM. A series of calculation equations that are able to consider the tunnel shape were derived and the effect of different parameters on the lining internal forces and shape of a quasi-rectangular tunnel was also investigated.

Despite the rapid development of quasi-rectangular tunnels, the research works mentioned above did not present how to analyze the internal forces of the quasi-rectangular tunnels excavated in rocks following the HB criterion [6,7,8]. In fact, tunnels often had to be built to connect two places in order to promote economic development, for example, the ongoing Sichuan–Tibet Railway project in China and the railway tunnel project between Germany and Italy across the Alps. As an increasing number of quasi-rectangular tunnels are adopted in the construction of mountain railway projects, it has become necessary to calculate the internal forces of this kind of tunnel excavated in rock masses following the HB failure criterion.

The aim of this manuscript is to study the internal forces of a quasi-rectangular tunnel lining built in the rock masses obeying HB failure criterion, and to provide a way to design the quasi-rectangular tunnels excavated in the rock masses following the HB failure criterion, which will be able to facilitate the development of this kind of tunnel.

The HRM was proposed by Oreste [9] for dimensioning the tunnel lining. This method was adopted to evaluate the internal forces of a circular segmental tunnel by Do et al. [10], who presented that the HRM is able to give reliable results but with less calculation time. Hence, this manuscript evaluates the lining internal forces of a quasi-rectangular tunnel built in the rock masses obeying HB criterion by means of the HRM. By transforming the HB strength parameters of the rocks into equivalent strength parameters of the MC criterion, it is realized to calculate the lining internal forces of a quasi-rectangular tunnel built in the rock masses obeying the HB criterion by means of the HRM. The method is successfully verified by comparison with values of a model test. The influence of different tunnel depths, H, different rock unconfined compressive strengths, ${\sigma }_{ci}$, and the geological strength parameter GSI on the lining internal forces of quasi-rectangular tunnel in terms of the bending moments and normal forces is then investigated. The obtained results showed that the parameters H, ${\sigma }_{ci}$, and GSI have a significant effect on the internal forces of the quasi-rectangular tunnel lining, which cannot be neglected in tunnel design. The present work is able to give a theoretical reference for designing the quasi-rectangular tunnel in the rock masses following the HB criterion.

2. Generalized HB Criterion and Equivalent MC Parameters

2.1. Generalized HB Criterion

Hoek et al. [11] introduced a generalized HB failure criterion to calculate the rock strength, which is:

$${\sigma }_1={\sigma }_3+{\sigma }_{ci}{(m_b\frac{{\sigma }_3}{{\sigma }_{ci}}+s)}^a$$

(1)

in which ${\sigma }_1$ is the major principal stress; ${\sigma }_3$ represents the minor principal stress; ${\sigma }_{ci}$ means the uniaxial compressive strength of the rock at failure (0.01 MPa $\le {\sigma }_{ci}\le$ 200 MPa); $m_b$, a, and s are the rock parameters, which are given as follows:

$$m_b=m_i\exp (\frac{GSI-100}{28-14D})$$

(2)

$$s=\exp (\frac{GSI-100}{9-3D})$$

(3)

$$a=\frac{1}{2}+\frac{1}{6}[\exp \left(-\frac{GSI}{15}\right)-\exp \left(-\frac{20}{3}\right)]$$

(4)

where $m_i$ is a material constant of the rock, which is in the range of 5 to 35; D is the factor to describe the disturbance degree of the rock [12,13]. The value of D is in the range of 0 to 1, and it means the disturbance to the rock masses caused by the tunnel excavation is very little when $D=0$. As this study focused on the analysis of internal forces of tunnel lining, $D=0$ is considered in this study. The Geological Strength Index (GSI) is a parameter to describe the deformation and strength of rock [11]. Its values could be evaluated according to characterization of rock masses ($5\le \mathrm{GSI}\le 100$). In addition, the Young’s modulus of rock mass $E_{rm}$ (MPa) could be calculated as follows [14]:

$$E_{rm}=1000(1-\frac{D}{2})\sqrt{\frac{{\sigma }_{ci}}{100}}·{10}^{\left(\frac{GSI-10}{40}\right)}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}\hspace{1em}{\sigma }_{ci}\le 100 \mathrm{MPa}$$

(5)

$$E_{rm}=1000(1-\frac{D}{2})·{10}^{(\frac{GSI-10}{40})}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{if}\hspace{1em}{\sigma }_{ci}>100 \mathrm{MPa}$$

(6)

Therefore, the shear modulus of rock mass $G_{rm}$ (MPa) could then be calculated by:

$$G_{rm}=\frac{E_{rm}}{2\left(1+v_{rm}\right)}$$

(7)

in which $v_{rm}$ is the Poisson’s ratio of rock mass (in the range of 0.1~0.3), and $v_{rm}$ = 0.3 is considered in the following analysis [15].

2.2. Equivalent MC Parameters

The friction angle $\phi$ and cohesion c of surrounding ground are the strength parameters which are usually needed to be considered in the tunnel design. Hoek [15] proposed the formulas to obtain the equivalent MC parameters c and $\phi$ for the rock masses obeying HB failure criterion, as follows:

$$c^{\prime }=\frac{{\sigma }_{ci}\left[\left(1+2a\right)s+\left(1-a\right){\mathrm{m}}_b{\sigma }_{3n}^{\prime }\right]{\left(s+{\mathrm{m}}_b{\sigma }_{3n}^{\prime }\right)}^{a-1}}{\left(1+a\right)\left(2+a\right)\sqrt{1+\left[6a{\mathrm{m}}_b{\left(s+{\mathrm{m}}_b{\sigma }_{3n}^{\prime }\right)}^{a-1}\right]/\left[\left(1+a\right)\left(2+a\right)\right]}}$$

(8)

$${\phi }^{\prime }=\arcsin \left[\frac{6a{\mathrm{m}}_b{\left(s+{\mathrm{m}}_b{\sigma }_{3n}^{\prime }\right)}^{a-1}}{2\left(1+a\right)\left(2+a\right)+6a{\mathrm{m}}_b{\left(s+{\mathrm{m}}_b{\sigma }_{3n}^{\prime }\right)}^{a-1}}\right]$$

(9)

in which ${\sigma }_{3n}^{\prime }={\sigma }_{3max}^{\prime }/{\sigma }_{ci}$. As for the deep-buried tunnels, ${\sigma }_{3max}^{\prime }$ is estimated by:

$$\frac{{\sigma }_{3max}^{\prime }}{{\sigma }_{cm}^{\prime }}=0.47{\left(\frac{{\sigma }_{cm}^{\prime }}{\gamma H}\right)}^{-0.94}$$

(10)

where H is the buried depth of the tunnel; $\gamma$ is the unit weight of rock. It should be noted that $\gamma H$ in Equation (10) should be replaced by the horizontal stress when the vertical stress is less than the horizontal stress; ${\sigma }_{cm}^{\prime }$ represents the strength of rock, which can be expressed by the following equation:

$${\sigma }_{cm}^{\prime }={\sigma }_{ci}·\frac{\left[{\mathrm{m}}_b+4s-a\left({\mathrm{m}}_b-8s\right)\right]{\left(\frac{{\mathrm{m}}_b}{4}+s\right)}^{a-1}}{2\left(1+a\right)\left(2+a\right)}$$

(11)

In the present work, the lining internal forces of the quasi-rectangular tunnel built in the rock masses obeying the HB failure criterion are analyzed by means of the HRM. The equivalent MC parameters used in the HRM method are calculated by Equations (8) and (9).

3. HRM

The HRM can simulate the interaction between the structure and rock mass around the tunnel through the “Winkler” springs. Because of the finite number of spring connections of the structure with the rock, it is named hyperstatic. The structure–rock interaction in this method is able to consider the mechanical characteristics of the surrounding rock. This interaction effects to a great extent the stress state in the tunnel lining. Oreste [9] proposed the HRM for dimensioning the tunnel lining. Based on the results of this method, the parameters that influence the dimensioning technique of the lining structure were presented. The calculation in the HRM was performed by developing a FEMSUP code based on a FEM framework. Oreste [9] performed an extensive parametric analysis using this FEMSUP code to verify the impact of different parameters on the dimensioning of the tunnel lining. Therefore, the rough indications can be given in the preliminary stages of a tunnel design. Based on the model of Oreste [7], Do et al. [10] presented an improved HRM to investigate the behavior of the segmental tunnel lining, and the parametric analyses were conducted to evaluate the lining behavior. Their study showed that the results of HRM were in good agreement with that of the numerical model FLAC in regard to the tunnel location. It should be noted that the design of a quasi-rectangular tunnel built in the rock masses following the Hoek–Brown failure criterion was not considered by Oreste [9] and Do et al. [10].

The HRM can be used to calculate the deformation and internal forces of the tunnel lining in a fast way, and it is particularly suitable for tunnel design [9,10]. As shown in Figure 1, it is assumed that the tunnel lining support is composed of a finite number of beam elements and those elements are connected to each other by nodes. The rock–structure interaction is therefore built by the normal and shear springs. The active loads can be transferred to the tunnel lining by those springs in the HRM. In Figure 1, σ_h is the horizontal loads; σ_v is the vertical loads; k_s means the shear stiffness of springs; k_n means the normal stiffness of springs; X and Y are the global Cartesian coordinates; EA and EJ represent the normal and bending stiffness of the support, respectively.

Since the rock–structure interaction is simulated by the springs distributed over the nodes, the element stress of the lining can be calculated on the basis of the node displacements. In other words, the internal forces of the whole tunnel structure can be calculated after the displacements of the nodes obtained. Notice that the number of elements have an impact on the calculated results. However, when the number of elements exceeds a certain value, the calculation results vary slightly, but the calculation time increases significantly. Therefore, taking the calculation accuracy and time into account, the number of elements in the present work is taken as 360.

The detailed introduction of HRM method is not presented here to avoid excessive repetition. For more details, one could refer to Do et al. [10].

3.1. Interaction of Rock Mass and Structure

As the tunnel lining is composed of the beam elements and the connection between the lining and rock masses is connected by springs, the rock–structure interaction in the HRM method is then established through springs (normal and shear).

The pressure-deformation ($p-\delta$) relation is nonlinear (hyperbolic) [9], which is shown in Figure 2. Therefore, the reaction pressure can be calculated by the following equation:

$$p=p_{lim}·\left(1-\frac{p_{lim}}{p_{lim}+{\eta }_0·\delta }\right)$$

(12)

in which $p_{lim}$ and ${\eta }_0$ represent the maximum reaction pressure and the initial stiffness of the rock mass, respectively. Specifically, the initial normal stiffness of rock mass ${\eta }_{n,0}$ is calculated by the following formula mentioned in Do et al. [10]:

$${\eta }_{n,0}=\beta \frac{1}{1+v_s}·\frac{E_s}{R}$$

(13)

where $\beta$ is a dimensionless parameter and it depends on the lining geometry [10]; $R$ is the radius of the tunnel lining; $E_s$ and $v_{\mathrm{s}}$ are the Young’s modulus and the Poisson’s ratio of the rock mass, respectively.

The shear stiffness can be calculated as follows [16]:

$${\eta }_s=0.5\frac{{\eta }_n}{1+v_s}$$

(14)

The maximum normal reaction pressure $p_{n,lim}$ is calculated by the following equation:

$$p_{n,lim}=\frac{2·c·cos\phi }{1-sin\phi }+\frac{1+sin\phi }{1-sin\phi }·\frac{{\sigma }_h+{\sigma }_v}{2}·\frac{v_s}{1-v_s}$$

(15)

Clearly, $p_{n,lim}$ depends on the cohesion c, friction angle φ, Poisson’s ratio $v_{\mathrm{s}}$, and external loads σ_v and σ_h. Furthermore, the maximum shear reaction pressure $p_{s,lim}$ can be given as follows:

$$p_{s,lim}=\frac{{\sigma }_h+{\sigma }_v}{2}·tan\phi$$

(16)

The stiffness $k_{n,i}$ and $k_{s,i}$ of spring could be calculated as follows:

$$k_{n,i}={\eta }_{n,i}^{*}·\left[\frac{\left(L_{i-1}+L_i\right)}{2}·1\right]=\frac{p_{n, lim}}{{\delta }_{n,i}}·\left(1-\frac{p_{n,lim}}{p_{n,lim}+{\eta }_{n,0}·{\delta }_{n,i}}\right)·\frac{\left(L_{i-1}+L_i\right)}{2}$$

(17)

$$k_{s,i}={\eta }_{s,i}^{*}·\left[\frac{\left(L_{i-1}+L_i\right)}{2}·1\right]=\frac{p_{s, lim}}{{\delta }_{s,i}}·\left(1-\frac{p_{s,lim}}{p_{s,lim}+{\eta }_{s,0}·{\delta }_{s,i}}\right)·\frac{\left(L_{i-1}+L_i\right)}{2}$$

(18)

in which $L_i$ represents the length of element $i$; ${\eta }^{*}$ is the apparent stiffness of surrounding rock mass, which could be given by the $p/\delta$ ratio, as follows:

$${\eta }^{*}=\frac{p_{lim}}{\delta }·\left(1-\frac{p_{lim}}{p_{lim}+{\eta }_0·\delta }\right)$$

(19)

3.2. Active External Loads

The vertical load ${\sigma }_v$ and active horizontal load σ_h could be calculated as follows:

$${\sigma }_v=\gamma H$$

(20)

$$\sigma =K_0·{\sigma }_v$$

(21)

where $K_0$ is the coefficient of lateral earth pressure.

4. Comparison and Validation

In this section, a comparison of the results with a full-scale loading test of a quasi-rectangular tunnel [4] is given to evaluate the accuracy of the present HRM. The parameters used in the comparison are adopted from the full-scale loading test of Zhang et al. [4], in which the thickness t of the lining is 0.5 m; the Young’s modulus E of the lining is 29.5 GPa; the unit weight γ of the rock mass is 18 kN/m³; the parameters of HB failure criterion are ${\sigma }_{ci}=$ 30 MPa, $m_i=$ 20, GSI = 60, and D = 0; and K₀ = 0.6. In addition, the quasi-rectangular tunnel cross-section is shown in Figure 3.

Table 1. Comparison of the HRM and results of Zhang et al. [4] with different tunnel depths H.

Burial Depth H/m	Max. Bending Moment M (kN.m)			Max. Normal Force N (kN)
	Zhang et al. (2019)	HRM	Error (%)	Zhang et al. (2019)	HRM	Error (%)
5	458.6	458.4	−0.04	994.1	979.7	−1.47
10	847.9	847.3	−0.07	1422.2	1440.1	1.24
15	1240.6	1239.8	−0.06	1951.6	1952.4	0.04

gives the comparison of the results in terms of the maximum bending moment M (kN.m) and the maximum normal force N (kN) of the tunnel lining considering different tunnel depths H. It could be found from Table 1 that the outcomes calculated by the present HRM are very close to the ones of Zhang et al. [4]. The relative error is less than 1.5%. This agreement means that the HRM method is valid to evaluate the internal forces of the quasi-rectangular tunnel lining.

5. Parametric Analysis

This section gives the internal forces of a quasi-rectangular tunnel considering different tunnel depths H, different rock unconfined compressive strengths ${\sigma }_{ci}$, and the geological strength parameter GSI. The influence of different H,${\sigma }_{ci}$, and GSI on the internal forces of a quasi-rectangular lining is investigated. Notice that the positive direction of the structural internal forces (normal force N, bending moment M) considered in the following analysis is displayed in Figure 4, and θ is the angle between the vertical axis of the cross-section and the element axis.

5.1. Impact of Tunnel Depth H on the Internal Forces of a Quasi-Rectangular Tunnel

The influence of tunnel depth H on the internal forces of quasi-rectangular tunnel is studied herein with t = 0.5 m, E = 29.5 GPa, γ = 18 kN/m³, ${\sigma }_{ci}=$ 50 MPa, GSI = 60, $m_i=$ 10, D = 0, and K₀ = 0.6. The values H = 10 m, 25 m, 40 m are taken into consideration. The distributions of internal forces of the quasi-rectangular tunnel lining considering different tunnel depths H are shown in Figure 5.

Figure 5 shows that the tunnel depth H has a great impact on the lining internal forces of the quasi-rectangular tunnel. Since the increase in H causes the increase in external loads applied to the lining structure, the bending moment M and normal force N increase with the increase in H. The maximum absolute bending moment occurs at the invert arch of the lining; however, the maximum absolute normal force appears at the sidewall of the lining.

5.2. Influence of ${\sigma }_{ci}$ on the Internal Forces of a Quasi-Rectangular Tunnel

The impact of the rock unconfined compressive strength ${\sigma }_{ci}$ on the internal forces of the quasi-rectangular tunnel lining is given in this section, and the parameters shown in Section 5.1 are also adopted in the analysis, except γ = 25 kN/m³ and H = 15 m. The values of ${\sigma }_{ci}=$ 25 MPa, 50 MPa, and 75 MPa are taken into consideration. The internal forces are shown in Figure 6 considering various ${\sigma }_{ci}$.

It can be found from Figure 6 that the values of the bending moment M decrease with the increase in ${\sigma }_{ci}$. In contrast, the normal forces N increase with the increase in ${\sigma }_{ci}$. It should be noted that the impact of ${\sigma }_{ci}$ on N is more significant than that of M at the tunnel crown and invert arch. In addition, similar to Figure 5, the maximum absolute bending moment occurs at the invert arch of the lining. The maximum absolute normal force of approximately 2.58 MN/m appears at the sidewall of the lining.

5.3. Influence of GSI on the Internal Forces of a Quasi-Rectangular Tunnel

The effect of geological strength parameter GSI on the internal forces of a quasi-rectangular lining is presented in Figure 7, corresponding to t = 0.5 m, E = 29.5 GPa, γ = 25 kN/m³,$m_i=$ 10, ${\sigma }_{ci}=$ 50 MPa, H = 15 m, D = 0, and K₀ = 0.6. The values of GSI close to 100 correspond to excellent-quality rock masses, while its values close to 10 correspond to very poor-quality rock masses. In addition, Carranza-Torres [17] found that a rock mass with a disturbed structure has a GSI close to 30. Herein, GSI values equal to 30, 60, and 90 are considered in the following study as the rock surrounding the excavated tunnel could not remain intact [18,19,20,21,22].

Figure 7 describes that the GSI has a great impact on M; however, it has no clear significant influence on N. The bending moments M decrease as the GSI increases. In contrast, the normal forces N increase with the increase in GSI. Specifically, the magnitude of bending moment M at the tunnel invert arch decreases from 1.92 MN·m/m to 0.87 MN.m/m as the GSI increases from 30 to 90. Similarly, the magnitude of normal force N at the tunnel invert arch increases from 1.59 MN/m to 2.51 MN/m. The reason is that the higher GSI values give better rock mass surface conditions and lower discontinuities, and the self-stability and the integrity of the excavated rock masses are better. Similarly, the maximum absolute bending moment M occurs at the invert arch of the lining, and the maximum absolute normal force appears at the sidewall of the lining.

6. Conclusions

By transforming the strength parameters of the rock that obey the HB failure criterion into equivalent Mohr–Coulomb (MC) strength parameters, the lining internal forces of a quasi-rectangular tunnel built in the rock masses obeying HB failure criterion were studied by means of the HRM. This method is easy for model construction and high computational efficiency which permits a large number of simulations within a short time. The present work is verified by comparing with the values of a model test. The influence of different tunnel depths, H, different rock unconfined compressive strengths, ${\sigma }_{ci}$, and the geological strength parameter GSI on the internal forces of a quasi-rectangular tunnel is investigated. Several conclusions are obtained as follows:

(1)

The bending moments M of the quasi-rectangular tunnel lining increase as the tunnel depth H increases but decrease as the ${\sigma }_{ci}$ and GSI increase.

(2)

The normal forces N of the quasi-rectangular tunnel lining increase as the H, ${\sigma }_{ci}$, and GSI increase.

(3)

The maximum absolute bending moment M of the quasi-rectangular tunnel lining occurs at the invert arch of lining, and the maximum absolute normal force appears at the sidewall of the lining for different values of H, ${\sigma }_{ci}$, and GSI.

In conclusion, the parameters H, ${\sigma }_{ci}$, and GSI have a significant effect on the lining internal forces of a quasi-rectangular tunnel. The present work provides a theoretical reference for designing the quasi-rectangular tunnel support structure in the rock masses following the HB criterion, which can facilitate the development of quasi-rectangular tunnels.