Research on Rock Resistant Coefficient in Lining Tunnels Based on the Generalized Hoek-Brown Failure Criterion

Rock resistant coe ﬃ cient is a key parameter in rock engineering and can re ﬂ ect the resistance of surrounding rocks. According to the load-structure method, there are many di ﬀ erent formulas in the determination of rock resistant coe ﬃ cient using elasticity or plasticity theory. Based on the generalized Hoek-Brown failure criterion, this paper develops a closed-form formula for calculating the rock resistant coe ﬃ cient. The sensibility of di ﬀ erent parameters on the rock resistant coe ﬃ cient is also analyzed. The fact shows that the formula is sensitive to a , σ c , and K d comparing with m b and s especially when they are in a low value, laying a foundation for the determination of rock resistant coe ﬃ cient in similar lining tunnels.


Introduction
Geomaterials have an inherent ability of resisting load, and the ability can be quantified by a coefficient named as subgrade reaction coefficient in soil mechanics or rock resistant coefficient in rock mechanics. Rock resistant coefficient can reflect the physical and mechanical properties of surrounding rock, and it is a key parameter in the design of tunnel lining. During the designing of a project, there are many different methods in the determination of rock resistant coefficient [1][2][3][4][5]. Xu (1993) and Qian (1955) proposed their formulas using the elasticity theory [6,7]. Tu et al. (2008) develops a formula for calculating rock resistant coefficient for underwater tunnel based on Mohr-Column yield criterion [8]. Cai et al. (2007) gave a common method to calculate the resistance coefficient of adjoining rock for pressure tunnels by use of the unified strength theory [9]. Zhu et al. (2011) proposed a formula for calculating rock resistant coefficient based on SMP failure criteria [10].
However, there is not a formula based on the generalized Hoek-Brown failure criterion which has been widely used in rock engineering as a semiempirical method.  pointed out that the most commonly used criteria may not be valid for many types of rock, particularly for a jointed rock mass [11], For such rock, the Hoek-Brown failure criterion has been proved to be more suitable. For this consideration, an analytical solution of calculating the rock resistant coefficient with the generalized Hoek-Brown failure criterion is proposed in this paper.

Generalized Hoek-Brown Criterion
The Hoek-Brown failure criterion for rock masses is widely accepted and has been applied in a large number of rock engineering projects around the world.  introduced their failure criterion in an attempt to provide input data for the analyses required for the design of underground excavations in hard rock [12][13][14]. The criterion was derived from the results of research on the brittle failure of intact rock by Hoek (1968) and on the modeling of jointed rock mass behaviour by . According to the latest modification, the generalized Hoek-Brown failure criterion is expressed as follows by Hoek et al. (2002) [15]: where σ 1 and σ 3 are the major and minor principal stresses, respectively; σ ci is the uniaxial compressive strength of the intact rock; m b is a reduced value of the material constant m i and is given by where s and a are constants for the rock mass given by the following relationships: In the equations above, m i is a material constant for the intact rock, GSI is the Geological Strength Index of the rock mass, and D is a rock mass disturbance factor. The ranges of GSI and D are normally as follows: 10 ≤ GSI ≤ 100 and 0 ≤ D ≤ 1. The value of GSI = 10 represents an extremely poor quality of rock mass, and D = 1 corresponds to a highly disturbed rock mass. For GSI = 100, the value of the constant a is equal to 0.5. In this case, the generalized failure criterion can degenerate to the original Hoek-Brown failure criterion.

Analytical Solution for Rock Resistant Coefficient
In order to obtain the rock resistant coefficient, a simplified model is shown in Figure 1. A circular opening excavated in a continuous, homogeneous, isotropic, initially elastic rock mass subjected to a hydrostatic stress p 0 . The opening surface is subjected to an internal pressure p, and the material behavior used in this study is Hoek-Brown failure criterion. Under normal circumstances, there is a plastic zone surrounded by an elastic zone. For some special cases, there is a loose zone close to the lining, which can only transfer radial stress. In this paper, considering the timely support in most of practical engineering, the loose zone is neglected.
Assuming that the displacement of tunnel surface is u 0 , according to the Winkler hypothesis, the rock resistant coefficient can be expressed as

Analytical Solutions of Stresses and the Radius of Plastic
Zone. It is assumed that rock resistance is of radial uniform distribution around the tunnel. For the tunnel is long enough, it can be considered as an axisymmetric, plane strain problem. σ 1 and σ 3 are equal to circumferential and radial stresses σ θ and σ r , respectively. According to the generalized Hoek-Brown failure criterion, relationship between σ r and σ θ can be written as follows: The radial stress σ R at the interface of elastic and plastic zone may be obtained by combining the relationship between σ r and σ θ in elastic and plastic state.
For generalized Hoek-Brown failure criterion, it is impossible to give analytical solutions of Equation (6). Thus, an approximate closed-form solution for σ R can be obtained by the Newton-Raphson method. When f ðxÞ = 0, the corresponding formula is The initial value for σ R0 is determined when a = 0:5.
According Equation (5), the resulting solution σ R can be expressed as where In fact, the above approximate solution has been developed by Sofianos (2003).
In plastic zone, the stresses are governed by failure criterion. According to the equation of equilibrium in polar coordinates,

Geofluids
Substitution of Equation (5) in Equation (10) leads to the following equation: This is a simple differential equation with an initial condition of σ r| r=r 0 = p, the result solution may be expressed as follows: The circumferential σ θ can be given by substituting Equation (12) in Equation (5).
The radius of plastic zones can be deprived by solving the equation obtained by substituting r = R 0 and σ r = σ R in Equation (12). The solution is expressed as 3.2. Analytical Solutions of Displacement. In the plastic region, total radial and tangential strains, ε r and ε θ , can be decomposed into elastic and plastic parts.
The strains can be expressed in terms of inward radial displacement u by Kyung and Yong (2006) et al. [16].
By using the small deformation theory and a linear flow rule, the plastic parts of radial and tangential strains may be related for the plane strain condition as For the Hoek-Brown failure criterion, the minimum of the dilation parameter is K d = 1 corresponding to a nondilating rock, while the maximum practical value is assumed to be equal to K d assoc which corresponding to the associated plasticity at r = R 0 , the expression is given by Brown et al. (1983) [17].
Combing Equations (14)- (17) to Equation (18), the following differential equation can be expressed as Equation (20) can be solved by using the following boundary condition for the radial displacement u R at the elastic plastic interface: where E is Young's modulus and ν is Poisson's ratio of the rock mass.
The solution of Equation (20) can be expressed as follows by Sharan (2008) [18]: and ε e r and ε e θ are expressed as where The integration of Equation (22) can lead to an expression for the radial displacement in the plastic region as 3.3. Determination of Rock Resistant Coefficient. The radial displacement at the tunnel surface can be given when r = r 0 .
Substituting Equation (26) into Equation (4), the rock resistant coefficient can be expressed as 3 Geofluids

Parameter Sensitivity Analysis
In order to illustrate the effects of different rock parameters of the generalized Hoek-Brown failure criterion on the rock resistant coefficient K, the sensibility study will be presented. Zhang and Zhu (1993) presented a method in sensitivity analysis [19]. There is a system F = f ðx 1 , x 2 , ⋯ ⋯ , x n Þ, where x i ði = 1, 2, ⋯ ⋯ , nÞ is the parameter of this system. Considering a reference state X * = ðx 1 * , x 2 * , ⋯ ⋯ , x n * Þ, with a character of F * = f ðX * Þ, let each parameter changed in its range, the tendency, and degree of the difference between F and F * is sensitivity analysis. For a parameter x i * , the sensitivity function can be written as According to the properties, data of rocks in underground opening mostly taken from the literatures by Sofianos (2003), Hoek and Brown (1997), and Canranza (2004) [20][21][22], the adopted parameters for the example study cover a wide range, and they are presented in Table 1.
According to Table 1 Table 2.
In fact, the sensitivity of σ 0 , p, E, r 0 , and v is simple and is of no difference comparing to other failure criterions such as Mohr-Column. In this paper, we focus on the characteristics of parameters in generalized Hoek-Brown failure criterion.
For the dilation parameter K d , it ranges from 1 to K d assoc . In the reference state, the maximum of K d can be calculated as 3.72. Then, the relationship of dilation parameter K d and rock resistant coefficient K can be seen in Figure 2. According to Equation (27), the sensitivity function of dilation parameter K d was drawn in Figure 2. Using the similar method, sensitivity factors of other parameters m b , s, and a can be obtained. Figure 2 shows the effects of different parameters on rock resistant coefficient K, the results show that rock resistant coefficient K increases with m b and σ c , while decreasing with a and K d . For parameters s, the range of rock resistant coefficient is very small. It changes a little big with a fluctuation of parameters K d and a. The relation between K and K d is homogeneous; however, the relation between K and σ c is heterogeneous. In the design of a tunnel, we need to pay more attention to a, K d , σ c , and m b ; also, it is very important to determine the range of different parameters. According to the equations of different parameters in the generalized Hoek-Brown failure criterion, it can be concluded that the sensitivity of rock resistant coefficient has close relationship with rock quality [23][24][25]. When the rock quality is good, the variability of rock resistant coefficient is small. On the contrary, it is very sensitive when the rock quality is poor, which means that a more accurate geology analysis is in demand. This is consistent with the engineering practice.

Conclusions
Rock resistant coefficient is a key parameter in the design of underground structures. Based on the generalized Hoek-Brown failure criterion, this paper develops a closed-form formula for calculating the rock resistant coefficient, and some useful conclusions are drawn as follows: (1) A new formula of rock resistant coefficient is proposed in closed form, and this formula is more suitable for a jointed rock mass than other formulas (2) According to the sensibility study of parameters, it can be concluded that the sensitivity factor of a, σ c , and K d is bigger than m and s (3) Sensitivity of rock resistant coefficient has close relationship with rock quality; all these parameters except s are highly sensitive when the rock quality is poor, which lays a foundation for the determination of rock resistant coefficient in similar lining tunnels

Data Availability
The data are available and explained in this article; readers can access the data supporting the conclusions of this study.

Disclosure
I would like to declare on behalf of my co-authors that the work described is original research and has not been previously published.

Conflicts of Interest
Authors declare no conflict of interest.