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

Authors: Shiyue Zhang; Ruiyuan Gao; Qingxiang Meng; Xiaohu Zhang
Journal: Geofluids
 (2021)

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## Abstract

Rock resistant coefficient is a key parameter in rock engineering and can reflect the resistance of surrounding rocks. According to the load-structure method, there are many different formulas in the determination of rock resistant coefficient 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 coefficient. The sensibility of different parameters on the rock resistant coefficient is also analyzed. The fact shows that the formula is sensitive toa, σc, and Kd comparing with mb and s especially when they are in a low value, laying a foundation for the determination of rock resistant coefficient in similar lining tunnels.


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## Body

## 1. 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 [1–5]. 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. Hoek and Brown (1980) 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.


## 2. 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. Hoek and Brown (1980) introduced their failure criterion in an attempt to provide input data for the analyses required for the design of underground excavations in hard rock [12–14]. 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 Brown (1980). According to the latest modification, the generalized Hoek-Brown failure criterion is expressed as follows by Hoek et al. (2002) [15]:
(1)σ1=σ3+σcimbσ3σci+sa,where σ1 and σ3 are the major and minor principal stresses, respectively; σci is the uniaxial compressive strength of the intact rock; mb is a reduced value of the material constant mi and is given by
(2)mb=miGSI−10028−14D,where s and a are constants for the rock mass given by the following relationships:
(3)s=expGSI−1009−3D,a=12+16e‐GSI/15−e−20/3.In the equations above,mi 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.


## 3. Analytical Solution for Rock Resistant Coefficient

In order to obtain the rock resistant coefficient, a simplified model is shown in Figure1. A circular opening excavated in a continuous, homogeneous, isotropic, initially elastic rock mass subjected to a hydrostatic stress p0. 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.Figure 1
A simple calculation model of a tunnel.Assuming that the displacement of tunnel surface isu0, according to the Winkler hypothesis, the rock resistant coefficient can be expressed as
(4)K=pu0.


### 3.1. 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.σ1 and σ3 are equal to circumferential and radial stresses σθ and σr, respectively. According to the generalized Hoek-Brown failure criterion, relationship between σr and σθ can be written as follows:
(5)σθ=σr+σcimbσrσci+sa.The radial stressσR at the interface of elastic and plastic zone may be obtained by combining the relationship between σr and σθ in elastic and plastic state.
(6)2σ0−σR=σcimbσRσci+sa.For generalized Hoek-Brown failure criterion, it is impossible to give analytical solutions of Equation (6). Thus, an approximate closed-form solution for σR can be obtained by the Newton-Raphson method. When fx=0, the corresponding formula is
(7)xk+1=xk−fxkf′xk.The initial value forσR0 is determined when a=0.5.
(8)σR0=σ0+18mbσci−σcimb2σci+16σ0mb+16sσci.According Equation (5), the resulting solution σR can be expressed as
(9)σR=σR0+2σ0−σR0−σciC1a2+ambC1a−1,where C1=mbσR0/σci+s.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,(10)dσrdr+σr−σθr=0.Substitution of Equation (5) in Equation (10) leads to the following equation:
(11)dσrdr=σcirmbσrσci+sa.This is a simple differential equation with an initial condition ofσr∣r=r0=p, the result solution may be expressed as follows:
(12)σr=mb/σcip+s1−a+1−amblnr/r01/1−a−smb/σci.The circumferentialσθ can be given by substituting Equation (12) in Equation (5).The radius of plastic zones can be deprived by solving the equation obtained by substitutingr=R0 and σr=σR in Equation (12). The solution is expressed as
(13)R0=r0expmb/σciσR+s1−a−mb/σcip+s1−a1−amb.


### 3.2. Analytical Solutions of Displacement

In the plastic region, total radial and tangential strains,εr and εθ, can be decomposed into elastic and plastic parts.
(14)εr=εre+εrp,(15)εθ=εθe+εθp.The strains can be expressed in terms of inward radial displacementu by Kyung and Yong (2006) et al. [16].
(16)εr=dudr,(17)εθ=ur.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(18)εrp+Kdεθp=0.For the Hoek-Brown failure criterion, the minimum of the dilation parameter isKd=1 corresponding to a nondilating rock, while the maximum practical value is assumed to be equal to Kdassoc which corresponding to the associated plasticity at r=R0, the expression is given by Brown et al. (1983) [17].
(19)Kdassoc=1+mb2mbσR/σci+s.Combing Equations (14)–(17) to Equation (18), the following differential equation can be expressed as
(20)dudr+Kdur=εre+Kdεθe.Equation (20) can be solved by using the following boundary condition for the radial displacement uR at the elastic plastic interface:
(21)uR=R01+νEσ0−σR,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]:
(22)u=r−Kd∫Rrr−Kdεre+Kdεθedr+uRR0rKd,and εre and εθe are expressed as
(23)εre=1+νE1−2νC+Dr2,εθe=1+νE1−2νC−Dr2,where
(24)C=σR−σ0R02−p−σ0r02R02−r02,D=R0r0p−σRR02−r02.The integration of Equation (22) can lead to an expression for the radial displacement in the plastic region as
(25)u=1+νEr−KdC1−2νR0Kd+1−rKd+1−DR0Kd−1−rKd−1+uRR0rKd.


### 3.3. Determination of Rock Resistant Coefficient

The radial displacement at the tunnel surface can be given whenr=r0.
(26)u0=1+νEr0−KdC1−2νR0Kd+1−r0Kd+1−DR0Kd−1−r0Kd−1+uRR0r0Kd.Substituting Equation (26) into Equation (4), the rock resistant coefficient can be expressed as
(27)K=p1+ν/Er0−KdC1−2νR0Kd+1−r0Kd+1−DR0Kd−1−r0Kd−1+uRR0/r0Kd.


## 3.1. 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.σ1 and σ3 are equal to circumferential and radial stresses σθ and σr, respectively. According to the generalized Hoek-Brown failure criterion, relationship between σr and σθ can be written as follows:
(5)σθ=σr+σcimbσrσci+sa.The radial stressσR at the interface of elastic and plastic zone may be obtained by combining the relationship between σr and σθ in elastic and plastic state.
(6)2σ0−σR=σcimbσRσci+sa.For generalized Hoek-Brown failure criterion, it is impossible to give analytical solutions of Equation (6). Thus, an approximate closed-form solution for σR can be obtained by the Newton-Raphson method. When fx=0, the corresponding formula is
(7)xk+1=xk−fxkf′xk.The initial value forσR0 is determined when a=0.5.
(8)σR0=σ0+18mbσci−σcimb2σci+16σ0mb+16sσci.According Equation (5), the resulting solution σR can be expressed as
(9)σR=σR0+2σ0−σR0−σciC1a2+ambC1a−1,where C1=mbσR0/σci+s.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,(10)dσrdr+σr−σθr=0.Substitution of Equation (5) in Equation (10) leads to the following equation:
(11)dσrdr=σcirmbσrσci+sa.This is a simple differential equation with an initial condition ofσr∣r=r0=p, the result solution may be expressed as follows:
(12)σr=mb/σcip+s1−a+1−amblnr/r01/1−a−smb/σci.The circumferentialσθ can be given by substituting Equation (12) in Equation (5).The radius of plastic zones can be deprived by solving the equation obtained by substitutingr=R0 and σr=σR in Equation (12). The solution is expressed as
(13)R0=r0expmb/σciσR+s1−a−mb/σcip+s1−a1−amb.


## 3.2. Analytical Solutions of Displacement

In the plastic region, total radial and tangential strains,εr and εθ, can be decomposed into elastic and plastic parts.
(14)εr=εre+εrp,(15)εθ=εθe+εθp.The strains can be expressed in terms of inward radial displacementu by Kyung and Yong (2006) et al. [16].
(16)εr=dudr,(17)εθ=ur.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(18)εrp+Kdεθp=0.For the Hoek-Brown failure criterion, the minimum of the dilation parameter isKd=1 corresponding to a nondilating rock, while the maximum practical value is assumed to be equal to Kdassoc which corresponding to the associated plasticity at r=R0, the expression is given by Brown et al. (1983) [17].
(19)Kdassoc=1+mb2mbσR/σci+s.Combing Equations (14)–(17) to Equation (18), the following differential equation can be expressed as
(20)dudr+Kdur=εre+Kdεθe.Equation (20) can be solved by using the following boundary condition for the radial displacement uR at the elastic plastic interface:
(21)uR=R01+νEσ0−σR,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]:
(22)u=r−Kd∫Rrr−Kdεre+Kdεθedr+uRR0rKd,and εre and εθe are expressed as
(23)εre=1+νE1−2νC+Dr2,εθe=1+νE1−2νC−Dr2,where
(24)C=σR−σ0R02−p−σ0r02R02−r02,D=R0r0p−σRR02−r02.The integration of Equation (22) can lead to an expression for the radial displacement in the plastic region as
(25)u=1+νEr−KdC1−2νR0Kd+1−rKd+1−DR0Kd−1−rKd−1+uRR0rKd.


## 3.3. Determination of Rock Resistant Coefficient

The radial displacement at the tunnel surface can be given whenr=r0.
(26)u0=1+νEr0−KdC1−2νR0Kd+1−r0Kd+1−DR0Kd−1−r0Kd−1+uRR0r0Kd.Substituting Equation (26) into Equation (4), the rock resistant coefficient can be expressed as
(27)K=p1+ν/Er0−KdC1−2νR0Kd+1−r0Kd+1−DR0Kd−1−r0Kd−1+uRR0/r0Kd.


## 4. Parameter Sensitivity Analysis

In order to illustrate the effects of different rock parameters of the generalized Hoek-Brown failure criterion on the rock resistant coefficientK, the sensibility study will be presented. Zhang and Zhu (1993) presented a method in sensitivity analysis [19]. There is a system F=fx1,x2,⋯⋯,xn, where xii=1,2,⋯⋯,n is the parameter of this system. Considering a reference state X∗=x1∗,x2∗,⋯⋯,xn∗, with a character of F∗=fX∗, let each parameter changed in its range, the tendency, and degree of the difference between F and F∗ is sensitivity analysis. For a parameter xi∗, the sensitivity function can be written as
(28)axi∗,Δxi=ΔF/F∗Δxi/xi∗.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) [20–22], the adopted parameters for the example study cover a wide range, and they are presented in Table 1.Table 1
Basic parameters of the generalized Hoek-Brown criterion used for example study.
Rock typeQuality of rock massGSIσc(MPa)E (GPa)vamsAVery good75150420.20.510.20.062BAverage508090.250.512.010.0039CVery poor30201.40.30.520.6570.0004DVery poor—3038.00.30.3—0.55EVery poor1050.2240.30.5850.4020.0000454According to Table1, the ranges of different parameters are E=0.22−42GPa, ν=0.2−0.3, σc=5−150MPa, mb=0.5−16, s=0.0000454−0.062, and a=0.5−0.585. In order to analyze the sensitivity of different parameters, reference parameters can be assumed in Table 2.Table 2
The parameters of reference state.
E(GPa)vσ0(MPa)p(MPa)r0(m)σc(MPa)KdmbsaK(MPa/m)90.252023801.220.0030.5245.8In fact, the sensitivity ofσ0, p, E, r0, 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 parameterKd, it ranges from 1 to Kdassoc. In the reference state, the maximum of Kd can be calculated as 3.72. Then, the relationship of dilation parameter Kd and rock resistant coefficient K can be seen in Figure 2. According to Equation (27), the sensitivity function of dilation parameter Kd was drawn in Figure 2. Using the similar method, sensitivity factors of other parameters mb, s, and a can be obtained.Figure 2
Sensitivity of mechanical parameters on rock resistant coefficient.
(a)
Good quality rock(b)
Average quality rock(c)
Poor quality rockFigure2 shows the effects of different parameters on rock resistant coefficient K, the results show that rock resistant coefficient K increases with mb and σc, while decreasing with a and Kd. For parameters s, the range of rock resistant coefficient is very small. It changes a little big with a fluctuation of parameters Kd and a. The relation between K and Kd is homogeneous; however, the relation between K and σc is heterogeneous. In the design of a tunnel, we need to pay more attention to a, Kd, σc, and mb; 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 [23–25]. 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.


## 5. 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 ofa, σc, and Kd is bigger than m and s(3)
Sensitivity of rock resistant coefficient has close relationship with rock quality; all these parameters excepts are highly sensitive when the rock quality is poor, which lays a foundation for the determination of rock resistant coefficient in similar lining tunnels



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