1 s2.0 S1674775522000336 Main
1 s2.0 S1674775522000336 Main
1 s2.0 S1674775522000336 Main
Face stability analysis of circular tunnels in layered rock masses using the
upper bound theorem
Jianhong Man, Mingliang Zhou*, Dongming Zhang, Hongwei Huang, Jiayao Chen
Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai,
200092, China
a r t i c l e i n f o a b s t r a c t
Article history: An analysis of tunnel face stability generally assumes a single homogeneous rock mass. However, most
Received 11 July 2021 rock tunnel projects are excavated in stratified rock masses. This paper presents a two-dimensional (2D)
Received in revised form analytical model for estimating the face stability of a rock tunnel in the presence of rock mass stratifi-
3 October 2021
cation. The model uses the kinematical limit analysis approach combined with the block calculation
Accepted 21 December 2021
Available online 23 February 2022
technique. A virtual support force is applied to the tunnel face, and then solved using an optimization
method based on the upper limit theorem of limit analysis and the nonlinear HoekeBrown yield cri-
terion. Several design charts are provided to analyze the effects of rock layer thickness on tunnel face
Keywords:
Face stability
stability, tunnel diameter, the arrangement sequence of weak and strong rock layers, and the variation in
Rock tunnel rock layer parameters at different positions. The results indicate that the thickness of the rock layer,
Layered rock masses tunnel diameter, and arrangement sequence of weak and strong rock layers significantly affect the tunnel
Upper bound solution face stability. Variations in the parameters of the lower layer of the tunnel face have a greater effect on
HoekeBrown criterion tunnel stability than those of the upper layer.
Ó 2022 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by
Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/
licenses/by-nc-nd/4.0/).
1. Introduction Executive, 1996). When the tunnel face collapses, a large amount
of rock mass flows into the tunnel, resulting in many casualties,
Layered rock masses are typical natural geological settings, ac- property losses, damage to tunnel support structures (Wang et al.,
counting for approximately 66.7% of the land area. Hence, various 2019; Huang et al., 2021), construction period delays, and other
engineering construction activities often encounter layered rock issues. Therefore, face stability analysis is of great importance for
masses. A growing number of rock tunnel projects involve complex tunnel engineering in layered rock masses.
geological environments (Anagnostou et al., 2014; Chen et al., 2021; Over the past several decades, the stability of tunnel faces in
Zhao et al., 2021), most of which have tunnel faces excavated in homogenous rock mass/soil has been extensively analyzed by
layered rock masses (Zhang and Zhou, 2017; Zhang et al., 2021a). experimental tests, numerical simulations, and analytical solutions.
Several studies have suggested that tunnel face failure, especially Many scholars have studied the failure shape and mechanism of
collapse, is often inextricably linked to the stratification of the tunnel faces through the centrifuge model (Idinger et al., 2011;
layered rock masses (Arnáiz Ronda et al., 2003; Babendererde et al., Chen et al., 2018; Weng et al., 2020) and 1-g physical model (Kirsch,
2006; Anagnostou and Zingg, 2013; Zhou et al., 2021). Tunnel 2010; Chen et al., 2015; Liu et al., 2018). While experimental tests
collapse accidents in China accounted for about 55.6% of con- can accurately simulate the failure phenomenon of a tunnel face, it
struction accidents from 2006 to 2016, most of which were due to is difficult to test the stability of a tunnel face under complex
the instability of the tunnel face (Zhang et al., 2018). Construction geological conditions. Three types of numerical simulation method
experience indicates that most tunnel collapse accidents can be are widely used in the stability analysis of tunnel faces: the finite
attributed to the instability of the tunnel face (Health and element method (FEM) (Vermeer et al., 2002; Li et al., 2009; Kim
and Tonon, 2010; Ukritchon et al., 2017), discrete element
method (DEM) (Maynar and Rodriguez, 2005; Chen et al., 2011;
* Corresponding author. Jiang and Yin, 2014; Zhang et al., 2021b), and FEM-DEM coupling
E-mail address: zhoum@tongji.edu.cn (M. Zhou). method (Long and Tan, 2020; Yin et al., 2020). These numerically-
Peer review under responsibility of Institute of Rock and Soil Mechanics, Chi- based methods can simulate complex stratum environment and
nese Academy of Sciences.
https://doi.org/10.1016/j.jrmge.2021.12.023
1674-7755 Ó 2022 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-
NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
J. Man et al. / Journal of Rock Mechanics and Geotechnical Engineering 14 (2022) 1836e1848 1837
changes in monitoring objects during the construction process. and is extensively used for rocks with varying degrees of fracture.
However, using any of these numerical methods to conduct the face The generalized HoekeBrown criterion reported by Hoek and
stability analysis of a typical tunnel setting comes at a high Brown (2019) can be written as follows:
computational cost, and takes a long time to complete the n
simulation. s3
s1 ¼ s3 þ sc mb þs (1)
In contrast, an analytical solution for face stability analysis can sc
quickly produce design charts and can conveniently be used on-
site. Analytical solutions typically have two types: the limit equi- where s1 and s3 are respectively the maximum and minimum
librium method and the upper bound solution. The wedge model principal stresses at failure, and sc is the uniaxial compressive
(Horn, 1961), wedge-prism model (Perazzelli et al., 2014; Paternesi strength (UCS) of intact rock. The HoekeBrown parameters (mb, s
et al., 2017; Zou et al., 2019; Zhang et al., 2020a), and triangular base and n) are determined by the rock material constant mi, the
prism model (Oreste and Dias, 2012) are common models using the disturbance coefficient d, and the geological strength index GSI by
limit equilibrium method, which do not consider the constitutive the following formulations:
law of the material and have low computational accuracy. There-
fore, the upper bound solution has been gradually adopted as an GSI 100
mb ¼ mi exp (2a)
efficient method for tunnel face stability analysis, which has a va- 28 14d
riety of hypothesized face failure mechanisms, e.g. the multi-block
failure mechanism (Chen, 1975; Davis et al., 1980; Leca and GSI 100
s ¼ exp (2b)
Dormieux, 1990; Subrin and Wong, 2002; Mollon et al., 2010, 9 3d
2011a, b; Zou et al., 2019; Li et al., 2020; Zhou et al., 2020). Most
current studies have focused on the face stability in homogeneous
1 1 GSI 20
geomaterials. Only a few attempts have been made to account for n¼ þ exp exp (2c)
2 6 15 3
the partial collapse of soil and the spatial variability of soil-based
stratum (Pan and Dias, 2018a, b; Zou and Qian, 2018; Li and Yang,
2020; Zhang et al., 2021c, d). To the authors’ knowledge, no
analytical solutions have been proposed for the face stability 2.2. Equivalent MohreCoulomb parameters
analysis of tunnels in layered rock masses during excavation. Thus,
this study proposes an upper bound-based analytical approach to In the upper bound solution, the internal energy dissipation
conduct the face stability analysis of circular tunnels in layered rock along the discontinuous surface is often calculated by the Mohre
masses. The HoekeBrown yield criterion was adopted to better Coulomb parameters of cohesion c and frictional angle 4. Howev-
represent the rock mass failure behavior, and the equivalent Mohre er, these two parameters are not used in the HoekeBrown failure
Coulomb model parameters were obtained for the analytical solu- criterion. Hence, it is necessary to perform equivalent conversions
tion. The geological strength index, unconfined compressive between the HoekeBrown parameters and the MohreCoulomb
strength, and rock material constants vary with the properties of parameters.
each rock layer, resulting in segmented spiral failure surfaces. The There are two main ways to convert the HoekeBrown param-
proposed model is composed of multiple logarithmic spiral failure eters into the MohreCoulomb parameters (cohesion c and fric-
surfaces. tional angle 4). One of such methods is the tangent method, which
The rest of the paper is organized as follows. The nonlinear can linearize the nonlinear failure criterion and obtain the equiv-
failure criterion of the rock masses and parameter equivalent alent parameters (Pan and Dias, 2018a; Zhang et al., 2020b). The
method are first presented in Section 2. Section 3 shows the deri- other is the direct equivalent method, which incorporates engi-
vation details of the proposed analytical solution. In Section 4, the neering experience, and has proven highly reliable when tested in
analytical approach is validated by comparison with the numerical practice (Wang et al., 2021). The direct equivalent method can also
calculation results in layered rock cases and other analytical solu- reflect the variation of parameters with the depth of the tunnel so
tions. In Section 5, the analytical approach is used to conduct a that the computed result is more representative of the site condi-
comprehensive stability analysis of circular tunnel faces in layered tion (Hoek et al., 2002). Therefore, this study uses the direct
rock masses with different geometries and material properties. equivalent method to derive the upper bound solution.
Finally, the major findings of this study and possible future studies The equivalent cohesion c and frictional angle 4 proposed by
are presented in Section 6. The calculation results obtained in this Hoek et al. (2002) are expressed as follows:
paper may be used as a reference for future tunnel excavation
design. sc ½ð1 þ 2nÞs þ ð1 nÞmb s3n ðs þ mb s3n Þn1
c¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h i. ffi
2. Nonlinear failure criterion ð1 þ nÞð2 þ nÞ 1 þ 6mb nðs þ mb s3n Þn1 ½ð1 þ nÞð2 þ nÞ
where g is the unit weight of the rock mass; H is the buried depth of logarithmic spiral collapse model in a homogeneous rock mass
the tunnel, being the distance between the top of the tunnel and configuration. Combining the double-logarithmic spiral with the
the ground surface; and scm is the rock mass strength, which can be approach of considering a multi-layer structure with progressive
written as failure (Qin et al., 2017), this paper establishes a progressive
collapse model of layered rock masses under two-dimensional (2D)
sc ½mb þ 4 nðmb 8sÞðmb =4 þ sÞn1 conditions. The failure surfaces thus consist of multiple double-
scm ¼ (5)
2ð1 þ nÞð2 þ nÞ logarithmic spirals, which can account for the failure of different
rock layers according to their material properties (Fig. 1). The failure
It is noteworthy that the equivalent method mentioned above surfaces of the proposed solution are composed of four logarithmic
does not consider the stratification of rock masses, especially in the spirals. From a mathematical geometry perspective, each loga-
solution process of Eq. (4). Layered rock masses have always been a rithmic spiral is determined by two angles with respect to the
complicated problem, thus certain simplifications are needed in the vertical direction (the starting and ending angles). However,
analytical solution. The simplified method is performed by comparing the upper and lower boundaries of the failure area, the
assuming that the properties of all rock layers are the same when angles corresponding to points A and C (relative to the vertical di-
solving the equivalent parameters of a specific rock layer. rection) can have two possible cases: q2 < q3 and q3 < q2. Therefore,
Hoek et al. (2002) converted the HoekeBrown parameters to the two collapse models are proposed, i.e. Cases 1 and 2, as shown in
MohreCoulomb parameters c and 4 using curve fitting, which Fig. 1.
adopts the principle of balancing the areas above and below the
MohreCoulomb plot. Therefore, the equivalent MohreCoulomb
parameters inevitably have calculation differences. However, the
3.2. Limit analysis and calculation method in layered rock mass
exhibited calculation difference has been proven acceptable in a
previous study (Wang et al., 2021).
According to Fig. 1, the diameter of the circular tunnel is denoted
as D, the thickness of the overburden layer is denoted as H, the
3. The upper bound solution in layered rock masses thickness of the upper layer of the tunnel face is denoted as h1, and
the thickness of the lower layer of the tunnel face is denoted as h2.
3.1. Analytical model description Based on the proposed failure models in Fig. 1, there are four log-
arithmic spiral surface lines: BC, CD, DE, and AE. For the failure
The upper bound solution combines the kinematically admis- mechanism, each failure surface is assumed to rotate with a con-
sible velocity field with the corresponding yield conditions, flow stant angular velocity u around point O, where the lengths of OA,
rules, and boundary conditions to study the stability of geotech- OB, OC, and OD are ra, rb, rc and rd, respectively. Meanwhile, the
nical structure. It is difficult to analyze the tunnel face stability due angles between OA, OB, OC, OD, and OE and the vertical direction are
to the complex mechanical behavior of layered rock mass. However, denoted as q2, q1, q3, q4, and q5, respectively. Based on the associated
Dong and Anagnostou (2013, 2014) and Anagnostou et al. (2014) flow rule of the upper bound theory of limit analysis, at any point
pointed out that in most cases, the complexity of mechanical on the logarithmic spiral failure lines, the frictional angle of each
behavior does not increase the complexity of engineering. From a layer is the same as the angle between the velocity direction and
practical point of view, the simplified model is still appropriate for the corresponding tangent direction of the failure lines. 4ti and
engineering purposes. Consequently, this paper uses simplified (i ¼ 0, 1, 2) are denoted as the frictional angle and the cohesion of
continuum modeling to consider the stratification of rock mass in the overburden layer, the upper layer of the tunnel face, and the
front of the tunnel face (Xu et al., 2017). The following assumptions lower layer of the tunnel face, respectively. As shown in Fig. 2, the
were made to simplify the geological and mechanical complica- logarithmic spiral curves AE, BC, CD, and DE can be written as
tions of the problem: follows:
(1) The stratum consists of three rock mass layers: the over- AE : r1 ðqÞ ¼ ra exp½ðq q2 Þtan 4t0 (6a)
burden layer, the upper tunnel face layer, and the lower
tunnel face layer. The exposed tunnel face is assumed to lie BC : r2 ðqÞ ¼ rb exp½ðq1 qÞtan 4t2 (6b)
within the upper and lower tunnel face layers.
(2) The excavated tunnel face within the upper and lower rock
layers is assumed to have a perfect circular shape.
CD : r3 ðqÞ ¼ rc exp½ðq3 qÞtan 4t1 (6c)
(3) The rock layer is assumed to be an ideal elastoplastic material
that conforms to the associated flow rule. Each rock layer is DE : r4 ðqÞ ¼ rd exp½ðq4 qÞtan 4t0 (6d)
homogeneous, and the dilation angle of each point along the
sliding surface is equal to the equivalent frictional angle of where q is a variable denoting the rotational angle. According to the
the rock medium. geometric relationship in Fig. 1, ra, rb, rc and rd can be expressed as
(4) The failure surface of each rock layer is continuous on the the functions of q2, q1, q3 and q4 as follows:
layer interface, and the associated flow rule also conforms to
the layer interface. sin q1 ðh1 þ h2 Þ
ra ¼ (7a)
(5) Each hypothesized sliding block is regarded as a rigid body, sinðq2 q1 Þ
and the strain within each block is not considered so that
energy dissipation only occurs on the failure surface. sin q2 ðh1 þ h2 Þ
rb ¼ (7b)
sinðq2 q1 Þ
To better analyze the influence of the stratification character-
istics of rock masses on the stability of rock tunnel face, an sin q2 cos q1 ðh1 þ h2 Þ h2
improved failure model is proposed, with two cross-layers and one rc ¼ (7c)
sinðq2 q1 Þcos q3 cos q3
cover layer. Subrin and Wong (2002) proposed a double-
J. Man et al. / Journal of Rock Mechanics and Geotechnical Engineering 14 (2022) 1836e1848 1839
Fig. 1. Improved failure model composed of multiple double-logarithmic spirals: (a) Parameters of each rock strata; and (b, c) Longitudinal sections of the failure surfaces in Case 1
(q2 < q3) and Case 2 (q3 < q2), respectively.
Fig. 2. 2D calculation models of (a) Case 1 (q2 < q3) and (b) Case 2 (q3 < q2).
where v and Gdq represent the tangent velocity at the center of The expressions of the parameters in the above equations are
gravity of the micro-element body and the gravity of the micro- shown in Appendix A. It is worth noting that, when calculating the
element body, respectively. work rate of a specific region, the parameters used are only related
According to the geometric relationship in Fig. 2, the auxiliary to the rock layer where the calculation region is located.
variables q11 and rf can be expressed as follows: Secondly, the work rate of the virtual support force on the
tunnel face can be written as follows:
sinðq2 q1 Þ h1
q11 ¼ arccot cot q2 þ (10a)
sin q2 sin q1 h1 þ h2 Zq2 .
PT ¼ sT vT lT cos qdq ¼ sT rb2 u 1 sin 2 q1 sin 2 q2 2 (15)
sin q2 sin q1 q1
rf ¼ ðh þ h2 Þ (10b)
sinðq2 q1 Þsin q11 1
where vT represents the tangent velocity of the micro-element
The work rate of the self-weight of region BFC can be written as body around point O for line AB, and lT dq represents the unit
follows: length of line AB.
Finally, the internal energy dissipation along the failure lines can
Zq3 be represented by the energy dissipation along the failure surfaces:
PBFC ¼ vG sin qdq ¼ ug2 rb3 f5 rb3 f6 rc3 f7
½OBCDOBFDOFC
q1 Zq5
(11) PAE ¼ c0 vD cos 40 lD dq ¼
q2
For Case 1, the region AFCD is composed of regions IDC and AFCI,
for which the work rates of the self-weight are calculated as
follows: 1
uc0 ra2 fexpf2½ðq5 q2 Þtan 4t0 g 1g (16a)
2 tan 40
Zq4
PIDC ¼ vG sin qdq ¼ ug1 rc3 f8 rd3 f9 (12a) Zq5
½ODCDOID
q3 PED ¼ c0 vD cos 40 lD dq ¼
q4
Zq11 Zq3 1
PAFCI ¼ vG sin qdq þ vG sin qdq uc0 rd2 f expf2½ðq4 q5 Þtan 40 g þ 1 g (16b)
2 tan 40
½DOBF ½DOFC
q1 q11
Zq2 Zq3 Zq4
1
vG sin qdq vG sin qdq PDC ¼ c1 vD cos 41 lD dq ¼ uc1 rc2 ,
½DOAB ½DOAI 2 tan 41
q1 q2 q3
¼ ug1 rc3 f10 ra3 f11 rb3 f12 f exp½2ðq3 q5 Þtan 41 þ exp½2ðq3 q4 Þtan 41 g (16c)
(12b)
Zq3
PAFCD ¼ PAICD þ PIFC (12c) PCB ¼ c2 vD cos 42 lD dq ¼
For Case 2, the region AFCD is composed of regions IFC and AICD, q1
for which the work rates of the self-weight are calculated as 1
uc2 rb2 f exp½2ðq1 q3 Þtan 42 þ 1 g (16d)
follows: 2 tan 42
mathematical optimization problem, which can search the best using the same parameters of the proposed method are compared
upper bound solution by optimizing the objective function of Eq. with the analytical results obtained by Senent et al. (2013) (see
(18) under the following constraints: Table 1). The differences in computed face stability caused by the
9 conversion of HoekeBrown parameters to their equivalent Mohre
0 < q1 < q11 < q2 < q3 < q4 < p=2 ðCase 1Þ >
> Coulomb parameters are shown in Table 1. Those results are ob-
>
0 < q1 < q11 < q3 < q2 < q4 < p=2 ðCase 2Þ >
>
> tained by the limit analysis and FLAC3D (fast Lagrangian analysis of
=
q4 < q5 < p continua in 3 dimensions) simulations performed by Senent et al.
(19)
ra < rf < rb >
> (2013). It is noteworthy that the simulation results of FLAC3D
>
>
rd < rc < rb >
>
; directly use the HoekeBrown parameters. There are differences
ra < rd between the analytical results in this paper and those given by
Several efficient nonlinear optimization methodologies have Senent et al. (2013), which are acceptable according to the com-
been developed in recent years, such as the sequence quadratic parison conducted by Zhang et al. (2020b) and Senent et al. (2013).
iterative algorithm, second-order cone programming, and genetic This also shows that it is feasible to use the parameter conversion in
algorithm (Lysmer, 1970; Zhang et al., 2015). The sequence this study.
quadratic iterative algorithm and second-order cone programming The face stability analysis in the layered rock mass is compared
can efficiently solve the optimal solution, but when faced with with the numerical analysis using the finite element software
complex nonlinear problems, it is easy for them to fall into a local ABAQUS. Fig. 4 illustrates the model geometry used for numerical
optimal solution, thus missing the global optimal solution. How- simulation of the tunnel face stability, which only shows half the
ever, the genetic algorithm uses a probabilistic mechanism for symmetrical model. The numerical model of the rock runnel has a
global search to effectively avoid such problems, and finds the diameter of 10 m and a buried depth of 30 m. The size of the nu-
global optimal solution in the sense of probability. The genetic al- merical model is taken as 40 m in the x-direction, 50 m in the y-
gorithm is based on biological evolution, which has the advantages direction, and 70 m in the z-direction. The model contains
of good convergence, short computation time, and high robustness. approximately 51,730 zones and 57,429 nodes. Hence, boundary
Therefore, this study uses the genetic algorithm to find the optimal effects can be avoided for these dimensions. The model’s boundary
combination of q1, q2, q3, and q4 for the global minimum solution of conditions are given by fixed displacements at the bottom and the
the objective function (virtual support force) using MATLAB soft- lateral perimeter of the model.
ware. The flow diagram for performing the stability analysis of the A linear perfectly elastoplastic constitutive model based on the
tunnel face in this paper is illustrated in Fig. 3. MohreCoulomb failure criterion (with an associated flow rule) is
applied to the rock masses. The process of collapse simulation is
implemented according to the sequence of in situ stress balance e
4. Validation of analytical model rock mass excavation e virtual support force e gradually
decreasing the support force until the tunnel face collapses. The
The proposed model is simplified to consider a homogeneous detailed steps were described by Vermeer et al. (2002), and the
rock layer. To validate the proposed model, the results obtained equivalent cohesion c and frictional angle 4 were obtained by Hoek
et al. (2002). The corresponding rock Young’s modulus can also be
obtained through the alternative equation proposed by Hoek and
Diederichs (2006). The tunnel concrete lining is simulated with
shell structural elements with a Young’s modulus of 10 GPa, a
Poisson’s ratio of 0.2, a density of 2500 kg/m3, and a thickness of
0.4 m.
In this section, the tunnel face of the calculation model is
assumed to be in a two-layer rock mass, with the thickness of each
layer being 5 m. The numerical analysis considers two types of rock
masses: very poor rock mass (Case A) and relatively better rock
mass (Case B). The material properties are listed in Table 2. The
equivalent cohesion c, equivalent frictional angle 4, and the cor-
responding equivalent Young’s modulus proposed by Hoek et al.
(2002) and Hoek and Diederichs (2006) are also shown in
Table 2. The values are reserved to three decimal places to ensure
the accuracy of the equivalent parameters as much as possible.
Table 1
Comparison of collapse pressure obtained in this work with analytical results gained
by Senent et al. (2013).
Limit FLAC3D
analysis (a) (b)
h1 and the lower layer of the tunnel face h2 are determined by the
computational cases.
5.1. Design charts with two different collapse models (Cases 1 and
2)
Fig. 5. (a) Comparison of numerical simulation and analytical results for two cases; and (b) Comparison of failure mechanisms computed with the upper-bound solution and the
numerical model (unit: m).
Table 3
Basic parameters of each rock layer.
tunnel face stability. Hence, the rock quality of the lower layer
Rock layer Thickness sc (MPa) m GSI d g (kN/m3) controls the stability of the tunnel face.
Overburden H ¼ 20 m 1 5 10 0 25 This section also investigates the influence of the rock layer
Upper layer h1 1 5 10 0 25 arrangement on the tunnel face stability. It needs to be explained
Lower layer h2 1 5 10 0 25 here that strength and weakness here are relative. Two types of
layer arrangements are investigated: one with m1 ¼ 5e25 and
m2 ¼ 15, and another with m2 ¼ 5e25 and m1 ¼ 15. As shown in
Fig. 11, to either the left or right of the intersection point, the sta-
thickness ratio. However, as the upper layer rock mass quality
bility of the tunnel face with the upper-weak and lower-strong
improves, the parameter sc1 becomes the most sensitive to changes
arrangement is more stable than that of the lower-weak and
in the thickness ratio, and this influence becomes greater as the
upper-strong arrangement. Therefore, the arrangement of the weak
tunnel diameter increases. Therefore, it can be concluded that the
and the strong rock layers has an important influence on the sta-
stability of a tunnel face is significantly affected by the size of the
bility of the tunnel face. In practice, it is necessary to judge the
tunnel diameter and the distribution of the rock layer thickness.
stability of a tunnel face according to the arrangement of weak and
strong layers.
5.4. Design charts with different rock layer parameters Fig. 12 shows the results obtained from 820 calculated exam-
ples. It was assumed that the tunnel face’s upper and lower rock
In this part, the thicknesses of the upper and lower rock layers of layer parameters change simultaneously, mainly because the rock
the tunnel face are set to 5 m to eliminate the influence of the masses are cut into finite blocks by a weak plane (persistent joint
thickness of the rock layers. The following analysis investigates the plane). Note that selection of the parameters primarily corresponds
influence of the geological strength index GSI and the rock material to the poor-quality rock masses, where face instability problems are
constant m on the stability of the tunnel face when the other pa- more likely to be found in engineering practice. Accordingly,
rameters are kept constant. The normalized virtual support force sc =ðgDÞ values vary from 4 to 100 (equivalent to sc between 1 MPa
sT =ðgi DÞ is plotted against the normalized UCS sci =ðgi DÞ. As shown and 25 MPa), typical of soft rocks to very soft rocks. GSI values are
in Fig. 10, the normalized virtual support force declines nonlinearly taken between 10 and 30, which are characteristic values of very
as sci =ðgi DÞ increases from 4 to 100, and the nonlinear behavior poor-quality rock masses. As shown in Fig. 12, the virtual support
gradually weakens. The first column in Fig. 10 shows that, as the force gradually decreases to zero when the rock mass quality in-
parameter sci =ðgi DÞ increases, the influence of changes in the creases, which means that the tunnel face can maintain a self-
parameter mi on the virtual support force gradually weakens. stable state. However, comparison with Fig. 10 shows that the
However, for the geological strength index GSIi, the second column stability of the tunnel face will be overestimated if the stratification
of Fig. 10 indicates that the influence of changes in GSIi on the of the rock mass and the differences in the properties of the upper
virtual support force gradually increases as sci =ðgi DÞ increases. and lower rock layers are not considered. Therefore, the calculation
More importantly, the comparison shows that the changes in the method proposed in this paper is more suitable for determining the
parameters of the lower rock layer have a greater influence on the stability state of a tunnel face in layered rock.
1844 J. Man et al. / Journal of Rock Mechanics and Geotechnical Engineering 14 (2022) 1836e1848
Fig. 6. Influence of varying parameters on the tunnel face stability under two collapse models: (a) m1, (b) GSI1, and (c) sc1.
Fig. 7. Influence of thickness on the virtual support force with variations in (a) m1, (b) GSI1, and (c) sc1.
J. Man et al. / Journal of Rock Mechanics and Geotechnical Engineering 14 (2022) 1836e1848 1845
(1) For layered rock masses with different layer thicknesses, the
geological strength index GSI1, rock material constant m1, and
UCS sc1 significantly influence the stability of tunnel face
characterized by the normalized virtual support force, which
varies nonlinearly with increases in GSI1, m1 and sc1 : The
nonlinear decreasing trend weakens with a decrease in the
thickness of the upper layer of the tunnel face. As the rock
mass quality improves, the key factor that controls the tunnel
face stability changes from GSI1 to sc1 .
(2) The tunnel diameter and the distribution of the rock layer
thickness significantly influence the stability of tunnel face.
The larger the tunnel diameter, the greater the proportion of
heavily fractured rock masses, the more unfavorable the
stability of the tunnel face. The arrangement of the weak and
Fig. 8. The average gradient varying with rock parameters.
strong rock layers has an important influence on the tunnel
face stability. The research found that the tunnel face in a
lower-weak and upper-strong composite layer fails more
6. Conclusions easily than that in an upper-weak and lower-strong com-
posite layer.
This study extends the upper bound limit analysis method to the (3) The influence of the parameters at different positions on the
face stability of a tunnel excavated in layered rock. The influence of tunnel face stability was also investigated. As expected, the
rock layer characteristics on the stability of the tunnel face is normalized virtual support force declines nonlinearly with
considered, and a series of dimensionless parameter charts is pre- an increase in sci =ðgi DÞ. The nonlinear decreasing trend
sented using the upper bound limit analysis method. These charts weakens as the rock quality increases. In addition, the face
Fig. 9. Influence of parameter variations on the stability of tunnel face under different tunnel diameters.
1846 J. Man et al. / Journal of Rock Mechanics and Geotechnical Engineering 14 (2022) 1836e1848
Fig. 10. Influence of the rock material constants for a layered rock on the virtual support force: (a) m0; (b) GSI0; (c) m1; (d) GSI1; (e) m2; and (f) GSI2.
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Declaration of competing interest Chen, R.P., Tang, L.J., Ling, D.S., Chen, Y.M., 2011. Face stability analysis of shallow
shield tunnels in dry sandy ground using the discrete element method. Comput.
The authors declare that they have no known competing Geotech. 38 (2), 187e195.
Chen, R.P., Tang, L.J., Yin, X.S., Chen, Y.M., Bian, X.C., 2015. An improved 3D wedge-
financial interests or personal relationships that could have
prism model for the face stability analysis of the shield tunnel in cohesionless
appeared to influence the work reported in this paper. soils. Acta Geotech. 10 (5), 683e692.
Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier, Amsterdam, The
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The work described in this paper is supported by the Key
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MOST of China (Grant No. 2016RA4059) and the Science and ence on Computational Methods in Tunneling and Subsurface Engineering.
Aedificatio Publishers, Freiburg, Germany, pp. 267e274.
Technology Project of Yunnan Provincial Transportation Depart-
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ment (No. 25 of 2018). the short-term response of squeezing ground to tunnel excavation. In: Com-
puter Methods and Recent Advances in Geomechanics: Proceedings of the 14th
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nouveau mécanisme de rupture 3D. C. R. Méc. 330 (7), 513e519 (in French). Dr. Mingliang Zhou is currently serving as a postdoctoral
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sional undrained tunnel face stability in clay with a linearly increasing shear neering, Tongji University, China. He is a member of the
strength with depth. Comput. Geotech. 88, 146e151. Machine Learning and Big Data (TC309) Group of the In-
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Wang, F.Y., Zhou, M.L., Zhang, D.M., Huang, H.W., Chapman, D., 2019. Random MSc, and PhD degrees from the University of Cambridge,
evolution of multiple cracks and associated mechanical behaviors of segmental UK. His research interest is risk assessment using
tunnel linings using a multiscale modeling method. Tunn. Undergr. Space advanced techniques and control of geotechnical infra-
Technol. 90, 220e230. structure. Utilizing recent advances in machine learning-
Wang, H.T., Liu, C., Liu, P., Zhang, X., Yang, Y., 2020. Prediction of the required based technology, he has established a research reputa-
supporting pressure for a shallow tunnel in layered rock strata based on 2D and tion using computer vision and deep learning techniques
3D upper bound limit analysis. Adv. Civ. Eng. 2020, 6261917. to solve practical geotechnical problems. He has been
Wang, Q., Xu, M., Zhang, Y., Cen, X., Chang, X., 2021. Mechanical parameters of deep- invited to present his work in more than 10 international
buried coal goaf rock mass based on optimized GSI quantitative analysis. Adv. and national level academic conferences, and has published more than 40 journal
Civ. Eng. 2021, 9935860. papers.