# Upper Bound Solution of Surrounding Rock Pressure for Deep Cavity Using Nonlinear Hoek–Brown Failure Criterion

Authors: Cheng-yang Wang; He-lin Fu; Jun-zhe Liu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The linear failure criterion is generally adopted in the stability problem of geotechnical engineering, whereas the experiments have indicated that there is a nonlinear relationship between the maximum and minimum principal stresses in weak surrounding rock. According to the characteristics of weak rock, the failure mechanism of deep cavity is constructed by combining the nonlinear Hoek–Brown failure criterion and the upper bound theorem of limit analysis. The upper bound solution of the surrounding rock pressure was deduced using the tangent method. The results show that the surrounding rock pressure of deep cavity is affected by surrounding rock grade, cavity depth, and section size. Especially, the influence of the disturbance factor is quite obvious. Upper bound solution based on nonlinear Hoek–Brown failure criterion can fully take the influence of parameters on surrounding rock pressure. Therefore, this method is more scientific than the linear failure criterion to calculate the surrounding rock pressure.


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## Body

## 1. Introduction

The stability problem of deep cavity has been highlighted greatly and continuously in the field of the underground engineering, whereas the problem that needs to be solved urgently is how to accurately obtain the surrounding rock pressure when the deep cavern is destroyed [1, 2]. Currently, the analytic methods for solving the surrounding rock pressure mainly include the limit equilibrium method and the limit analysis method, both of which have been illustrated widely in literature. However, the former does not consider the constitutive relationship of rock, while the latter does by adopting the orthogonal flow rule. Therefore, the upper bound theory is more rigorous than the limit equilibrium method. Especially, there is no need to study the whole process of rock’s elastic-plastic deformation, but directly to pay attention to the ultimate failure state of rock. Thus, its calculation process is simple and has been regarded as one of the most effective approaches to solve the surrounding rock pressure problem in the deep cavity [3–5].In recent years, some scholars have adopted the limit analysis method to study the surrounding rock pressure of deep cavity. According to the results of the model test, Davis et al. [6] proposed four failure mechanisms for underground cavity, which included the arch collapse mode and vault collapse mode bordered wall, and obtained the upper bound solution of the surrounding rock pressure by the limit analysis method. Takemura et al. [7] constructed the failure mechanism of the deep cavity using the centrifugal model test. The failure model is composed of 5 rigid blocks, and the limit analysis method is used to solve the surrounding rock pressure of the underground cavity under the anisotropic soil conditions. The results were compared with the measured values, and the validity of the results has been verified. Yang and Yang [8] proposed the destruction mechanism of underground chambers composed of n blocks. According to the requirement of calculation accuracy, the upper bound solution of surrounding rock pressure under different number of blocks was obtained. Through comparison, the destruction characteristics of deep cavity can be better illustrated with more rigid blocks, and accordingly, the surrounding rock pressure is more accurate. However, the above research studies are applied to shallow buried cavity where the Mohr–Coulomb failure criterion is used and more suitable for soil mass.For deep cavity, Atkinson and Potts [9] established the failure mode of circular which were based on the results of model tests, obtained the upper bound solution of the surrounding rock pressure using the limit analysis method, and the correctness of the results was verified. And the curve failure mechanism of the deep cavity based on the limit analysis method was proposed by Fraldi and Guarracino [10–12], which the Hoek–Brown failure criterion and the variational method were applied to solve the analytic expression of the curve and got the surrounding rock pressure under the limit state of the vault collapse. Yang and Yang [8] combined the limit analysis theory with the variational principle, deduced the expression of the surrounding rock pressure of deep rectangle cavity and specified the corresponding failure range. In addition, Huang et al. [13] took the influence of pore water based on the previous research studies into account and got a reasonable result. The limit analysis method was used to calculate the surrounding rock pressure of deep buried circular cavity. Compared with the existing research results, the correctness of the calculation results was verified. Qin et al. [2] considered the multilayer rock mass and used the upper bound method to study the influence of the variation of the parameters of the Hoek–Brown failure criterion on the surrounding rock pressure of the deep buried rectangular cavity. The vault of deep cavity was only considered and assumed to obey a curve function fx in the abovementioned studies, and yet, the side wall was ignored in the analysis. However, it could not accord with the conditions in reality.Therefore, by referring to the previous research results, the multiblock and logarithmic spiral curves are used to construct the failure mechanism of deep cavity in this paper. Based on the upper bound theorem of limit analysis and Hoek–Brown failure criterion, the vault and side wall damage state is investigated to determine the rock pressure when the deep cavity collapses and provide the scientific basis for the design in engineering.


## 2. Hoek–Brown Nonlinear Failure Criterion and Tangent Method

In 1980, Brown proposed the initial Hoek–Brown failure criterion on the basis of Hoek’s experimental research studies on rock plastic behaviors. Subsequently, Hoek made a further revision to the failure criterion and put forward the modified generalized Hoek–Brown criterion. It is expressed as follows [14, 15]:(1)σ1′=σ3′+σcimbσ3′σci+Sα,where σ1′ and σ3′ are the maximum and minimum effective principal stress of rock mass at critical failure, respectively; σci is the uniaxial compressive strength of rock; α is the physical index relating to the integrity of rock; mb and S indicate the dimensionless parameters related to geotechnical materials, which characterizes the integrity of the rock. The above parameters can be calculated through the GSI index; the specific process is as follows [16–18]:(2)mb=miexpGSI−10028−14D,S=expGSI−1009−3D,α=12+16+e−GSI/15−e−20/3.For the upper limit solution of the deep cavity under the Hoek–Brown failure criterion, it is necessary to use the tangent method to obtain the counterforce of the support. Among them, the correlations betweenct and φt can be written as follows [19, 20]:(3)ctσci=cosφt2ma1−sinφt2sinφta/1−a−tanφtm1+sinφtama1−sinφt2sinφta/1−a+smtanφt.In the above formula, the nonlinear shear strength indexφt can be optimized by the least energy principle when the upper bound is calculated. The ct is obtained by Equation (3) after the determination of φt.


## 3. Building Destruction Mechanism

During the excavation process of the deep cavity, the collapse easily happens due to the untimely support or weak strength. Based on the existing research results [8, 10] and the requirement of the upper bound theorem of limit analysis, the failure mode of deep cavity was structured, as shown in Figure 1. The width of the chamber is l, and the height is h. The wedge ABGOG1B1 at the top of the cavity is collapsing vertically and downwards at a rate of v0. The BGC and B1G1C1 segments, respectively, rotate failure around G and G1 points. The triangles CGD, DGE, and EGF and C1G1D1, D1G1E1, and E1G1F1 would undergo translational failure. The surrounding rock pressure of vault and side wall are q and e, respectively, and e=Kq.Figure 1
Diagrammatic sketch of circular sandwich failure mechanism.The velocity vector diagram corresponding to this failure mechanism is shown in Figure2.Figure 2
Velocity vector diagram.Calculation of speed and velocity discontinuity’s length:(4)V1=V0.Based on the analysis of the velocity field in the shear zone of the circular arc radiation, the following can be obtained:(5)V2=V1⋅eα1tan2φ=V0⋅eα1tan2φ.From Figure2 and geometric triangle relationship can be obtained:(6)V23=sinα2cos2φV2,V3=cos2φ−α2cos2φV2,V4=cos2φ−α3cos2φV3,V34=sinα3cos2φV3.The recurrence relation of various velocity discontinuities is as follows:(7)GE=cosα4GF,EF=sinα4GF,GD=cosα3GE,DE=sinα3GE,GC=cosα2GD,CD=sinα2GD,GB=GC,AO=AB⋅cosφ+BG⋅sinφ,AB=BGtanφ+GOsinφ,GO=l2,GF=h.


## 4. Calculation of Surrounding Rock Pressure

According to the upper bound theorem of limit analysis, the external power and internal energy dissipation rate can be obtained when the deep cavity collapses. If the external power and internal energy dissipation rate are equal, the surrounding rock pressure at the collapse of the deep cavity can be obtained.


### 4.1. Calculation of External Power

#### 4.1.1. Power of Gravity

Area of each rigid block:(8)12SABGOG1B1=SABGO=12BG⋅AB+12AO⋅GO,SGBC=∫0α112GB2dθ′,SGCD=12GC⋅CD,SGDE=12GD⋅DE,SGEF=12GE⋅EF.The gravitational power generated by each rigid slider is as follows.Quadrilateral rigid blockABGO:(9)W1=γ⋅SABGO⋅V0=γ⋅12BG⋅AB+12AO⋅GO⋅V0=γ⋅V0⋅h2⋅f1,f1=12cosα2cosα3cosα4⋅cosα2cosα3cosα4tanφ+l2hsinφ+cosα2cosα3cosα4tanφ+l2hsinφ⋅cosφ+cosα2cosα3cosα4⋅sinφ⋅l4h.Sector regionGBC:The power calculation of circular shear zoneBGC is shown in Figure 3.Figure 3
Schematic diagram of the power calculation of circular shear zoneBGC.(10)
W
2
=
γ
∫
0
α
1
1
2
⋅
G
B
2
⋅
d
θ
′
⋅
V
0
e
θ
′
tan
2
φ
sin
π
2
−
θ
′
=
γ
⋅
V
0
⋅
h
2
⋅
f
2
.Among them,(11)f2=12⋅cos2α2cos2α3cos2α41+tan22φsinα1+tan2φcosα1eα1tan2φ−tan2φ.Triangular rigid blockGCD:(12)W3=γSGCDV2cosπ2+φ−α2−α3−α4=γ⋅12GC⋅CD⋅V2⋅sinα2+α3+α4−φ=γh2V0f3.Among them,(13)f3=12sinα2⋅cosα2⋅cos2α3⋅cos2α4⋅sinα2+α3+α4−φ⋅eα1tan2φ.Triangular rigid blockGDE:(14)W4=γ⋅SGDE⋅V3⋅cosπ2+φ−α1−α2=γ⋅12GD⋅DE⋅V3⋅sinα3+α4−φ=γh2V0f4.Among them,(15)f4=12sinα3⋅cosα3⋅cos2α4⋅cos2φ−α2cos2φ⋅eα1tan2φ⋅sinα3+α4−φ.Triangular rigid blockGEF:(16)W5=γ⋅SGEF⋅V4⋅cosπ2+φ−α4=γ⋅12GE⋅EF⋅V4⋅sinα4−φ=γh2V0f5.Among them,(17)f5=12sinα4⋅cosα4⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅sinα4−φ.Power of gravity:(18)Wsoil=γh2V0f1+f2+f3+f4+f5.


#### 4.1.2. Power of Supporting Force

(19)
W
T
=
−
q
⋅
l
2
⋅
V
0
−
e
⋅
h
⋅
V
4
⋅
sin
α
4
−
φ
,
=
−
q
⋅
l
2
⋅
V
0
−
K
q
⋅
h
⋅
cos
2
φ
−
α
3
⋅
cos
2
φ
−
α
2
cos
2
2
φ
e
α
1
tan
2
φ
⋅
V
0
⋅
cos
α
4
−
φ
,
=
−
q
V
0
h
f
6
.Among them,(20)e=Kq,f6=l2h+K⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅cosα4−φ.


#### 4.1.3. Total External Power

The total external power is equal to the sum of the deadweight power of the surrounding rock and the power of the supporting counterforce. That is,(21)Wext=Wsoil+WT=γh2V0f1+f2+f3+f4+f5−qhV0f6.


### 4.2. Internal Energy Dissipation Power

Energy dissipation along discontinuity lineAB:(22)DAB=c⋅AB⋅V0⋅cosφ=h⋅cosα2⋅cosα3⋅cosα4tanφ+l2sinφ⋅c⋅V0⋅cosφ=c⋅V0⋅h⋅f7.Among them,(23)f7=cosα2⋅cosα3⋅cosα4tanφ+l2hsinφ⋅cosφ.Energy dissipation of circular shear surfaceBC and shear zone GBC:(24)DBC=c⋅BG⋅V1⋅cosφtan2φeα1⋅tan2φ−1=chV0f8,f8=cosα4cosα3cosα2⋅cosφtan2φeα1⋅tan2φ−1,DGBC=c⋅BG⋅V1⋅cosφsin2φeα1⋅tan2φ−1=chV0f9,f9=cosα4cosα3cosα2⋅cosφsin2φeα1⋅tan2φ−1.Energy dissipation along discontinuity lineCD:(25)DCD=c⋅CD⋅V2⋅cosφ=chV0f10,f10=sinα2cosα3cosα4⋅eα1tan2φ⋅cosφ.Energy dissipation along discontinuity lineGD:(26)DGD=c⋅GD⋅V23⋅cosφ=chV0f11,f11=sinα2cosα3cosα4cosφcos2φeα1tan2φ.Energy dissipation along discontinuity lineDE:(27)DDE=c⋅DE⋅V3⋅cosφ=chV0f12,f12=cos2φ−α2sinα3cosα4cosφcos2φeα1tan2φ.Energy dissipation along discontinuity lineGE:(28)DGE=c⋅GE⋅V34⋅cosφ=chV0f13,f13=cosφcosα4sinα3cos2φ−α2cos22φeα1tan2φ.Energy dissipation along discontinuity lineEF:(29)DEF=c⋅EF⋅V4⋅cosφ=chV0f14,f14=sinα4cosφcos2φ−α3cos2φ−α2cos22φeα1tan2φ.Total internal energy dissipation:(30)Dint=chV0⋅f7+f8+f9+f10+f11+f12+f13+f14.


### 4.3. Calculation of Supporting Counterforce

According to the principle of virtual power,(31)Wext=Dint.The expression of surrounding rock pressure is(32)q=γ⋅h⋅f1+f2+f3+f4+f5−c⋅f7+f8+f9+f10+f11+f12+f13+f14f6.Above all of the formulas,h is the height of the cavity, measured in meter. l represents the width of the cavity, measured in meter. q is the vertical supporting pressure, and e is the lateral supporting pressures of which the unit is kPa. φ is the internal friction angle, measured in degree. γ is the unit weight of soil, measured in kg/m3. c is the cohesion, measured in kPa. K is the lateral pressure coefficient. V0, V1, V2, V3, and V4 are the speed in the velocity field. α1, α2, α3, and α4 are the geometrical variables which determine the shape of the failure mechanism whose unit is degree. W is the external power, Wexit is the external total power, Wsoil is the weight power of surrounding rock, WT is the power of support of the antiforce, and the unit is watt. W1, W2, W3, W4, and W5 are variables. D is the dissipation power of internal energy, and f0, f1, f2, …, f14 are variables.The constraint condition of Equation (32) is(33)α1+α2+α3+α4=π2+φt,Invoking the command in the Matlab software optimization toolbox, it will generate a set of angle data under the constraint condition (33). According to the formula of the surrounding rock pressure, an upper bound solution q is obtained; then, by adjusting the variable parameter values, numerous q values are obtained, in which the maximum q is the upper bound solution of the surrounding rock pressure.


## 4.1. Calculation of External Power

### 4.1.1. Power of Gravity

Area of each rigid block:(8)12SABGOG1B1=SABGO=12BG⋅AB+12AO⋅GO,SGBC=∫0α112GB2dθ′,SGCD=12GC⋅CD,SGDE=12GD⋅DE,SGEF=12GE⋅EF.The gravitational power generated by each rigid slider is as follows.Quadrilateral rigid blockABGO:(9)W1=γ⋅SABGO⋅V0=γ⋅12BG⋅AB+12AO⋅GO⋅V0=γ⋅V0⋅h2⋅f1,f1=12cosα2cosα3cosα4⋅cosα2cosα3cosα4tanφ+l2hsinφ+cosα2cosα3cosα4tanφ+l2hsinφ⋅cosφ+cosα2cosα3cosα4⋅sinφ⋅l4h.Sector regionGBC:The power calculation of circular shear zoneBGC is shown in Figure 3.Figure 3
Schematic diagram of the power calculation of circular shear zoneBGC.(10)
W
2
=
γ
∫
0
α
1
1
2
⋅
G
B
2
⋅
d
θ
′
⋅
V
0
e
θ
′
tan
2
φ
sin
π
2
−
θ
′
=
γ
⋅
V
0
⋅
h
2
⋅
f
2
.Among them,(11)f2=12⋅cos2α2cos2α3cos2α41+tan22φsinα1+tan2φcosα1eα1tan2φ−tan2φ.Triangular rigid blockGCD:(12)W3=γSGCDV2cosπ2+φ−α2−α3−α4=γ⋅12GC⋅CD⋅V2⋅sinα2+α3+α4−φ=γh2V0f3.Among them,(13)f3=12sinα2⋅cosα2⋅cos2α3⋅cos2α4⋅sinα2+α3+α4−φ⋅eα1tan2φ.Triangular rigid blockGDE:(14)W4=γ⋅SGDE⋅V3⋅cosπ2+φ−α1−α2=γ⋅12GD⋅DE⋅V3⋅sinα3+α4−φ=γh2V0f4.Among them,(15)f4=12sinα3⋅cosα3⋅cos2α4⋅cos2φ−α2cos2φ⋅eα1tan2φ⋅sinα3+α4−φ.Triangular rigid blockGEF:(16)W5=γ⋅SGEF⋅V4⋅cosπ2+φ−α4=γ⋅12GE⋅EF⋅V4⋅sinα4−φ=γh2V0f5.Among them,(17)f5=12sinα4⋅cosα4⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅sinα4−φ.Power of gravity:(18)Wsoil=γh2V0f1+f2+f3+f4+f5.


### 4.1.2. Power of Supporting Force

(19)
W
T
=
−
q
⋅
l
2
⋅
V
0
−
e
⋅
h
⋅
V
4
⋅
sin
α
4
−
φ
,
=
−
q
⋅
l
2
⋅
V
0
−
K
q
⋅
h
⋅
cos
2
φ
−
α
3
⋅
cos
2
φ
−
α
2
cos
2
2
φ
e
α
1
tan
2
φ
⋅
V
0
⋅
cos
α
4
−
φ
,
=
−
q
V
0
h
f
6
.Among them,(20)e=Kq,f6=l2h+K⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅cosα4−φ.


### 4.1.3. Total External Power

The total external power is equal to the sum of the deadweight power of the surrounding rock and the power of the supporting counterforce. That is,(21)Wext=Wsoil+WT=γh2V0f1+f2+f3+f4+f5−qhV0f6.


## 4.1.1. Power of Gravity

Area of each rigid block:(8)12SABGOG1B1=SABGO=12BG⋅AB+12AO⋅GO,SGBC=∫0α112GB2dθ′,SGCD=12GC⋅CD,SGDE=12GD⋅DE,SGEF=12GE⋅EF.The gravitational power generated by each rigid slider is as follows.Quadrilateral rigid blockABGO:(9)W1=γ⋅SABGO⋅V0=γ⋅12BG⋅AB+12AO⋅GO⋅V0=γ⋅V0⋅h2⋅f1,f1=12cosα2cosα3cosα4⋅cosα2cosα3cosα4tanφ+l2hsinφ+cosα2cosα3cosα4tanφ+l2hsinφ⋅cosφ+cosα2cosα3cosα4⋅sinφ⋅l4h.Sector regionGBC:The power calculation of circular shear zoneBGC is shown in Figure 3.Figure 3
Schematic diagram of the power calculation of circular shear zoneBGC.(10)
W
2
=
γ
∫
0
α
1
1
2
⋅
G
B
2
⋅
d
θ
′
⋅
V
0
e
θ
′
tan
2
φ
sin
π
2
−
θ
′
=
γ
⋅
V
0
⋅
h
2
⋅
f
2
.Among them,(11)f2=12⋅cos2α2cos2α3cos2α41+tan22φsinα1+tan2φcosα1eα1tan2φ−tan2φ.Triangular rigid blockGCD:(12)W3=γSGCDV2cosπ2+φ−α2−α3−α4=γ⋅12GC⋅CD⋅V2⋅sinα2+α3+α4−φ=γh2V0f3.Among them,(13)f3=12sinα2⋅cosα2⋅cos2α3⋅cos2α4⋅sinα2+α3+α4−φ⋅eα1tan2φ.Triangular rigid blockGDE:(14)W4=γ⋅SGDE⋅V3⋅cosπ2+φ−α1−α2=γ⋅12GD⋅DE⋅V3⋅sinα3+α4−φ=γh2V0f4.Among them,(15)f4=12sinα3⋅cosα3⋅cos2α4⋅cos2φ−α2cos2φ⋅eα1tan2φ⋅sinα3+α4−φ.Triangular rigid blockGEF:(16)W5=γ⋅SGEF⋅V4⋅cosπ2+φ−α4=γ⋅12GE⋅EF⋅V4⋅sinα4−φ=γh2V0f5.Among them,(17)f5=12sinα4⋅cosα4⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅sinα4−φ.Power of gravity:(18)Wsoil=γh2V0f1+f2+f3+f4+f5.


## 4.1.2. Power of Supporting Force

(19)
W
T
=
−
q
⋅
l
2
⋅
V
0
−
e
⋅
h
⋅
V
4
⋅
sin
α
4
−
φ
,
=
−
q
⋅
l
2
⋅
V
0
−
K
q
⋅
h
⋅
cos
2
φ
−
α
3
⋅
cos
2
φ
−
α
2
cos
2
2
φ
e
α
1
tan
2
φ
⋅
V
0
⋅
cos
α
4
−
φ
,
=
−
q
V
0
h
f
6
.Among them,(20)e=Kq,f6=l2h+K⋅cos2φ−α3⋅cos2φ−α2cos22φeα1tan2φ⋅cosα4−φ.


## 4.1.3. Total External Power

The total external power is equal to the sum of the deadweight power of the surrounding rock and the power of the supporting counterforce. That is,(21)Wext=Wsoil+WT=γh2V0f1+f2+f3+f4+f5−qhV0f6.


## 4.2. Internal Energy Dissipation Power

Energy dissipation along discontinuity lineAB:(22)DAB=c⋅AB⋅V0⋅cosφ=h⋅cosα2⋅cosα3⋅cosα4tanφ+l2sinφ⋅c⋅V0⋅cosφ=c⋅V0⋅h⋅f7.Among them,(23)f7=cosα2⋅cosα3⋅cosα4tanφ+l2hsinφ⋅cosφ.Energy dissipation of circular shear surfaceBC and shear zone GBC:(24)DBC=c⋅BG⋅V1⋅cosφtan2φeα1⋅tan2φ−1=chV0f8,f8=cosα4cosα3cosα2⋅cosφtan2φeα1⋅tan2φ−1,DGBC=c⋅BG⋅V1⋅cosφsin2φeα1⋅tan2φ−1=chV0f9,f9=cosα4cosα3cosα2⋅cosφsin2φeα1⋅tan2φ−1.Energy dissipation along discontinuity lineCD:(25)DCD=c⋅CD⋅V2⋅cosφ=chV0f10,f10=sinα2cosα3cosα4⋅eα1tan2φ⋅cosφ.Energy dissipation along discontinuity lineGD:(26)DGD=c⋅GD⋅V23⋅cosφ=chV0f11,f11=sinα2cosα3cosα4cosφcos2φeα1tan2φ.Energy dissipation along discontinuity lineDE:(27)DDE=c⋅DE⋅V3⋅cosφ=chV0f12,f12=cos2φ−α2sinα3cosα4cosφcos2φeα1tan2φ.Energy dissipation along discontinuity lineGE:(28)DGE=c⋅GE⋅V34⋅cosφ=chV0f13,f13=cosφcosα4sinα3cos2φ−α2cos22φeα1tan2φ.Energy dissipation along discontinuity lineEF:(29)DEF=c⋅EF⋅V4⋅cosφ=chV0f14,f14=sinα4cosφcos2φ−α3cos2φ−α2cos22φeα1tan2φ.Total internal energy dissipation:(30)Dint=chV0⋅f7+f8+f9+f10+f11+f12+f13+f14.


## 4.3. Calculation of Supporting Counterforce

According to the principle of virtual power,(31)Wext=Dint.The expression of surrounding rock pressure is(32)q=γ⋅h⋅f1+f2+f3+f4+f5−c⋅f7+f8+f9+f10+f11+f12+f13+f14f6.Above all of the formulas,h is the height of the cavity, measured in meter. l represents the width of the cavity, measured in meter. q is the vertical supporting pressure, and e is the lateral supporting pressures of which the unit is kPa. φ is the internal friction angle, measured in degree. γ is the unit weight of soil, measured in kg/m3. c is the cohesion, measured in kPa. K is the lateral pressure coefficient. V0, V1, V2, V3, and V4 are the speed in the velocity field. α1, α2, α3, and α4 are the geometrical variables which determine the shape of the failure mechanism whose unit is degree. W is the external power, Wexit is the external total power, Wsoil is the weight power of surrounding rock, WT is the power of support of the antiforce, and the unit is watt. W1, W2, W3, W4, and W5 are variables. D is the dissipation power of internal energy, and f0, f1, f2, …, f14 are variables.The constraint condition of Equation (32) is(33)α1+α2+α3+α4=π2+φt,Invoking the command in the Matlab software optimization toolbox, it will generate a set of angle data under the constraint condition (33). According to the formula of the surrounding rock pressure, an upper bound solution q is obtained; then, by adjusting the variable parameter values, numerous q values are obtained, in which the maximum q is the upper bound solution of the surrounding rock pressure.


## 5. Result Analysis

Under the nonlinear Hoek–Brown failure criterion, the upper bound theorem is used to analyze the influence of related parameters on the surrounding rock pressure of deep cavity. According to the failure mechanism and method mentioned above, the calculation results are as follows.


### 5.1. Influence ofγ and K on Surrounding Rock Pressure

To investigate the influence ofγ and K on the surrounding rock pressure, the parameters are assumed as follows: Cavity width l=10m, height h=10m, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength σci=400kPa, disturbance factor D = 0, and γ is the soil gravity, which had taken the values of 16, 18, 20, 22, 24, and 26 kN/m3, respectively, in the experiment. Similarly, the coefficient K was taken 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4, respectively. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction together, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 1 and 2 and Figures 4 and 5. The tables and figures present that the surrounding rock pressures q and e increase with the rising of soil gravity γ, and the pressure of surrounding rock q decreases, the surrounding rock pressure e increases with the rising of coefficient K. It shows that the buried depth has certain influence on the surrounding rock pressure.Table 1
Upper bound solution of the surrounding rock pressureq at different γ and K.
γ
kN/m
3
q (kPa)
K = 0.4
K = 0.6
K = 0.8
K = 1.0
K = 1.2
K = 1.4
16
112.1
84.0
67.2
56.1
48.2
42.2
18
153.9
115.7
92.8
77.6
66.6
58.4
20
201.9
152.3
122.5
102.5
88.1
77.3
22
255.8
193.6
156.0
130.7
112.5
98.8
24
315.5
239.5
193.3
162.1
139.7
122.7
26
380.9
289.9
234.3
196.8
169.6
149.1Table 2
Upper bound solution of the surrounding rock pressuree at different γ and K.
γ
kN/m
3
e (kPa)
K = 0.4
K = 0.6
K = 0.8
K = 1.0
K = 1.2
K = 1.4
16
44.8
50.4
53.8
56.1
57.8
59.1
18
61.5
69.4
74.2
77.6
80.0
81.8
20
80.7
91.4
98.0
102.5
105.7
108.2
22
102.3
116.2
124.8
130.7
135.0
138.3
24
126.2
143.7
154.6
162.1
167.6
171.8
26
152.3
173.9
187.5
196.8
203.6
208.8Figure 4
Effect ofγ and K on the surrounding rock pressure q.Figure 5
Effect ofγ and K on the surrounding rock pressure e.The graph is drawn from the above table, as follows.


### 5.2. Influence of Chambers Section Size on Surrounding Rock Pressure

Likewise, to study the influence of chambers section size on the surrounding rock pressure, the parameters are soil gravityγ=20kN/m3, coefficient K = 1.0, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength σci=400kPa, disturbance factor D = 0; the value of chamber width l and height h was taken as 5, 6, 7, 8, 9, and 10 m, respectively, in the experiment to conduct further analysis. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction together, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 3 and 4 and Figures 6 and 7. The tables and figures present that the surrounding rock pressures q and e increase with the rising of the chamber width l and height h, and the effect is obvious. Based on the research and above illustration, it is obvious that the chambers section size has a great influence on the pressure of surrounding rock.Table 3
Upper bound solution of the surrounding rock pressureq at different l and h.
l (m)
q (kPa)
h = 5 m
h = 6 m
h = 7 m
h = 8 m
h = 9 m
h = 10 m
5
15.0
19.4
25.1
32.9
42.2
53.0
6
20.3
24.9
31.2
39.7
49.8
61.3
7
26.6
31.3
38.3
47.4
58.3
70.5
8
33.9
38.7
46.3
56.1
67.5
80.4
9
42.0
47.0
55.2
65.5
77.6
91.1
10
50.9
56.2
64.9
75.8
88.3
102.5Table 4
Upper bound solution of the surrounding rock pressuree at different l and h.
l (m)
e (kPa)
h = 5 m
h = 6 m
h = 7 m
h = 8 m
h = 9 m
h = 10 m
5
15.0
19.4
25.1
32.9
42.2
53.0
6
20.3
24.9
31.2
39.7
49.8
61.3
7
26.6
31.3
38.3
47.4
58.3
70.5
8
33.9
38.7
46.3
56.1
67.5
80.4
9
42.0
47.0
55.2
65.5
77.6
91.1
10
50.9
56.2
64.9
75.8
88.3
102.5
The graph is drawn from the above table, as follows.Figure 6
Effect ofl and h on the surrounding rock pressure q.Figure 7
Effect ofl and h on the surrounding rock pressure e.


### 5.3. Influence of GSI andmi on Surrounding Rock Pressure

To study the influence of GSI andmi on the surrounding rock pressure, the parameters are soil gravity γ=20kN/m3, coefficient K = 1.0, cavity width l=10m, height h=10m, uniaxial compressive strength σci=400kPa, disturbance factor D = 0, and the value of geological strength index GSI was taken as 10, 15, 20, 25, 30, and 35, respectively; rock mass constant value mi was taken as 5, 10, 15, 20, 25, and 30, respectively. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 5 and 6 and Figures 8 and 9.Figure 8
Effect of GSI andmi on the surrounding rock pressure q.Figure 9
Effect of GSI andmi on the surrounding rock pressure e.Table 5
Upper bound solution of the surrounding rock pressureq at different GSI and mi.
GSI
q (kPa)
m
i = 5
m
i = 10
m
i = 15
m
i = 20
m
i = 25
m
i = 30
10
1117.3
505.8
301.4
199.8
140.0
101.3
15
896.2
394.9
227.7
145.3
97.5
67.4
20
718.9
306.0
169.0
102.5
64.9
43.4
25
575.5
234.2
122.2
69.2
41.7
25.0
30
458.6
176.1
85.1
40.7
27.3
16.1
35
362.6
129.0
56.2
29.3
15.5
6.7Table 6
Upper bound solution of the surrounding rock pressuree at different GSI and mi.
GSI
e (kPa)
m
i = 5
m
i = 10
m
i = 15
m
i = 20
m
i = 25
m
i = 30
10
1117.3
505.8
301.4
199.8
140.0
101.3
15
896.2
394.9
227.7
145.3
97.5
67.4
20
718.9
306.0
169.0
102.5
64.9
43.4
25
575.5
234.2
122.2
69.2
41.7
25.0
30
458.6
176.1
85.1
40.7
27.3
16.1
35
362.6
129.0
56.2
29.3
15.5
6.7The graph is drawn from the above table, as follows.When coefficientK=1.0, the surrounding rock pressure e=q. The tables and figures show that the surrounding rock pressure q and e decrease with the increasing of the geological strength index GSI and rock mass constant mi, and the influence is apparent. It could be concluded that the quality of surrounding would have a significant and direct influence on the pressure of surrounding rock.


### 5.4. Influence ofσci and D on Surrounding Rock Pressure

To study the influence ofσci and D on the surrounding rock pressure, the parameters are soil gravity γ=20kN/m3, coefficient K = 1.0, cavity width l=10m, height h=10m, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength had taken the value of 200, 400, 600, 800, 1000, and 1200 kPa, and the disturbance factor D had taken the value of 0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively, to conduct further analysis. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 7 and 8 and Figures 10 and 11.Figure 10
Effect ofσci and D on the surrounding rock pressure q.Figure 11
Effect ofσci and D on surrounding rock pressure e.Table 7
Upper bound solution of the surrounding rock pressureq at different σci and D.
σ
ci
kPa
q (kPa)
D = 0
D = 0.2
D = 0.4
D = 0.6
D = 0.8
D = 1.0
200
262.5
400.6
647.8
1148.6
2366.8
6292.6
400
102.5
177.2
314.9
595.8
1278.2
3473.7
600
49.9
98.1
193.5
393.1
879.8
2443.0
800
30.0
59.2
130.4
285.7
668.5
1897.5
1000
18.0
39.6
92.2
218.8
536.2
1556.1
1200
12.0
25.5
67.1
173.0
445.0
1320.9Table 8
Upper bound solution of the surrounding rock pressuree at different σci and D.
σ
ci
kPa
e (kPa)
D = 0
D = 0.2
D = 0.4
D = 0.6
D = 0.8
D = 1.0
200
262.5
400.6
647.8
1148.6
2366.8
6292.6
400
102.5
177.2
314.9
595.8
1278.2
3473.7
600
49.9
98.1
193.5
393.1
879.8
2443.0
800
30.0
59.2
130.4
285.7
668.5
1897.5
1000
18.0
39.6
92.2
218.8
536.2
1556.1
1200
12.0
25.5
67.1
173.0
445.0
1320.9The graph is drawn from the above table, as follows.When coefficientK=1.0, the surrounding rock pressure e=q. The tables and figures show that the surrounding rock pressure q and e decrease with the increasing of the uniaxial compressive strength σci, while the surrounding rock pressure q and e increase with the rising of disturbance factor D. And the effect is obvious, which reveals that the disturbance factor would influence the pressure of surrounding rock directly.


## 5.1. Influence ofγ and K on Surrounding Rock Pressure

To investigate the influence ofγ and K on the surrounding rock pressure, the parameters are assumed as follows: Cavity width l=10m, height h=10m, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength σci=400kPa, disturbance factor D = 0, and γ is the soil gravity, which had taken the values of 16, 18, 20, 22, 24, and 26 kN/m3, respectively, in the experiment. Similarly, the coefficient K was taken 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4, respectively. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction together, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 1 and 2 and Figures 4 and 5. The tables and figures present that the surrounding rock pressures q and e increase with the rising of soil gravity γ, and the pressure of surrounding rock q decreases, the surrounding rock pressure e increases with the rising of coefficient K. It shows that the buried depth has certain influence on the surrounding rock pressure.Table 1
Upper bound solution of the surrounding rock pressureq at different γ and K.
γ
kN/m
3
q (kPa)
K = 0.4
K = 0.6
K = 0.8
K = 1.0
K = 1.2
K = 1.4
16
112.1
84.0
67.2
56.1
48.2
42.2
18
153.9
115.7
92.8
77.6
66.6
58.4
20
201.9
152.3
122.5
102.5
88.1
77.3
22
255.8
193.6
156.0
130.7
112.5
98.8
24
315.5
239.5
193.3
162.1
139.7
122.7
26
380.9
289.9
234.3
196.8
169.6
149.1Table 2
Upper bound solution of the surrounding rock pressuree at different γ and K.
γ
kN/m
3
e (kPa)
K = 0.4
K = 0.6
K = 0.8
K = 1.0
K = 1.2
K = 1.4
16
44.8
50.4
53.8
56.1
57.8
59.1
18
61.5
69.4
74.2
77.6
80.0
81.8
20
80.7
91.4
98.0
102.5
105.7
108.2
22
102.3
116.2
124.8
130.7
135.0
138.3
24
126.2
143.7
154.6
162.1
167.6
171.8
26
152.3
173.9
187.5
196.8
203.6
208.8Figure 4
Effect ofγ and K on the surrounding rock pressure q.Figure 5
Effect ofγ and K on the surrounding rock pressure e.The graph is drawn from the above table, as follows.


## 5.2. Influence of Chambers Section Size on Surrounding Rock Pressure

Likewise, to study the influence of chambers section size on the surrounding rock pressure, the parameters are soil gravityγ=20kN/m3, coefficient K = 1.0, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength σci=400kPa, disturbance factor D = 0; the value of chamber width l and height h was taken as 5, 6, 7, 8, 9, and 10 m, respectively, in the experiment to conduct further analysis. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction together, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 3 and 4 and Figures 6 and 7. The tables and figures present that the surrounding rock pressures q and e increase with the rising of the chamber width l and height h, and the effect is obvious. Based on the research and above illustration, it is obvious that the chambers section size has a great influence on the pressure of surrounding rock.Table 3
Upper bound solution of the surrounding rock pressureq at different l and h.
l (m)
q (kPa)
h = 5 m
h = 6 m
h = 7 m
h = 8 m
h = 9 m
h = 10 m
5
15.0
19.4
25.1
32.9
42.2
53.0
6
20.3
24.9
31.2
39.7
49.8
61.3
7
26.6
31.3
38.3
47.4
58.3
70.5
8
33.9
38.7
46.3
56.1
67.5
80.4
9
42.0
47.0
55.2
65.5
77.6
91.1
10
50.9
56.2
64.9
75.8
88.3
102.5Table 4
Upper bound solution of the surrounding rock pressuree at different l and h.
l (m)
e (kPa)
h = 5 m
h = 6 m
h = 7 m
h = 8 m
h = 9 m
h = 10 m
5
15.0
19.4
25.1
32.9
42.2
53.0
6
20.3
24.9
31.2
39.7
49.8
61.3
7
26.6
31.3
38.3
47.4
58.3
70.5
8
33.9
38.7
46.3
56.1
67.5
80.4
9
42.0
47.0
55.2
65.5
77.6
91.1
10
50.9
56.2
64.9
75.8
88.3
102.5
The graph is drawn from the above table, as follows.Figure 6
Effect ofl and h on the surrounding rock pressure q.Figure 7
Effect ofl and h on the surrounding rock pressure e.


## 5.3. Influence of GSI andmi on Surrounding Rock Pressure

To study the influence of GSI andmi on the surrounding rock pressure, the parameters are soil gravity γ=20kN/m3, coefficient K = 1.0, cavity width l=10m, height h=10m, uniaxial compressive strength σci=400kPa, disturbance factor D = 0, and the value of geological strength index GSI was taken as 10, 15, 20, 25, 30, and 35, respectively; rock mass constant value mi was taken as 5, 10, 15, 20, 25, and 30, respectively. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 5 and 6 and Figures 8 and 9.Figure 8
Effect of GSI andmi on the surrounding rock pressure q.Figure 9
Effect of GSI andmi on the surrounding rock pressure e.Table 5
Upper bound solution of the surrounding rock pressureq at different GSI and mi.
GSI
q (kPa)
m
i = 5
m
i = 10
m
i = 15
m
i = 20
m
i = 25
m
i = 30
10
1117.3
505.8
301.4
199.8
140.0
101.3
15
896.2
394.9
227.7
145.3
97.5
67.4
20
718.9
306.0
169.0
102.5
64.9
43.4
25
575.5
234.2
122.2
69.2
41.7
25.0
30
458.6
176.1
85.1
40.7
27.3
16.1
35
362.6
129.0
56.2
29.3
15.5
6.7Table 6
Upper bound solution of the surrounding rock pressuree at different GSI and mi.
GSI
e (kPa)
m
i = 5
m
i = 10
m
i = 15
m
i = 20
m
i = 25
m
i = 30
10
1117.3
505.8
301.4
199.8
140.0
101.3
15
896.2
394.9
227.7
145.3
97.5
67.4
20
718.9
306.0
169.0
102.5
64.9
43.4
25
575.5
234.2
122.2
69.2
41.7
25.0
30
458.6
176.1
85.1
40.7
27.3
16.1
35
362.6
129.0
56.2
29.3
15.5
6.7The graph is drawn from the above table, as follows.When coefficientK=1.0, the surrounding rock pressure e=q. The tables and figures show that the surrounding rock pressure q and e decrease with the increasing of the geological strength index GSI and rock mass constant mi, and the influence is apparent. It could be concluded that the quality of surrounding would have a significant and direct influence on the pressure of surrounding rock.


## 5.4. Influence ofσci and D on Surrounding Rock Pressure

To study the influence ofσci and D on the surrounding rock pressure, the parameters are soil gravity γ=20kN/m3, coefficient K = 1.0, cavity width l=10m, height h=10m, geological strength index GSI = 20, rock mass constant mi=20, uniaxial compressive strength had taken the value of 200, 400, 600, 800, 1000, and 1200 kPa, and the disturbance factor D had taken the value of 0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively, to conduct further analysis. According to the failure mechanism constructed in this paper which combined translational destruction and rotational destruction, the upper bound solution of the surrounding rock pressure obtained by the limit analysis method is shown in Tables 7 and 8 and Figures 10 and 11.Figure 10
Effect ofσci and D on the surrounding rock pressure q.Figure 11
Effect ofσci and D on surrounding rock pressure e.Table 7
Upper bound solution of the surrounding rock pressureq at different σci and D.
σ
ci
kPa
q (kPa)
D = 0
D = 0.2
D = 0.4
D = 0.6
D = 0.8
D = 1.0
200
262.5
400.6
647.8
1148.6
2366.8
6292.6
400
102.5
177.2
314.9
595.8
1278.2
3473.7
600
49.9
98.1
193.5
393.1
879.8
2443.0
800
30.0
59.2
130.4
285.7
668.5
1897.5
1000
18.0
39.6
92.2
218.8
536.2
1556.1
1200
12.0
25.5
67.1
173.0
445.0
1320.9Table 8
Upper bound solution of the surrounding rock pressuree at different σci and D.
σ
ci
kPa
e (kPa)
D = 0
D = 0.2
D = 0.4
D = 0.6
D = 0.8
D = 1.0
200
262.5
400.6
647.8
1148.6
2366.8
6292.6
400
102.5
177.2
314.9
595.8
1278.2
3473.7
600
49.9
98.1
193.5
393.1
879.8
2443.0
800
30.0
59.2
130.4
285.7
668.5
1897.5
1000
18.0
39.6
92.2
218.8
536.2
1556.1
1200
12.0
25.5
67.1
173.0
445.0
1320.9The graph is drawn from the above table, as follows.When coefficientK=1.0, the surrounding rock pressure e=q. The tables and figures show that the surrounding rock pressure q and e decrease with the increasing of the uniaxial compressive strength σci, while the surrounding rock pressure q and e increase with the rising of disturbance factor D. And the effect is obvious, which reveals that the disturbance factor would influence the pressure of surrounding rock directly.


## 6. Conclusion

(1)
The nonlinear Hoek–Brown failure criterion is applied to the upper bound of limit analysis by the tangent method to construct the circular arc failure mechanism of deep cavity. According to the principle of virtual power, the analytic expression of surrounding rock pressure could be deduced, and the upper bound solution of the surrounding rock pressure of deep cavity could be obtained.
(2)
The surrounding rock pressures increase obviously with the rising of soil gravityγ, tunnel width l, and tunnel height h. Besides, the pressure of surrounding rock q decreases with the rising of coefficient K, whereas the change rule of the surrounding rock pressure e is quite the opposite.
(3)
The surrounding rock pressures decrease significantly with the increase of GSI,mi and σci, while the surrounding rock pressure increase greatly with the rising of D.



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*Source: 1490848-2018-09-23.xml*