# Plastic Failure Analysis of Roadway Floor Surrounding Rocks Based on Unified Strength Theory

Authors: Wenxiang Zheng; Qingwei Bu; Yaoqing Hu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Taking the whole surrounding rock of an excavation roadway as the research object, the elastoplastic failure mechanics analysis of the surrounding rock body of the excavated roadway under three-directional in situ stresses is carried out by using the knowledge of generalized plane strain problems and unified strength theory, and the equations are derived for the elastic-plastic zone stress together with the plastic fracture range of the roadway floor surrounding rock under three-directional in situ stresses. At the same time, by means of the conclusion of mechanical analysis and the results of in situ detection of in situ stress in the Changcun mining area, the stability of the roadway floor surrounding rock was analyzed. The analysis reveals the influence on the stability of the roadway floor surrounding rock between the spatial relationship with the different in situ stresses and the roadway layout; meanwhile, it calculates the range of the plastic failure zone and the stress value of the #1 roadway floor in the S6 mining area of Changcun mine, which provides a reliable theoretical mechanical reference for research on roadway floor heave control technology.


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## Body

## 1. Introduction

Generally, the single shear strength criterion is used for the elastic-plastic mechanics solution of roadway surrounding rocks, such as the Mohr–Coulomb criterion or the Hook–Brown criterion [1–5]. Although the elastic-plastic failure of the roadway surrounding rocks can be considered as a plane strain problem, in the field engineering practice, the roadway surrounding rock is mostly in the triaxial compression state because of the distribution of residual stresses, the hinge of the fractured rock, and the existence of the supporting structure [6–10]. Owing to the single shear strength criterion, which considers the maximum principal stress and the minimum principal stress and completely neglects the influence of the intermediate principal stress, there is a great difference between the deformation and the actual situation of the surrounding rock of the plastic zone; in the mechanical analysis, the stress on the roadway section is only considered and the effect of axial stress is ignored [11–16]. Recently, many researchers have focused on studying the influence of the roadway axial principal stress to the plastic analysis and time-dependent change in the plastic zone of the deep circular roadway and hope the results will fit the specific reality better. For example, Zhao et al. [17–19] studied complex elastoviscoplastic behaviors of rock subjected to triaxial stress state and proposed a new EVP creep model to study time-dependent change in the plastic zone. In 1985, Yu et al. [20, 21] put forward a unified strength theory and a double shear strength theory based on the concept of double shear. Under the condition of hydrostatic pressure, the elastic-plastic distribution of the circular roadway obtained by the unified strength theory is analyzed. Compared with the Mohr–Coulomb strength theory, the parameters of the roadway axial stress and strength theory have an impact on the roadway surrounding rock plastic zone radius, stress distribution, and radial displacement [22–25]. This paper utilizes the uniform strength theory as the criteria of plastic failure of roadway surrounding rock and analyzes the plastic mechanics of the surrounding rock of a circular roadway under the triaxial stress condition [26, 27]. Figure 1 is a flowchart for the research.Figure 1
The flowchart for the research.


## 2. The Mechanical Analysis of Plastic Failure Zone Range and Stress of Roadway Floor Surrounding Rock under Three-Directional In Situ Stresses

Assuming that the compressive stress is positive, while the tensile stress is negative, the unified strength theory can be expressed as(1)F=λσ1−eσ2+σ31+e=σt,σ2≤λσ1+σ31+λ,(2)F′=λσ1+eσ21+e−σ3=σt,σ2≥λσ1+σ31+λ,where σ1 is the maximum principal stress, σ2 is the intermediate principal stress, σ3 is the minimum principal stress, e is the effect of the intermediate principal stress and the normal stress acting on the failure surface of the material, λ is the tension and compression ratio of the material, and σt is the uniaxial tensile strength of the material [28–30].For rock materials, if (1) and (2) are represented by the principal stress, rock cohesion C, and rock internal friction angle φ, they can be written as(3)F=σ11−sinφ−eσ2+σ31+e1+sinφ=2Ccosφ,σ2≤σ1+σ32−σ1−σ32sinφ,F′=σ1+eσ21+e1−sinφ−σ31+sinφ=2Ccosφ,σ2≥σ1+σ32−σ1−σ32sinφ.


### 2.1. The Solution of Stress Equation in Plastic Failure Zone of Roadway Floor Surrounding Rock Based on Unified Strength Theory

Assuming that the strain of the plastic zone surrounding rock mass satisfiesεV=0, the relationship among radial stress σrp, tangential stress σθp, and axial stress σzp in the plastic zone can be described by(4)σzp=12σrp+σθp.In the roadway surrounding rock mass around, tangential stressσθp is the largest and radial stress σrp is the smallest; according to the brief description of the theory of uniform strength, we can see that if the three principal stresses in the plastic zone satisfy σ2≥σ1+σ3/2−σ1−σ3/2sinφ, then (5) is established:(5)σθp+eσzp1+e1−sinφ−σrp1+sinφ=2Ccosφ.Substituting (4) into (5):(6)σθp=Kpσrp+Kc,where(7)Kp=2+2sinφ+3esinφ+e2+e1−sinφ,Kc=4C1+ecosφ2+e1−sinφ.In the plastic failure zone, when the roadway surrounding rock is in the ultimate equilibrium stress state, the stress of the plastic zone is basically unrelated with the stress of the original rock. The equivalent floor shape is an axisymmetric roadway, so the stress distribution of the plastic fracture zone of the surrounding rock has an axisymmetric shape. When the body force is not taken into account, the axisymmetric equilibrium equation for the plane strain problem of the plastic zone is expressed as(8)rdσrpdr+σrp−σθp=0,τrθp=0.Combined with (6), it can be rewritten as(9)dσrpσrp+Kc/Kp−1=Kp−1rdr.Using a differential equation, (10) is derived:(10)lnσrp+KcKp−1=Kp−1lnr+Q.According to the boundary conditionr=R0, σrp=Pi (Pi is the support force):(11)Q=1Kp−1lnPiKp−1+Kc−lnR0.Hence, the following is obtained:(12)σrp=Pi+KcKp−1rR0Kp−1−KcKp−1,σθp=KpPi+KcKp−1rR0Kp−1−KcKp−1,σy1p=Kp+12Pi+KcKp−1rR0Kp−1−KcKp−1,τrθp=0.


### 2.2. The Solution of Plastic Failure Zone Range of Roadway Floor Surrounding Rock

The calculation form of the plastic fractured zoneS is as follows:(13)S=rp+fθ.In ther=S position, to facilitate the calculation of the above formula, r−2, r−4, rKp−1, and rKp+1 need to be simplified. Therefore, expanding the polynomial and omitting the all after three items of polynomials, they are expressed as(14)r−2=S−2=rp−2−2fθrp−3,r−4=S−4=rp−4−4fθrp−5,rKp+1=SKp+1=rpKp+1+Kp+1fθrpKp,rKp−1=SKp−1=rpKp−1+Kp−1fθrpKp−2.The stress equations for the plastic fracture zone and the elastic zone of the roadway floor surrounding rock under inhomogeneous stress can be described by (14).The stress equation of the plastic fracture zone is(15)σrp=Pi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,σθp=KpPi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,σyp=Kp+12Pi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,τrθp=0.The stress equations of the elastic zone is(16)σre=Px+Pz21−arp−2−2fθrp−3+Px−Pz2cos2θ1−2brp−2−2fθrp−3+crp−4−4fθrp−5,σθe=Px+Pz21+arp−2−2fθrp−3−Px−Pz2cos2θ1+crp−4−4fθrp−5,σye=Py−μbPx−Pzcos2θrp−2−2fθrp−3,τrθe=τθre=−Px−Pz2sin2θ1+brp−2−2fθrp−3−crp−4−4fθrp−5.The tangential stress and radial stress equations are analyzed in light of the equality of stress continuous conditions, equal stress, and are represented by the formulas(17)N1R0Kp−1rpKp−1−KcKp−1=Px+Pz21−arp−2,KpN1R0Kp−1rpKp−1−KcKp−1=Px+Pz21+arp−2,(18)N1R0Kp−1Kp−1fθrpKp−2=Px+Pz22fθarp−3+Px−Pz2cos2θ1−2brp−2−2fθrp−3+crp−4−4fθrp−5,KpN1R0Kp−1Kp−1fθrpKp−2=Px+Pz2−2fθarp−3−Px−Pz2cos2θ1+crp−4−4fθrp−5,where N=Pi+Kc/Kp−1.When the boundary position isτrθeS=τθreS=τrθpS=τθrpS=0,(19)H=1+brp−2−2fθrp−3−crp−4−4fθrp−5=0.According to (18), previous formulas are rewritten as(20)N1R0Kp−1Kp−1fθrpKp−2=Px+Pz22fθarp−3+Px−Pz2cos2θ2−H−brp−2−2fθrp−3,KpN1R0Kp−1Kp−1fθrpKp−2=Px+Pz2−2fθarp−3−Px−Pz2cos2θ2−H+brp−2−2fθrp−3.Combining (18)–(20), a new expression is obtained, as follows:(21)Kp−1Px+Pz+2Kc/Kp−1Kp+1Kp−1fθ=−2fθPx+PzPx+PzKp−1+2KcPx+PzKp+1−Px−Pzcos2θ2−2Hrp,fθ=2Pz−Pxcos2θrpPx+Pz+2Kc/Kp−1Kp−1,where rp=R0Px+Pz+2Kc/Kp−1/Kp+1N1/Kp−1, and N, Kp, and Kc are the same as above.The equation of plastic fracture zone rangeS of roadway floor surrounding rock is expressed as (22) according to (13):(22)S=Px+Pz+2Kc/Kp−1Kp+1Pi+Kc/Kp−11/Kp−1R0−2Px−Pzcos2θPx+Pz+2Kc/Kp−1Kp−1Px+Pz+2Kc/Kp−1Kp+1Pi+Kc/Kp−11/Kp−1R0.When the intermediate principal stress is neglected (i.e.,e=0), the equation is consistent with the Rubinite equation; when Px=Pz, the equation is consistent with the Castner equation.According to the unified strength theory, the equations of plastic fracture zone range and stress of roadway floor surrounding rock are derived under three-directional in situ stresses, the influence of middle principal stress is considered, and the elastoplastic plane strain problem of roadway floor surrounding rock is studied by using the previous single shear strength criterion (such as Mohr–Coulomb criterion and Hook–Brown criterion), so that it is better suited to the practice. But there are the following shortcomings. It still takes the prototype roadway as the research object, and assumes that the surrounding rock is a uniform continuous medium, and does not consider the discontinuity of the surrounding rock and the shear effect of the surrounding rock after entering the plastic state. Many of the formulas obtained are less verified in the field. There is still a certain error in the result.


## 2.1. The Solution of Stress Equation in Plastic Failure Zone of Roadway Floor Surrounding Rock Based on Unified Strength Theory

Assuming that the strain of the plastic zone surrounding rock mass satisfiesεV=0, the relationship among radial stress σrp, tangential stress σθp, and axial stress σzp in the plastic zone can be described by(4)σzp=12σrp+σθp.In the roadway surrounding rock mass around, tangential stressσθp is the largest and radial stress σrp is the smallest; according to the brief description of the theory of uniform strength, we can see that if the three principal stresses in the plastic zone satisfy σ2≥σ1+σ3/2−σ1−σ3/2sinφ, then (5) is established:(5)σθp+eσzp1+e1−sinφ−σrp1+sinφ=2Ccosφ.Substituting (4) into (5):(6)σθp=Kpσrp+Kc,where(7)Kp=2+2sinφ+3esinφ+e2+e1−sinφ,Kc=4C1+ecosφ2+e1−sinφ.In the plastic failure zone, when the roadway surrounding rock is in the ultimate equilibrium stress state, the stress of the plastic zone is basically unrelated with the stress of the original rock. The equivalent floor shape is an axisymmetric roadway, so the stress distribution of the plastic fracture zone of the surrounding rock has an axisymmetric shape. When the body force is not taken into account, the axisymmetric equilibrium equation for the plane strain problem of the plastic zone is expressed as(8)rdσrpdr+σrp−σθp=0,τrθp=0.Combined with (6), it can be rewritten as(9)dσrpσrp+Kc/Kp−1=Kp−1rdr.Using a differential equation, (10) is derived:(10)lnσrp+KcKp−1=Kp−1lnr+Q.According to the boundary conditionr=R0, σrp=Pi (Pi is the support force):(11)Q=1Kp−1lnPiKp−1+Kc−lnR0.Hence, the following is obtained:(12)σrp=Pi+KcKp−1rR0Kp−1−KcKp−1,σθp=KpPi+KcKp−1rR0Kp−1−KcKp−1,σy1p=Kp+12Pi+KcKp−1rR0Kp−1−KcKp−1,τrθp=0.


## 2.2. The Solution of Plastic Failure Zone Range of Roadway Floor Surrounding Rock

The calculation form of the plastic fractured zoneS is as follows:(13)S=rp+fθ.In ther=S position, to facilitate the calculation of the above formula, r−2, r−4, rKp−1, and rKp+1 need to be simplified. Therefore, expanding the polynomial and omitting the all after three items of polynomials, they are expressed as(14)r−2=S−2=rp−2−2fθrp−3,r−4=S−4=rp−4−4fθrp−5,rKp+1=SKp+1=rpKp+1+Kp+1fθrpKp,rKp−1=SKp−1=rpKp−1+Kp−1fθrpKp−2.The stress equations for the plastic fracture zone and the elastic zone of the roadway floor surrounding rock under inhomogeneous stress can be described by (14).The stress equation of the plastic fracture zone is(15)σrp=Pi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,σθp=KpPi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,σyp=Kp+12Pi+KcKp−11R0Kp−1rpKp−1+Kp−1fθrpKp−2−KcKp−1,τrθp=0.The stress equations of the elastic zone is(16)σre=Px+Pz21−arp−2−2fθrp−3+Px−Pz2cos2θ1−2brp−2−2fθrp−3+crp−4−4fθrp−5,σθe=Px+Pz21+arp−2−2fθrp−3−Px−Pz2cos2θ1+crp−4−4fθrp−5,σye=Py−μbPx−Pzcos2θrp−2−2fθrp−3,τrθe=τθre=−Px−Pz2sin2θ1+brp−2−2fθrp−3−crp−4−4fθrp−5.The tangential stress and radial stress equations are analyzed in light of the equality of stress continuous conditions, equal stress, and are represented by the formulas(17)N1R0Kp−1rpKp−1−KcKp−1=Px+Pz21−arp−2,KpN1R0Kp−1rpKp−1−KcKp−1=Px+Pz21+arp−2,(18)N1R0Kp−1Kp−1fθrpKp−2=Px+Pz22fθarp−3+Px−Pz2cos2θ1−2brp−2−2fθrp−3+crp−4−4fθrp−5,KpN1R0Kp−1Kp−1fθrpKp−2=Px+Pz2−2fθarp−3−Px−Pz2cos2θ1+crp−4−4fθrp−5,where N=Pi+Kc/Kp−1.When the boundary position isτrθeS=τθreS=τrθpS=τθrpS=0,(19)H=1+brp−2−2fθrp−3−crp−4−4fθrp−5=0.According to (18), previous formulas are rewritten as(20)N1R0Kp−1Kp−1fθrpKp−2=Px+Pz22fθarp−3+Px−Pz2cos2θ2−H−brp−2−2fθrp−3,KpN1R0Kp−1Kp−1fθrpKp−2=Px+Pz2−2fθarp−3−Px−Pz2cos2θ2−H+brp−2−2fθrp−3.Combining (18)–(20), a new expression is obtained, as follows:(21)Kp−1Px+Pz+2Kc/Kp−1Kp+1Kp−1fθ=−2fθPx+PzPx+PzKp−1+2KcPx+PzKp+1−Px−Pzcos2θ2−2Hrp,fθ=2Pz−Pxcos2θrpPx+Pz+2Kc/Kp−1Kp−1,where rp=R0Px+Pz+2Kc/Kp−1/Kp+1N1/Kp−1, and N, Kp, and Kc are the same as above.The equation of plastic fracture zone rangeS of roadway floor surrounding rock is expressed as (22) according to (13):(22)S=Px+Pz+2Kc/Kp−1Kp+1Pi+Kc/Kp−11/Kp−1R0−2Px−Pzcos2θPx+Pz+2Kc/Kp−1Kp−1Px+Pz+2Kc/Kp−1Kp+1Pi+Kc/Kp−11/Kp−1R0.When the intermediate principal stress is neglected (i.e.,e=0), the equation is consistent with the Rubinite equation; when Px=Pz, the equation is consistent with the Castner equation.According to the unified strength theory, the equations of plastic fracture zone range and stress of roadway floor surrounding rock are derived under three-directional in situ stresses, the influence of middle principal stress is considered, and the elastoplastic plane strain problem of roadway floor surrounding rock is studied by using the previous single shear strength criterion (such as Mohr–Coulomb criterion and Hook–Brown criterion), so that it is better suited to the practice. But there are the following shortcomings. It still takes the prototype roadway as the research object, and assumes that the surrounding rock is a uniform continuous medium, and does not consider the discontinuity of the surrounding rock and the shear effect of the surrounding rock after entering the plastic state. Many of the formulas obtained are less verified in the field. There is still a certain error in the result.


## 3. The Stability Numeral Case Analysis of Roadway Floor Surrounding Rock

Due to the roadway layout and the limitation of its use, it is inevitable that the roadway’s spatial position and the three-dimensional principal stress of the in situ stress form a certain spatial relationship. The three-dimensional principal stress magnitude and orientation of the underground rock strata can be obtained by making use of the geodesic technique as well as the known conditions of the original rock stress. Moreover, the stability of the roadway floor is analyzed according to the different in situ stress environments and the spatial relationship of the roadway layout. Figure2 is the numerical model of the roadway in a random direction.Figure 2
The numerical model of the roadway in a random direction.The following are the assumed initial conditions: the three-dimensional principal stress of the in situ stress areP0x = P0y = P0z = 10 Pa, the roadway rock mass cohesion is C=1MPa, the internal friction angle is φ=30°, the unified strength theory coefficient is e=0, the range of the plastic failure zone of roadway surrounding rock is S, the support resistance in the unsupported state is 0 MPa, and the radius of the equivalent circular roadway is R0=2.5m.When the initial in situ stress satisfiesP0x > P0y = P0z, P0x = 20 MPa and β=0°, the three-dimensional stress state of roadway surrounding rock, the plastic failure zone, and the plastic failure zone boundary stress distribution with the increase of α are as shown in Figure 3.Figure 3
The three-dimensional stress states and change rules of the stress distribution in the surrounding rock plastic zone and plastic zone boundary with the increase ofα. (a) Three-dimensional stress state change, (b) plastic failure zone distribution changes, (c) plastic failure zone boundary radial stress distribution change, (d) plastic failure zone boundary tangential stress distribution change.
(a)
(b)
(c)
(d)It can be seen from Figure3 that the horizontal coordinate of the roadway section and the horizontal stress of the in situ stress with the change of α, and the stress state of the vertical stress Pz do not change. However, a stress state transition between the horizontal stress component Px of the roadway section and the axial horizontal stress component Py is produced, which causes a marked stress redistribution of the roadway floor surrounding rock, and the changes of the floor surrounding rock is obvious in the plasticity. When the initial in situ stress satisfies P0x > P0y = P0z,, the horizontal stress component Px of the roadway section gradually decreases and converts into P0y, and the axial horizontal stress component Py of the roadway gradually increases and converts into P0x with the increase of α. The plastic fracture distribution of the roadway floor is significantly reduced, while the plastic fracture deformation of the both sides of the surrounding rock is slightly increased. It is shown that a reduction of the horizontal stress component Px of the roadway section and an increase of the axial horizontal principal stress component Py of the roadway have the gain effect to the stability of the roadway floor surrounding rock. If α is close to 0°, the horizontal stress component Px of the roadway section is basic approaches the maximum stress component P0x of the in situ stress component; at this time, the plastic failure distribution of the floor rock mass is the largest, and the stress distribution of the surrounding rock is the most severe. Moreover, the most difficult to control is the floor surrounding rock. Therefore, it is concluded that the horizontal stress component of the roadway section should be avoided in accordance with the maximum stress component in the three-dimensional stress state, and the support intensity of the roadway floor should be increased when the initial vertical stress is greater than the maximum vertical stress, so as to guarantee the safety and stability of the roadway surrounding rock.When the initial in situ stress satisfiesP0x > P0y = P0z, P0x = 20 MPa and α=0°, the changes in the three-dimensional stress state of roadway surrounding rock, the plastic failure zone, and the plastic failure zone boundary stress distribution as β increases are as shown in Figure 4.Figure 4
The three-directional stress states and change rules of the stress distribution in the surrounding rock plastic zone and plastic zone boundary with the increase ofβ. (a) Three-dimensional stress state change, (b) plastic failure zone distribution changes, (c) plastic failure zone boundary radial stress distribution change, and (d) plastic failure zone boundary tangential stress distribution change.
(a)
(b)
(c)
(d)It can be seen from Figure4 that the change of the inclination angle β between the roadway’s axis and the horizontal plane and the stress state of the horizontal stress component Px of the roadway section do not change; however, a stress state transition between the roadway vertical stress component Pz and the roadway axial horizontal stress component Py is produced, which causes a marked stress redistribution on both sides of the roadway surrounding rock, and the changes in both sides of the roadway surrounding rock are obvious in the plasticity. When the initial in situ stress satisfies P0z > P0x = P0y, the roadway vertical stress component Pz gradually decreases to convert to P0y, and the roadway axial horizontal stress component Py gradually increases to convert to P0z as β increases. Although the destruction of the floor rock is slightly increased, the distribution of plastic fractures in both sides of the roadway surrounding rock is significantly reduced; the advantage is that the span of the roadway floor surrounding rock is obvious reduced. In addition, as the model span decreases, the amount of load, the transverse stress and shear stress in the beam, and the deflection will be significantly reduced; it is beneficial to the stable bearing of the roadway floor surrounding rock. This result indicates that the decrease of the roadway vertical stress component Pz and the increase of the roadway horizontal stress component Py make the span of the roadway floor surrounding rock to reduce. This phenomenon presents a gain effect to the stability of roadway floor surrounding rock. If β is close to 0°, the vertical stress component Pz of the roadway section basically approaches the maximum stress component P0x of the in situ stress component. Under these conditions, the plasticity failure zone of the floor surrounding rock is slightly reduced, but the span of the roadway floor surrounding rock is increased, which may bring about a drastic change in the stress distribution of floor surrounding rock. Moreover, the floor heave cannot be effectively controlled, and the support structure is unstable due to the increase of the size of the surrounding rock. Therefore, it is also important to pay attention to the three-dimensional stress state of the in situ stress to support the inclined roadway, according to the stress distribution of the surrounding rock and the distribution characteristics of the plastic fracture, and the spatial position relationship of the roadway should be combined to guide the roadway floor heave to ensure the safety and stability of the floor surrounding rock.


## 4. The Mechanical Analysis of Plastic Failure Zone Range and Stress of the #1 Roadway Floor in the S6 Mining Area of Changcun Mine

The test results of the geomechanics and in situ stresses of the roadway surrounding rock in Changcun mine show that the three-dimensional principal stress of the in situ stresses areP0x = 11.31 MPa, P0y = 5.92 MPa, and P0z = 8.36 MPa, the average compressive strength of the surrounding rock mass is 6.77 MPa (cohesion C = 2.2 MPa, internal friction angle φ = 24°), the included angle of the roadway’s horizontal plane is α = 6.1°, and the roadway incidence is β = 6°. Further, when the unified strength theory coefficient is e = 0, destruction of the surrounding rock is the most dangerous. Supposing that the radius of the equivalent circular roadway is R0 = 3.05 m, the plastic failure zone range S and the stress values of the floor surrounding rock of the #1 return airway in the S6 mining area of Changcun mine are calculated according to the previous mechanical analysis.


### 4.1. The Primary Rock Stress Calculation of Floor Surrounding Rock

(23)
P
x
P
x
y
P
z
z
P
y
x
P
y
P
y
z
P
z
x
P
z
y
P
z
=
11.31
cos
2
6.1
°
+
5.92
sin
2
6.1
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
cos
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
sin
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
cos
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
cos
2
6
°
+
8.63
sin
2
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
−
8.63
cos
6
°
sin
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
sin
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
−
8.63
cos
6
°
sin
6
°
13.19
sin
2
6.1
°
+
5.92
cos
2
6.1
°
sin
2
6
°
+
8.63
cos
2
6
°
.By calculating, theprimary rock stresses are(24)Px=11.25MPa,Pv=6.01MPa,Pz=8.60MPa.


### 4.2. The Solutions of Plastic Failure Zone RangeS and Stress Value of Floor Surrounding Rock

Here,(25)Kc=2×2.2cos24°1−sin24°=6.77,Kp=1+sin24°1−sin24°=2.37,N=6.772.37−1=4.94,rp=3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1=4.65m.Furthermore,(26)S=3.05×11.25+8.06+2×6.77/2.37−12.37+1×4.941/2.37−1−2×11.25−8.60cos2θ11.25+8.60+26.77/2.37−12.37−1×3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1,S=3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1−2×11.25−8.60cos2θ11.25+8.60+2×6.77/2.37−12.37−1×3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1.σrp=6.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1,σθp=2.376.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1,σyp=2.37+126.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1.Whenθ=270°, it can be calculated that the radius of the plastic fracture zone range is S = 5.26 m, the floor failure depth is 3.51 m (the distance from the center of the equivalent circular roadway to the actual floor is 1.75 m), the boundary radial stress peak value in the floor plastic failure zone is σrp=5.02MPa, the boundary shear stress peak value in the floor plastic failure zone is σθp=18.65MPa, and the boundary axial stress peak value in the floor plastic failure zone is σyp=11.86MPa.


## 4.1. The Primary Rock Stress Calculation of Floor Surrounding Rock

(23)
P
x
P
x
y
P
z
z
P
y
x
P
y
P
y
z
P
z
x
P
z
y
P
z
=
11.31
cos
2
6.1
°
+
5.92
sin
2
6.1
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
cos
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
sin
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
cos
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
cos
2
6
°
+
8.63
sin
2
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
−
8.63
cos
6
°
sin
6
°
11.31
−
5.92
cos
6.1
°
sin
6.1
°
sin
6
°
11.31
sin
2
6.1
°
+
5.92
cos
2
6.1
°
−
8.63
cos
6
°
sin
6
°
13.19
sin
2
6.1
°
+
5.92
cos
2
6.1
°
sin
2
6
°
+
8.63
cos
2
6
°
.By calculating, theprimary rock stresses are(24)Px=11.25MPa,Pv=6.01MPa,Pz=8.60MPa.


## 4.2. The Solutions of Plastic Failure Zone RangeS and Stress Value of Floor Surrounding Rock

Here,(25)Kc=2×2.2cos24°1−sin24°=6.77,Kp=1+sin24°1−sin24°=2.37,N=6.772.37−1=4.94,rp=3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1=4.65m.Furthermore,(26)S=3.05×11.25+8.06+2×6.77/2.37−12.37+1×4.941/2.37−1−2×11.25−8.60cos2θ11.25+8.60+26.77/2.37−12.37−1×3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1,S=3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1−2×11.25−8.60cos2θ11.25+8.60+2×6.77/2.37−12.37−1×3.05×11.25+8.60+2×6.77/2.37−12.37+1×4.941/2.37−1.σrp=6.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1,σθp=2.376.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1,σyp=2.37+126.772.37−113.052.37−14.652.37−1+2.37−1×4.652.37−2fθ−6.772.37−1.Whenθ=270°, it can be calculated that the radius of the plastic fracture zone range is S = 5.26 m, the floor failure depth is 3.51 m (the distance from the center of the equivalent circular roadway to the actual floor is 1.75 m), the boundary radial stress peak value in the floor plastic failure zone is σrp=5.02MPa, the boundary shear stress peak value in the floor plastic failure zone is σθp=18.65MPa, and the boundary axial stress peak value in the floor plastic failure zone is σyp=11.86MPa.


## 5. Conclusions

(1)
This paper takes the excavation roadway overall surrounding rock as the research object and uses the generalized plane strain problem and uniform strength theory to derive for the equations of the elastic-plastic zone stress together with the plastic fracture range of the roadway floor surrounding rock under three-directional in situ stresses.
(2)
From the analysis of the influence of the stress environment to the roadway floor stability, it is concluded that the horizontal stress component of the roadway section should be avoided in accordance with the maximum stress component in the three-dimensional stress state, and the support strength of the roadway floor should be increased when the initial vertical stress is greater than the maximum vertical stress, so as to ensure the safety and stability of the roadway floor surrounding rock. It is also important to pay attention to the three-dimensional stress state of the in situ stress to support the inclined roadway, according to the stress distribution of the surrounding rock and the distribution characteristics of the plastic fracture, and the spatial position relationship of the roadway should be combined to guide the roadway floor heave to ensure the safety and stability of the floor surrounding rock.
(3)
From the mechanical analysis, the following conclusions are obtained: the range of the plastic failure zone range of roadway floor rock mass isS = 5.26 m, the depth of the floor plastic failure is 3.51 m, the boundary radial stress peak value in the floor plastic failure zone is σrp=5.02MPa, the boundary shear stress peak value in the floor plastic failure zone is σθp=18.65MPa, and the boundary axial stress peak value in the floor plastic failure zone is σyp=11.86MPa. The above information provides a reliable mechanical reference for research on prevention and control technology of the roadway bottom drum.



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*Source: 7475698-2018-07-26.xml*