# Application of Homogenized Load-Bearing Ring Hypothesis in Roadway Supporting

Authors: Quntao Zhang; Jianwei Zheng; Xiaodong Sun; Wenzhou Li; Kang Yi; Yukai Fu; Jiulin Shi; Shuai Wang; Biao Liu; Qi Wang
Journal: Shock and Vibration
 (2022)

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## Abstract

Artificial support system takes advantages of self-bearing capacity of surrounding rock mass in roadway support, which is one of the important supporting ideas in modern times. The implementation way of an artificial support system is worthy of deep investigation. In this study, an HLBR hypothesis was proposed with comprehensive consideration of mechanical properties of surrounding rock mass based on the reinforced arch theory and axial variation theory. A model of bearing structure in surrounding rock mass was constructed, and the stress state on the bearing structure was analyzed. It was believed that when the axial ratio (k) of this bearing structure is equal to the lateral stress factor (λ) (which is the optimal axial ratio), the tangential stress on the bearing boundary reaches the minimum and is in uniform distribution. At this moment, the roadway and the supporting structure can make good use of the self-carrying capacity of surrounding rock mass, which is conducive to lowering the stress distribution level of surrounding rock mass. According to prestress diffusion characteristics of bolt, some ways to achieve HLBR were proposed, which were the optimal axial ratio design, difference support and local grouting modification, and finally performance of the original support design in a belt roadway of the mining district IV of a mine in Shanxi Province. A new supporting scheme was provided according to the HLBR hypothesis. In addition, the FLAC3D strain-softening model was modified by C++, which was used to interpret the reasonability of the suggested supporting scheme. The proposed HLBR hypothesis discloses ways to use self-capacity of surrounding rock mass. This study provides a new idea for roadway sectional design and roadway support design, which are beneficial to maintaining the stability of surrounding rock mass in roadways.


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## Body

## 1. Introduction

Stability control of surrounding rock mass in a roadway has been one of the key problems against safety recycling of underground coal resources in China. With the increase of mining depth, geological environment and stress environment where the roadway locates in become more and more complicated, accompanied by increasing difficulties in corresponding roadway support and serious threats to operation safety of underground workers [1, 2]. Based on theoretical analysis, laboratory analysis, and field studies, many scholars have investigated stability of surrounding rock mass from the perspectives of action of stress and strengthening of surrounding rock mass. The action of stress is to propose the corresponding control theories by analyzing stress that induces roadway deformation, such as the classical circular roadway based on elastic theory [3] and stress analysis of oval roadway [4–8]. The strengthening of surrounding rock mass means increasing the bearing capacity of surrounding rock mass through artificial measures, which covers classical collapse arch theory [9, 10] and surrounding rock mass strengthening theory based on self-stability ability of surrounding rock mass as well as the collaborative action of artificial support and surrounding rock mass [11–14]. These theories solve the support problem of some roadway to some extent. Among the theories, the energy supporting theory [15] analyzes the action mechanism of support from the perspective of energy equilibrium of surrounding rock mass. The collapse arch theory [16] emphasizes the self-stability of surrounding rock mass. The hanging theory [17] belongs to the scope of the passive support mechanism of bolt. The reinforced arch theory [18], combined with arch theory and surrounding rock mass theory, can be viewed as strengthening physical mechanical properties of surrounding rock mass and forming a good bearing structure. Yang et al. believed that existing supporting technologies can be mainly divided into three types [19], including positive internal support of surrounding rock mass, passive surface support of roadway [20], and modified support [21, 22]. Gao et al. [23] analyzed the coal rock mass fractal evolution caused by mining at different depths and provided a foundation for roadway supporting. Kang et al. [24] proposed the concept of support stress field and emphasized the collaboration mechanism among supporting stress field, mining-induced stress, and primary rock stress field. They believed that the application of prestressed bolt and cable can make surrounding rock mass on roadway surface recover to three-direction stress states to some extent and increase the bearing capacity of surrounding rocks. The axial variation theory was proposed by Yu [25, 26], and the self-stabilization balance ring theory was proposed by Huang et al. [27]. The loosing ring theory [28, 29] believed that the artificial supporting object was the bulking force during the formation of loosing ring of surrounding rock mass. Zhang et al. [30] carried out a simulation analysis of four different strengthened U-shaped steel structures and found that the U-shaped steel structure with beams and longitudinal beams can offer good support to uniformly distributed loads of the roadway.Existing roadway supporting theories and corresponding supporting methods [31–36] lay solid foundations for stability control of surrounding rock mass. With the increase of research depth, science researchers emphasize the self-bearing effect of surrounding rock mass, but relevant deep studies on ways to achieve self-bearing effect of surrounding rock mass are needed. Based on the reinforced arch and axial variation theory, this study proposed the homogenized load-bearing ring (HLBR) hypothesis with comprehensive consideration of mechanical properties of surrounding rock mass and introduced ways to achieve self-bearing capacity of surrounding rock mass. Subsequently, some optimization suggestions for the supporting design of the belt roadway of mining district IV of a mine in Shanxi Province were proposed according to the HLBR hypothesis. Moreover, the new supporting design was verified through a numerical simulation method.


## 2. Concept of the HLBR

The space with certain functions which is excavated in strata is called roadway (chamber). Excavation of roadway breaks the equilibrium state of the primary rock stress field within a certain range and induces redistributions of stress in surrounding rock mass, thus forming stress gradients. The roadway may develop deformation failure when the secondary stress in local positions of a roadway edge exceeds the ultimate strength of surrounding rock mass. Without external support, the surrounding rock mass is destroyed and separated from the matrix. Geometric morphology of roadway section is changed, thus changing the distribution of stress field. Roadway failure stops when the stress at surrounding rock mass boundary is lower than the ultimate strength of surrounding rock mass in the roadway. The mutual feedback process between geometrical morphology and boundary stress of roadway section is called self-stabilization (self-organized equilibrium) of surrounding rock mass in roadways. Based on abundant field observation, the final geometric morphology when surrounding rock mass reaches self-stabilization after failure can be abstracted into an “ellipse” with a certain axial ratio (the best axial ratio) or “similar ellipse.” In other words, the (similar) elliptical section with the best axial ratio has good self-bearing (self-stabilization) performances under specific stress field conditions. Therefore, if the roadway is designed into an elliptical section with the best axial ratio, the stress field distribution of surrounding rock mass is optimized in favor of stability control of surrounding rock mass. Theoretically, it can develop the self-bearing (self-stability) capacity of surrounding rock mass well. However, practical design of roadway section shall integrate various influencing factors, such as construction technological level, maximum equipment size in roadway, and usage as well as service life of roadways. Geometric morphology of roadway section must be compromised to some extent, thus resulting in the abovementioned uneven stress gradients.During roadway support, the supporting stress which is applied to surrounding rock mass artificially will be diffused in surrounding rock mass and form a load-bearing structure in the roadway. Surrounding rock mass in this load-bearing structure is in the three-direction stress states, and it can bear relatively high external loads. With consideration of the physical and mechanical properties of surrounding rock mass, this study proposed the HLBR hypothesis of surrounding rock mass in the roadway based on reinforced arch theory and axial variation theory. The load-bearing ring refers to a cyclic bearing structure in the artificial supporting system that undertakes external loads formed in surrounding rock mass. This load-bearing structure has a large bearing capacity, and its range and bearing characteristics are influenced by stress environment, roadway section, and artificial support. When the axial ratio of load-bearing ring is close to the best axial ratio of the roadway in the local geological and stress field environment, the bearing capacity of surrounding rock mass can be fully used. Under this circumstance, the load-bearing ring formed in surrounding rock mass is called an HLBR. Moreover, the goal of reasonable support to roadway is to promote the HLBR with the optimal axial ratio in the surrounding rock mass to bear most external loads. The HLBR is composed of ultimate external bearing boundary and ultimate internal bearing boundary. The ultimate external bearing boundary refers to the maximum inscribed ellipse envelope curve that connects ends of bolts, while the ultimate internal bearing boundary refers to the minimum ellipse envelope curve, which is connected to the roadway section. The equivalent bearing section refers to the curved surface formed by the centerline of the HLBR profile. The real roadway section is called an internal section. Distributed zone of prestress bolt refers to the range in the rock mass, generated by prestress bolt. Superposed stress field of prestress bolt refers to the superimposed part between stress zone generated by different bolts. Bare roadway stabilization boundary refers to the boundary of roadway which can maintain stability without artificial supporting. The HLBR model of surrounding rock mass in the roadway is shown in Figure1.Figure 1
Model of HLBR.


## 3. Mechanical Analysis of the HLBR

### 3.1. Description of Mechanical Model of the HLBR

A model of roadway surrounding rock mass was constructed based on elastic mechanics. Necessary and sufficient conditions to form the HLBR in roadway surrounding rock mass and the mechanical behaviors of this ring under external loads were analyzed. Bearing boundary conditions of surrounding rock mass were simplified. Moreover, five basic hypotheses which conform to elastic mechanics were proposed for surrounding rock mass. Stress conditions of the load-bearing ring of roadway surrounding rock mass are shown in Figure2. In Figure 2, 2ao and 2bo are long axis and short axis of the external bearing boundary, while 2ai and 2bi are long axis and short axis of the internal bearing boundary. The elliptical load-bearing ring was simplified, and it was equivalent to the analysis of mechanical behaviors of equivalent ellipse (equivalent section) under two-way loading conditions. The intersection points between the equivalent section in Figure 2 and the X-axis are X and X`, whereas the intersection points between the equivalent section and the Y-axis are Y and Y`.Figure 2
Mechanical model of simplified load-bearing ring in roadway surrounding rock mass.


### 3.2. Deduction of Mechanical Model of the HLBR

Stress expression at different points of the equivalent section (equivalent ellipse) boundary, which is formed in roadway surrounding rock mass under two-way loads, was set up by using a function of complex variables [37]:(1)σθξ0=sinh2ξ01+e−2ξ0cosh2ξ0−cos2θ−1e2ξ0Ph+sinh2ξ0e−2ξ0−1cosh2ξ0−cos2θ+1e2ξ0Pv,where ξ0 is the distance from the center of equivalent ellipse to boundaries. θ is the direction angle between the research points and long axis of equivalent ellipse. σθ(ζ0) is the tangential stress on any point of boundary of equivalent ellipse. Pv is the vertical stress that the equivalent ellipse bears. Ph is the horizontal principal stress that the equivalent ellipse bears.The auxiliary equations are shown in equations (2)–(7), where 2a is the long axis of equivalent ellipse, 2b is the short axis of equivalent ellipse, and 2c is the focal length of equivalent ellipse. Besides, there is c2 = a2-b2:(2)coshξ0=ac,(3)sinhξ0=bc,(4)sinh2ξ0=2coshξ0sinhξ0=2abc2=e2ξ0−e−2ξ02,(5)cosh2ξ0=coshξ02+sinhξ02=a2+b2c2=e2ξ0+e−2ξ02,(6)e2ξ0=cosh2ξ0+sinh2ξ0=a+b2c2=a+ba−b,(7)e−2ξ0=cosh2ξ0−sinh2ξ0=a−b2c2=a−ba+b.Leta = kb, where k refers to the axial ratio of stress ellipse (k > 0 and k≠1). Let Ph = λPv, where λ is a lateral stress factor (λ > 0). Equation (1) can be rewritten based on equations (2)–(7), thus getting the expression (8) of equivalent ellipse boundary stress. Moreover, σθ can be viewed as a function of θ(θ∈[0, 2π]):(8)σθξ0=2kPvk−λk2sin2θ+cos2k−1+Pvk+1λ−1k−1.For (8), derivatives of θ were calculated at two sides of the equation, thus getting the expression of maximums:(9)∂σθξ0∂θ=2kPvλ−kk−1×1−k2sin2θk2sin2θ+cos2θ2.It can be seen from (9) that maximums can be calculated when θ = 0 and θ = π/2. According to symmetry, the maximums can be gained when θ = π and θ = 3π/2. In other words, the intersection points between stress ellipse and coordinate axis (X, X`, Y, and Y′) are maximum locations. Therefore, θ = 0, θ = π, θ = π/2, and θ = 3π/2 were brought into equation (8), thus getting stress maximums of the stress ellipse on the horizontal and vertical axes (σθ(ξ0) (-) and σθ(ξ0) (⊥)) as follows:(10)σθξ=ξ0−=2k+1−λPv,σθξ=ξ0⊥=2λ+kλ−kkPv.


### 3.3. Mechanical Analysis of the HLBR

Types of tangential stress and maximum location on the equivalent ellipse under different relationships between axial ratio of equivalent ellipse and lateral stress factor are shown in Table1.Table 1
Types of tangential stress and maximum location on the equivalent ellipse.
Function relationship betweenλ and kX-axisY-axisPosition of extremumType of tress inX pointType of tress inX′ pointType of tress in Y pointType of tress in Y′ pointk < λλ > 2k + 1Tensile stressTensile stressCompressive stressCompressive stressYλ = 2k + 100Compressive stressCompressive stressYk < λ < 2k + 1Compressive stressCompressive stressCompressive stressCompressive stressYk = λk = λCompressive stressCompressive stressCompressive stressCompressive stressX = Yλ< kk/(2 + k) < λ < kCompressive stressCompressive stressCompressive stressCompressive stressXλ = k/(2 + k)Compressive stressCompressive stress00Xλ < k/(2 + k)Compressive stressCompressive stressTensile stressTensile stressXBased on the determined types of tangential stress and maximum locations on equivalent ellipse, tensile tress and zero stress can only be developed onX and X` when λ ≥ 1. Similarly, tensile stress and zero stress are developed on Y and Y′ only when λ ≤ 1. The relation (qualitative trend description) between changes of lateral stress factor and changes of tangential stress factor on boundary is shown in Figure 3. The distribution characteristics of tangential stress concentration factor on the ellipse boundary with a fixed constant axial ratio under different lateral stress factors are shown in Figure 3(a). Aiming to make the rules clear, the axial ratio and lateral stress factors suppose that k = 3 and λ = 0.2, 0.4, 0.6, 1, 2, 3, 4, 5, 7, 8, and 10. The distribution characteristics of tangential stress concentration factor on the ellipse boundary under a fixed constant lateral factor and different axial ratios when λ < 1 are shown in Figure 3(b), in which it supposes that λ = 0.5 and k = 0.3, 0.4, 0.5, 0.8, 1.5, 2, 3, and 4. The distribution characteristics of tangential stress concentration factor on the ellipse boundary under a fixed constant lateral factor and different axial ratios when λ > 1 are shown in Figure 3(c), in which λ = 5 and k = 0.9, 1.5, 2, 3, 4, 5, 7, and 8.Figure 3
Distribution characteristics of tangential stress concentration factor on stress elliptic boundary under different conditions. (a) k = 3, (b)λ = 0.5, and (c) λ = 5.
(a)(b)(c)Given a fixed axial ratio in Figure3(a), surrounding rock mass on roof and floor of roadway changed from tensile failure to compressive failure with the increase of lateral stress coefficient, while the surrounding rock mass on two sides of the roadway changed from compressive failure to tensile failure. Given a fixed lateral stress factor, the failure mode of surrounding rock mass on roof and floor of roadway with the increase of axial ratio shall be determined according to specific situations. It can be seen from Figure 3(b) that surrounding rock mass on roof and floor of roadway changes from compressive failure to tensile failure, but two sides of the roadway have been bearing compressive stress, without tensile stress. With the increase of k, the compressive stress on two sides of roadway increases. It can be seen from Figure 3(c) that surrounding rock mass on two sides of roadway changes from tensile failure to compressive failure. The surrounding rock mass on the roof and floor has been under the compressive stress. Meanwhile, the compressive stress decreases with the increase of k.Tensile strength of surrounding rock mass is far smaller than the compressive strength. LetRt be the tensile strength of surrounding rock mass in a roadway. In this section, only longitudinal axis is analyzed. Six stress states at Y and Y` can be gained by analyzing the relationship between σθ(ξ0) (⊥) and tensile strength of surrounding rock mass (Rt). These six stress states are strong tensile stress state (I), ultimate tensile stress state (II), weak tensile stress state (III), zero-stress state (IV), compressive stress state (V), and equal stress state (). When σθ(ξ0) (⊥) at Y and Y′ is higher than the tensile strength (Rt) of surrounding rock mass (k<2λ/Rt/Pv−λ+1), Y and Y′ develop tensile failure, which can be called the strong tensile stress state (I). When k=2λ/Rt/Pv−λ+1, Y and Y′ are at the ultimate tensile stress state (II). When σθ(ξ0) (⊥)> Rt (k<2λ/Rt/Pv−λ+1), the vertical axis end develops no tensile failure, which can be viewed as a weak tensile stress state (III). When σθ(ξ0) (⊥) = 0 (k=2λ/1−λ), Y and Y′ are at the zero-stress state (IV). When σθ(ξ0) (⊥) is compressive stress (k<2λ/1−λ), Y and Y′ are at the compressive stress state (V). When σθ(ξ0) (⊥) is a compressive stress and it is equal to σθ(ξ0) (-) (k = λ), all points on stress ellipse boundary have equal stress and are all in a compressive stress state, which is called the equal stress state (VI).The above 6 stress states of Y and Y′ were analyzed comprehensively. Stress environmental optimization of roadway surrounding rock mass shall focus on reduction of stress level and prevention of tensile stress based on strong compressive strength and poor tensile strength of surrounding rock mass. Therefore, equal stress state (VI) is the ideal stress state of load-bearing structure. Hence, it can be verified theoretically that stress on load-bearing ring boundary is minimum and in uniform distribution when k = λ, thus enabling us to realize the structure of HLBR.


## 3.1. Description of Mechanical Model of the HLBR

A model of roadway surrounding rock mass was constructed based on elastic mechanics. Necessary and sufficient conditions to form the HLBR in roadway surrounding rock mass and the mechanical behaviors of this ring under external loads were analyzed. Bearing boundary conditions of surrounding rock mass were simplified. Moreover, five basic hypotheses which conform to elastic mechanics were proposed for surrounding rock mass. Stress conditions of the load-bearing ring of roadway surrounding rock mass are shown in Figure2. In Figure 2, 2ao and 2bo are long axis and short axis of the external bearing boundary, while 2ai and 2bi are long axis and short axis of the internal bearing boundary. The elliptical load-bearing ring was simplified, and it was equivalent to the analysis of mechanical behaviors of equivalent ellipse (equivalent section) under two-way loading conditions. The intersection points between the equivalent section in Figure 2 and the X-axis are X and X`, whereas the intersection points between the equivalent section and the Y-axis are Y and Y`.Figure 2
Mechanical model of simplified load-bearing ring in roadway surrounding rock mass.


## 3.2. Deduction of Mechanical Model of the HLBR

Stress expression at different points of the equivalent section (equivalent ellipse) boundary, which is formed in roadway surrounding rock mass under two-way loads, was set up by using a function of complex variables [37]:(1)σθξ0=sinh2ξ01+e−2ξ0cosh2ξ0−cos2θ−1e2ξ0Ph+sinh2ξ0e−2ξ0−1cosh2ξ0−cos2θ+1e2ξ0Pv,where ξ0 is the distance from the center of equivalent ellipse to boundaries. θ is the direction angle between the research points and long axis of equivalent ellipse. σθ(ζ0) is the tangential stress on any point of boundary of equivalent ellipse. Pv is the vertical stress that the equivalent ellipse bears. Ph is the horizontal principal stress that the equivalent ellipse bears.The auxiliary equations are shown in equations (2)–(7), where 2a is the long axis of equivalent ellipse, 2b is the short axis of equivalent ellipse, and 2c is the focal length of equivalent ellipse. Besides, there is c2 = a2-b2:(2)coshξ0=ac,(3)sinhξ0=bc,(4)sinh2ξ0=2coshξ0sinhξ0=2abc2=e2ξ0−e−2ξ02,(5)cosh2ξ0=coshξ02+sinhξ02=a2+b2c2=e2ξ0+e−2ξ02,(6)e2ξ0=cosh2ξ0+sinh2ξ0=a+b2c2=a+ba−b,(7)e−2ξ0=cosh2ξ0−sinh2ξ0=a−b2c2=a−ba+b.Leta = kb, where k refers to the axial ratio of stress ellipse (k > 0 and k≠1). Let Ph = λPv, where λ is a lateral stress factor (λ > 0). Equation (1) can be rewritten based on equations (2)–(7), thus getting the expression (8) of equivalent ellipse boundary stress. Moreover, σθ can be viewed as a function of θ(θ∈[0, 2π]):(8)σθξ0=2kPvk−λk2sin2θ+cos2k−1+Pvk+1λ−1k−1.For (8), derivatives of θ were calculated at two sides of the equation, thus getting the expression of maximums:(9)∂σθξ0∂θ=2kPvλ−kk−1×1−k2sin2θk2sin2θ+cos2θ2.It can be seen from (9) that maximums can be calculated when θ = 0 and θ = π/2. According to symmetry, the maximums can be gained when θ = π and θ = 3π/2. In other words, the intersection points between stress ellipse and coordinate axis (X, X`, Y, and Y′) are maximum locations. Therefore, θ = 0, θ = π, θ = π/2, and θ = 3π/2 were brought into equation (8), thus getting stress maximums of the stress ellipse on the horizontal and vertical axes (σθ(ξ0) (-) and σθ(ξ0) (⊥)) as follows:(10)σθξ=ξ0−=2k+1−λPv,σθξ=ξ0⊥=2λ+kλ−kkPv.


## 3.3. Mechanical Analysis of the HLBR

Types of tangential stress and maximum location on the equivalent ellipse under different relationships between axial ratio of equivalent ellipse and lateral stress factor are shown in Table1.Table 1
Types of tangential stress and maximum location on the equivalent ellipse.
Function relationship betweenλ and kX-axisY-axisPosition of extremumType of tress inX pointType of tress inX′ pointType of tress in Y pointType of tress in Y′ pointk < λλ > 2k + 1Tensile stressTensile stressCompressive stressCompressive stressYλ = 2k + 100Compressive stressCompressive stressYk < λ < 2k + 1Compressive stressCompressive stressCompressive stressCompressive stressYk = λk = λCompressive stressCompressive stressCompressive stressCompressive stressX = Yλ< kk/(2 + k) < λ < kCompressive stressCompressive stressCompressive stressCompressive stressXλ = k/(2 + k)Compressive stressCompressive stress00Xλ < k/(2 + k)Compressive stressCompressive stressTensile stressTensile stressXBased on the determined types of tangential stress and maximum locations on equivalent ellipse, tensile tress and zero stress can only be developed onX and X` when λ ≥ 1. Similarly, tensile stress and zero stress are developed on Y and Y′ only when λ ≤ 1. The relation (qualitative trend description) between changes of lateral stress factor and changes of tangential stress factor on boundary is shown in Figure 3. The distribution characteristics of tangential stress concentration factor on the ellipse boundary with a fixed constant axial ratio under different lateral stress factors are shown in Figure 3(a). Aiming to make the rules clear, the axial ratio and lateral stress factors suppose that k = 3 and λ = 0.2, 0.4, 0.6, 1, 2, 3, 4, 5, 7, 8, and 10. The distribution characteristics of tangential stress concentration factor on the ellipse boundary under a fixed constant lateral factor and different axial ratios when λ < 1 are shown in Figure 3(b), in which it supposes that λ = 0.5 and k = 0.3, 0.4, 0.5, 0.8, 1.5, 2, 3, and 4. The distribution characteristics of tangential stress concentration factor on the ellipse boundary under a fixed constant lateral factor and different axial ratios when λ > 1 are shown in Figure 3(c), in which λ = 5 and k = 0.9, 1.5, 2, 3, 4, 5, 7, and 8.Figure 3
Distribution characteristics of tangential stress concentration factor on stress elliptic boundary under different conditions. (a) k = 3, (b)λ = 0.5, and (c) λ = 5.
(a)(b)(c)Given a fixed axial ratio in Figure3(a), surrounding rock mass on roof and floor of roadway changed from tensile failure to compressive failure with the increase of lateral stress coefficient, while the surrounding rock mass on two sides of the roadway changed from compressive failure to tensile failure. Given a fixed lateral stress factor, the failure mode of surrounding rock mass on roof and floor of roadway with the increase of axial ratio shall be determined according to specific situations. It can be seen from Figure 3(b) that surrounding rock mass on roof and floor of roadway changes from compressive failure to tensile failure, but two sides of the roadway have been bearing compressive stress, without tensile stress. With the increase of k, the compressive stress on two sides of roadway increases. It can be seen from Figure 3(c) that surrounding rock mass on two sides of roadway changes from tensile failure to compressive failure. The surrounding rock mass on the roof and floor has been under the compressive stress. Meanwhile, the compressive stress decreases with the increase of k.Tensile strength of surrounding rock mass is far smaller than the compressive strength. LetRt be the tensile strength of surrounding rock mass in a roadway. In this section, only longitudinal axis is analyzed. Six stress states at Y and Y` can be gained by analyzing the relationship between σθ(ξ0) (⊥) and tensile strength of surrounding rock mass (Rt). These six stress states are strong tensile stress state (I), ultimate tensile stress state (II), weak tensile stress state (III), zero-stress state (IV), compressive stress state (V), and equal stress state (). When σθ(ξ0) (⊥) at Y and Y′ is higher than the tensile strength (Rt) of surrounding rock mass (k<2λ/Rt/Pv−λ+1), Y and Y′ develop tensile failure, which can be called the strong tensile stress state (I). When k=2λ/Rt/Pv−λ+1, Y and Y′ are at the ultimate tensile stress state (II). When σθ(ξ0) (⊥)> Rt (k<2λ/Rt/Pv−λ+1), the vertical axis end develops no tensile failure, which can be viewed as a weak tensile stress state (III). When σθ(ξ0) (⊥) = 0 (k=2λ/1−λ), Y and Y′ are at the zero-stress state (IV). When σθ(ξ0) (⊥) is compressive stress (k<2λ/1−λ), Y and Y′ are at the compressive stress state (V). When σθ(ξ0) (⊥) is a compressive stress and it is equal to σθ(ξ0) (-) (k = λ), all points on stress ellipse boundary have equal stress and are all in a compressive stress state, which is called the equal stress state (VI).The above 6 stress states of Y and Y′ were analyzed comprehensively. Stress environmental optimization of roadway surrounding rock mass shall focus on reduction of stress level and prevention of tensile stress based on strong compressive strength and poor tensile strength of surrounding rock mass. Therefore, equal stress state (VI) is the ideal stress state of load-bearing structure. Hence, it can be verified theoretically that stress on load-bearing ring boundary is minimum and in uniform distribution when k = λ, thus enabling us to realize the structure of HLBR.


## 4. Applications and Numerical Simulation Analysis of the HLBR

To elaborate the role of HLBR in supporting structure, a case study based on the supporting design of a belt roadway of the mining district IV of a mine in Shanxi Province was carried out. A numerical simulation analysis of this belt roadway was performed by FLAC3D, and the design scheme of the primary roadway supporting structure to surrounding rock mass was evaluated. A new supporting design scheme was proposed based on the HLBR hypothesis. Its performance was compared with that of the primary support design.


### 4.1. Model Background and Primary Support Design

The advancing section in the study area was 5.30 m wide and 3.55 m high. Stability of roadway surrounding rock mass was maintained by a compound supporting scheme of “bolt + cable + metal mesh + steel ladder beam.” The row space of bolts on roof (MSGLW, 335/22 × 2400) is 800 mm × 800 mm, and the row space of bolts on two sides (MSGLW, 335/22 × 2400) is 1000 mm × 800 mm. Cable (SKP, 18.9 × 8300) was in three-flower distribution, and metal mesh used the graticules (50 mm × 50 mm) made of Φ10 iron wires. The steel ladder beam (Φ14 mm rebar) was 60 mm wide, and it was covered on the roof and two sides of the roadway. However, no prestress was applied to the bolt and cable in the supporting design. Instead, they were installed by end anchoring mode. Prestress on the bolt and cable was measured 5–10 kN and 20–30 kN, respectively. Low-strength ordinary supporting plate was applied to the bolt and cable.


### 4.2. Research on Roadway Geostress

A geostress measurement system was installed in the study area [38]. Measuring positions and measuring results are shown in Figure 4. According to measurement results, the maximum and minimum horizontal principal stress (σm and σl) are 9.04 MPa and 5.22 MPa, and the vertical stress (σv) is 8.40 MPa. Besides, the included angle between the maximum horizontal principal stress and belt roadway in mining district IV is NE20°. The coordinate transformation is shown in equations (11) and (12) [37], where σm is the maximum horizontal principal stress under the old coordinate system; σl is the minimum principal stress under the old coordinate system; σml is the tangential stress under the old coordinate system, and there is σml = 0; σx is the maximum horizontal principal stress under the new coordinate system; σy is the minimum principal stress under the new coordinate system; σy is the minimum principal stress under the new coordinate system; and θ is the deflection angle of the coordinate system. Measurement results of geostress were brought into equations (11) and (12), and stresses on two sides of the roadway were gained: σx = 8.59 Mpa and σy = 5.67 MPa. Therefore, the lateral stress factor in the roadway was λ = σx/σv = 1.02 and k = 1.49:(11)σx=12σm+σl+12σm−σlcos2θ+σmlsin2θ,(12)σy=12σm+σl−12σm−σlcos2θ−σmlsin2θ.Figure 4
Measurement location and maximum horizontal stress direction of geostress in the study area.The primary support design in the study area applied the low-level prestress on bolts and cable. The primary support design was analyzed based on the HLBR hypothesis. It found that the primary support design failed to form effective positive support to load-bearing structure in surrounding rock mass, and it was only passive support. Hence, the axial ratio of bearing structure in the study area is equal to the axial ratio in the primary sectional design (k = 1.49) of the roadway. Moreover, the axial ratio of final self-stabilization boundaries was k = λ = 1.02 under the influences of stress redistribution. Therefore, depth of the plastic region on roof and floor during the self-stabilization of belt roadway was higher than the development depth of the plastic region on two sides of the roadway.


### 4.3. Optical Borehole Observation in Roadway Surrounding Rock Mass

Optical borehole observation was implemented in the belt roadway of mining district IV. Measurement results are shown in Figure5. According to analysis, they were developed with fractured sandy mudstone within 0–1.3 m of roadway roof, medium sandstone within 1.3–4.5 m of roadway, fractures within 1.3–1.5 m of roadway, and sandy mudstone with high integrity within 4.5–7.0 m of roadway. Coal fractures since 0–0.7 m of roadway surface on sides of roadway are loose. Roadway deformation was analyzed by considering that rock mass is a homogeneous elastic material. According to the symmetric principle, failure depths of roadway roof and floor were 1.5 m, and the failure depth on two sides was 0.7 m after the roadway was drilled. The real axial ratio of roadway after the development of plastic region in the surrounding rock mass was kn = (2.65 + 0.7)/(1.75 + 1.5) = 1.03, indicating a certain degree of self-stabilization of roadway under loads. The stress distribution on roadway surrounding rock mass was optimized, during which the artificial supporting system developed limited effect. The actual deformation features of the roadway basically conform to the above theoretical analysis.Figure 5
Borehole optical observation in roof and coal seam side.


### 4.4. Simulation on Stress Evolution of the HLBR

A numerical simulation analysis of the study area was carried out by using FLAC3D. Width, axial length, and vertical height of the model were set at 50 m, 30 m, and 47 m, respectively. The roof strata within 23 m above the coal seam were simulated, and the floor 20 m below the coal seam was rock strata. The simulation model is shown in Figure6. The distance from the boundaries of the model to the excavation boundary exceeds 5 times of the maximum radius of the roadway. Lithological and physical mechanical parameters of rock mass on roof and floor of the roadway are shown in Table 2. The model set boundary conditions and initial conditions based on geostress test results. According to strain-softening model analysis of coal seam and rock mass, linear approximation was implemented according to [39] in the process of strain softening. In other words, the internal frictional angle is constant in the strain-softening process. Among strength parameters, only cohesion degenerates with the increase of plastic shearing strain since softening variable has different definitions; the calculation formula of softening variable in the FLAC3D strain-softening model was modified by C++ and recompiled again, which is shown in equations (12) and (13) [39]:(13)c=c01.001−tanh100ηtanh10a,η=γp=ε1p−ε3p,where c is the cohesion, c0 is the initial cohesion, and a is the fitting parameter which ranges between 0.29 and 0.34 for sedimentary rocks. In this study, the mean value is 0.315. η is the softening variable, and γp is the plastic shearing strain. ε1p and ε3p are maximum and minimum plastic principal strains, respectively.Figure 6
Simulation model.Table 2
Physicomechanical performance of coal seams and their surrounding rock mass.
LithologyBulk modulus (GPa)Shear modulus (GPa)Initial cohesion (MPa)Internal friction angle (°)Dilation angle (°)Fine-grained sandstone5.633.156.0127.910.3Mudstone2.951.622.1827.312.3Medium sandstone7.854.568.5329.89.7Coal1.450.761.863713.1Siltstone4.653.365.9723.28.2Sandy mudstone5.853.114.3326.715.6Simulation results of roadway surrounding rock mass under different supporting systems are shown in Figure7. Without supporting system, the load-bearing structure formed by sectional shape of roadway surrounding rock mass is shown in Figure 7(a). The load-bearing structure of roadway surrounding rock mass in the roadway surrounding rock mass is shown in Figure 7(b). According to the comparison between Figure 7(a) and Figure 7(b), the primary supporting design fails to promote changes in load-bearing structural morphology of surrounding rock mass effectively due to the low prestress. According to different supporting philosophy in implementation of HLBR, the roadway support can be designed by combining sectional design and surrounding geostress distribution in the study area. Specifically, the length and prestress level of bolt on roof and floor are higher than those on two sides. Length of cable on roof can be lowered appropriately, but large prestress shall be applied on it. Moreover, a certain length of high-prestress cable shall be installed at the end corners of roadway. The suggested supporting scheme in Figure 7(c) is set as follows: length of bolt on roof and floor = 3.0 m, designed prestress level = 70 kN, and row space = 800 mm × 800 mm; length of cable = 6.4 m, prestress level = 300 kN, and three-flower distribution; length of cable at the end corner = 4.4 m and prestress level = 200 kN; length of bolt on two sides of roadway is adjusted to 2.0 m and the prestress level is designed 20 kN, accompanied by a row space of 800 mm × 800 mm. The load-bearing structure formed by the suggested supporting scheme in roadway surrounding rock mass is shown in Figure 7(c). According to the analysis, the suggested supporting scheme changes the geometric morphology of bearing structure effective in the roadway surrounding rock mass and promotes the formation of a homogenized load-bearing structure in the roadway surrounding rock mass. Displacement changes of roadway under no supporting scheme, primary supporting design, and the suggested supporting scheme are shown in Figures 8(a), 8(b), and 8(c). It can be seen from the comparison of Figures 8(a), 8(b), and 8(c) that the roadway displacement under the suggested supporting scheme is the minimum, indicating that the load-bearing structure under the suggested supporting scheme can offer strong supports to the roadway surrounding rock mass and thereby lower displacement of roadway.Figure 7
Load-bearing structure and tangential stress distribution of roadway surrounding rock mass under different conditions.Figure 8
Displacement of roadway surrounding rock mass under different conditions.


### 4.5. Implementation Ways of the HLBR

The HLBR hypothesis is formed by evolutionary characteristics of collapse boundaries at natural equilibrium of surrounding rock mass based on the reinforced arch theory. When the axial ratio of load-bearing ring formed by the artificial supporting system in the roadway surrounding rock mass is equal to the optimal axial ratio of the roadway under local geology and stress conditions, the self-bearing (self-stabilization) ability of roadway surrounding rock mass can be used well in theory. As a result, stress field distribution of roadway surrounding rock mass is optimized to be in favor of stability control. Based on the above studies and other scholars’ description of prestress diffusion of bolt, this study believed that the HLBR can be achieved by the following three ways, including optical axial ratio design, difference support, and local grouting modification.


### 4.6. Optimal Axial Ratio Design

Based on the above analysis of stress states at Y and Y′, all boundaries of roadway undertake relatively small compressive stresses, and these stress values are equal. The lateral stress factor (λ) of the region where roadway locates in is used as an index to determine the geometric morphology and size of roadway section. With consideration to practical demands of field investigation, decisions on geometric morphology and axial ratio of internal section shall be made. During the design of roadway sectional dimension under the premise of roadway functions, attention shall be paid to try to use the axial ratio (k) of the internal ellipse as the lateral stress factor on roadway surrounding rock mass (λ) but avoid the roadway sectional design in which the principal stress direction on roadway surrounding rock mass is perpendicular with long axis of the roadway. This design can make full use of self-bearing capacity of surrounding rock mass and promote roadway surrounding rock mass maintaining a relatively small stress state.


### 4.7. Difference Support

The axial ratio of load-bearing ring can be equal to the lateral stress factor through the bolt (cable) arrangement in different support, thus constructing an HLBR structure in roadway surrounding rock mass. Difference support is to install bolts (cables) of different lengths, different prestress levels, and row spaces (roof, floor, and two sides) on roadway surrounding rock mass to form an HLBR with the optimal axial ratio:(1)
Whenk < λ, the stress condition can be divided into three types, namely, k < 1 > λ, 1 < k < λ, and k < λ < 1. It can be seen from equations (10) and (11) that Y and Y` have been bearing compressive stresses, whereas X and X′ were easy to have tensile stresses. Therefore, load-bearing structure on two sides of roadway surrounding rock mass was constructed by increasing prestress level and length of bolts on two sides as well as decreasing the row space. The detailed length, prestress level, and row space of bolts shall be decided according to specific conditions. When k > λ, the stress condition also can be divided into three types, namely, k > λ > 1, k > 1 < λ, and 1 < k > λ. Similarly, it can be seen from equations (10) and (11) that Y and Y′ are easy to develop tensile stress. In the same way, a load-bearing structure on roof and floor of roadway surrounding rock mass also can be constructed by increasing prestress level on roof and floor, length of bolts, and row space.(2)
It is easy to have high-stress concentration on the corner of the section center in the roadway. Supplementary support with high-prestress bolts (cables) at the corner is needed. On the one hand, the bolts (cable) at the end corner can control the corner deformation of the roadway. On the other hand, they can maintain the correlation between roof and floor and stress diffusion range at two sides, forming a complete load-bearing structure. Axial ratio (k) of equivalent section can be adjusted by controlling the length and row space of bolts (cables) on the roadway section. Consequently, the equivalent section presents a state of equal stresses (k = λ).(3)
A full sectional support is needed due to the strong stress and unique geological conditions of the roadway in order to assure a closed stress diffusion range of roof-sides-floor of roadway. Such closed complete sectional support is in favor of forming a complete HLBR structure in roadway surrounding rock mass and preventing large roadway deformation due to inadequate bearing capacity and stress releasing in local regions. If support to floor is simplified, it is inevitable to have certain heaving on floor. When the roadway lies in good geological conditions and heaving floor can meet demands for safety production, the acceptable simple support to the floor can be designed from perspectives of economy and construction speed.


### 4.8. Local Grouting Modification

Priestess field of bolt is formed by the nut torque which is applied at end of bolts in roadway surrounding rock mass. Many scholars have carried out abundant theoretical and experimental studies on the range and morphology of prestress field in the roadway surrounding rock mass [40–44]. Results demonstrated that the integrity degree of roadway surrounding rock mass (fracture development degree) and its mechanical properties determine the formation of prestress field. When there is a development of low-strength fractures in roadway surrounding rock mass, primary support and grouting reinforcement in local areas shall be installed in time during the tunnel advancing, followed by application of prestress on surrounding rock mass.Grouting measures can fill in primary and secondary fractures in surrounding rock mass, thus increasing the integrity and overall strength of roadway surrounding rock mass and realizing the goal of physical modification to surrounding rock mass. On the one hand, increasing the integrity and strength of roadway surrounding rock mass promotes the formation of a prestress field by single bolt in surrounding rock mass and a prestress overlapping field by multiple bolts, thus making surrounding mass in the prestress overlapping field bear three-direction loads and achieving a higher load-bearing capacity. On the other hand, it can lower loss of prestress on bolt, so that the anchor road can maintain a certain diffusion range of prestress field for a long period. This is beneficial to the long-term bearing capacity of roadway surrounding rock mass. Local grouting modification in roadway surrounding rock mass can promote the formation and long-term stability of prestress field caused by bolts. Applying prestress again to the surrounding rock mass after grouting modification is a compensation for the lost prestress of bolts due to surrounding rock mass deformation. It not only assures the formation of internal prestress diffusion range of bolts in surrounding rock mass but also is conducive to the formation and effective maintaining of the HLBR in roadway surrounding rock mass.


## 4.1. Model Background and Primary Support Design

The advancing section in the study area was 5.30 m wide and 3.55 m high. Stability of roadway surrounding rock mass was maintained by a compound supporting scheme of “bolt + cable + metal mesh + steel ladder beam.” The row space of bolts on roof (MSGLW, 335/22 × 2400) is 800 mm × 800 mm, and the row space of bolts on two sides (MSGLW, 335/22 × 2400) is 1000 mm × 800 mm. Cable (SKP, 18.9 × 8300) was in three-flower distribution, and metal mesh used the graticules (50 mm × 50 mm) made of Φ10 iron wires. The steel ladder beam (Φ14 mm rebar) was 60 mm wide, and it was covered on the roof and two sides of the roadway. However, no prestress was applied to the bolt and cable in the supporting design. Instead, they were installed by end anchoring mode. Prestress on the bolt and cable was measured 5–10 kN and 20–30 kN, respectively. Low-strength ordinary supporting plate was applied to the bolt and cable.


## 4.2. Research on Roadway Geostress

A geostress measurement system was installed in the study area [38]. Measuring positions and measuring results are shown in Figure 4. According to measurement results, the maximum and minimum horizontal principal stress (σm and σl) are 9.04 MPa and 5.22 MPa, and the vertical stress (σv) is 8.40 MPa. Besides, the included angle between the maximum horizontal principal stress and belt roadway in mining district IV is NE20°. The coordinate transformation is shown in equations (11) and (12) [37], where σm is the maximum horizontal principal stress under the old coordinate system; σl is the minimum principal stress under the old coordinate system; σml is the tangential stress under the old coordinate system, and there is σml = 0; σx is the maximum horizontal principal stress under the new coordinate system; σy is the minimum principal stress under the new coordinate system; σy is the minimum principal stress under the new coordinate system; and θ is the deflection angle of the coordinate system. Measurement results of geostress were brought into equations (11) and (12), and stresses on two sides of the roadway were gained: σx = 8.59 Mpa and σy = 5.67 MPa. Therefore, the lateral stress factor in the roadway was λ = σx/σv = 1.02 and k = 1.49:(11)σx=12σm+σl+12σm−σlcos2θ+σmlsin2θ,(12)σy=12σm+σl−12σm−σlcos2θ−σmlsin2θ.Figure 4
Measurement location and maximum horizontal stress direction of geostress in the study area.The primary support design in the study area applied the low-level prestress on bolts and cable. The primary support design was analyzed based on the HLBR hypothesis. It found that the primary support design failed to form effective positive support to load-bearing structure in surrounding rock mass, and it was only passive support. Hence, the axial ratio of bearing structure in the study area is equal to the axial ratio in the primary sectional design (k = 1.49) of the roadway. Moreover, the axial ratio of final self-stabilization boundaries was k = λ = 1.02 under the influences of stress redistribution. Therefore, depth of the plastic region on roof and floor during the self-stabilization of belt roadway was higher than the development depth of the plastic region on two sides of the roadway.


## 4.3. Optical Borehole Observation in Roadway Surrounding Rock Mass

Optical borehole observation was implemented in the belt roadway of mining district IV. Measurement results are shown in Figure5. According to analysis, they were developed with fractured sandy mudstone within 0–1.3 m of roadway roof, medium sandstone within 1.3–4.5 m of roadway, fractures within 1.3–1.5 m of roadway, and sandy mudstone with high integrity within 4.5–7.0 m of roadway. Coal fractures since 0–0.7 m of roadway surface on sides of roadway are loose. Roadway deformation was analyzed by considering that rock mass is a homogeneous elastic material. According to the symmetric principle, failure depths of roadway roof and floor were 1.5 m, and the failure depth on two sides was 0.7 m after the roadway was drilled. The real axial ratio of roadway after the development of plastic region in the surrounding rock mass was kn = (2.65 + 0.7)/(1.75 + 1.5) = 1.03, indicating a certain degree of self-stabilization of roadway under loads. The stress distribution on roadway surrounding rock mass was optimized, during which the artificial supporting system developed limited effect. The actual deformation features of the roadway basically conform to the above theoretical analysis.Figure 5
Borehole optical observation in roof and coal seam side.


## 4.4. Simulation on Stress Evolution of the HLBR

A numerical simulation analysis of the study area was carried out by using FLAC3D. Width, axial length, and vertical height of the model were set at 50 m, 30 m, and 47 m, respectively. The roof strata within 23 m above the coal seam were simulated, and the floor 20 m below the coal seam was rock strata. The simulation model is shown in Figure6. The distance from the boundaries of the model to the excavation boundary exceeds 5 times of the maximum radius of the roadway. Lithological and physical mechanical parameters of rock mass on roof and floor of the roadway are shown in Table 2. The model set boundary conditions and initial conditions based on geostress test results. According to strain-softening model analysis of coal seam and rock mass, linear approximation was implemented according to [39] in the process of strain softening. In other words, the internal frictional angle is constant in the strain-softening process. Among strength parameters, only cohesion degenerates with the increase of plastic shearing strain since softening variable has different definitions; the calculation formula of softening variable in the FLAC3D strain-softening model was modified by C++ and recompiled again, which is shown in equations (12) and (13) [39]:(13)c=c01.001−tanh100ηtanh10a,η=γp=ε1p−ε3p,where c is the cohesion, c0 is the initial cohesion, and a is the fitting parameter which ranges between 0.29 and 0.34 for sedimentary rocks. In this study, the mean value is 0.315. η is the softening variable, and γp is the plastic shearing strain. ε1p and ε3p are maximum and minimum plastic principal strains, respectively.Figure 6
Simulation model.Table 2
Physicomechanical performance of coal seams and their surrounding rock mass.
LithologyBulk modulus (GPa)Shear modulus (GPa)Initial cohesion (MPa)Internal friction angle (°)Dilation angle (°)Fine-grained sandstone5.633.156.0127.910.3Mudstone2.951.622.1827.312.3Medium sandstone7.854.568.5329.89.7Coal1.450.761.863713.1Siltstone4.653.365.9723.28.2Sandy mudstone5.853.114.3326.715.6Simulation results of roadway surrounding rock mass under different supporting systems are shown in Figure7. Without supporting system, the load-bearing structure formed by sectional shape of roadway surrounding rock mass is shown in Figure 7(a). The load-bearing structure of roadway surrounding rock mass in the roadway surrounding rock mass is shown in Figure 7(b). According to the comparison between Figure 7(a) and Figure 7(b), the primary supporting design fails to promote changes in load-bearing structural morphology of surrounding rock mass effectively due to the low prestress. According to different supporting philosophy in implementation of HLBR, the roadway support can be designed by combining sectional design and surrounding geostress distribution in the study area. Specifically, the length and prestress level of bolt on roof and floor are higher than those on two sides. Length of cable on roof can be lowered appropriately, but large prestress shall be applied on it. Moreover, a certain length of high-prestress cable shall be installed at the end corners of roadway. The suggested supporting scheme in Figure 7(c) is set as follows: length of bolt on roof and floor = 3.0 m, designed prestress level = 70 kN, and row space = 800 mm × 800 mm; length of cable = 6.4 m, prestress level = 300 kN, and three-flower distribution; length of cable at the end corner = 4.4 m and prestress level = 200 kN; length of bolt on two sides of roadway is adjusted to 2.0 m and the prestress level is designed 20 kN, accompanied by a row space of 800 mm × 800 mm. The load-bearing structure formed by the suggested supporting scheme in roadway surrounding rock mass is shown in Figure 7(c). According to the analysis, the suggested supporting scheme changes the geometric morphology of bearing structure effective in the roadway surrounding rock mass and promotes the formation of a homogenized load-bearing structure in the roadway surrounding rock mass. Displacement changes of roadway under no supporting scheme, primary supporting design, and the suggested supporting scheme are shown in Figures 8(a), 8(b), and 8(c). It can be seen from the comparison of Figures 8(a), 8(b), and 8(c) that the roadway displacement under the suggested supporting scheme is the minimum, indicating that the load-bearing structure under the suggested supporting scheme can offer strong supports to the roadway surrounding rock mass and thereby lower displacement of roadway.Figure 7
Load-bearing structure and tangential stress distribution of roadway surrounding rock mass under different conditions.Figure 8
Displacement of roadway surrounding rock mass under different conditions.


## 4.5. Implementation Ways of the HLBR

The HLBR hypothesis is formed by evolutionary characteristics of collapse boundaries at natural equilibrium of surrounding rock mass based on the reinforced arch theory. When the axial ratio of load-bearing ring formed by the artificial supporting system in the roadway surrounding rock mass is equal to the optimal axial ratio of the roadway under local geology and stress conditions, the self-bearing (self-stabilization) ability of roadway surrounding rock mass can be used well in theory. As a result, stress field distribution of roadway surrounding rock mass is optimized to be in favor of stability control. Based on the above studies and other scholars’ description of prestress diffusion of bolt, this study believed that the HLBR can be achieved by the following three ways, including optical axial ratio design, difference support, and local grouting modification.


## 4.6. Optimal Axial Ratio Design

Based on the above analysis of stress states at Y and Y′, all boundaries of roadway undertake relatively small compressive stresses, and these stress values are equal. The lateral stress factor (λ) of the region where roadway locates in is used as an index to determine the geometric morphology and size of roadway section. With consideration to practical demands of field investigation, decisions on geometric morphology and axial ratio of internal section shall be made. During the design of roadway sectional dimension under the premise of roadway functions, attention shall be paid to try to use the axial ratio (k) of the internal ellipse as the lateral stress factor on roadway surrounding rock mass (λ) but avoid the roadway sectional design in which the principal stress direction on roadway surrounding rock mass is perpendicular with long axis of the roadway. This design can make full use of self-bearing capacity of surrounding rock mass and promote roadway surrounding rock mass maintaining a relatively small stress state.


## 4.7. Difference Support

The axial ratio of load-bearing ring can be equal to the lateral stress factor through the bolt (cable) arrangement in different support, thus constructing an HLBR structure in roadway surrounding rock mass. Difference support is to install bolts (cables) of different lengths, different prestress levels, and row spaces (roof, floor, and two sides) on roadway surrounding rock mass to form an HLBR with the optimal axial ratio:(1)
Whenk < λ, the stress condition can be divided into three types, namely, k < 1 > λ, 1 < k < λ, and k < λ < 1. It can be seen from equations (10) and (11) that Y and Y` have been bearing compressive stresses, whereas X and X′ were easy to have tensile stresses. Therefore, load-bearing structure on two sides of roadway surrounding rock mass was constructed by increasing prestress level and length of bolts on two sides as well as decreasing the row space. The detailed length, prestress level, and row space of bolts shall be decided according to specific conditions. When k > λ, the stress condition also can be divided into three types, namely, k > λ > 1, k > 1 < λ, and 1 < k > λ. Similarly, it can be seen from equations (10) and (11) that Y and Y′ are easy to develop tensile stress. In the same way, a load-bearing structure on roof and floor of roadway surrounding rock mass also can be constructed by increasing prestress level on roof and floor, length of bolts, and row space.(2)
It is easy to have high-stress concentration on the corner of the section center in the roadway. Supplementary support with high-prestress bolts (cables) at the corner is needed. On the one hand, the bolts (cable) at the end corner can control the corner deformation of the roadway. On the other hand, they can maintain the correlation between roof and floor and stress diffusion range at two sides, forming a complete load-bearing structure. Axial ratio (k) of equivalent section can be adjusted by controlling the length and row space of bolts (cables) on the roadway section. Consequently, the equivalent section presents a state of equal stresses (k = λ).(3)
A full sectional support is needed due to the strong stress and unique geological conditions of the roadway in order to assure a closed stress diffusion range of roof-sides-floor of roadway. Such closed complete sectional support is in favor of forming a complete HLBR structure in roadway surrounding rock mass and preventing large roadway deformation due to inadequate bearing capacity and stress releasing in local regions. If support to floor is simplified, it is inevitable to have certain heaving on floor. When the roadway lies in good geological conditions and heaving floor can meet demands for safety production, the acceptable simple support to the floor can be designed from perspectives of economy and construction speed.


## 4.8. Local Grouting Modification

Priestess field of bolt is formed by the nut torque which is applied at end of bolts in roadway surrounding rock mass. Many scholars have carried out abundant theoretical and experimental studies on the range and morphology of prestress field in the roadway surrounding rock mass [40–44]. Results demonstrated that the integrity degree of roadway surrounding rock mass (fracture development degree) and its mechanical properties determine the formation of prestress field. When there is a development of low-strength fractures in roadway surrounding rock mass, primary support and grouting reinforcement in local areas shall be installed in time during the tunnel advancing, followed by application of prestress on surrounding rock mass.Grouting measures can fill in primary and secondary fractures in surrounding rock mass, thus increasing the integrity and overall strength of roadway surrounding rock mass and realizing the goal of physical modification to surrounding rock mass. On the one hand, increasing the integrity and strength of roadway surrounding rock mass promotes the formation of a prestress field by single bolt in surrounding rock mass and a prestress overlapping field by multiple bolts, thus making surrounding mass in the prestress overlapping field bear three-direction loads and achieving a higher load-bearing capacity. On the other hand, it can lower loss of prestress on bolt, so that the anchor road can maintain a certain diffusion range of prestress field for a long period. This is beneficial to the long-term bearing capacity of roadway surrounding rock mass. Local grouting modification in roadway surrounding rock mass can promote the formation and long-term stability of prestress field caused by bolts. Applying prestress again to the surrounding rock mass after grouting modification is a compensation for the lost prestress of bolts due to surrounding rock mass deformation. It not only assures the formation of internal prestress diffusion range of bolts in surrounding rock mass but also is conducive to the formation and effective maintaining of the HLBR in roadway surrounding rock mass.


## 5. Discussion

Three ways, including optical axial ratio design, difference support, and local grouting modification, are proposed to reach the goal of roadway supported by the HLBR hypothesis. Firstly, optical axial ratio design refers to the design progress of roadway; the idea of optical axial ratio design should be concerned when designing the roadway section. Difference support means that ellipse-shaped superposed stress field could be achieved by different support parameter. Local grouting modification is the common method to strengthen the roadway surrounding rock mass. It is demonstrated that a good use of self-carrying capacity of surrounding rock mass can be achieved by the three ways mentioned above, which is conducive to lowering the stress distribution level of surrounding rock mass and maintaining the stability of roadway.


## 6. Conclusions

In addition, the conclusions should be simplified.(1)
With comprehensive consideration of mechanical properties of roadway surrounding rock mass, this study proposes an HLBR hypothesis based on reinforced arch theory and axial variation theory. This hypothesis points out that the axial ratio (k) of load-bearing ring formed by an artificial supporting system in roadway surrounding rock mass is equal to or similar to the ratio between horizontal principal stress and vertical principal stress (λ, which is the optimal axial ratio) in the roadway. Stress on this load-bearing ring is the minimum and uniform distribution, thus forming an HLBR. The roadway and supporting structure can make full use of self-bearing capacity of roadway surrounding rock mass, which is beneficial to stress distribution of roadway surrounding rock mass.(2)
Given a fixed axial ratio, surrounding rock mass on roof and floor changes from tensile failure to compressive failure, while the surrounding rock mass on two sides changes from compressive failure to tensile failure with the increase of lateral stress factor. When the lateral stress factor is fixed, there are two situations with the increase of the axial ratio of the roadway.(1)
Whenλ < 1, surrounding rock mass on roof and floor changes from compressive failure to tensile failure, but no tensile stress is developed on surrounding rock mass on two sides of the roadway. Compressive stress on two sides of the roadway increases with the increase of k.(2)
Whenλ > 1, the surrounding rock mass on two sides of the roadway changes from tensile failure to compressive failure, and the surrounding rock mass on roof and floor has been bearing compressive stress. Compressive stress on roof and floor decreases with the increase of k.(3)
There are three ways to form an HLBR, which are optimal axial ratio design, difference support, and local grouting modification. The optimal axial ratio design means that the axial ratio of internal ellipse of roadway section shall try to be equal to lateral stress coefficient in the roadway. Difference support promotes the formation of HLBR in roadway surrounding rock mass by designing different lengths, prestress level, and row space (roof, floor, and two sides) of bolt. Local grouting modification is to make physical modification to surrounding rock mass by grouting measures, and prestress is applied again to assure the formation of stress diffusion range in surrounding rock mass. This is beneficial to the formation of an HLBR.



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*Source: 1038231-2022-05-24.xml*