Achieving a comprehensive and accurate understanding of the anchor reinforcement mechanism and a quantitative evaluation of the surrounding rock’s stability for an anchored underground cavern can provide an important theoretical basis for supporting and excavating the cavern. First, the composite bearing structure composed of the anchor and surrounding rock was defined as the surrounding rock reinforcement body by using the homogenization method, and a new method for evaluating the stability of surrounding rock by the surrounding rock reinforcement body deformation and damage degree was proposed. Second, based on the anchor reinforcement effect, the expression of the physicalmechanical parameters of the surrounding rock reinforcement body was deduced, and the analytical solution of stress and displacement of the surrounding rock reinforcement body was obtained. Finally, the stability coefficient of surrounding rock indicating the degree of the surrounding rock reinforcement body damage was defined. The research showed that the theoretical solution agreed well with the results of the numerical simulation, and the difference between the theoretical solution and the monitoring value was less than 10%, which verified the reliability of the method and the results of this paper. The design of the length and spacing of the anchor followed the principle of long but sparse and short but dense, and the pretightening force of the anchor and the stability coefficient of surrounding rock varied linearly. The analytical solution of this paper provides a theoretical reference for understanding the mechanism of anchor support and provides a quantitative evaluation method for the stability of surrounding rock. Compared with the traditional support design, the theory of this paper gives full play to the selfstability of the surrounding rock and the strength of the anchor, which is conducive to saving support costs and avoiding the construction limitations in some projects.
With the rapid development of infrastructure construction in China, a large number of underground projects have been built in various fields, such as hydropower, civil engineering, and mining. The stability of surrounding rock in an underground cavern is the key to the construction of largescale underground projects, and the anchor support is an important means of ensuring the stability of the underground cavern. The wholly grouted anchor constrains surrounding rock deformation through its own stiffness, strength, and transferring stress by anchoring interfaces, such as a mortar or resin anchoring agent, which strengthens the rock mass, and has been widely used in the field of geotechnical engineering [
To study the mechanical effects of the wholly grouted anchor support, several studies have been conducted through theoretical analyses, numerical simulations, and experimental means. Freeman proposed the neutral point theory by observing the stress process of the anchor and the distribution of the stress along the anchor length [
Moreover, through experiments, numerical simulations, and mechanical methods, scholars have gradually realized that the anchor support cannot only improve the stress state of the surrounding rock (supporting effect) but can also improve the strength index of the anchorage body (reinforcing effect). Therefore, it is considered that the anchor support improves the surrounding rock’s mechanical parameters and the surrounding rock cohesion but has little effect on the surrounding rock’s internal friction angle [
The above scholars, through different research ideas, have analyzed the anchorage mechanism of the wholly grouted anchor, established the elasticplastic analysis model of the surrounding rock of the cavern under the anchor support, and obtained rich research results. However, there are still some shortcomings: (1) most of the calculation models only consider the support effect of the anchor, neglecting the reinforcement effect. (2) When considering the reinforcement effect of the anchor, the mechanical parameters of the equivalent material are obtained by the average distribution of the physical and mechanical parameters of the anchor and the surrounding rock, which ignores the mutual coupling effect between the anchor and the surrounding rock. (3) The intermediate principal stress effect and the dilatancy characteristics of the rock mass are less considered in the model calculation. Therefore, on the basis of previous studies, this paper regards the anchor as a type of reinforcement measure and regards the composite of the anchor and the anchored rock as a bearing structure, which is defined as the surrounding rock reinforcement body. The physical and mechanical parameters of the surrounding rock reinforcement body are obtained under the coupling of the anchor and the anchored rock mass. Considering the intermediate principal stress effect and dilatancy characteristics of the rock mass, the analytical solutions of the stress and the displacement of the surrounding rock reinforcement body and deep surrounding rock are derived. Compared with the FLAC^{3D} numerical simulation results and actual monitoring data, the rationality of the method and the reliability of the calculation results are verified.
An underground cavern excavation leads to the redistribution of the surrounding rock stress. When the circumferential stress of the cave wall is greater than the compressive strength of the surrounding rock, the surrounding rock begins to plastically yield, and the width of the plastic zone gradually extends from the cave wall to the deep part of the surrounding rock. To increase the integrity and strength of the surrounding rock and to achieve the purpose of supporting surrounding rock, tunnels, chambers, roadways, quarries, and other projects, an anchor is often used as a support form. The traditional surrounding rock loose circle theory states that the anchorage section of an anchor must extend into a certain range of the surrounding rock’s elastic zone to play the role of anchor support and ensure the stability of the surrounding rock when the anchor parameters are designed. In the actual project, it is found that the stability of the underground cavern actually depends on the stability of the surrounding rock within the reinforcement range of the anchor.
Therefore, to determine the reasonable anchor support parameters and quantitatively evaluate the stability of the anchored rock mass, the homogenization method is adopted to macroscopically consider the rock mass and anchor complex as a continuous, homogeneous and isotropic surrounding rock reinforcement body. As a new supporting structure, the surrounding rock reinforcement body constrains the deformation of the deep surrounding rock.
After cavity excavation, the plastic yield range of surrounding rock extends from the cave wall to the deep section. When the deformation is stable, the plastic zone of surrounding rock is formed with a certain width, as shown in Figure
Surrounding rock damage evolution and anchor reinforcement: (a) plastic zone width after excavation of the cavern; (b) anchor length is greater than the plastic zone width; (c) anchor length is less than the plastic zone width; (d, e) the surrounding rock reinforcement body damage zone.
Under the initial ground stress, a part of the surrounding rock reinforcement body yields plastically. The plastic zone of the surrounding rock reinforcement body is defined as the surrounding rock reinforcement body failure zone, and the elastic zone of the surrounding rock reinforcement body is defined as the surrounding rock reinforcement body stability zone. The physical properties of the surrounding rock reinforcement body are similar to those of the rock, such as the bulk density, specific gravity, porosity, and water absorption. Compared with the physicalmechanical parameters of surrounding rock, the anchor support improves the deformation parameters and the strength parameters of the surrounding rock reinforcement body.
Once the anchorage body is equivalent to the surrounding rock reinforcement body, the anchor support form is transformed into the surrounding rock reinforcement body support form, and the surrounding rock reinforcement body is also deformed under the surrounding rock pressure. The mechanical model for reflecting the coordinated deformation between deep surrounding rock and the surrounding rock reinforcement body is established, as shown in Figure
Mechanical model.
In Figure
For a deep buried cavern, the initial ground stress can be assumed to be equally isotropic (
There are many discontinuous surfaces of different sizes in the rock mass (except largescale faults and weak interlayers), but the distribution of these discontinuous surfaces can be approximated as random, and their impact on the rock mass are not very significant as a whole. Therefore, to meet the basic assumptions of the medium in elastoplastic mechanics, the surrounding rock and the surrounding rock reinforcement body are assumed to be continuous, homogeneous, and isotropic elastomers.
The time of the cavern excavation and support is much smaller than the time of the cavern operation, so it can be assumed that the cavern excavation and support are completed instantaneously.
The principle of modern support is based on the joint bearing of surrounding rock and supporting structure. The composite bearing structure composed of the anchor, and anchored rock mass is analyzed, which can fully exert the selfsupporting capacity of surrounding rock. For anchors anchored into the interior of the rock mass, the material properties of the anchor and the rock mass are different so that the deformation (trend) of the two under the same stress field is different. The deformation of the surrounding rock is restrained by the anchor, and the anchor also generates an axial force and deformation due to the deformation of the surrounding rock. Therefore, this paper establishes the theory of the surrounding rock reinforcement body support, which is based on the stability of a composite bearing structure for the anchor support design of surrounding rock. Compared with the traditional support theory, the difference in the surrounding rock reinforcement body support theory lies in the design basis, design method, and support effect evaluation. The details are as follows.
The traditional support theory relies on the plastic zone of the surrounding rock and the radius of the loose rock of the surrounding rock to design the anchor parameters. For soft rock roadways, the radius of the plastic zone is larger, and the required length of the anchor is longer, which increases the construction time and support cost. However, the surrounding rock reinforcement body support theory is based on the stability of the composite which consists of the anchor and the anchored rock mass to design the anchor parameters, which are applicable to any working condition, and can effectively avoid construction difficulties and save support costs.
The traditional support theory first designs the length of the anchor and then designs the spacing between the anchors according to engineering experience and the construction scheme. It is not conducive to adjust the spacing between the anchors and the length of the anchor, and the strength of the anchor cannot be fully utilized. The surrounding rock reinforcement body support theory can simultaneously consider the anchor length and the spacing between the anchors to design the supporting parameters and repeatedly adjust the anchor parameters to obtain a reasonable supporting strength, which can meet the construction requirements and optimize the support cost.
The traditional support theory judges the stability of surrounding rock based on the axial force of the anchor and the deformation of the surrounding rock. The surrounding rock reinforcement body support theory first determines whether the anchor is yielding and then determines the stability of the surrounding rock according to the degree of deformation of the surrounding rock reinforcement body. The discriminating method is simple and can comprehensively consider the selfstability of the surrounding rock and the anchor support strength.
To obtain the deformation parameters of the surrounding rock reinforcement body, first, the support force of the anchor is transformed into the additional volume force of the surrounding rock acting on the interior of the anchored rock mass. Then, the mechanical model of the anchored cavity is established, as shown in Figure
Elastic analysis of the surrounding rock deformation under (a) anchor support and (b) the surrounding rock reinforcement body support.
Under the same initial ground stress as the former, the model of the coordinated deformation mechanics of surrounding rock and the surrounding rock reinforcement body is established and analyzed to obtain the radial displacement of the inner and outer edges of the reinforcement body, as shown in Figure
Studies have shown that the shear stress is 0 in a certain part of the anchor when the anchor and surrounding rock coordinate deform, where the neutral position of the anchor is located. Near the tunnel wall, the surrounding rock has a large deformation and a greater effect on the anchor, and the anchor interface produces an interface shear stress directed towards the center of the tunnel, and the reaction force of the anchor to the surrounding rock points to the distal end of the anchor. Near the distal end of the anchor, the surrounding rock is less deformed and is also affected by the pulling force at the proximal end of the anchor, and the deformation of the surrounding rock is less than the deformation of the anchor. Therefore, the anchor interface generates an interface shear stress directed towards the distal end of the anchor, and the reaction force of the anchor to the surrounding rock points towards the proximal end of the anchor, as shown in Figure
Coordination deformation between the anchor and surrounding rock [
The interfacial shear stress of the anchor is generated by the relative displacement of the anchor and surrounding rock, and the overall displacement of the anchor can be represented by the displacement of surrounding rock at the neutral point. The interfacial shear stress of the wholly grouted anchor can be expressed as
Rock type 


Hard rock  5.00∼10.00 
Soft rock  1.50∼3.00 
Weathered rock  1.00∼2.00 
Mudstone  1.20∼2.50 
Diluvium sand  0.40∼0.70 
Gravel  0.40∼0.70 
Diluvium clay  0.40∼1.00 
Shock layer sand  0.05∼0.20 
The stress analysis of a single anchor is as follows:
The axial force distribution of the anchor is
The support force of the anchor to surrounding rock is converted into the additional volume force of surrounding rock [
In Figure
In Figure
The boundary conditions are as follows: when
Bringing the boundary conditions into equations (
Bringing the constants
Bringing
As shown in Figure
The boundary conditions are as follows: when
The equations
When the anchor is not supported,
The internal friction angle of the surrounding rock reinforcement body is determined by the internal friction angle of the anchor, the internal friction angle of the rock, and the stress state on the friction surface. If the stress states of the anchor and the anchored rock mass are the same, the internal friction angle of the reinforcement body can be obtained according to the area equivalent principle. Because the total area of the anchor in the anchorage area is small, the internal friction angle of the reinforcement body is approximately equal to the internal friction angle of the rock mass before being anchored [
The increase in the cohesion of the surrounding rock reinforcement body is caused by two aspects. One aspect is that the lateral action of the anchor increases the shear strength of the fracture surface, and the other part is that the vertical action of the anchor exerts a certain pressure on surrounding rock, which improves the stress state of the rock mass and increases its nondeformability. The compressive stress applied to the surrounding rock is decomposed into the fracture surface, as shown in Figure
Influence of the anchor on the cohesive force of the surrounding rock reinforcement body: (a) the wedgeshaped unit of the anchor and surrounding rock; (b) anchor vertical action; (c) anchor lateral action.
Experiments have shown that the maximum principal stress direction of the anchor is perpendicular to the anchor [
In Figure
In Figure
The cohesion of the surrounding rock reinforcement body at
Some scholars have proposed a homogenization method which means that the surrounding rock is regarded as a homogeneous elastomer, and the physicomechanical parameters that vary from place to place are converted into equal parameters. By the homogenization method, the cohesion of the surrounding rock reinforcement body can be expressed as
When the anchor is not supported, the anchor length
To consider the influence of the intermediate principal stress
The equilibrium differential equation of surrounding rock in the plastic zone of the reinforcement body can be expressed as
Through the simultaneous equations (
It is known from elastic mechanics that the stress and the displacement expressions of surrounding rock of the elastic zones I and II can be expressed as
The boundary conditions are as follows: when
By bringing the boundary conditions into equations (
At the elastoplastic interface, the radial stress
When
The rock mass material has dilatancy characteristics, the volumetric strain of the plastic zone is not equal to 0, and the plastic zone satisfies the following nonassociated flow rules [
The total strain of the plastic zone can be regarded as the superposition of the elastic strain and the plastic strain of the plastic zone:
By bringing the geometric equations
The paper deals with the plane strain problem, so the elastic strain in the plastic zone still satisfies the generalized Hooke’s law:
By bringing the boundary condition
First, it is necessary to ensure that the displacement of the cave wall is not greater than the maximum displacement allowed in the cave wall and that it does not affect the normal use of the cavity. Second, it must be ensured that the anchor does not yield and that the maximum shear stress of the anchor does not exceed the shear strength of the anchoring agent. If the anchor yield or the maximum shear stress of the anchor is greater than that of the shear strength of the anchoring agent, the bearing structure of the surrounding rock reinforcement body cannot be formed. Finally, it must be ensured that the bearing structure of the surrounding rock reinforcement body does not fully yield. Even if a part of the surrounding rock reinforcement body has yielded, it can constrain the surrounding rock deformation and controls the surrounding rock stability.
If
If
For the above two cases, it must be guaranteed:
When the initial ground stress is small or the anchor support strength is high, the surrounding rock reinforcement body is in an entirely elastic state, which indicates that surrounding rock has good stability. When the initial ground stress is large or the anchor support strength is very low, the surrounding rock reinforcement body is in an entirely plastic state, which indicates that the surrounding rock stability is poor, and the surrounding rock is about to become or has become unstable. Under a certain initial ground stress, reasonable anchor support parameters will cause the surrounding rock reinforcement body to partially yield, but it still can constrain the deformation of surrounding rock and better control the stability of surrounding rock. Therefore, the surrounding rock stability coefficient
When
When
When
The radius of a circular cavern excavation is
Anchor support parameters.
Parameter  Value 


210 

0 

20 

10 

2.4 

1.0 

1.0 

335 
Based on the example of the appeal project, the analytical solution is obtained by using the theory of this paper, and the FLAC^{3D} finite difference program is used for numerical simulation. The analytical solution is compared with the numerical solution, and then the above are compared with the actual monitoring data to verify the rationality of the deduced analytical formula.
The physicalmechanical parameters of the surrounding rock reinforcement body are calculated by bringing the surrounding rock parameters and anchor parameters into equations (
Physicalmechanical parameters of the surrounding rock reinforcement body.
Parameter  Value 


1.510 

0.297 

1.077 

30 
Bringing the surrounding rock parameters and anchor parameters into equations (
It can be known from equation (
Bringing the integral constant obtained above into equation (
It can be seen from equation (
It is known from equations (
It can be seen from Table
The influence of the cavern excavation on surrounding rock is 3∼5 times the width or height of the cavern excavation, so the size of the model is 60 m × 60 m × 1 m. Due to the model's axis symmetry, only a quarter of the mesh model is selected for calculations. The left boundary of the model is constrained by the displacement of the
Numerical calculation model: (a) model size and boundary conditions; (b) anchor arrangement.
Figure
Comparison of the theoretical solution and numerical simulation results: (a) stress comparison; (b) displacement comparison.
In the actual project, the initial design of the anchor spacing is 1.0 × 1.0 m. After actual monitoring, it is determined that the maximum axial force of the anchor is far from the ultimate tensile strength, and the anchor strength is not fully exerted. Therefore, the anchor spacing is adjusted to 1.2 × 1.2 m, and a pretightening force of 100 kN is applied to the anchor. The displacement of the tunnel wall and the axial forces of the anchor under the new support scheme are monitored. To further verify the rationality of the theory, the theoretical solutions under the two support schemes are compared with the actual monitoring data and numerical simulation results, as shown in Table
Comparison of results.
Anchor length (m)  Anchor spacing (m)  The displacement of the tunnel wall (mm)  The axial force of the anchor (kN) 



Theoretical value  Numerical solution  Monitoring value  Theoretical value  Numerical solution  Monitoring value  
Scheme 1  2.4  1.0 × 1.0  47.40  44.61  43.78  100.54  107.21  97.32  0.216 
Scheme 2  2.4  1.2 × 1.2  47.36  44.52  43.14  190.48  192.14  186.23  0.218 
It can be seen from Table
Figure
Influence of the anchor length and spacing on the stability of surrounding rock.
Figure
Influence of the anchor pretightening force and anchor diameter on the stability of the surrounding rock.
Figure
Influence of the rock’s physicalmechanical parameters on the stability of the surrounding rock: (a) elastic modulus; (b) Poisson’s ratio; (c) cohesion; (d) internal friction angle.
Figure
Influence of the rock’s dilatancy angle and intermediate principal stress on the surrounding rock deformation: (a) radius of the surrounding rock reinforcement body damage zone; (b) radial displacement of the tunnel wall.
In order to further verify the rationality and superiority of the theory, the parameters used in the calculation and analysis are taken from the examples used in the articles of Park [
Comparison of results from different authors.
Park and Kim [ 
Ogawa and Lo [ 
Reed [ 
FLAC^{3D}  Meng et al. [ 
This paper  

Wall displacement without anchor support (mm)  50.233  23.902  17.885  64.090  17.238  16.523 
Wall displacement with anchor support (mm)  —  —  —  4.545  5.696  5.347 
Plastic zone radius of surrounding rock without anchor support (m)  5.561  5.509  4.697  5.500  5.561  5.506 
Plastic zone radius of surrounding rock with anchor support  —  —  —  3.000  4.028  3.921 
It can be seen from Table
When there is an anchor support, the displacement of the cave wall calculated in this paper is 16.523 mm, and the displacement of the cave wall calculated by Meng is 17.238 mm. The calculation result in this paper is slightly smaller than that of Meng. The reason is that when the physicomechanical parameters of the anchored rock mass are calculated in this paper, the mechanical coupling effect between the anchor and the surrounding rock is considered. However, Meng only averages the surrounding rock parameters and anchor parameters by area to obtain the physical and mechanical parameters of the composite. Therefore, the calculation process of this paper considers the coordinated deformation of the anchor and the surrounding rock, and the calculation result is closer to the actual situation.
Using the idea of homogenization, the composite of the anchor and the anchored rock mass is regarded as a supporting structure, which is defined as the surrounding rock reinforcement body. A method of the surrounding rock reinforcement body support is proposed, and a mechanical model of the coordinated deformation of the surrounding rock reinforcement body and deep rock mass is established.
By analyzing the mutual coupling effect between the wholly grouted anchor and the anchored rock mass, the expression of the physicalmechanical parameters of the surrounding rock reinforcement body is derived. After the elastoplastic analysis of the mechanical model is conducted, the stress and displacement expressions of the surrounding rock are derived, and the displacement of the tunnel wall, the radius of the surrounding rock reinforcement body damage zone, and the maximum axial force of the anchor are obtained. A method for evaluating the stability of surrounding rock by the degree of damage of the surrounding rock reinforcement body is proposed, and the stability coefficient of surrounding rock is also defined.
Research presented in this paper shows that the anchor support increases the elastic modulus and cohesion of the surrounding rock by 0.67% and 7.70%, respectively, and reduces the Poisson's ratio of the surrounding rock by 1.00%. The anchor support has the greatest influence on the cohesion of the surrounding rock reinforcement body. Under different support schemes, the theoretical calculations agree well with the numerical simulation results, and the difference from the actual monitoring values is less than 10%, which verifies the rationality of the surrounding rock reinforcement body theory and shows the accuracy of the method.
Research presented in this paper also shows that, to control the stability of surrounding rock, the design of the anchor length and spacing should follow the principle of long but sparse and short but dense. However, when the length of the anchor increases to a certain value or the anchor spacing reduces to a certain value, the surrounding rock stability coefficient changes little. Therefore, the parameter optimization of the length and spacing of the anchor should be considered to save the amount of anchors and avoid material waste. The pretightening force of the anchor and the surrounding rock stability coefficient show a linear change law, and its law is not affected by other anchor parameters. However, when the pretightening force increases to a certain value, the anchor loses the supporting effect due to yielding.
The cohesion and internal friction angle of rock have a great influence on the stability of surrounding rock, but the elastic modulus and Poisson’s ratio of the rock mass have little influence on the stability of surrounding rock. The larger the intermediate principal stress value is, the better the selfsupporting capacity of surrounding rock is, which is beneficial for optimizing the anchor support parameters and reducing the support cost. However, the displacement of the tunnel wall and the axial force of the anchor increase sharply as the dilatancy angle of surrounding rock increases, which may cause the anchor to yield and lose its supporting effect, so it is necessary to increase the support strength to ensure the surrounding rock’s stability. Therefore, when designing anchor support parameters and evaluating the surrounding rock’s stability, the dilatancy characteristics and intermediate principal stress effects of the surrounding rock should be comprehensively considered.
Through comparison with numerical simulation, field measurement, and existing theory, the theory and calculation method of this paper are more suitable for actual working conditions. It can provide a theoretical reference for further understanding the reinforcement mechanism of the anchor, provide a new method for the support design of bolts, and quantitatively evaluate the stability of the roadways under the reinforcement of the anchors.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was financially supported by projects (no. 51508462 and 11402195) supported by the National Natural Science Foundation of China and a project (no. 2013JK0961) supported by the Shaanxi Provincial Department of the Education Funding Project.