1 Introduction

The stability of roadways is a necessary condition for ensuring the safety and efficient production of coal mines. The stability of the surrounding rock of underground roadways is weakened, posing a considerable threat to the safety of coal mines [1, 2]. High stress and mining disturbances lead to increasingly complex stress environments [3]. The rock mass deteriorates and destabilizes due to changes in the stress environment [4, 5]. The stress gradient produced by the variation in stress [6] and support stress compensation are the factors that affect the stability of roadway.

The stress gradient refers to the ratio of the stress to the unit distance inside the surrounding rock. Instability or rock damage often occurs in areas with significant stress gradients. WASANTHA et al [7] proposed that there was a gradient change in the stress peak under the influence of the strain rate. ZUO et al [8] and WENG et al [9] reported that the stress gradient changes after excavation, which results in roadway failure. ZHAO et al [10] noted that excavation leads to gradient stress inside the surrounding rock in the radial direction of the tunnel. JIANG et al [11] noted that the adjacent stress zone with a large difference in the surrounding rock is the main area where rock bursts occur. ZHANG et al [12] established a zonal fracture energy damage failure criterion based on strain gradient for analyzing the damage to the surrounding rock. UKRITCHON et al [13] discussed the influence of the linear shear strength gradient ratio on tunnel stability and proposed an approximate solution for the stability of the surrounding rock. SINHA et al [14, 15] reported the bilinear nature of the stress gradient in the yield zone, and noted that Bieniawksi’s strength gradient equation has a limited applicable scope. MARK and IANNACCHIONE [16] estimated the stress gradient at the limit load based on different pillar strength formulas. WILSON [17] hypothesized that the stress within the plastic yield zone increases exponentially. Therefore, the stress gradient should be considered in the safe control of surrounding rock.

Reasonable prestress can ensure support efficiency throughout the whole life of the roadway, avoiding low efficiency in the early stages and failure in later stages. The new austrian tunnelling method (NATM) can make full use of the self-supporting capacity of the surrounding rock [18]. HOEK et al [19] explained the effect of rock bolts on broken rock masses, that is, the range of action of each rock bolt overlaps to form a large-scale interaction zone. To reduce the damage caused by stress gradients to the surrounding rock, active support is applied to regulate the stress field. DONG et al [20] proposed that a loose-zone is generated by the interaction between stress and rock. ZHU et al [21] analyzed the stress distribution of the surrounding rock considering different lateral pressure coefficients, and reported that the stability of the surrounding rock can be achieved by regulating the stress. SONG et al [22, 23] noted that the range of overlying strata that significantly affects stress is limited and controllable at any buried depth. GONG et al [24] noted that prestressed support can generate a support stress field to compensate for the mining stress field. PINTO et al [25] reported that the permanent load is related only to a constant load and prestress in civil engineering, but excessive prestress should be avoided. KANG [26] noted that prestressed rock bolts can effectively control the initial deformation of roadways. LI et al [27] proposed that the distribution of the support stress field was closely related to the prestress and support density. WANG et al [28] reported that the prestress of rock bolts effectively improved the bearing capacity of the surrounding rock. The stability of the surrounding rock is positively correlated with the strength of the rock mass and the stress gradient. Support compensation is beneficial to the design of a reasonable prestress [29]. Prestressing is an important parameter in active support. The application of prestress enables the formation of stress areas in advance [30]. Prestress can effectively reduce the deformation of the lining in a cavern [31].

The support stress field generated by prestressed support affects the redistribution of mining-induced stress, and the stress gradient affects the failure mode of the surrounding rock. To maintain the integrity and self-supporting capacity of the surrounding rock, and reduce the degradation of the rock caused by stress gradients, stress compensation is proposed based on active support. Conventional strength criteria such as the Mohr-Coulomb and Hoek-Brown criteria, which are commonly used in rock engineering, have a narrow range of application [32, 33]. The stress gradient expressions considering different strength criteria are not the same; therefore, solving a more widely applicable stress gradient-based unified strength theory is necessary [34, 35].

Overall, the stress gradient of roadway surrounding rock and reasonable prestress of the anchor were determined. By solving the stress gradient based on unified strength theory, the reasonable prestress of the anchor can be determined, providing an idea with a wider scope of application for quantifying the support of surrounding rock.

2 Stress compensation effect

The whole life of a roadway involves excavation unloading, stress redistribution and a support compensation regulation process. A reasonable prestressed support is one of the key factors for ensuring the stability of the surrounding rock in the balance of the support stress field and the mine-induced stress field. HE et al [36] and YANG et al [37] proposed the support theory of the excavation compensation method, which adopts a high pre-tightening force support strategy to restore and mobilize the strength of a rock mass, but relatively few studies quantify the degree of compensation exist. In this work, a reasonable prestress is applied to control the stress field via stress compensation. Stress compensation can effectively balance and compensate for the stress of the surrounding rock and improve the stability of roadways.

As shown in Figure 1, the stress of the roadway is simplified as a plane model [38], and the stress of the surrounding rock is expressed by the principal stress in three directions. The stress of the surrounding rock is redistributed before and after support is applied in the roadway, and the stability of the surrounding rock changes. The radial stress of the surrounding rock decreases with excavation, whereas the circumferential stress increases. Applying support compensation increases the radial stress and decreases the circumferential stress. The difference between them decreases, and the generation and development of plastic zones are inhibited in the surrounding rock effectively.

Figure 1全屏查看

Principle of stress compensation and regulation

Active support can effectively improve the stress distribution and stability of the surrounding rock. Different degrees of active support have different effects on stress field compensation, as shown in Figure 2. The regulation for active support to stress field compensation is a dynamic process, and the stress field changes with support prestress compensation. The integrity of the surrounding rock is directly related to the distribution of the stress field. The range of broken surrounding rock gradually decreases as the compensation level increases, the stress concentration and rock deformation are reduced, and the integrity is enhanced.

Figure 2全屏查看

Stress compensation and peak transfer of surrounding rock support in roadway

3 Analytical solution for stress gradient of surrounding rock

3.1 Unified strength theory and stress gradient

Plastic yield and failure occur when the stress exceeds the material elastic limit of the surrounding rock. To accurately determine the stress state of the surrounding rock and provide reasonable support, it is necessary to conduct elastoplastic stress analysis to obtain a more practical and applicable strength criterion. The classic strength criteria commonly used for rock include the Mohr-Coulomb criterion (M-C), the Hoek-Brown criterion (H-B), and the Drucker-Prager criterion (D-P). The influence of intermediate principal stresses is ignored in the M-C and H-B strength criteria [39, 40]. The M-C strength criterion assumes elastic-plastic materials with perfect plasticity after yielding, which is inaccurate in practice. Compared with the M-C and H-B strength criteria, the D-P strength criterion considers the influence of intermediate principal stresses and the effect of hydrostatic pressure. The D-P model can also better describe the nonlinear characteristics of materials [41].

Compared with the D-P strength criterion, the unified strength criterion is a series of yield and failure criteria, with a unified mechanical model and unified mathematical expressions. It can be applied to a variety of materials, reflecting the influence of intermediate principal stress on the yield and failure of rock materials [42, 43]. This finding indicates that rock failure is caused by the maximum shear stress and the corresponding normal stress [44]. Unified strength theory can not only better match the actual results by adjusting the intermediate principal stress, but also has a piecewise linear expression that is convenient to apply. The general form of unified strength theory is as follows [35]:

(1) (2)

The material parameters β and C can be obtained from the tensile test () and compression test ():

(3)

where β and C are the material parameters; b is the intermediate principal stress parameter, reflecting the influence on material failure; n is the nonlinear coefficient of failure criterion; α is the tension-compression strength ratio of material; σ_t is the tensile strength of material. Different values of b correspond to different series of strength criteria. The effect of the strength criterion [45, 46] can lead to differences in the strength results of rock considering different strength criteria. Choosing different strength criteria with different intermediate principal stress parameters based on the actual conditions can effectively avoid the adverse consequences of underestimating or overestimating the mechanical properties of the surrounding rock. An appropriate and unified yield criterion is highly important for support design and optimization in roadway surrounding rock [47, 48].

The stress gradient of roadway surrounding rock is a key parameter, which describes the change of internal stress of surrounding rock with distance or depth. The stress gradient can be expressed by a mathematical derivative, that is, the partial derivative of the stress with respect to the position. In the Cartesian coordinate system, the stress function of each point in the surrounding rock of the roadway is set as σ(x, y, z) which represents a scalar field. The partial derivatives in the x, y and z directions are obtained for the scalar field, and the stress gradient can be obtained. The gradient of the stress field is a vector field :

(4)

The stress gradient function is defined as:

(5)

3.2 Elastic-plastic analytical solution for roadway surrounding rock

Underground engineering is common in coal mine tunnels, transportation and hydropower projects. Tunnels are relatively long along the axial direction and subjected to loads that do not change along the axial direction. Therefore, the stress analysis of a tunnel is simplified to a plane strain problem [38].

Taking a circular roadway as an example, the stress gradient of the surrounding rock of an underground roadway and rational support prestress are determined. A mechanical model is established to perform elastoplastic stress analysis for a roadway (Figure 3), with the following basic assumptions: the roadway section is a circle of infinite length; the surrounding rock is continuous, uniform and isotropic; and the burial depth of the roadway is more than 20 times the radius of the roadway. The lateral pressure coefficient is 1, r₀ is the radius of the roadway, P₀ is the in-situ stress, Pi is the support force, r_p is the radius of the plastic zone, σ_p is the radial stress at the junction of the plastic zone and the elastic zone, and are respectively the displacement of plastic zone and elastic zone, and are respectively the radial stress and circumferential stress of plastic zone and elastic zone, and are respectively the radii of the elastic zone.

Figure 3全屏查看

Mechanical model of circular tunnel excavation

Unified strength theory has various expressions [49]. For rock materials, it can be expressed via the principal stresses , , and , cohesion and rock internal friction angle :

When ,

(6)

When ,

(7)

There are only four stress components and because in plane strain problems, and the three principal stresses are as follows:

(8)

where is between and , that is , then [49]. An intermediate coefficient m is introduced to represent the relationship between and :

(9)

The value of m can be appropriately selected based on tests. In the case of plane strain, the value of m cannot be greater than 1 [50].

(10)

When the surrounding rock is in a plastic state, the value of m tends to be 1, so:

(11)

For convenience, the value of m is set to 1. The applicable assumption is as follows:

(12)

Therefore, Eq. (6) is taken as the expression of unified strength theory. Substituting Eq. (10) into Eq. (6) yields:

(13)

In rock mechanics, tensile stress is negative, whereas compressive stress is positive in the direction of stress. Therefore,

(14)

Then, Eq. (6) can be transformed into:

(15)

In the model, are orthogonal and are identified as the principal stresses in three directions. The elastic-plastic stress gradient solution for the model is as follows:

1) Plastic zone

The circumferential stress is the highest, and the radial stress is the lowest in the surrounding rock. It is as follows:

, , (16) (17)

Without the body force, the equilibrium equation for the plane strain problem is as follows:

(18)

Substituting Eqs. (16) and (17) into Eq. (14) yields:

(19)

If

(20)

then

(21)

Substituting Eq. (21) into Eq. (18) yields:

(22)

Taking logarithms on both sides of the equation yields:

(23)

where is a function without . The following is known from the model boundary conditions (, ):

(24)

Substituting Eq. (24) into Eq. (23) yields:

(25)

Combining Eqs. (17), (21) and (25), the stress is as follows:

(26)

The stress gradient is as follows:

(27)

Under static water pressure conditions, the relationship between the stress and the original rock stress at the boundary between the plastic and elastic zones is as follows:

(28)

Combining Eqs. (21) and (28), the stress at the boundary () is as follows:

(29)

Combining Eqs. (26) and (29), the radius of the plastic zone is as follows:

(30)

2) Elastic zone

For the surrounding rock in the elastic zone (), the stress is as follows:

(31)

Combining Eqs. (29) and (31), the stress is as follows:

(32)

Setting , the stress is as follows:

(33)

The stress gradient is as follows:

(34)

3) Displacement in the elastic-plastic zone

For the surrounding rock in the elastic zone (), the displacement given by ZHENG [51] is as follows:

(35)

where G is the shear modulus.

Combining Eqs. (29) and (35), the displacement is as follows:

(36)

For the surrounding rock in the plastic zone (), assume that the rock mass has a strain of zero in the plastic zone. The displacement is as follows:

(37) (38)

Combining Eqs. (37) and (38), is obtained. Another expression is as follows:

(39)

where A is an undetermined coefficient. The deformation coordination condition is given at the boundary ():

(40)

Combining Eqs. (35), (39) and (40), the coefficient A is obtained as follow:

(41)

Substituting Eq. (41) into Eq. (39), the displacement of the plastic is as follows:

(42)

The displacements at the boundary () of the elastic-plastic zone is equal.

3.3 Verification and analysis

1) Verification via reference

The comparison between the analytical solution calculated in this paper and the examples in Ref. [52] is shown in Figure 4. Data filtering was performed to remove data with large deviations, and the average relative errors were 1.99%, 2.73%, 2.72% and 2.51% at b=1, 0.5, 0.25 and 0, respectively.

Figure 4全屏查看

Stress comparison of the analytical solution and examples in the literature: (a) Results with different b; (b) Relative difference between the analytical solution and example results

2) Verification via numerical simulation

ABAQUS with the embedded Mohr-Coulomb criterion was used to establish a model of a circular roadway with size of 60 m×60 m (length×width). The boundary was fixed. The rock parameters are shown in Table 1, where E is elastic modulus, μ is Poisson ratio, and c is cohesion the values were measured by field test. The model is used to verify the elastoplastic analytical solution when b=0.

Table 1全屏查看

Rock mechanical parameters

E/GPa	G/GPa	μ	c/MPa
0.4	0.145	0.39	0.34
φ/(°)	r0/m	P0/MPa	Pi/MPa
30	2.0	15	0, 0.1, 0.2, 0.5

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A comparison of the plastic radii of the surrounding rock determined via the numerical simulation and the analytical solution is shown in Figure 5. The numerical simulation indicates that the plastic radius is 7.409 m. The plastic radius solved via the analytical solution is 7.276 m. The relative error between the two is 1.828%, which may be caused by the model size and grid number of the model. The larger the model size is, the smaller the error.

Figure 5全屏查看

Circumferential stress of the surrounding rock and its fitting results with the analytical solution

The radial stress of the surrounding rock and its fitting results with the analytical solution are shown in Figure 6. The trend of the radial stress in the surrounding rock is basically consistent, with similar peak values. The average relative error is 2.71%. The fitting results are influenced by the size of the ABAQUS model. The larger the model size is, the better the fitting results.

Figure 6全屏查看

Radial stress of the surrounding rock and its fitting results with the analytical solution

The numerical solution and theoretical analytical solution of radial stress for roadway surrounding rock considering different prestress are compared and analyzed (Figure 7). The radial stress of the surrounding rock increases with the increase of prestress. The numerical solution is basically consistent with the theoretical analytical solution, and the relative error between the two is 2.38%.

Figure 7全屏查看

Radial stress of surrounding rock under different support forces (Pi)

3) Verification via an in situ test

The burial depth of the 1512 working face in a coal mine is approximately 400 m, and the surrounding rock of the roadway is dominated by mudstone and sandy mudstone. The rock mechanical parameters obtained by field sampling test are shown in Table 2. The in situ measured maximum horizontal stress is 9.41-10.13 MPa, the vertical stress is approximately 10 MPa, and the lateral pressure coefficient is approximately 1.

Table 2全屏查看

Mechanical parameters of the roadway surrounding rock in the working face

E/GPa	G/GPa	μ	c/MPa	φ/(°)	r0/m	P0/MPa	Pi/MPa
3.5	4.13	0.25	0.4	30	2.0	10	0.173

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Based on the mechanical parameters of the surrounding rock, the elastic-plastic analytical solution is applied to calculate the radius of the plastic zone. Because the calculation neglects b (b=0) or exaggerates b (b=1), resulting in the intermediate principal stress deviating from the actual value, b=0.4-0.6 is selected. The calculation results show that the radius of the plastic zone is 4.0-4.2 m, which means that the plastic range of the surrounding rock is 2.0-2.2 m, as shown in Figure 8.

Figure 8全屏查看

Comparison of the theoretical plastic zone range and the borehole images

The method of borehole imaging is used to analyze rock fracture development. The surrounding rock of the roadway can be divided into three zones: the broken zone, the fractured zone, and the intact zone. The plastic zone mainly includes the broken zone (0-2.0 m) and the fractured zone (2.0-2.4 m), which means that the plastic zone is approximately 2.4 m. Due to the limitations of the site conditions, there are some deviations between the actual measurements and the theoretical results.

The correctness of the analytical solutions for the surrounding rock based on unified strength theory was checked. The results verify the practicality and accuracy of the elastic-plastic analytical solution for the surrounding rock based on the unified strength theory.

3.4 An approximate stress solution for non-circular roadway surrounding rock

Compared with the circular roadway, the solution of the plane elastic problem for the non-circular roadway is much more complicated. To make it easier to apply in engineering, the equivalent radius method is applied to stress of non-circular roadway, that is, a virtual circle with the same area as the non-circular roadway is used to simplify the non-circular roadway [53, 54]. The radius of the virtual circle represents the equivalent radius of the non-circular roadway. Using the equivalent radius as the feature size, the stress distribution can be analyzed more accurately to ensure the safety of the project.

The modified calculation formula of virtual circle radius of non-circular roadway [55] is as follows:

(43)

where is the equivalent radius of the non-circular roadway; k^* is the correction coefficient of different shape roadways [56] (Table 3); S^* is non-circular roadway cross-sectional area.

Table 3全屏查看

Correction coefficient of roadways with different shapes

Ellipse	Arch	Square	Rectangle	Trapezium
1.05	1.10	1.15	1.20	1.20

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4 Results and discussion

The intermediate principal stress parameter represents the degree of influence of the intermediate principal stress on the surrounding rock. Unified strength theory with parameter b ranging 0-1 can be transformed into a series of failure criteria [44, 57].

When b=0, the M-C criterion ignores the intermediate principal stress. When 0<b1, it is a series of strength criteria that reflect the effects of different intermediate principal stresses [58]. When b=1, it is the twin shear strength theory, which can further deduce the D-P criterion.

The value of b is related to the shear strength, tensile strength and compressive strength of the surrounding rock. The influence of different b values and prestresses on roadway stability was studied to explore more applicable strength criteria and support prestress.

4.1 Stress evolution law of the surrounding rock

The stress law of the roadway surrounding rock with different b values and without support is shown in Figure 9. The variation trend of the radial stress with different b values is consistent and increases as the b value increases. The stress difference for different b values first increases and then decreases along the radial direction, with a large difference within 5 times the radius of the roadway. In the far-field elastic zone (), the stress difference approaches 0. In the plastic zone, the stress gradient tends to increase. The stress gradient gradually decreases after it enters the elastic zone. In the far-field elastic zone, the stress gradient approaches 0. The strength criterion corresponding to b=0 underestimates the radial stress in the plastic zone of the surrounding rock.

Figure 9全屏查看

Stress and stress gradient of the surrounding rock with different b values and without support: (a) Radial stress; (b) Circumferential stress

As shown in Figure 9(b), the circumferential stress of the surrounding rock is affected by the plastic zone, and the laws in the elastic and plastic zones are significantly different. In the plastic zone, the circumferential stress is similar to the radial stress and increases with increasing b value. As the b value approaches 1, the difference in peak circumferential stress gradually decreases; that is, the influence of the b value on the peak circumferential stress weakens. In the elastic zone, the trends of the circumferential stress and radial stress are opposite and decrease with increasing b value. The absolute value of the stress gradient gradually decreases, and the stress gradient in the far-field elastic zone approaches 0.

To analyze the influence of prestress on the stress of surrounding rock, b=0.5 is taken as an example, as shown in Figure 10. The radial stress increases with increasing prestress. In the plastic zone, the circumferential stress increases with increasing prestress and decreases in the elastic zone. The peak stress is equal, and the stress increment at different prestress levels gradually decreases. The greater the prestress is, the closer the peak stress to the roadway center, and the smaller the stress concentration zone.

Figure 10全屏查看

Relationship between the surrounding rock stress and radial distance with different support forces (b=0.5): (a) Radial stress; (b) Circumferential stress

Taking b=0.5 as an example, the effect of prestress on the radial stress gradient was analyzed, as shown in Figure 11. With increasing prestress, the stress gradient gradually increases in the plastic zone, and the increment of the peak stress gradient gradually decreases. This results in a gradual weakening of the support effect. Therefore, a reasonable prestress can ensure the stability of the surrounding rock.

Figure 11全屏查看

Variations in the radial stress gradient of the surrounding rock with different support forces (b=0.5): (a) Radial stress gradient; (b) Peak radial stress gradient variation

4.2 Displacement analysis of surrounding rock

Figure 12 shows the variation in the displacement of the surrounding rock with different b values and prestresses. As shown in Figure 12(a), as b approaches 1, the displacement gradually decreases. At the roadway sidewalls, the displacement difference for different b values is the largest. This indicates that the results with the M-C criterion (b=0) are relatively conservative and that the surrounding rock is predicted to be severely damaged. Correspondingly, the criterion of b=1 underestimates the deformation of the surrounding rock.

Figure 12全屏查看

Variation in the displacement of the roadway with different b values: (a) With increasing radial distance; (b) With increasing prestress

At the roadway sidewalls, the roadway displacement varies with the prestress for different b values, as shown in Figure 12(b). When the prestresses are the same, the displacement of the roadway decreases as the b value increases, this indicates that the intermediate principal stress can effectively reduce the range of the plastic zone. The effect of the strength criterion is not significant. The displacement decreases with increasing prestress, the increment decreases, and the prestress effect weakens.

4.3 Plastic zone of surrounding rock

Figure 13 shows the influence of b values on the plastic zone. The radius of the plastic zone decreases with increasing b value. This indicates that the intermediate principal stress can effectively inhibit the development of the plastic zone. When b=0, the radius of the plastic zone is the largest. This result is relatively conservative, but the cost increases. When b=1, the radius of the plastic zone is the smallest. The result is relatively dangerous.

Figure 13全屏查看

Influence of b values on the plastic zone

Figure 14 shows the law of the plastic zone with different prestresses. As the prestress increases, the radius of the plastic zone gradually decreases within a certain range. This finding indicates that the plastic zone does not continue to decrease with increasing the prestress. Ideally, when the prestress is equal to the horizontal stress, there is no plastic zone in the surrounding rock of the roadway.

Figure 14全屏查看

Law of the plastic zone with different support forces

The intermediate principal stress parameter affects the stability of the surrounding rock. The strength criterion is more realistic when the b value is estimated based on actual conditions. The application of prestress enhances the stability of the surrounding rock. Within a certain range, an increase in prestress is beneficial to the stability of the surrounding rock.

5 Determination of prestress

The effect of support on the surrounding rock gradually weakens and tends to be constant with increasing prestress. Ideally, the triaxial stress state reappears under the support. Therefore, a reasonable prestress can ensure the stability of the surrounding rock [59]. A reduction in cost and improvement in efficiency are achieved.

The prestress anchorage mainly improves the stability and bearing capacity of the surrounding rock by applying the active prestress. As shown in Figure 15, there is a continuous and overall distribution of compressive stress in the anchorage section in the roof with support. The area where each rock bolt plays a role is superimposed continuously. The rock bolt has the effect of resisting sliding, resisting separation and resisting shear on the surrounding rock. The stress state of the surrounding rock is improved by the rock bolts, and three-dimensional stress state of that is restored. The sliding, shear and tensile deformation of the sidewalls are limited by lateral support force generated by the rock bolt. The sliding and separation of the roof and floor are resisted by rock bolts.

Figure 15全屏查看

Relation between the surrounding rock and rock bolts: (a) Compression arch; (b) Mechanical model of rock bolt support [26]; (c) Support changes the stress state of the roof; (d) Support changes the stress state of the sidewall

For analysis, b=0.5 is taken as an example. The spacing and volume of the rock microelements are smaller than those in the mining damage state. The interaction force between the microelements increases, resulting in an increase in the stress gradient. The stress gradient of the surrounding rock with different prestresses is shown in Figure 16(a). The radial stress gradient is 0.71 MPa/m at the sidewalls without support. The zone with a radial stress gradient greater than 0.71 MPa/m is selected as the stress influence zone. The initial stress gradient at the sidewalls, the peak stress gradient, the stress influence zone and the radius of the plastic zone are plotted in Figure 16(b). The point of the peak stress gradient coincides with the elastic-plastic interface of the surrounding rock. The stress gradient gradually increases with increasing stress in the plastic zone, and the stress gradient gradually decreases in the elastic zone.When the prestress increases from 0 to 4 MPa, the peak stress gradient gradually increases from 3.02 to 7.10 MPa/m, and the increment of the peak stress gradient gradually decreases. The peak stress gradient position shifts toward the sidewalls, the radius of the plastic zone decreases from 5.64 to 2.40 m, and the displacement decreases from 0.47 to 0.08 m. The range of stress concentration gradually decreases from 9.12 to 5.16 m. This finding indicates that the prestress can effectively reduce the range of surrounding rock deterioration and significantly improve the stability of the surrounding rock [60]. The increment of the displacement and plastic zone area was discussed to evaluate the balance between the prestress and support effects (Figures 17 and 18).

Figure 16全屏查看

Evolution of the stress gradient with different support forces: (a) Stress gradient distribution; (b) Key-point analysis

Figure 17全屏查看

Relationships between the support forces and displacement increment: (a) ; (b)

Figure 18全屏查看

Relationships between the support forces and plastic zone area increment: (a) ; (b)

The displacement of the roadway is the largest without support. To evaluate the support effect and stability of the surrounding rock after support compensation, the displacement increment is defined:

(44)

The displacement increment characterizes the effect of the support system with increasing prestress. The displacement increment gradually decreases as the prestress increases. The larger the displacement increment is, the more significant the effect of the support system on the surrounding rock. To more easily determine the correlation between prestress and the support effect, is used to represent the effect of the support system during the increase in prestress.

(45)

The support effect weakens and the surrounding rock becomes more stable as increases. Similarly, the change in the plastic zone area of the surrounding rock with different prestresses was discussed to further indicate the effect of the support system and solve for a reasonable prestress.

As the prestress increases, the displacement and plastic zone area of the surrounding rock gradually decrease, and the effect of the support system gradually weakens. At the point of displacement balance, the prestress is 0.51 MPa, and the displacement of the surrounding rock is 40.5% lower than that in the condition without support. The deformation compensation ratio of the roadway is 40.5%, and the area of the plastic zone is reduced by 46.4%. After the point of displacement balance, the effect of prestress is significantly weakened. At the balance point of the plastic zone area, the prestress is 0.46 MPa. At this time, the displacement of the surrounding rock is reduced by 38.2%, and the plastic zone area is reduced by 43.7%.

The balance point of the plastic zone area appears before the displacement, which indicates that the prestress first controls the plastic zone before controlling the displacement. When the plastic zone of the surrounding rock tends to be stable, the surrounding rock is still subject to small displacement under far-field stress conditions. The plastic zone area gradually decreases, and the deformation compensation ratio of the roadway gradually increases. The correlation between the two is shown in Figure 19.

Figure 19全屏查看

Relationships between the plastic zone and deformation compensation ratio with different support forces

Prestress compensation is positively correlated with the stress gradient of the surrounding rock, as shown in Figure 20. The increase in the compensation coefficient gradually decreases. The relationship between the compensation coefficient and the prestress is obtained as follows:

(46)

Figure 20全屏查看

Stress gradient compensation coefficient of the surrounding rock

where is the compensation coefficient, and is the support force.

The support system of the roadway is mainly composed of trays, rock bolts and cable bolts. The stress of the support system on the surrounding rock is achieved by applying a prestress to the bolts. The support strength required for the stability of the surrounding rock is assumed to be entirely undertaken by the rock bolts and cable bolts. In practical engineering applications, the force of the support system is equal to the ratio of the point load to the effective support area.

(47)

where is the sum of the point loads of rock bolts and cable bolts; is the effective support area; n is the number of bolts; F is the bolt point load (prestress); and are the arranged spacings between rows.

The fraction of prestress applied to the rock bolts and cable bolts is as follows:

(48)

where is the yield load of the bolts.

In summary, it is possible to calculate the optimal prestress and compensation coefficient, providing a reference for support design under the same or similar conditions.

The stress gradient solution for the surrounding rock in this study, which is based on unified strength theory, is derived from an isostatic circular tunnel model with a lateral pressure coefficient of 1. The elastoplastic solution and stress gradient solution of circular/deformed tunnels of unequal pressure under the influence of lateral pressure will be explored.

6 Conclusions

A reasonable prestress and stress gradient are beneficial to the stability of roadway surrounding rock. The nonuniform stress field of a roadway produces a stress gradient, which leads to deformation and stress changes in the surrounding rock. The stress field is regulated by applying prestress compensation to the support system. It maintains and strengthens the self-bearing capacity and stability of the surrounding rock.

1) The elastoplastic analytical solution based on unified strength theory has wider applicability. Using unified strength theory, the mechanics analytical solution in the plastic zone and the elastic zone under plane strain conditions is derived. A stress gradient solution is proposed. It provides a method and idea for stress analysis of roadways with different shapes. The stress, displacement, and plastic zone of the surrounding rock obtained from the analytical solution agree with those of the reference, numerical simulation and in situ data; this verifies the correctness and practicality of the solution.

2) The relationships between the stress gradient and the roadway surrounding rock damage are studied. The intermediate principal stress parameter and prestresses are positively correlated with the stress gradient in the surrounding rock and negatively correlated with the plastic zone and displacement.

3) Reasonable prestress can ensure the stability of the surrounding rock. The displacement increment and plastic zone area increment are established to characterize the support effect under different prestress values. The balance point of the plastic zone appears prior to that of the displacement, with a deformation compensation ratio of 40.5% and a plastic zone area reduction of 46.4%. A compensation relationship between the prestress and stress gradient is obtained, and a compensation coefficient is proposed to quantitatively determine the applied value of the prestress.

4) The results provide an approach for designing and economical surrounding rock support.