Analytical Solution of a Circular Opening considering Nonuniform Pressure and Its Engineering Application

Based on the Mohr–Coulomb criterion, a new analytical solution of a circular opening under nonuniform pressure was presented, which considered rock dilatancy effect and elastic-brittle-plastic failure characteristics. In the plastic zone, the attenuation of Young’s modulus was considered using a radius-dependent model (RDM), and solution of the radius and radial displacement of plastic zone was obtained. (e results show that many factors have important impact on the response of the surrounding rock, including lateral pressure coefficient, dilation coefficient, buried depth, and Young’s modulus attenuation. Under nonuniform pressure condition, the distribution of plastic zone and deformation around the opening show obvious nonuniform characteristic: with the increasing of lateral pressure coefficient, the range of plastic zone and deformation decrease gradually at side, while they increase at roof and floor, and the location of the maximum value of support and surrounding rock response curve transfers from side to roof. Based on the analytical results and engineering practice, an optimization method of support design was proposed for the circular opening under nonuniform pressure.


Introduction
It is important to predict the distribution of stress and deformation accurately for a deep circular opening, which is the basis for evaluating the stability of surrounding rock and reliability of support design in deep mining and other underground engineering. Many efforts have been paid on this topic, based on the linear Mohr-Coulomb (M-C) criterion or nonlinear Hoek-Brown (H-B) criterion, considering ideal elastic-plastic model (EPM), elastic-brittle-plastic model (EBM), or strain-softening model [1][2][3][4][5][6][7][8][9][10][11]. In the previous researches, the stress was usually assumed to be uniform, and the influence of lateral pressure coefficient on the stressstate distribution of surrounding rock was ignored. However, many engineering practices indicate that the gravity stress of the overlying strata and the tectonic stress usually lead to the nonuniform stress distribution in the surrounding rock [12][13][14][15][16]. Some theoretical analysis and numerical simulation results indicate that the lateral pressure coefficient has a remarkable influence on the stress state of surrounding rock [17][18][19]. erefore, the research results without considering the effect of lateral pressure coefficient are inconsistent with the engineering conditions. In the present research, a mechanical model for circular opening with nonuniform confining pressure is first proposed. en, the impacts of the lateral pressure coefficient on the distribution of plastic zone in the surrounding rock and the response curve of the support system were analyzed.
When the surrounding rock undergoes plastic deformation, the mechanical properties of the rock mass may change due to rock damage, such as the attenuations of internal friction angle, cohesion, Young's modulus, and Poisson's ratio. As for Young's modulus, at present, there are two methods to define the attenuation process for the postpeak rock. In the first method, Young's modulus of the postpeak rock is determined by confining pressure and minimum principal stress, which is named the pressure-dependent Young's modulus model (PDM) [20][21][22]. In the other method, Young's modulus decreases with some function relationship along the radius direction of the surrounding rock, which is called the radius-dependent Young's modulus model (RDM) [23][24][25][26]. e RDM is regarded as the extension of PDM. It is thought that the close relationship between Young's modulus attenuation and rock fracture can explain efficiently the large deformation problems of surrounding rock in the deep underground engineering [27]. In the present work, the RDM is adopted to account for the effect of Young's modulus attenuation. In addition, some scholars regard the surrounding rock as viscoelastic model [28][29][30][31][32][33] and deduce a series of simplified formulas for lining pressure and tunnel convergence, which provides an effective method for predicting the time-varying displacement and stress field of surrounding rock of deepburied circular tunnel. Wu and Shao [34][35][36] proposed using flexible layer to deal with large deformation of tunnel in viscoelastic rock mass. eory and engineering practice shows that flexible layer has a good absorption effect on rock rheological deformation.
Based on the proposed mechanical model, the closedform solutions of stress, deformation, and radius of plastic zone of the surrounding rock in circle opening with nonuniform confining pressure was obtained. e influence of the lateral pressure coefficient, dilation coefficient, and Young' modulus attenuation on the response of the surrounding rock was studied.
en, according to the engineering practices in different geological conditions, a support design method was proposed considering nonuniform stress state.  Figure 1, a circular opening with a radius of r 0 is excavated in the uniform, isotropic, and continuous rock media. e internal support pressure (p in ) acts on the location of r 0 , and the vertical stress (σ 0 ) and the lateral pressure (λσ 0 ) act at the outer boundaries, where λ is the lateral pressure coefficient. e stress of surrounding rock redistributes after excavation. When the stress reaches the yield strength of the rock mass, plastic deformation will occur, and the radius of plastic zone is represented by r j in Figure 1.

Analytical Solution of the Circular Opening
In the paper, the linear M-C yield criterion was adopted as yield condition of surrounding rock, and it is expressed as [5,8,23] where σ 1 and σ 3 are the maximum stress and minimum principal stress, respectively, Y � 2c cos φ/(1 − sin φ), ξ � (1 + sin φ)/(1 − sin φ), and c and φ are the cohesion and internal friction angle of the rock, respectively. For the plane strain problem of the circular tunnel, the hoop stress (σ θ ) and radial stress (σ r ) are the maximum and minimum principal stress, respectively [17], which means that σ 1 � σ θ and σ 3 � σ r . en, equation (1) could be represented as Considering the elastic-brittle characteristic of rock mass, there is a stress drop after the peak strength [5,9,10], after which the rock mass undergoes postpeak-flow state. In the postpeak region, the residual rock parameters are used (residual cohesion c r , residual internal frictional angle φ r , and residual Poisson's ratio ] r ).

Attenuation of Young's Modulus.
e mechanical behavior of surrounding rock is closely related with its damage in the plastic zone. e higher damage usually leads to larger deformation ability. en, the attenuation of Young's modulus should be considered in the plastic zone. In general, the damage of the rock mass can be expressed by the attenuation of Young's Modulus. Based on previous researches [23,24], a radius-dependent Young's modulus attenuation model (RDM) with a power function is adopted here as follows: where E and E r are the initial and residual Young's modulus, respectively, α � log (E/E r )/log(r j /r 0 ), if α � 0 and E � E r . Equation (3) shows that the attenuation of Young's modulus is not only related to the initial and residual Young's modulus but also closely related to the radius of the plastic zone. Compared with the solution presented by Zhang and Ewy [23,24], the solution of this paper is obtained under nonuniform pressure, and the influence of lateral pressure coefficient on the surrounding rock state is considered. erefore, the radius of plastic zone in different positions around the surrounding rock varies, and the attenuation degree of Young's modulus is also nonuniform around the opening.
According to equation (3), at the boundaries of the plastic zone, Young's modulus should be e strain-displacement relationship is For the plane strain problem, the constitute equation, i.e., Hooke's law, is where ] is Poisson's ratio.

Solution of Stress and Deformation in Elastic Zone.
To solve the problems of a circular opening under nonuniform stress field, the stress conditions at boundaries are divided into two parts [37], as shown in Figure 2. e first part (Figure 2(a)) is uniform compressive stress field with a value of σ u 0 � (1 + λ)σ 0 /2; and the second one ( Figure 2(b)) is the stress field with horizontal tensile stress and vertical compressive stress, the value is σ n 0 � (1 − λ)σ 0 /2. In addition, R represents an infinite position that the stress state is equal to in situ stress.

Solution of Uniform Compressive Stress Field.
Computational mechanics model of uniform compressive stress field is presented in Figure 2(a). Based on elastic theory, the stress of the elastic zone under uniform pressure is obtained on the consideration of stress boundary conditions σ re � p y on elastic-plastic interface r � r j (where p y is the radial stress in elastic-plastic interface): where the subscript notation e represents the elastic zone. Figure 2(b), for a point in the surrounding rock with the position of r � R, the stress at the point is original rock stress, and the stress boundary conditions are σ x � − (1 − λ)σ 0 /2, σ y � (1 − λ)σ 0 /2, and τ xy � 0. So the stress expression in polar the coordinate is as follows:

Solution of Stress Field with Horizontal Tensile Stress and Vertical Compressive Stress. As shown in
Equation (9) gives the outer boundary conditions at the position of r � R. And the internal boundary condition under r � r j is shown as follows: From equations (9) and (10), using the semi-retro-solution method [37], the stress function in the elastic zone is Substituting equation (11) into the compatible equation in the polar coordinate system, i.e., ((z 2 /zr 2 ) + (1/r) + (z/zr) + (1/r 2 ) + (z 2 /zθ 2 ))F � 0, the expression of stress function is determined as follows: where A 1 , A 2 , A 3 , and A 4 are all constants. e stress component is solved by equation (12), combined with the boundary condition expressions (9) and (10). e result is as follows: By superposing the two stress fields mentioned above, the stress equation of elastic zone of a circular opening under nonuniform stress field can be obtained:

Advances in Civil Engineering
When λ � 1, equation (14a) will be transferred into the stress expression in the elastic zone under uniform pressure, as given in equation (14b), which is consistent with the results of other researchers [5]: Substituting equations (7) and (14a) into equation (6), the displacement of the elastic zone on consideration of stress instantaneous release is where When r � r j , the radial displacement at elastoplastic interface is As the surrounding rock is at the state of critical yield stress at elastoplastic interface, its circumferential and radial stresses both satisfy the yield criterion at peak point. en, substituting equation (14a) into equation (2), the radial contact stress of elastoplastic zone with M-C criterion is When λ � 1, equation (17) can be transferred into radial contact stress in the elastoplastic zone with uniform pressure as given in equation (18), which is the same with the result given by Park [5]:

Stress and Radius of Plastic
Zone. e stress of the plastic zone should satisfy the yield criterion and equilibrium equation under the postpeak state. By combining equations (2) and (5) with the stress boundary condition σ rp � p in in the position of r � r 0 , the expression of stress can be deduced as follows:.

Plastic zone
Elastic zone Plastic zone Elastic zone  Advances in Civil Engineering where the subscript notation p represents the plastic zone, From equations (14a) and (14b), it shows that the radius of the plastic zone should be determined first to obtain the closed analytical solution of the stress in the surrounding rock. According to the boundary conditions at the elastoplastic interface, r � r j , and (σ rp ) r�rj � p y 1 , the radius of the plastic zone is proposed by combining equations (17) and (19):

Displacement of Plastic Zone.
It is assumed that the circumferential strain and radial strain in the plastic zone are both composed of elastic and plastic strain, and the total strain is where ε rp and ε θp are the total radial and circumferential strain, respectively; ε p r and ε p θ are the radial and circumferential plastic strain, respectively; and ε e r and ε e θ are the radial and circumferential elastic strain, respectively.
In any cases, the plastic deformation of rock should satisfy the flow rule, which is determined by the plastic potential function. Corresponding to the Mohr-Coulomb yield criterion, the following potential function is used: where β r � (1 + sin ψ)/(1 − sin ψ) and ψ is the dilation angle of rock. If the dilation angle is not equal to the frictional angle, a nonassociated flow rule is used. Based on plastic potential theory, the circumferential and radial plastic stresses in plastic zone are where λ p is the parameter related to plastic strain. Substituting equation (22) into equation (23), the plastic strain in the plastic zone should satisfy the following relationship: And substituting equations (21) and (24) into equation (6), the differential equation of radial displacement of the plastic zone is where g(r) � ε e r + β r ε e θ . According to radial displacement continuous condition u rp � u r j in the position of r � r j , the radial displacement of the plastic zone is obtained by solving equation (25): According to equation (26), it may be seen that the displacement of the plastic zone is closely related to the elastic stain g(r). By taking into account Young's modulus attenuation along the radial direction, the elastic strain at a specific position in the plastic zone can be expressed as follows: Substituting equation (27) into equation g(r) � ε e r + β r ε e θ , the expression of g(r) is Substituting equation (19) into (28), equation g (r) with M-C criterion can be obtained: where From equations (26) and (29), it is known that the deformation of surrounding rock is closely related with Young's modulus in the plastic zone. In order to analyze the influence of different attenuation functions of Young's modulus on the rock deformation, based on the RDM in equation (3), three expressions for Young's modulus attenuation are proposed: (i) Case 1: when α � 0 and E � E (r) � E r , Young's modulus of rock in the plastic zone is the initial one, which means that the effect of Young's modulus attenuation on rock deformation is ignored (ii) Case 2: when α � 0 and E ≠ E (r) � E r , Young's modulus of rock in the plastic zone is the residual one, which is a constant Advances in Civil Engineering (iii) Case 3: when α > 0 and E (r) � E r (r/r 0 ) α , Young's modulus of rock in the plastic zone decreases, as defined in RDM For Cases 1 and 2, substituting equations (29) into (26), the displacement of the plastic zone without considering the RDM is where and f 2 (r j ) � (r β r +1 j /1 + β r ). Similarly, with Case 3, substituting equations (3) and (29) into (26), the displacement of the plastic zone considering the RDM is where

Model Validation and Comparison.
Comparing with the closed-form solutions in the literature, the presented work takes lateral pressure coefficient and Young's modulus attenuation into consideration. When λ � 1, α � 0, and v � v r , the model provides solution for the condition of uniform stress and constant Young's modulus, which has been studied by Park [5]. To verify the accuracy of the proposed model, results of the proposed model and Park's solution were compared, as shown in Figure 3. e geometric and physical parameters are listed in Table 1. As shown in Figure 3, when λ � 1, α � 0, and v � v r � 0.2, the proposed model in the present study provides the same result as that of Park's research. erefore, the result of Park's research is a special case of present work.

Effect of Lateral Pressure Coefficient on the Distribution of Plastic Zone.
e elastoplastic radial contact stress and the radius of the plastic zone under nonuniform pressure can be obtained by equations (17) and (19). e distribution of the plastic zone is shown in Figure 4, which shows that the lateral pressure coefficient has important influence on the distribution of the plastic zone. When 0 < λ < 1, the ranges of the plastic zone in side walls are larger than those in the roof and floor; when λ > 1, the results are on the contrary. Besides, when the lateral pressure coefficient increases, the ranges of the plastic zone in side walls shrink, while those in roof and floor increase.

Effect of Lateral Pressure Coefficient on the Surface Displacement of Surrounding Rock.
e radial displacement on the surface of opening under nonuniform pressure was obtained by equations (30) and (31), as shown in Figure 5. It shows that the lateral pressure coefficient has remarkable effects on the surface displacement. With the increase of lateral pressure coefficient, the surface displacement decreases at the sides of opening and increases around the roof. For example, when β r � 2 and λ increases gradually from 0.8 to 1.2∼1.5, the displacement at roof increases by 1.07 mm to 1.37 mm, which is 202 to 258 percent; while the displacement at side wall decreases by approximately 0.39 mm to 0.67 mm, which is 30 to 52 percent. Meanwhile, when 0 < λ < 1, the surface displacement at side wall is larger than at roof and floor; however, when λ > 1, the results are on the contrary.
From Figure 5, it is also known that the surface displacement increases greatly with the increase of the dilation coefficient. For example, when λ � 0.8 and β r changes from 1 to 1.5∼2, the surface displacement at roof, spandrel (60°) and side wall increase about 0.06 to 0.13 mm, 0.11 to 0.24 mm, and 0.25 to 0.63 mm, respectively, which vary by 15 to 32.5, 22.91 to 50, and 37.31 to 94.03 percent, respectively. We may conclude that the dilation coefficient also has great influence on surface displacement of surrounding rock.

Effect of the Buried Depth on Maximum Surface Displacement of Surrounding Rock.
e maximum surface displacement of surrounding rock is given in Figure 6. It may be seen that the surface displacement increases with the burial depth by exponential law. For example, when λ � 1.2 and σ 0 varies from 1 to 1.2∼1.8 MPa, the displacement increases from 3.3 to 20.7 mm, which means a great influence of the buried depth on the deformation of surrounding rock. It is important to note that the location of the maximum displacement is also different with different lateral pressure coefficients. When 0 < λ < 1，the location of the maximum surface displacement occurs in two sides (0°); otherwise, when λ > 1, the maximum displacement distributes in the roof and floor (90°or 270°).

Effect of Lateral Pressure Coefficient on Response Curve between Surrounding Rock and Support.
According to the analytical solution, the response curve between the surrounding rock and support can be deduced (Figure 7). It can be seen that the lateral pressure coefficient influences the response curve in two aspects. First, that response curve varies at different sites under specific lateral pressure coefficient. Second, the location of the maximum value of response curve transfers from side (0°) to roof (90°) with the increasing of lateral pressure coefficient, which is in accordance with the deformation of surrounding rock.
As shown by the above analysis, the influence of lateral pressure coefficient should be taken into full consideration for the design of the primary support under nonuniform pressure, and the reinforced support should be carried out in the key parts to prevent rock instability. Table 2 illustrates the critical support resistance of each position when the surrounding rock begins to show the plastic state. If p in > p yc , it means that the part of surrounding rock is in the elastic state, whose response curve could be 6 Advances in Civil Engineering   Advances in Civil Engineering 7 obtained by equation (15); on the contrary, if p in < p yc , it means the part of surrounding rock is in the plastic state, whose response curve could be obtained by equation (30) or (31).

Effect of Young's Modulus Attenuation on Radial Displacement of Plastic
Zone. e change law of radial displacement of the plastic zone with the increasing radius is shown in Figure 8 in different cases of Young's modulus attenuation. It is known that Young's modulus attenuation has great influence on radial displacement of the plastic zone when using RDM. For example, at side position (0°), comparing Case 2 and Case 3 with Case 1, the surface displacement of plastic zone (uE/σ 0 r 0 ) increases 9.91 and 2.77, respectively, which is increased by 103 and 29 percent. erefore, it indicates that the damage of surrounding rock in Case 2 is largest, while the one in Case 1 is smallest.

Effect of Young's Modulus Attenuation on the Response Curve between Surrounding Rock and Support.
e response curves between surrounding rock and support are shown in Figure 9 in different cases of Young's modulus attenuation. It shows that when Young's modulus attenuation is ignored (Case 1), the deformation of the surrounding rock and support force will be underestimated; however, when a residual value of Young's modulus is assumed in the plastic u 0 is the radial displacement at the surface of tunnel. Note. As the displacement at the surface of surrounding rock is small, the result is scaled up by 1 : 400 to give a more intuitive illustration. Advances in Civil Engineering Table 2: e changes of critical support resistance (p yc ) when the surrounding rock changes from elastic to plastic state. zone (Case 2), the deformation of the surrounding rock and support force will be overestimated. e RDM (Case 3) seems to give more reasonable results and is recommended for design of support parameters and stability analysis of surrounding rock in the circular tunnel.

Support Design for Opening under Nonuniform Pressure
According to the above analysis, it is known that the lateral pressure coefficient is the main reason for nonuniform distribution of the plastic zone and deformation of surrounding rock. When 0 < λ < 1, the range of the plastic zone and deformation presents larger at side than roof positions, while when λ > 1, the result is on the contrary. e rule was also verified well in many engineering practices in deep openings [12][13][14][15][16], as shown in Table 3. Many support techniques of deep opening under nonuniform pressure were put forward, such as bolt-grouting support, bolt-grouting and flexible arch combined support, and bolt-mesh-cable and bolt-grouting combined support. e principles of support design can be concluded in three aspects as follows: (1) Allocate the key parts (where the surrounding rock fails easily) and provide reinforced support in these areas. Due to the nonuniform deformation around the opening induced by the nonuniform initial pressure, the deformation of the support structure is inhomogeneous, which may cause buckling failure of the support. erefore, reinforced support around the key parts is needed to control the nonuniform deformation of surrounding rock and make the support structure bear uniform pressure [38]. Specifically speaking, when 0 < λ < 1, the key locations are around the side wall, and when λ < 1, the key locations are around the roof and floor. (2) Improve the self-bearing capacity of surrounding rock by grouting reinforcement. e grouting reinforcement can increase the strength of rock mass and make it response elastically. Based on the results of present research, the distribution of grouting boreholes can be optimized to account for the nonuniform plastic zone extent.  Combining engineering practices and theoretical analysis results, an optimization method for support design was proposed for deep circular opening under nonuniform pressure. e specific optimization flow is shown in Figure 10.

Conclusions
e mechanical model of a circular opening with nonuniform pressure was proposed, which takes into account the nonassociated flaw rule and Young's modules attenuation. e closed-form solutions of stress, deformation, and radius of the plastic zone of surrounding rock were obtained. e  Advances in Civil Engineering influences of some factors including lateral pressure coefficient, dilation coefficient, buried depth, and Young's modules attenuation on the state of surrounding rock were analyzed. e following conclusions can be drawn: (1) An analytical solution of a circular opening with nonuniform pressure was obtained. Compared with the results under uniform pressure in the literature, the proposed analytical solution is verified. (2) e lateral pressure coefficient has remarkable influence on the distribution of the plastic zone. When 0 < λ < 1, the ranges of the plastic zone around side walls are larger than those of the roof and floor; when λ > 1, the results are on the contrary. Besides, with the increasing of lateral pressure coefficient, the range of the plastic zone in side walls shrinks gradually, while that in the roof and floor increases. (3) e lateral pressure coefficient has an impact on surrounding rock deformation and response curve of the support. With the consideration on the effect of lateral pressure coefficient, both the deformation and response curve show nonuniform characteristic around the tunnel. (4) e surface displacement of surrounding rock increases with dilation coefficient and buried depth. erefore, their effect on deformation of surrounding rock should be taken a full account on support design. (5) Young's modulus attenuation has important influence on radial displacement of plastic zone. When the attenuation effect is ignored, it will cause underestimation for the deformation of surrounding rock. e RDM can be more effective on the design of support parameters and stability analysis of surrounding rock in deep circular opening.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.