A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts

Featured Application: The proposed model can be employed as a useful tool in the preliminary design and stability analysis of a tunnel reinforced by fully grouted passive bolts. Abstract: The use of fully grouted passive bolts as a reinforcement technique has been widely applied to improve the stability of tunnels. To analyze the behaviors of passive bolts and rock mass in a deep circular tunnel, a new semi-analytical solution is presented in this work based on the ﬁnite di ﬀ erence method. The rock mass was assumed to experience elastic–brittle–plastic behavior, and the linear Mohr–Coulomb criterion and the nonlinear generalized Hoek–Brown criterion were employed to govern the yielding of the rock mass. The interaction and decoupling between the rock mass and bolts were considered by using the spring–slider model. To simplify the analysis process, a bolted tunnel was divided into a bolted region and an unbolted region, while the contact stress at the bolted–unbolted interface and the rigid displacement of the bolts were obtained using two boundary conditions in combination with the bisection method. Comparisons show that the results obtained using the proposed solution agree well with those from the commercial numerical software and the in situ test. Finally, parametric analyses were performed to examine the e ﬀ ects of various reinforcement parameters on the tunnel’s stability. The proposed solution provided a fast but accurate estimation of the behavior of a reinforced deep circular tunnel for preliminary design purposes.


Introduction
Rock bolts have been used in underground engineering to stabilize rock masses for more than 100 years [1]. At present, many kinds of rock bolts have been developed, among which, fully grouted passive bolts are the most popular type and have been widely employed due to their economical and convenient nature. The analytical solution for a deep circular tunnel reinforced with passive bolts in a hydrostatic field is a classical problem in underground engineering. Although the practicability of the analytical solutions is limited by their assumptions, they can be employed to quickly assess the mechanical behaviors of the rock mass and reinforcement and can be used for preliminary design analyses or the pre-dimensioning of the reinforcement [2,3].
(2) The rock mass experiences elastic-brittle-plastic behavior, and the yielding of the rock mass is governed by the linear Mohr-Coulomb criterion or the nonlinear generalized Hoek-Brown criterion.
(3) The passive bolts with length lb are installed uniformly around the tunnel. In the transverse section perpendicular to the axis of the tunnel, the angle between two adjacent bolts is ω. In the longitudinal section parallel to the axis of the tunnel, the spacing between two adjacent bolts is lz. Therefore, the tributary area of each bolt is assumed to be lzrω (r denotes the radial distance in the rock mass to the tunnel center).

Tunnel Face Effect during Tunneling
As shown in Figure 1, a support pressure p is applied on the tunnel periphery by the tunnel face effect, and its value gradually decreases to 0 with the advance of the tunnel face. Therefore, due to the tunnel face effect, the behavior of a deep tunnel during tunneling is essentially a threedimensional problem. The bolts are installed behind the tunnel face, where the support pressure is βp0 (β is the coefficient of the tunnel face effect) and the corresponding rock displacement is u ini r (the initial radial displacement of the rock mass), while the bolts are yet to work at this moment. Then, the support pressure decreases during the advance of the tunnel, which leads to an increase of the radial displacement of rock mass ur and the interaction between the rock mass and bolts. The tunnel face effect can be approximated using the β-method.
The β-method consists of two steps: (1) the tunnel is excavated and a support pressure βp0 is applied to the tunnel periphery, and (2) the bolts are installed and the stress βp0 is shared by the ground and the bolts during the advance of the tunnel. The magnitude of β is related to the distance between the installation position and the tunnel face, where a large distance is associated with a small β and vice versa. The magnitude of β must be known a priori, and its value strongly depends on the ground condition, excavation method, etc. The value of β can range from 0.2 to 0.8 [2,30].

Equation of Equilibrium
For this axisymmetric problem, the equilibrium equation of the rock mass in an unbolted deep circular tunnel is:

Tunnel Face Effect during Tunneling
As shown in Figure 1, a support pressure p is applied on the tunnel periphery by the tunnel face effect, and its value gradually decreases to 0 with the advance of the tunnel face. Therefore, due to the tunnel face effect, the behavior of a deep tunnel during tunneling is essentially a three-dimensional problem. The bolts are installed behind the tunnel face, where the support pressure is βp 0 (β is the coefficient of the tunnel face effect) and the corresponding rock displacement is u ini r (the initial radial displacement of the rock mass), while the bolts are yet to work at this moment. Then, the support pressure decreases during the advance of the tunnel, which leads to an increase of the radial displacement of rock mass u r and the interaction between the rock mass and bolts. The tunnel face effect can be approximated using the β-method.
The β-method consists of two steps: (1) the tunnel is excavated and a support pressure βp 0 is applied to the tunnel periphery, and (2) the bolts are installed and the stress βp 0 is shared by the ground and the bolts during the advance of the tunnel. The magnitude of β is related to the distance between the installation position and the tunnel face, where a large distance is associated with a small β and vice versa. The magnitude of β must be known a priori, and its value strongly depends on the ground condition, excavation method, etc. The value of β can range from 0.2 to 0.8 [2,30].

Equation of Equilibrium
For this axisymmetric problem, the equilibrium equation of the rock mass in an unbolted deep circular tunnel is: where σ r and σ θ are the radial and tangential stresses, respectively.

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As shown in Figure 2a, an infinitesimal element of the bolted rock mass is subjected to a radial stress σ r , a tangential stress σ θ , and a tension force in the bolt F n . Supposing that the tension force can be spread uniformly around its tributary area l z rω, the equilibrium equation of an infinitesimal element of the bolted rock mass (Figure 2b) can be expressed as: Since the angle dθ is infinitesimal, sin(dθ/2) = dθ/2. According to Figure 2a, the axial force in an infinitesimal bolt element dF n = πd s τ s dr. Therefore, Equation (2) can be simplified to: where τ s is the shear stress at the bolt-rock interface and d s is the effective diameter of the bolts; see Section 2.7 for the determination of τ s and d s . where σr and σθ are the radial and tangential stresses, respectively. As shown in Figure 2a, an infinitesimal element of the bolted rock mass is subjected to a radial stress σr, a tangential stress σθ, and a tension force in the bolt Fn. Supposing that the tension force can be spread uniformly around its tributary area lzrω, the equilibrium equation of an infinitesimal element of the bolted rock mass (Figure 2b) can be expressed as: Since the angle dθ is infinitesimal, sin(dθ/2) = dθ/2. According to Figure 2a, the axial force in an infinitesimal bolt element dFn = πdsτsdr. Therefore, Equation (2) can be simplified to: where τs is the shear stress at the bolt-rock interface and ds is the effective diameter of the bolts; see Section 2.7 for the determination of τs and ds.

Mohr-Coulomb Criterion
The Mohr-Coulomb criterion is expressed as:

Mohr-Coulomb Criterion
The Mohr-Coulomb criterion is expressed as: where N = (1 + sin ϕ)/(1 − sin ϕ) and Y = 2ccos ϕ/(1 − sin ϕ) are strength parameters; c and ϕ are the cohesion and friction angle of the rock mass, respectively; their corresponding peak values are N p , Y p , c p , and ϕ p , respectively; and their corresponding residual values are N r , Y r , c r , and ϕ r , respectively.

Generalized Hoek-Brown Criterion
The generalized Hoek-Brown criterion [31] is expressed as: where σ ci is the unconfined compressive strength of the intact rock, and m b , s, and a are semi-empirical parameters that characterize the rock mass. Their corresponding peak values are σ ci,p , m b,p , s p , and a p , respectively, and their corresponding residual values are σ ci,r , m b,r , s r , and a r , respectively.

Plastic Potential Equation
The Mohr-Coulomb criterion is chosen to be the plastic potential function, as follows: where ψ is the dilatancy angle of the rock mass and K ψ = (1 + sin ψ)/(1 − sin ψ) is the dilatancy coefficient.

Equations of Strains and Displacements
For this axisymmetric problem, the radial strain ε r and tangential strain ε θ can be expressed in terms of the radial displacement u r :

In an Elastic Region
According to Hooke's law, the elastic radial and tangential strains can be obtained using: where G = E/2/(1 + ν), E and ν are the shear modulus, Young s modulus, and Poisson ratio of the rock mass, respectively. Considering Equations (8) and (10) and the relation between the radial and tangential stresses in an elastic region-i.e., σ θ + σ r = 2p 0 -the radial displacement u r can be expressed as:

In a Plastic Region
The radial and tangential strains are composed of the plastic strain components (ε p r and ε p θ ) and the elastic strain components (ε e r and ε e θ ), as follows: Based on the plastic potential equation Equation (6), the plastic tangential strain and radial strain are formulated using the plastic multiplier λ: Appl. Sci. 2020, 10, 4402 6 of 22 According to Equations (6), (14), and (15), the relation between the plastic tangential strain and the radial strain is: By incorporating Equations (7)-(10) and Equations (12)-(16), the differential equation for the radial displacement u r is: where:

Interaction between the Rock Mass and the Passive Bolts
The pull-out tests show that the load-displacement curve is non-linear because of the decoupling at the bolt-rock interface and the confining pressure influences the shear strength of the interface greatly [32,33]. The spring-slider model [25] was employed here to characterize the relationship between the relative displacement ∆u s and shear stress τ s at the bolt-rock interface, as shown in Figure 3.

Interaction between the Rock Mass and the Passive Bolts
The pull-out tests show that the load-displacement curve is non-linear because of the decoupling at the bolt-rock interface and the confining pressure influences the shear strength of the interface greatly [32,33]. The spring-slider model [25] was employed here to characterize the relationship between the relative displacement Δus and shear stress τs at the bolt-rock interface, as shown in Figure 3.   The shear stress τ s and tension force F n of the bolt can be expressed as: where: K s , c s , and ϕ s are the shear stiffness, cohesion, and friction angle at the rock-bolt interface, respectively; A b is the cross-sectional area of the bolt; u rig b and u elo b are the rigid displacement and the elongation of the bolt, respectively; d s is the effective diameter of the bolt, as shown in Figure 4, which may equal the diameter of the bolt, grout hole, or inside the grout because these interfaces are relatively weak in pull-out tests; and r b = R + l b is the radial distance from the far end of the bolt to the tunnel center. Note that K s , d s , c s , and ϕ s can be easily obtained from conventional pull-out tests [25].
Ks, cs, and φs are the shear stiffness, cohesion, and friction angle at the rock-bolt interface, respectively; Ab is the cross-sectional area of the bolt; u rig b and u elo b are the rigid displacement and the elongation of the bolt, respectively; ds is the effective diameter of the bolt, as shown in Figure 4, which may equal the diameter of the bolt, grout hole, or inside the grout because these interfaces are relatively weak in pull-out tests; and rb = R + lb is the radial distance from the far end of the bolt to the tunnel center. Note that Ks, ds, cs, and φs can be easily obtained from conventional pull-out tests [25]. As shown in Figure 5, the compression between the end plate and rock mass produces a tension force Fn(R) (Equation (20)) at the near end of the bolt and support pressure pep (Equation (21)) on the tunnel periphery. These two equations were used as boundary conditions in the calculation procedure (Appendix B): where Kep is the stiffness of the end plate, where Kep = 0 in the absence of the end plate. Note that, compared with the original spring-slider model [25], the effects of the bolt elongation and the end plate were considered in this work.  As shown in Figure 5, the compression between the end plate and rock mass produces a tension force F n(R) (Equation (20)) at the near end of the bolt and support pressure p ep (Equation (21)) on the tunnel periphery. These two equations were used as boundary conditions in the calculation procedure (Appendix B):

Stress-Strain Analysis of the Rock Mass and Passive Bolts Using the Finite Difference Method
where K ep is the stiffness of the end plate, where K ep = 0 in the absence of the end plate.
Ks, cs, and φs are the shear stiffness, cohesion, and friction angle at the rock-bolt interface, respectively; Ab is the cross-sectional area of the bolt; u rig b and u elo b are the rigid displacement and the elongation of the bolt, respectively; ds is the effective diameter of the bolt, as shown in Figure 4, which may equal the diameter of the bolt, grout hole, or inside the grout because these interfaces are relatively weak in pull-out tests; and rb = R + lb is the radial distance from the far end of the bolt to the tunnel center. Note that Ks, ds, cs, and φs can be easily obtained from conventional pull-out tests [25]. As shown in Figure 5, the compression between the end plate and rock mass produces a tension force Fn(R) (Equation (20)) at the near end of the bolt and support pressure pep (Equation (21)) on the tunnel periphery. These two equations were used as boundary conditions in the calculation procedure (Appendix B): where Kep is the stiffness of the end plate, where Kep = 0 in the absence of the end plate. Note that, compared with the original spring-slider model [25], the effects of the bolt elongation and the end plate were considered in this work.  Note that, compared with the original spring-slider model [25], the effects of the bolt elongation and the end plate were considered in this work.

Stress-Strain Analysis of the Rock Mass and Passive Bolts Using the Finite Difference Method
As shown in Figure 6, to simplify the analysis process, a bolted circular tunnel was divided into two parts: a bolted region and an unbolted region. The radial stress, tangential stress, and displacement at the bolted-unbolted interface are σ r(R+lb) , σ θ(R+lb) , and u r(R+lb) , respectively.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 22 As shown in Figure 6, to simplify the analysis process, a bolted circular tunnel was divided into two parts: a bolted region and an unbolted region. The radial stress, tangential stress, and displacement at the bolted-unbolted interface are σr(R+lb), σθ(R+lb), and ur(R+lb), respectively.

Stress-Strain Analysis in an Unbolted Region
Notice that an unbolted region can be treated as an unbolted circular tunnel; therefore, stresses and displacements in an unbolted region can be obtained by replacing p and R in the equations in Appendix A with σr(R+lb) and R+lb, respectively.

Stress-Strain Analysis in an Unbolted Region
Notice that an unbolted region can be treated as an unbolted circular tunnel; therefore, stresses and displacements in an unbolted region can be obtained by replacing p and R in the equations in Appendix A with σ r(R+lb) and R+l b , respectively.

Stress-Strain Analysis in a Bolted Region
In Figure 6, a bolted region is discretized into n annuluses with a uniform thickness ∆r, starting from the bolted-unbolted interface. The ith annulus is bounded by two circles of r = r i−1 and r = r i . The radial stress σ r(0) , tangential stress σ θ(0) , and radial displacement u r(0) at the outer boundary of the bolted region (r = r (0) = R+l b ) equal σ r(R+lb) , σ θ(R+lb) , and u r(R+lb) , respectively. The number of the rock mass annuluses n and inner radius r j of each annulus can be respectively expressed as: For the mechanical analysis of a bolted region, first, the elastoplastic state of the rock mass in the ith annulus is verified using Equation (4) (Mohr-Coulomb (M-C) rock mass) or Equation (5) (Hoek-Brown (H-B) rock mass); then, the stresses and displacements in an elastic annulus can be analyzed using the equations in Section 3.2.1; the stresses and displacements in a plastic annulus can be analyzed using the equations in Section 3.2.2; the behaviors of the passive bolts can be analyzed using the equations in Section 3.2.3; and the radius of the plastic region can be obtained in this process.

Stress-Strain Analysis in an Elastic Region
Considering Equation (3) and the relation between the radial and tangential stresses in an elastic region-i.e., σ θ + σ r = 2p 0 -the stresses at r = r (i) can be expressed in an incremental form by using the finite difference method, as follows: According to Equation (11), the radial displacement at r = r (i) is:

Stress-Strain Analysis in a Plastic Region
By incorporating Equations (4) and (5) into Equation (3), the stresses at r = r (i) can be obtained. For the M-C rock mass, the stresses at r = r (i) are: For the H-B rock mass, the stresses at r = r (i) are: According to Equation (17), the radial displacement at r = r (j) is:

Behaviors of the Passive Bolts
The shear stress at the bolt-rock interface (Equation (18)) and tension force in the bolts (Equation (19)) at r = r (i) can be respectively expressed as: where: The initial radial displacement u ini r(i) can be obtained by replacing p in the equations in Appendix A with βp 0 . Notice that, in Equations (20)

Verification
The proposed solution was programmed using the MATLAB language (R2016a, MathWorks, Natick, MA, USA), and the verification is detailed in this section.

Establishment of the Numerical Model
The commercial numerical simulation software FLAC 3D [34] was employed here to verify the accuracy of the proposed model. As shown in Figure 7, only a quarter of the circular tunnel was considered due to the symmetry of the problem. To simulate a deep tunnel under the hydrostatic and plane-strain condition, the outer boundary was 15R from the tunnel center, and its thickness was 1 m. Roller boundaries were set at the left and bottom boundaries, and the displacements in the tunnel axes directions were fixed. The hydrostatic stress field p 0 was applied to the outer boundary. The passive bolts were installed uniformly around the tunnel, which was simulated using the cable element in FLAC 3D . considered due to the symmetry of the problem. To simulate a deep tunnel under the hydrostatic and plane-strain condition, the outer boundary was 15R from the tunnel center, and its thickness was 1 m. Roller boundaries were set at the left and bottom boundaries, and the displacements in the tunnel axes directions were fixed. The hydrostatic stress field p0 was applied to the outer boundary. The passive bolts were installed uniformly around the tunnel, which was simulated using the cable element in FLAC 3D .

Mohr-Coulomb Criterion
The properties employed in this section are listed in Table 1, and the parameters of the rock mass and passive bolts were adopted from the works conducted by Bobet and Einstein [2] and Cui et al. [29], respectively. As shown in Figure 8, the results calculated using the proposed model and the numerical model were in excellent agreement in both the bolted and unbolted cases. Figure 8a,b indicates that the passive bolts displayed a significant reinforcement effectiveness on the rock mass.

Mohr-Coulomb Criterion
The properties employed in this section are listed in Table 1, and the parameters of the rock mass and passive bolts were adopted from the works conducted by Bobet and Einstein [2] and Cui et al. [29], respectively. As shown in Figure 8, the results calculated using the proposed model and the numerical model were in excellent agreement in both the bolted and unbolted cases. Figure 8a,b indicates that the passive bolts displayed a significant reinforcement effectiveness on the rock mass. Table 1. The parameters of rock mass and passive bolts [2,29].

Rock Mass
Passive Bolts

Generalized Hoek-Brown Criterion
The calculation parameters of the rock mass in Table 2 were taken from the work of Lee and Pietruszczak [35]. Except for Ks = 20 MPa, the properties of the passive bolts employed in this section were the same as in Table 1. The stresses and displacements of the rock mass, as well as the shear stress and tension force of the passive bolts obtained using the two methods, are plotted in Figure 9.

Generalized Hoek-Brown Criterion
The calculation parameters of the rock mass in Table 2 were taken from the work of Lee and Pietruszczak [35]. Except for K s = 20 MPa, the properties of the passive bolts employed in this section were the same as in Table 1. The stresses and displacements of the rock mass, as well as the shear stress and tension force of the passive bolts obtained using the two methods, are plotted in Figure 9. It can be seen that the results of the proposed solution show good agreement with those from FLAC 3D , and the reinforcement effectiveness of the passive bolts was significant.

Comparison with the In Situ Test Results from the Kielder Experimental Tunnel
The results calculated using the proposed solution were compared with the measurements carried out in the Kielder experimental tunnel. The data were mainly from Oreste and Peila [8]. The rock mass in which the tests were carried out was a mudstone (RMR (Rock Mass Rating) = 32 and Q (Quantitative Classification of Rock Mass) = 0.33) with a strata thickness of 8 m. The main properties of the rock mass and passive bolts are listed in Table 3. The values of Ks and Kep were assumed to be 70 MPa and 30 MN/m, respectively. The tension force and radial displacement after 10 days from the

Comparison with the In Situ Test Results from the Kielder Experimental Tunnel
The results calculated using the proposed solution were compared with the measurements carried out in the Kielder experimental tunnel. The data were mainly from Oreste and Peila [8]. The rock mass in which the tests were carried out was a mudstone (RMR (Rock Mass Rating) = 32 and Q (Quantitative Classification of Rock Mass) = 0.33) with a strata thickness of 8 m. The main properties of the rock mass and passive bolts are listed in Table 3. The values of K s and K ep were assumed to be 70 MPa and 30 MN/m, respectively. The tension force and radial displacement after 10 days from the initial bolt are potted in Figure 10. It can be observed that the calculation results agree well with the measured data. Table 3. The parameters of rock mass and passive bolts [8].

Rock Mass
Passive Bolts

Symbol (Unit) Value Symbol (Unit) Value
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 22 initial bolt are potted in Figure 10. It can be observed that the calculation results agree well with the measured data.

Discussion
The following examples are given to explain the reinforcement effectiveness and mechanical behaviors of passive bolts. The generalized Hoek-Brown criterion was employed and the properties for a poor rock mass were adopted from the study conducted by Osgoui and Oreste [19], as shown in Table 4. The normalized displacement ur(R)/u ub r(R) was employed to quantitatively evaluate the reinforcement effectiveness of the bolts, where ur(R) and u ub r(R) are the displacement at the tunnel periphery in the bolted and unbolted cases, respectively. Note that the smaller the normalized displacement, the better the reinforcement effectiveness.

Discussion
The following examples are given to explain the reinforcement effectiveness and mechanical behaviors of passive bolts. The generalized Hoek-Brown criterion was employed and the properties for a poor rock mass were adopted from the study conducted by Osgoui and Oreste [19], as shown in Table 4. The normalized displacement u r(R) /u ub r(R) was employed to quantitatively evaluate the reinforcement effectiveness of the bolts, where u r(R) and u ub r(R) are the displacement at the tunnel periphery in the bolted and unbolted cases, respectively. Note that the smaller the normalized displacement, the better the reinforcement effectiveness.

Comparison of the Solutions That Considered or Neglected Decoupling at the Bolt-Rock Interface
It was interesting to compare the results obtained from the models that considered and neglected the decoupling at the bolt-rock interface. Note that for the model that neglected the decoupling, the cohesion at the bolt-rock interface was fixed at ∞. The results plotted in Figure 11 show that the calculation results of the two models were identical before decoupling, and then the model that neglected the decoupling significantly overestimated the reinforcement effectiveness of the passive bolts, which may result in an unsafe design. Therefore, the decoupling at the bolt-rock interface should be considered in passive bolt designs.

Comparison of the Solutions that Considered or Neglected Decoupling at the Bolt-Rock Interface
It was interesting to compare the results obtained from the models that considered and neglected the decoupling at the bolt-rock interface. Note that for the model that neglected the decoupling, the cohesion at the bolt-rock interface was fixed at ∞. The results plotted in Figure 11 show that the calculation results of the two models were identical before decoupling, and then the model that neglected the decoupling significantly overestimated the reinforcement effectiveness of the passive bolts, which may result in an unsafe design. Therefore, the decoupling at the bolt-rock interface should be considered in passive bolt designs.

Effect of the End Plate and Shear Stiffness at the Bolt-Rock Interface
The results with different shear stiffnesses and with or without end plates are plotted in Figure  12. In the case without an end plate, Kep = 0 MN/m, while in the case with an end plate, Kep = 20 MN/m. As shown by the black solid line in Figure 12a, the normalized displacement decreased with the increasing shear stiffness before a critical value (about 20 MPa in this case), beyond which, the normalized displacement was almost constant because the shear stress at the bolt-rock interface increased with the increasing shear stiffness (see Figure 12b), while the decoupling at the bolt-rock

Effect of the End Plate and Shear Stiffness at the Bolt-Rock Interface
The results with different shear stiffnesses and with or without end plates are plotted in Figure 12. In the case without an end plate, K ep = 0 MN/m, while in the case with an end plate, K ep = 20 MN/m. As shown by the black solid line in Figure 12a, the normalized displacement decreased with the increasing shear stiffness before a critical value (about 20 MPa in this case), beyond which, the normalized displacement was almost constant because the shear stress at the bolt-rock interface increased with the increasing shear stiffness (see Figure 12b), while the decoupling at the bolt-rock interface occurred when the shear stress exceeds its capability. Moreover, Figure 12b shows that the maximum shear stress was generated at the end of bolts; therefore, the failure of the bolt-rock interface was more likely to start from the tunnel perimeter.
The red dashed line in Figure 12a illustrates that the end plate could effectively improve the reinforcement effectiveness of passive bolts by about 20% and avoids decoupling at the bolt-rock interface, which was due to the interaction between the end plate and rock mass exerting a support pressure on the tunnel periphery (see Figure 12c) and decreasing the shear stress (see Figure 12b). Figure 12c shows that the maximum tensile force in the case with an end plate was significantly larger than that in the other case, which implies that the tension failure of the bolts was more likely to occur in the case with an end plate.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22 interface occurred when the shear stress exceeds its capability. Moreover, Figure 12b shows that the maximum shear stress was generated at the end of bolts; therefore, the failure of the bolt-rock interface was more likely to start from the tunnel perimeter. The red dashed line in Figure 12a illustrates that the end plate could effectively improve the reinforcement effectiveness of passive bolts by about 20% and avoids decoupling at the bolt-rock interface, which was due to the interaction between the end plate and rock mass exerting a support pressure on the tunnel periphery (see Figure 12c) and decreasing the shear stress (see Figure 12b). Figure 12c shows that the maximum tensile force in the case with an end plate was significantly larger than that in the other case, which implies that the tension failure of the bolts was more likely to occur in the case with an end plate.

Effect of Bolt Length and Density
The normalized displacements with different bolt lengths and densities are plotted in Figure  13a, and the normalized bolt length lbn = lb/(r ub p − R) [27] was taken as the abscissa, where r ub p is the plastic radius in the unbolted case. Note that if 0 ≤ lbn ≤ 1, the bolts were entirely embedded in the plastic region; when lbn > 1.0, the bolts penetrated the elastic region. Figure 13a shows that the normalized displacement decreased with the increasing normalized bolt length until lbn = 1.3, whereas the longer bolts could barely take more effective reinforcement. This phenomenon can be explained as follows: once the bolt section within the elastic region provided effective anchorage, the reinforcement effectiveness could not be further improved by increasing the bolt length. Therefore,

Effect of Bolt Length and Density
The normalized displacements with different bolt lengths and densities are plotted in Figure 13a, and the normalized bolt length l bn = l b /(r ub p − R) [27] was taken as the abscissa, where r ub p is the plastic radius in the unbolted case. Note that if 0 ≤ l bn ≤ 1, the bolts were entirely embedded in the plastic region; when l bn > 1.0, the bolts penetrated the elastic region. Figure 13a shows that the normalized displacement decreased with the increasing normalized bolt length until l bn = 1.3, whereas the longer bolts could barely take more effective reinforcement. This phenomenon can be explained as follows: once the bolt section within the elastic region provided effective anchorage, the reinforcement effectiveness could not be further improved by increasing the bolt length. Therefore, the far end of the bolts should penetrate the elastic region to fully mobilize their effectiveness, while it was not necessary to over-extend the bolts beyond the range of the plastic region [25]. Furthermore, increasing the bolt density can result in better effectiveness (as shown by the red dashed line in Figure 13b). Figure 13b,c plots the tension force and shear stress with different bolt lengths and densities. The figures show that the tension force and shear stress both increased with increasing bolt length and decreasing bolt density, while a large length or small density could cause the tension failure of the bolt and decoupling at the bolt-rock interface. Therefore, a reasonable length and density should be considered in passive bolt designs.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 22 the far end of the bolts should penetrate the elastic region to fully mobilize their effectiveness, while it was not necessary to over-extend the bolts beyond the range of the plastic region [25]. Furthermore, increasing the bolt density can result in better effectiveness (as shown by the red dashed line in Figure  13b). Figure 13b,c plots the tension force and shear stress with different bolt lengths and densities. The figures show that the tension force and shear stress both increased with increasing bolt length and decreasing bolt density, while a large length or small density could cause the tension failure of the bolt and decoupling at the bolt-rock interface. Therefore, a reasonable length and density should be considered in passive bolt designs.

Conclusions
A new semi-analytical solution was established in this study for the mechanical analysis of fully grouted passive bolts and rock masses in tunnels. Comparisons and parameter analyses were conducted to assess the behavior of the proposed solution. The main conclusions are as follows: (1) Comparisons with numerical results and in situ results indicate that the proposed solution could give a good estimation of the behaviors of passive bolts and rock masses in a deep circular tunnel.
(2) Compared with the model that considered the decoupling at the rock-bolt interface, a model that neglected the decoupling may significantly overestimate the reinforcement effectiveness of passive bolts, which may result in an unsafe design.

Conclusions
A new semi-analytical solution was established in this study for the mechanical analysis of fully grouted passive bolts and rock masses in tunnels. Comparisons and parameter analyses were conducted to assess the behavior of the proposed solution. The main conclusions are as follows: (1) Comparisons with numerical results and in situ results indicate that the proposed solution could give a good estimation of the behaviors of passive bolts and rock masses in a deep circular tunnel.
(2) Compared with the model that considered the decoupling at the rock-bolt interface, a model that neglected the decoupling may significantly overestimate the reinforcement effectiveness of passive bolts, which may result in an unsafe design.
(3) The displacement of the rock mass decreased with increasing shear stiffness before a critical value, beyond which, decoupling at the bolt-rock interface occurred, resulting in the reduction of the reinforcement effectiveness.
(4) The interaction between the end plate and rock mass exerted a support pressure on the tunnel periphery and decreased the shear stress at the rock-bolt interface, thus effectively improving the reinforcement effectiveness and avoiding decoupling. The maximum tensile force in the bolts in the case with an end plate was significantly larger than that in the case without an end plate.
(5) To ensure the effectiveness and economy of passive bolts, the far end of the bolts should penetrate the elastic region, while it is not necessary to over-extend the bolts beyond the range of the plastic region. Increasing the bolt density can result in better effectiveness.
(6) The tension force and shear stress both increased with the increasing bolt length and decreasing bolt density, while a large length or small density may cause the tension failure of the bolt and decoupling at the bolt-rock interface.
m b,r p cr /σ ci,r + s r 1−a r − m b,r p/σ ci,r + s r Equation (A3) can be solved numerically using the Newton-Raphson method.