Preamble

Calculation Method for Support Force of Full-Length Prestressed Anchor Bolts Considering Shear Performance

Abstract: To investigate the shear performance of full-length prestressed anchor bolts against shear-slip bodies, this study proposes a method for calculating the minimum required support resistance based on shear-slip theory. The study derives the neutral point position and the support force calculation method for full-length prestressed anchor bolts, applying these findings to a tunnel case study to verify their rationality. The results indicate that calculating the minimum required resistance using the slip body area and surrounding rock classification is simpler and provides a higher safety margin compared to traditional methods. Within specified ranges, increasing the anchor bolt diameter shifts the neutral point toward the tail of the bolt, while increasing the prestress shifts the neutral point toward the tip. Although these two factors affect the neutral point differently, both lead to a proportional increase in the bolt's axial force and radial support resistance. When employing shotcrete-bolt support, the radial support resistance provided by the bolts is relatively small. It is recommended that anchor bolt parameters be designed based on the shear force at the slip surface; appropriately increasing the diameter and prestress can enhance shear performance while simultaneously improving radial support resistance. These research findings provide a theoretical reference for calculating the shear performance of full-length prestressed anchor bolts.

1 Introduction

In underground engineering, the stability of the surrounding rock is often maintained through the use of full-length prestressed anchor bolts. While traditional design methods focus primarily on the axial load-bearing capacity of these bolts, the shear performance at the interface of potential slip bodies is equally critical. Understanding how the neutral point—the location where the relative displacement between the bolt and the rock is zero—shifts under different loading conditions is essential for accurately determining the support force provided to the rock mass.

2 Calculation of Minimum Support Resistance for Slip Bodies

Based on shear-slip theory, the stability of a potential slip body depends on the balance between the driving forces (primarily gravity and tectonic stress) and the resisting forces provided by the rock's inherent strength and the installed support system.

Traditional methods for calculating support resistance often involve complex numerical simulations or empirical formulas that may not fully account for the specific geometry of the slip body. This study proposes a simplified approach that utilizes the area of the slip body and the specific parameters associated with the surrounding rock grade. By integrating the shear strength along the potential failure plane, the minimum required support resistance $P_{min}$ can be determined. This method offers a more straightforward calculation process and ensures a higher safety factor for engineering applications.

3 Neutral Point and Support Force of Full-Length Prestressed Anchor Bolts

The performance of a full-length prestressed anchor bolt is characterized by the distribution of axial stress along its length. A critical element in this distribution is the neutral point.

3.1 Determination of the Neutral Point

The position of the neutral point is influenced by the interaction between the bolt, the grout, and the surrounding rock. Through theoretical derivation, it is found that the neutral point position is sensitive to both the physical dimensions of the bolt and the applied initial tension.

Specifically, increasing the diameter of the anchor bolt increases its stiffness, which causes the neutral point to migrate toward the bolt tail (the end near the tunnel face). Conversely, increasing the initial prestress applied to the bolt shifts the neutral point toward the bolt tip (the end embedded deep in the rock).

3.2 Calculation of Support Force

The radial support resistance $P_r$ provided by the bolt is a function of its axial force distribution. The relationship can be expressed as a proportional increase: as the axial force increases due to rock deformation or higher prestress, the radial resistance exerted on the slip body increases accordingly.

However, in standard shotcrete-bolt support systems, the radial resistance contributed by the bolts alone is often lower than expected. This suggests that the primary mechanism of stabilization in high-shear environments may be the bolt's resistance to lateral displacement across the slip surface.

4 Case Study and Validation

The proposed calculation method was applied to a specific tunnel project to verify its practical utility. [TABLE:1] provides the geological parameters and bolt specifications used in the analysis.

[FIGURE:1]

The analysis of the tunnel case demonstrates that the calculated neutral point aligns with observed deformation patterns in the field. Furthermore, the results confirm that relying solely on the axial capacity of the bolts may lead to an underestimation of the required reinforcement if shear forces are dominant.

5 Conclusion and Recommendations

This study provides a refined method for calculating the support force of full-length prestressed anchor bolts by considering their shear performance. The main conclusions are as follows:

1. The method of calculating the minimum required resistance based on the slip body area and rock grade is more efficient and conservative than traditional empirical methods.
2. The neutral point of the bolt is dynamically affected by diameter and prestress. While their effects on the neutral point position are opposite, both parameters effectively enhance the bolt's axial and radial support capabilities.
3. In engineering practice, especially where shear slip is a primary concern, anchor bolt parameters should be designed based on the shear force values at the slip interface.
4. To optimize support performance, it is recommended to appropriately increase the bolt diameter and the applied prestress. This dual approach not only improves the shear resistance across potential failure planes but also significantly boosts the radial support resistance provided to the surrounding rock.

The findings presented here serve as a technical basis for the design and

关键词

Calculation method of support force of full-length prestressed bolt considering shear performance

Abstract: In response to the challenges of large deformations and high ground pressures encountered in soft rock tunnels, this paper investigates the mechanical behavior of full-length prestressed bolts. By integrating shear-slip theory with the mechanical characteristics of bolt-grout interfaces, a calculation method for support force is proposed that accounts for the shear performance of the anchoring system. The study analyzes the distribution of axial stress and shear stress along the bolt, identifying the location of the neutral point and its influence on the overall support resistance. The results provide a theoretical basis for the optimized design of bolt-shotcrete support systems in difficult geological conditions.

Keywords: soft rock tunnel; shear-slip theory; bolt-shotcrete support; neutral point; support resistance

1 Introduction

In the construction of tunnels through soft rock formations, the stability of the surrounding rock is often compromised by low strength, high plasticity, and significant rheological effects. Full-length bonded prestressed bolts have become a primary support element due to their ability to provide immediate active reinforcement and continuous resistance to rock mass deformation. However, traditional design methods often simplify the interaction between the bolt and the surrounding rock, frequently neglecting the complex shear transfer mechanisms at the interface.

The effectiveness of a bolt depends largely on the shear stress distribution along its length. As the surrounding rock deforms, the relative displacement between the rock and the bolt generates shear stresses, which in turn develop the axial force within the bolt. A critical feature of this interaction is the "neutral point"—the location where the relative displacement between the bolt and the rock is zero, and the shear stress reverses direction. Understanding the position of this neutral point and the shear performance of the anchoring interface is essential for accurately calculating the support force provided by the bolt system.

2 Shear-Slip Theory and Interface Mechanics

The mechanical behavior of the bolt-grout-rock interface is the core factor determining the reinforcement effect. According to shear-slip theory, the shear stress $\tau$ at the interface is a function of the relative displacement $u$ between the bolt and the borehole wall.

2.1 Interface Constitutive Model

The relationship between shear stress and slip can be characterized by a multi-stage constitutive model, typically involving elastic, softening, and residual stages. In the elastic stage, the shear stress increases linearly with displacement:

$$\tau = k_s \cdot u$$

where $k

1. School of Civil Engineering

Lanzhou Jiaotong University

730070 Lanzhou

China 2. China Construction Eighth Bureau Rail Transit Construction Co.

210046 Nanjing

China 3. China Railway Tunnel Bureau Group Co.

511458 Guangzhou

China

Abstract

In order to find out the shear resistance of the full-length prestressed bolt to the shear-slip body based on the shear-slip body theory a method to calculate the minimum support resistance of the slip body was proposed the neutral point position and supporting force calculation method was deduced and the ra- tionality of the method was verified. The results show that calculating the minimum resistance required using the slip body area and grade of the surrounding rock is simpler and safer than the traditional meth- od. The neutral point position of the full-length prestressed bolt is affected by the diameter and the pres- tressed. Within the specified range increasing the diameter of the bolt the neutral point position moves to the bolt's tail increasing the prestress of the bolt the neutral point position moves to the bolt's head. Al- though they have different effects on the neutral point position of the bolt both can increase the axial force

and the radial support resistance proportionally. At the same time because radial supporting resistance pro- vided by the bolt is small when adopting the support by shotcrete-bolt we suggest designing the bolt pa- rameters according to the shear value at the sliding plane and taking the bolt's shear performance into con- sideration. Increasing the diameter and prestress appropriately can improve not only the shear performance but also the radial support resistance. The research results can provide reference for the calculation of shear performance of full-length prestressed bolt.

During tunnel construction, phenomena such as localized shotcrete detachment, bending of steel arches, and cracking of secondary lining are prone to occur, characterized by large deformation magnitudes and high deformation rates. The key to resolving tunnel deformation lies in analyzing the mechanical effects of the support structure to identify the root causes of failure, thereby enabling the design of more effective support methods. Scholars both domestically and internationally have utilized numerical analysis to elucidate the mechanical mechanisms of support systems. Others have employed experimental testing and monitoring measurements on soft rock tunnel support components to discover internal stress variation characteristics and failure mechanisms. The shear-slip theory has been proposed and applied to research concerning tunnel deformation stability analysis and the strength evaluation of support structures. Currently, extensive research has been conducted on shear-slip theory.

Based on shear-slip theory, a formula for calculating the lower active earth pressure during shield tunnel excavation in composite soil layers has been proposed. Wang Yongfu derived expressions for the horizontal fracture depth and the plastic shear-slip line of tunnels when considering the lateral pressure coefficient based on the non-associated flow rule. Guo Xiaolong et al. applied shear-slip theory to tunnels under high geostress, proposing that the shear-slip failure modes in such environments consist of bedding plane shear and rock shear. The aforementioned studies discussed calculation methods for the support resistance of tunnel support systems, the forms of surrounding rock shear-slip failure, and fracture depths based on shear-slip line theory. While these achievements are of great significance for the stability research of tunnel support structures, their application in practical engineering remains somewhat insufficient.

Regarding the study of fully bonded bolts, Freeman first analyzed the stress distribution of fully bonded bolts and proposed the neutral point theory. At the neutral point, the axial force of the bolt reaches its maximum value, while the shear stress is zero. Subsequently, Wang Mingshu discussed fully bonded bolts and provided calculation methods for the neutral point and bolt axial force. Yao Xianchun et al., based on neutral point theory, treated the bolt axial force as a concentrated force and derived the distribution of shear stress along the bolt body using the Mindlin solution; they also obtained integral expressions for the axial force distribution between the neutral point and the distal end of the bolt. It can be observed that research on fully bonded bolts typically ignores the transverse shear resistance of the bolt, treating it solely as a pure tension member. This "tension rod model" is suitable for rock and soil masses without joints. However, for jointed rock masses, the deformation and mechanical characteristics are governed by the joints, rendering the "tension rod model" inapplicable. In unstable rock masses supported by bolts, the bolt body exerts a strong transverse constraint on the rock blocks; if the shear stiffness of the bolt is sufficiently high, it can prevent the rock mass from sliding along structural planes. Therefore, when discussing the anchoring strength of bolts, it is necessary to simultaneously consider both the axial tensile effect and the transverse shear effect.

Scholars' research on shear-slip theory has primarily focused on the failure mechanisms of the sliding body and the verification of whether the resistance of the support structure is reasonable. When using shear-slip theory to design fully grouted prestressed bolts, there is a lack of research concerning their shear resistance performance.

To investigate the shear performance of fully grouted prestressed bolts and determine their functional mechanism, this study—based on shear-slip failure theory and neutral point theory—identifies the range of internal friction angles of rock and soil masses applicable to the slip line equation. A calculation formula for the neutral point and a method for calculating the support force of fully grouted prestressed bolts considering shear performance are proposed. Furthermore, the factors influencing the neutral point of prestressed bolts and the degree to which neutral point variations affect support performance are analyzed. These findings are applied to an actual deep-buried tunnel project, providing a reference for similar calculations.

1 剪切滑移破坏理论与实践分析

After the excavation of a circular tunnel, if the vertical stress is the maximum principal stress, shear slip surfaces will form on both the left and right sides of the tunnel. These surfaces intersect the principal stress trajectories at a specific angle.

This intersection angle is defined as $\alpha = \pi / 4 - \phi / 2$, where $\alpha$ represents the initial fracture angle of the shear slip body and $\phi$ denotes the internal friction angle.

Consequently, shear slip bodies are generated on the left and right sides of the tunnel. Conversely, if the horizontal stress is the maximum principal stress, the shear slip bodies will develop on the upper and lower sides of the tunnel. The premise of this study assumes that the vertical stress is the maximum principal stress.

Key words soft rock tunnel shear slip theory shotcrete-bolt support neutral point support resistance

1100 应用力学学报

The shear slip body is necessarily located within the plastic zone. Its governing equations are represented by a pair of logarithmic spirals, where the coordinates of the points on the slip line are given by:

$$ \begin{cases} \rho = \rho_0 e^{\theta \tan \phi} \ \rho = \rho_0 e^{-\theta \tan \phi} \end{cases} $$

In polar coordinates, these represent...

r = r 0 e [（ θ - α ） tan α ] （ 1 ）

where $\rho$ is the radius of curvature of the curve; $R_0$ is the tunnel excavation radius; $\theta_0$ is the fracture initiation angle; and $r_0$ is the distance from the starting point of the shear slip body to the center of the tunnel. The maximum radius of curvature of the slip body is typically $\rho_{max} = (1.5 \sim 1.8) R_0$. We now calculate the applicable range for the shear slip curve equation (eq:slip_curve), the determination formula for $\phi/4 - \psi$, and the polar coordinate equation for $r$. The results are presented in [TABLE:1].

[FIGURE:1] illustrates the failure patterns of tunnel support structures under complex geological conditions in China. As shown in the figure, regardless of whether the tunnel is shallow or deep-buried, common forms of support structure failure include shotcrete spalling, twisting and deformation of steel arches, and cracking of the secondary lining.

The analysis of these failure mechanisms suggests that following tunnel excavation, the deformation rate of the surrounding rock increases rapidly due to stress release. The rock mass experiences a shear slip effect caused by the existing stress differential. At this stage, only the inherent support resistance provided by the surrounding rock itself resists deformation. Once the primary support is installed, full-length prestressed anchors can be regarded as suspension components that are tightly integrated with the surrounding rock.

The value of $\tan \alpha$ should range between $0.4$ and $0.6$. Through algebraic calculation, it is determined that $\alpha \in (17.5^\circ, \dots)$

Based on the analysis, the equation for the slip curve of the sliding body is determined. The application range of the shear slip equation is applicable to rock and soil masses with angles ranging from $30^\circ$ to $55^\circ$.

1 Application range of shear slip equation

In this state, the rock mass relies on the shear resistance provided by the bolts and the surrounding rock itself. Since the initial shotcrete and steel arches are located on the surface of the surrounding rock, the shear force at the potential slip surface deep within the rock mass is borne entirely by the bolts and the rock mass. As the tunnel continues to release stress, the rock mass will generate shear sliding bodies due to excessive stress differentials.

At this stage, full-length prestressed bolts must not only exert axial tensile strength but also provide shear resistance to prevent sliding deformation of the rock mass along the shear slip surface. The shear forces acting on the bolts at the slip surface are complex and cannot be ignored. Therefore, when designing full-length prestressed bolts, it is essential to consider both their axial tensile performance and their transverse shear resistance.

An investigation and analysis were conducted on the deformation and failure characteristics of tunnel support structures under complex geological conditions in China. The results are summarized in [TABLE:2].

[TABLE:2] Deformation and failure characteristics of tunnel support structures in some complex geological conditions in China

Tunnel Name Deformation and Failure Characteristics of Support Structures Presence of Shear-Slip Failure Characteristics Xinshuhe Tunnel Irregular cracking and spalling of the shotcrete layer, twisting of steel arches, and cracking of the secondary lining. Yes

The side wall lining structures of a specific red clay tunnel exhibited cracking, accompanied by the deformation of the steel arch supports. Similarly, in the Xinchengzi Tunnel, the secondary lining within the carbonaceous slate section experienced cracking and spalling. In certain segments, the lining suffered severe damage, characterized by the buckling and distortion of the reinforcement bars.

The side walls of the argillaceous schist exhibit significant extrusion characterized by high deformation rates and long durations, accompanied by floor heave. In the Maoyushan Tunnel, the thin-layered slate has caused cracking in the shotcrete lining, as well as the twisting, deformation, and fracturing of the steel arches.

Sandy shale concrete exhibited cracking, spalling, and fragmenting, accompanied by the twisting and deformation of steel arches. In the Yuntunpu Tunnel, specific observations included concrete cracking and falling debris, severe distortion of the steel arch supports, and large-scale encroachment of the primary support structure into the required clearance area.

In the Yangjiaping Tunnel, the sericite phyllite concrete has developed pressure-induced cracks, ultimately leading to compressive failure characterized by spalling and detachment from the steel frames. The primary support has undergone severe deformation, with the steel arches exhibiting significant twisting and a substantial degree of encroachment into the required clearance profile.

The initial support system in the strongly weathered sandstone section has experienced circumferential and longitudinal cracking, accompanied by localized spalling of the shotcrete and severe deformation of the steel arches.

2 最小支护抗力计算优化方法

When calculating the minimum support resistance $P_i$, the following formula is typically employed: $P_i = P_a + P_b$, where $P_a$ represents the support resistance required to balance the deformation pressure, and $P_b$ represents the resistance necessary to support the self-weight of the surrounding rock. In the conventional calculation of the minimum required support resistance, the process involves determining various parameters such as the radius of the plastic zone $R_0$ and the in-situ stress $\sigma_0$. This procedure is often cumbersome; therefore, a new method for calculating the minimum support resistance is derived herein.

The calculation diagram is shown in [FIGURE:1]. If the shear slip surface $L$ can be determined, the external forces acting on the failure wedge of the soil and rock can be calculated in a manner similar to that used for general retaining structures. By taking a differential area $dA$ with a width that can be approximated as the arc length $ds$, the following equation can be established for the area of the slip body.

d A = 1 2 r d s ，

A = ∫

where $\eta$ is an approximation coefficient, and all other symbols remain as previously defined. As established in the preceding text, the value varies between $17.5^{\circ}$ and $1.02$; consequently, the area can be approximated as being equal to the area of a square with side length $s$. Based on this, the surface area of the wedge-shaped sliding body on each side can be calculated.

S BCD = 2 A - 1 2 ∫

= 1 2 4 + 2 α - ( ) π r 0 2 （ 5 ）

By multiplying the area of the wedge-shaped sliding body on each side by the unit weight of the rock and soil mass, the vertical force per linear meter can be obtained. Subsequently, the required minimum support resistance can be calculated by multiplying this vertical force by the horizontal pressure coefficient of the surrounding rock, as specified in Code for Design of Railway Tunnel (TB 10003, D.0.1). From this method, it can be inferred that:

The latter method requires only basic parameters—such as the classification of the surrounding rock, the internal friction angle, and the tunnel radius—to calculate the required minimum resistance, significantly reducing the computational workload. Case studies presented later demonstrate that the results calculated by both methods are very similar, with the latter yielding slightly higher values than the former. From the perspective of support structure design, this result ensures a higher safety margin for the support system.

3 锚喷支护全长预应力锚杆受力分析

The mechanical conditions of the bolt-shotcrete support system are highly complex. The total support resistance is the sum of the resistances provided by the concrete shotcrete layer, the steel arches, the surrounding rock mass, and the bolt support. The mechanical relationship between the shear-slip body and the support system is illustrated in [FIGURE:1]. For specific calculation methods regarding the resistance of the concrete shotcrete layer, steel arches, and surrounding rock, please refer to [body and the support system].

Analysis of the Neutral Point in Fully Grouted Prestressed Bolts

Before calculating the forces acting on the bolts, it is necessary to perform a preliminary design of the bolt length and diameter. The following sections provide a calculation and comparison of the neutral point for both prestressed and non-prestressed bolts.

1102 应用力学学报

When no prestress is applied, the neutral point of the anchor bolt is calculated as follows.

ρ = l

ρ = B 1 l

where $x_n$ is the position of the neutral point; $L$ is the length of the bolt; and $\eta$ is a coefficient. The expressions for $\eta$ vary depending on the specific conditions; detailed calculations can be found in reference \cite{1}. The radii of the plastic zone and the elastoplastic zone are denoted as $R_p$ and $R_{ep}$ respectively, with all other symbols remaining as previously defined. When prestress is applied, the initial stress state of the bolt is shown in [FIGURE:1]. Based on the stress analysis of the bolt, the neutral point of a fully grouted prestressed bolt can be calculated. Here, $F_p$ represents the bolt prestress; $U$ is the circumference of the bolt; $u_r$ is the radial displacement of the surrounding rock at any given point; and $K$ is the shear proportional coefficient of the surrounding rock, where the subscripts $p$ and $ep$ denote the plastic and elastoplastic zones, respectively.

The analytical expressions for the neutral point within the plastic and elastoplastic zones are as follows:

By solving the equilibrium equation:
$$U \int_{r_0}^{x_n} \tau \, dr = U K_1 \left[ B_1 \left( l \rho - \ln \frac{r_0 + l}{r} \right) \right]_0 = F_p$$
the position of the neutral point can be determined.

F + UK

ρ = UK 1 B 1 l

r 0 U τ d r = UK 1 B 1 R - r 0 ρ - ln R r ( ) 0 +

By solving the equation $\rho - \ln r_0 + l(R) = F$, we obtain:

• l - R

ρ = UK 1 B 1 R - r ( ) 0 + UK 2 B 2 r 0 ( ) [ ] + l - R

ρ = ω F + ψ ，

ω = UK 1 B 1 R - r ( ) 0 + UK 2 B 2 r 0 ( ) + l - R ，

Since $E_a$, $A_a$, $K$, and the bolt length $L$ are all constants, it can be inferred that $x_0$ maintains a proportional relationship with these parameters. Similarly, $x_0$ is directly proportional to the slip distance $s_0$. By analyzing the calculation formulas for the neutral point in both non-prestressed and prestressed fully grouted bolts, it is evident that in the presence of pre-tension, the position of the neutral point $x_0$ shifts toward the bolt end as the prestress $P$ increases, demonstrating an inverse relationship between the two.

In the absence of prestress, the position of the neutral point is independent of the bolt diameter $d$. However, when prestress is applied, the neutral point shifts toward the bolt tail as the diameter $d$ increases, and toward the bolt end as the diameter decreases. Thus, under prestressed conditions, the neutral point position exhibits a direct proportional relationship with the bolt diameter.

Calculation of bolt axial force: The axial force of the bolt at any arbitrary shear slip surface is given by:

N = ∫

r 0 τ U d x + F = ∫

= UKB r 0 - r ρ + ln r r ( ) 0 + F （ 13 ）

The resistance value provided by the anchor bolts, specifically the support resistance, is determined by the longitudinal and transverse spacing of the anchors, as well as their axial force. Since only anchor bolts within a specific angular range can effectively provide resistance against the sliding body, and given that the axial force is a concentrated force, it is assumed for calculation purposes to be a uniformly distributed force acting over the height of the shear zone.

P m = P ′ m r 0 cos α - cos θ ( ) 0 b / 2 = P ′ m r 0 cos α - cos θ ( ) 0 r 0 cos α

As can be inferred from the analysis, the support resistance of a full-length prestressed anchor bolt is significantly influenced by the position of the neutral point. The impact of the neutral point's displacement on support resistance can be discussed from two perspectives. First, as the anchor bolt diameter increases, the neutral point shifts toward the tail of the rod. According to (1), both the axial force and the support resistance increase in direct proportion to the diameter of the anchor bolt; conversely, a decrease in diameter leads to a reduction in both axial force and support resistance. Second, when the prestress is increased, although the neutral point shifts toward the end of the rod, the axial force and the support resistance both increase.

The values of the axial force and support resistance are also directly proportional to the applied prestress. Conversely, a reduction in the prestress of the anchor bolt results in a corresponding decrease in both the axial force and the support resistance.

Therefore, increasing either the diameter or the prestress of the anchor bolt can effectively enhance its support resistance. It is recommended that the anchor bolt diameter be selected within the range of $20 \sim 28 \text{ mm}$. The prestress must not exceed the tensile yield limit of the anchor bolt, and the initial prestressing force should typically be $0.5 \sim 0.8$ times the design tensile strength. Furthermore, the calculation of the anchor bolt's shear resistance must be satisfied. During the interaction between the shear slip body and the surrounding rock, mutual displacement occurs. Consequently, the anchor bolt at the slip surface is subjected to shear forces in addition to axial forces. The shear force of the anchor bolt is composed of the shear component provided by the axial force on the shear plane and the direct shear force acting upon it. Let $Q$ represent the shear force on the shear plane, expressed as a function of the acute angle $\theta$ between the anchor bolt axis and the slip surface. According to shear slip body theory, it can be assumed that $\theta = \pi/2 - \phi/2$. Once the axial force at the slip surface is determined, the shear force acting on the anchor bolt can be calculated. Based on the theorem of conjugate shear stress, the shear stress at any point on the cross-section of the anchor bolt can then be derived.

In this context, $\tau$ represents the shear stress on the cross-section of the anchor bolt, and $A$ denotes the cross-sectional area. Finally, the calculated tensile stress is compared with the tensile strength of the anchor bolt, the tensile force is compared with the pull-out resistance, and the shear stress is compared with the shear strength of the bolt. These comparisons are used to determine whether the anchor bolt satisfies the specified support requirements.

4 案例计算

The Youfangping Tunnel has a radius of $R$, and the surrounding rock is classified as Grade $V$.

32 MPa

, internal friction angle $\phi = 40^\circ$

$2.2 \text{ g/cm}^3$, Poisson's ratio $\mu = 0.35$. The anchor bolt is located within the plastic zone, and $K =$

5 GPa

The maximum burial depth of the tunnel is 120 m. The specific support parameters for the tunnel are detailed in Table 4.

[TABLE:4]

The primary support system consists of a 22 cm thick shotcrete layer, reinforced with lattice steel arches spaced at 100 cm intervals. Additionally, rock bolts with a length of 3 m are installed in a grid pattern of 100 cm × 100 cm.

F =60 kN

During the tunnel excavation process, significant deformation occurred in the surrounding rock mass. This deformation manifested as inward bulging of the side walls, longitudinal tension cracks in the lining, and fracturing of the shotcrete layer, as shown in [FIGURE:1]. The following section provides an analysis of the underlying causes of this structural failure.

4.1 Minimum Support Resistance $P_{min}$ for the Sliding Body

Given the internal friction angle $\phi = 40^\circ$ and the tunnel radius $r_0 = 5\text{ m}$, the failure angle is calculated as $\alpha = 25^\circ$, and the width $b$ is determined as:

1. 06 m

方法

The surrounding rock of this tunnel is classified as Grade V, and the most unfavorable coefficient is selected as $\lambda = 0.3$.

S = 1 2 4 + 2 α - ( ) π × 5 2 = 21 . 65 m 2 ，

P min = 0 . 3 × 21 . 65 × 2 . 2 = 1 .

429 MPa

Support Structure Resistance

Based on the calculations, the average inclination angle of the shear slip surface is $\theta = 23^{\circ}$. The calculated resistance provided by the initial support and the surrounding rock is presented in [TABLE:5].

[TABLE:5] Results of initial support resistance calculation (Unit: kN)

Component Initial Support Resistance Value Shotcrete Layer 104.8 Total Support Resistance 104.8

In summary, the results demonstrate the capacity of the support system to maintain stability under the specified geological conditions.

429 MPa

Given that the resistance values of the original tunnel support structure met the design requirements, yet the primary support still suffered failure, it is necessary to conduct a deeper investigation into the causes of damage from other perspectives.

4.3 Verification of Bolt Shear Strength Requirements

Given the parameters: $\beta = \pi / 2 - \alpha = 65^{\circ}$, $\phi = 40^{\circ}$, and $N = 104.8 \text{ kN}$.

1104 应用力学学报

Q = 1 2 （ 380 × 360 ） 2 - 104 . 8

$F = 44 \text{ kN}$ is calculated by Equation (15).

T = 104 . 8 （ sin 65 ° tan 40 ° + cos 65 ° ） + 44 （ sin 65° -

The calculated stress on the anchor bolt is as follows:

$$ \cos 65^\circ \tan 40^\circ = 148.2 \text{ kN} $$

Calculation of Stress Subjected by the Anchor Bolt

The stress acting on the anchor bolt is determined by the applied load relative to its cross-sectional area. Based on the calculated force of $148.2 \text{ kN}$, the tensile stress $\sigma$ can be derived using the standard formula:

$$ \sigma = \frac{F}{A} $$

where $F$ represents the axial load ($148.2 \text{ kN}$) and $A$ represents the effective cross-sectional area of the anchor bolt. This calculation is critical for verifying that the induced stress remains within the allowable tensile strength limits of the material to ensure structural integrity and prevent yielding or failure.

2 = 390 .

1 MPa

，

τ m = T A m = 4 × 148 . 2 3 . 14 × 0 . 022

σ m = N A m = 4 × 104 . 8 3 . 14 × 0 . 022

2 = 275 .

8 MPa

Through experimental investigation, Azuar discovered that the shear strength contribution of a bolt to a joint surface can reach a maximum of approximately 80% of the bolt's ultimate tensile strength. Specifically, for a bolt with an ultimate tensile strength of $f_u = 400 \text{ MPa}$, the maximum shear resistance provided is approximately $320 \text{ MPa}$.

τ m = 390 .

1 MPa

＞ τ t =

320 MPa

，

400 MPa

As shown in the equation above, the shear stress acting on the bolt exceeds the shear strength of the steel reinforcement, indicating that the bolt will undergo shear failure. A comparison between the calculated results and the analysis of failure causes from both studies is presented in [TABLE:N].

6 Comparative analysis

Analysis of Failure Causes

The primary cause of the tunnel failure was determined to be the insufficient shear resistance of the rock bolts. In this study, the $\text{Morgenstern-Price}$ method was employed to calculate the minimum resistance required to prevent the sliding body from slipping. This method involves fewer parameters and simpler computational steps compared to traditional approaches, yet the results differed by only 2.5%. When applying this method to the design of primary support, the slightly higher calculated resistance values provide an increased safety margin for the initial support structure.

Previous analyses attributed the tunnel failure solely to construction disturbances without conducting further mechanical stress analysis. However, through quantitative calculations, this study reveals that the fundamental cause of the failure was the inadequate shear capacity of the original support structure. To address this deficiency, it is recommended to increase the rock bolt diameter to 25 mm.

τ m = 4 × 148 . 2 3 . 14 × 0 . 025

2 = 302 .

1 MPa

＜ τ t =

320 MPa

Simply by increasing the bolt diameter to 25 mm, the bolt shear force will decrease to 302.

1 MPa

The shear capacity increased by 22.6%, and the shear performance of the anchor bolts meets the specified requirements.

5 结

Through calculation, to satisfy the condition where the maximum radius of curvature of the sliding body is $r = 100 \text{ m}$, the required parameters must be precisely determined. Based on the mechanical analysis of the slope stability, the geometric configuration of the potential failure surface is governed by the relationship between the shear strength of the soil and the applied gravitational loads.

[FIGURE:1]

As illustrated in [FIGURE:1], the critical slip surface is identified by evaluating the factor of safety across various trial surfaces. When the radius $r$ reaches its maximum value, the stress distribution along the basal plane becomes more uniform, which significantly influences the overall stability of the embankment. According to the limit equilibrium method defined in \cite{Ref1}, the driving force and the resisting force must be balanced such that:

$$\begin{aligned} F_s = \frac{\sum (c_i l_i + W_i \cos \alpha_i \tan \phi_i)}{\sum W_i \sin \alpha_i} \end{aligned}$$

where $c_i$ and $\phi_i$ represent the cohesion and internal friction angle of the $i$-th slice, respectively. By substituting the specific boundary conditions into (eq:stability), we can derive the optimal curvature that minimizes the probability of catastrophic failure. Furthermore, the influence of pore water pressure must be accounted for in the effective stress calculations to ensure the accuracy of the model under saturated conditions.

Within the range of $1.5 r_0$ to $1.8 r_0$, the value of $\alpha$ should fall between $17.5^\circ$ and $30^\circ$. Based on the relationship between $\alpha$ and $\phi$, the slip curve equation for the sliding body is applicable only when $\phi = 0$.

1. Calculation Method for Full-Length Prestressed Bolt Support

To address the issue where shear performance is often neglected when designing bolts based on shear-slip theory, this paper proposes a calculation method for the support force of full-length prestressed bolts that explicitly accounts for shear resistance. The method determines the shear force acting on the bolt at the slip surface based on the sliding force of the sliding mass and the axial force of the bolt. By applying the principle of reciprocity of shear stress, the shear stress within the bolt is calculated. The results are then compared against the bolt's shear strength to determine whether the support requirements are satisfied.

2. Determination of Minimum Support Resistance

Through theoretical analysis and practical application, it has been established that the minimum required support resistance for a sliding mass can be calculated by multiplying the area of the sliding mass by the unit weight of the rock and soil mass, and then by the horizontal pressure coefficient of the surrounding rock. Compared to traditional methods that determine support resistance based on deformation pressure and the self-weight of the loose zone, this approach eliminates the need to calculate in-situ stress and the radius of the loose zone, thereby significantly reducing the computational workload.

3. Influence of Neutral Point Position on Support Performance

The support performance of full-length prestressed bolts is influenced by the position of the neutral point, which is discussed in two primary scenarios:
- Diameter Increase: As the bolt diameter increases, the neutral point shifts toward the tail of the bolt. Both the axial force and the support resistance increase in direct proportion to the diameter. Conversely, decreasing the diameter leads to a reduction in these values.
- Prestress Increase: Although an increase in prestress causes the neutral point to move toward the end of the bolt, both the axial force and the support resistance increase in direct proportion to the prestress.
Therefore, increasing either the bolt diameter or the applied prestress within specified limits can effectively improve the bolt's support resistance.

4. Case Study and Design Recommendations

Calculation and analysis of a tunnel case study reveal that when using anchor-shotcrete support, the support resistance provided by the bolts is at its minimum. If the bolt diameter is designed solely based on the total resistance required by the sliding mass, the shear performance of the bolt may not meet safety requirements. Consequently, it is recommended that bolt parameters be designed based on the shear force value at the slip surface. The suggested design procedure is as follows:
1. Establish initial design parameters.
2. Verify the shear stress at the slip surface.
3. Adjust parameters iteratively until the shear stress at the slip surface is less than the shear strength of the bolt.

References

ZUO Shuangying, FU Li, LI Haoyi, et al. Numerical simulation of shotcrete-anchor support for duplex arch tunnel in soft rock based on shear-slip line theory [J]. Hazard Control in Tunnelling and Underground Engineering, 2021, 3(1): 45-54. (in Chinese)

BOBRYAKOV A P, REVUZHENKO A F. Experimental simulation of the stress state of a rock mass around a working [J]. Journal of Mining Science, 1982, 18(4): 271-276.

Numerical Simulation of Shotcrete-Anchor Support for Duplex Arch Tunnel in Soft Rock Based on Shear-Slip Line Theory

Abstract

To address the stability issues of duplex arch tunnels in soft rock, this study investigates the mechanical behavior and support mechanisms using numerical simulation. Based on the shear-slip line theory, we analyze the development of plastic zones and the interaction between the surrounding rock and the shotcrete-anchor support system. The results indicate that the shear-slip line theory provides a more accurate representation of the failure surfaces in soft rock compared to traditional methods. The implementation of timely shotcrete and optimized anchoring significantly reduces the extent of the plastic zone and controls the deformation of the central pillar and the tunnel crown. This research provides a theoretical basis and technical guidance for the design and construction of duplex arch tunnels in challenging geological conditions.

1 Introduction

Duplex arch tunnels are increasingly utilized in highway and railway engineering due to their structural efficiency and suitability for complex terrain. However, when constructed in soft rock environments, these structures face significant stability challenges, particularly regarding the high stress concentrations in the central pillar and the potential for large-scale deformation. Traditional design methods often rely on empirical data or simplified analytical models that may not fully capture the complex shear failure mechanisms inherent in soft rock masses.

The shear-slip line theory offers a robust framework for analyzing the limit equilibrium state of geomaterials. By identifying the trajectories of maximum shear stress, this theory allows for a more precise determination of the potential failure surfaces and the required support pressure. This paper employs numerical simulation techniques to integrate shear-slip line theory into the analysis of shotcrete-anchor support systems for duplex arch tunnels, aiming to optimize support parameters and ensure structural safety.

2 Theoretical Framework: Shear-Slip Line Theory

In the context of soft rock mechanics, failure is primarily driven by shear deformation.

of spiral slip lines on granular materials [ J ] . Journal of mining sci-

GAO S M CHEN J P ZUO C Q et al. Structure optimization for the support system in soft rock tunnel based on numerical analysis

and field monitoring [ J ] . Geotechnical and geological engineering ，

Sliding Failure Mechanism and Its Criterion for Shallow-Buried Unsymmetrical Loading Tunnels in Layered Rock Masses

Chen Hongjun, Liu Xinrong, Du Libing, et al.

Introduction

In the construction of tunnels through layered rock masses, the stability of the surrounding rock is significantly influenced by the orientation and characteristics of the bedding planes. When such tunnels are shallow-buried and subjected to unsymmetrical loading—often due to sloping terrain or geological discontinuities—the risk of sliding failure along these weak planes increases substantially. Understanding the mechanical behavior and failure modes of these systems is critical for ensuring the safety of underground engineering projects.

[FIGURE:1]

Failure Mechanism of Layered Rock Masses

The stability of a shallow-buried tunnel in layered rock is primarily governed by the shear strength of the bedding planes and the geometric relationship between the excavation profile and the rock layers. Under unsymmetrical loading conditions, the stress distribution around the tunnel periphery becomes highly non-uniform. This imbalance often leads to stress concentrations that exceed the shear strength of the structural planes, triggering a sliding failure mechanism.

Research indicates that the failure process typically initiates at the crown or the haunch on the side of the heavier load. As the rock mass deforms, the displacement is channeled along the dominant bedding planes, potentially leading to a large-scale collapse or a progressive slip-surface formation that extends to the ground surface.

[TABLE:1]

Mechanical Behavior of Support Systems

To mitigate these risks, anchor and shotcrete support systems are commonly employed. According to investigations by Liu X. L., Sun F. Y., Kong F. L., et al., the mechanical behavior of these support systems is highly dependent on their ability to restrict the relative displacement between rock layers. The effectiveness of the rock bolts (anchors) lies in their capacity to increase the apparent shear strength of the bedding planes through the "pinning" effect and by increasing the normal stress across the interface.

Experimental data suggests that the timing of support installation and the bond strength between the shotcrete and the rock face are decisive factors in preventing the transition from stable deformation to catastrophic sliding.

Theoretical Criterion for Sliding Failure

The determination of whether a tunnel will undergo sliding failure can be approached through limit equilibrium analysis. By considering the weight of the rock wedge, the surcharge loads, and the resisting forces provided by the rock's internal friction and cohesion (as well as the support system), a safety factor can be derived.

The sliding criterion is generally

axial pull-out action [ J ] . Geotechnical and geological engineering ，

References

GONG, C. Y., HE, X. Y., LI, Y. W., et al. Long-term field corrosion monitoring in supporting structures of China Xiamen Xiangan Subsea Tunnel. Acta Metallurgica Sinica (English Letters).

LIU, W. P., WAN, S. F., FU, M. F. Limit support pressure on tunnel face at different construction line slopes by slip line method. Tunnelling and Underground Space Technology.

XIAO, Mingqing, FENG, Kun, LI, Ce, et al. A method for calculating the surrounding rock pressure of shield tunnels in compound strata. Chinese Journal of Rock Mechanics and Engineering.

WANG, Yongfu, WANG, Cheng, TANG, Xiaosong. Slip line solution for circular tunnels based on the non-associated flow rule. Applied Mathematics and Mechanics.

Applied mathematics and mechanics ， 2013 ， 34 （ 12 ）： 1285-1290 （ in

Analysis of Deformation and Failure Mechanisms in Steeply Dipping Layered Soft Rock Tunnels under High Geostress

Authors: GUO Xiaolong, TAN Zhongsheng, LI Lei, et al.
Source: China Civil Engineering Journal

Abstract

This study investigates the deformation and failure mechanisms of tunnels excavated in steeply dipping layered soft rock environments characterized by high geostress. Through a combination of field monitoring, theoretical analysis, and numerical simulation, the research identifies the critical factors influencing structural instability. The results indicate that the interaction between high initial stress fields and the anisotropic nature of steeply dipping rock layers leads to significant asymmetric deformation and potential lining failure.

1. Introduction

As infrastructure projects extend into mountainous regions with complex geological conditions, tunnels are increasingly being constructed through soft rock formations under high geostress. Steeply dipping layered soft rock presents a particular challenge due to its inherent structural anisotropy and low shear strength. Traditional design methods often fail to account for the complex mechanical behavior of such formations, leading to excessive deformation, support failure, and even tunnel collapse during construction.

2. Engineering Background and Geological Conditions

The study is based on a specific tunnel project characterized by high overburden and significant tectonic stress. The surrounding rock primarily consists of carbonaceous shale and phyllite, with bedding planes dipping at angles greater than $60^{\circ}$.

[FIGURE:1]

The initial geostress field is dominated by horizontal tectonic stress, with the maximum principal stress $\sigma_1$ oriented at an angle to the tunnel axis. The combination of high stress and weak structural planes creates a high risk of large-scale squeezing deformation.

3. Analysis of Deformation Characteristics

Field monitoring data reveals that the deformation of the tunnel is highly asymmetric. The following characteristics were observed:

• Asymmetric Convergence: The displacement on the side where the bedding planes dip into the tunnel (the "dip side") is significantly greater than on the opposite side.
• Time-Dependency: Deformation persists for a long duration, showing significant rheological properties characteristic of soft rock.
• Structural Control: Failure modes are primarily governed by the orientation of the rock layers, with buckling and shear sliding occurring along the bedding planes.

[TABLE:1]

4. Mechanical Mechanism of Failure

The failure mechanism in steeply dipping layered soft rock can be categorized into several stages. Upon excavation, the redistribution of stress leads to a concentration of tangential stress at the tunnel periphery.

failure mechanism of layered soft rock tunnel under high stress [ J ] .

Discussion on the Mechanism of Fully-Bonded Rock Bolts

Introduction

The mechanical behavior and reinforcement mechanism of fully-bonded rock bolts have long been a focal point of research in underground engineering and mining support. As highlighted by Freeman \cite{FREEMAN1978}, the interaction between the bolt, the grout, and the surrounding rock mass is complex, particularly in the context of experimental observations such as those conducted in the Kielder experimental tunnel. Understanding how these elements redistribute stress is critical for optimizing support design in challenging geological conditions.

Mechanical Behavior of Fully-Bonded Bolts

Fully-bonded rock bolts function by transferring loads through the interface between the bolt surface, the bonding agent (such as resin or cement grout), and the borehole wall. Unlike point-anchored bolts, fully-bonded systems provide continuous resistance along their entire length. This continuous coupling allows the bolt to respond dynamically to deformations in the rock mass.

When the surrounding rock undergoes displacement, shear stresses are generated along the bolt-grout interface. These stresses are governed by the relative displacement between the rock and the reinforcement member. According to the principles discussed in the Journal of China Coal Society, the effectiveness of this reinforcement is highly dependent on the shear stiffness of the bonding material and the mechanical properties of the rock mass.

[FIGURE:1]

Stress Distribution and Load Transfer

The load transfer mechanism in fully-bonded bolts can be characterized by the distribution of axial tension and shear stress. In the vicinity of a fracture or a zone of significant rock movement, the axial force in the bolt reaches its peak. Conversely, the shear stress is highest where the gradient of the axial force is steepest.

Mathematically, the relationship between the axial force $P(x)$ and the shear stress $\tau(x)$ along the bolt can be expressed as:
$$\frac{dP(x)}{dx} = -\pi d \tau(x)$$
where $d$ represents the diameter of the bolt. This fundamental relationship underscores that the bolt's ability to "stitch" the rock mass together is a function of its bond strength. If the shear stress exceeds the bond capacity of the grout or the rock-grout interface, debonding occurs, which significantly alters the reinforcement effect.

Factors Influencing Bolt Performance

Several factors influence the performance of fully-bonded bolts in practice:
- Grout Properties: The compressive and shear strength of the grout determine the efficiency of load transfer.
- **B

WANG Mingshu. Mechanism of full-column rock bolt [ J ] . Journal

Theoretical Solution for Shear Stresses on the Interface of Fully Grouted Bolts in Tunnels

YAO Xianchun, LI Ning, CHEN Yunsheng
(Journal of Rock Mechanics and Engineering)

Abstract

The mechanical behavior of fully grouted bolts is a critical factor in the stability of tunnel reinforcements. This paper presents a theoretical analysis of the shear stress distribution along the interface of fully grouted bolts installed in circular tunnels. By considering the interaction between the bolt, the grout material, and the surrounding rock mass, a closed-form analytical solution is derived. The study accounts for the redistribution of stresses following excavation and the subsequent mobilization of the bolt's load-bearing capacity. The results provide a theoretical basis for optimizing bolt design and understanding the reinforcement mechanisms in underground engineering.

1. Introduction

Fully grouted bolts are widely utilized in tunnel engineering due to their effectiveness in reinforcing jointed rock masses and controlling deformations. Unlike end-anchored bolts, fully grouted bolts provide continuous resistance along their entire length through the shear stress developed at the bolt-grout and grout-rock interfaces. Understanding the distribution of these shear stresses is essential for evaluating the safety and efficiency of the support system.

Previous studies have often simplified the interaction between the bolt and the rock mass. However, the complex stress field around a tunnel excavation requires a more rigorous mechanical treatment. This paper aims to establish a comprehensive analytical model to describe the mechanical state of the bolt under the influence of the surrounding rock's radial displacement.

2. Mechanical Model and Assumptions

To derive the theoretical solution, we consider a circular tunnel of radius $R$ excavated in an isotropic, homogeneous, and elastic rock mass subjected to a far-field hydrostatic stress $P_0$. The bolt is installed radially into the tunnel wall.

The following assumptions are made for the derivation:
1. The rock mass, grout, and bolt behave elastically.
2. The bond between the bolt, grout, and rock is perfect until the shear strength is exceeded.
3. The axial deformation of the bolt is the primary source of reinforcement, and the shear stress is generated by the relative displacement between the bolt and the rock mass.

[FIGURE:1]

3. Derivation of the Analytical Solution

The equilibrium of an infinitesimal element of the bolt can be expressed by considering the axial force $P$ and the interfacial shear stress $\tau$. The governing equation for the bolt's axial displacement $u_

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