Numerical Investigation of the Pull-Out and Shear Mechanical Characteristics and Support Effectiveness of Yielding Bolt in a Soft Rock Tunnel

Abstract

Conventional bolts frequently fail under large deformations due to stress concentration in soft rock tunnels. In contrast, yielding bolts incorporate energy-absorbing mechanisms to sustain controlled plastic deformation. This study employed FLAC3D to numerically investigate the pull-out, shear, and bending behaviors of yielding bolts, evaluating their support effectiveness in soft rock tunnels. Three-dimensional finite difference models incorporating nonlinear coupling springs and the Mohr–Coulomb criterion were developed to simulate bolt–rock interactions under multifactorial loading. Validation against experimental pull-out tests and field measurements confirmed the model accuracy. Under pull-out loading, the axial forces in yielding bolts decreased more rapidly along the bolt length, reducing stress concentration at the head. The central position of the maximum load-bearing capacity in conventional bolts fractured under tension, resulting in an hourglass-shaped axial force distribution. Conversely, the yielding bolts maintained yield strength for an extended period after reaching it, exhibiting a spindle-shaped axial force distribution. Parametric analyses reveal that bolt spacing exerts a greater influence on support effectiveness than length. This study bridges critical gaps in understanding yielding bolt behavior under combined loading and provides a validated framework for optimizing energy-absorbing support systems in soft rock tunnels.

1. Introduction

The stability of tunnels in soft rock masses, characterized by low strength, high deformability, and time-dependent creep behavior, remains a critical concern in geotechnical engineering, particularly in deep mining and transportation infrastructure projects [1,2]. These geological conditions often lead to large-scale convergence, stress redistribution, and progressive failure during and after excavation, necessitating support systems capable of accommodating significant deformations while maintaining structural integrity [3,4,5]. Among reinforcement techniques, rock bolting has long been a cornerstone strategy for stabilizing underground openings [6,7,8]. However, conventional bolts exhibit limited adaptability under high-strain conditions, frequently suffering brittle failure induced by stress concentration in squeezing grounds [9]. This limitation has driven the development of yielding bolts, which integrate energy-absorbing mechanisms to sustain controlled plastic deformation without losing load-bearing capacity [10]. Despite their growing application, the mechanical behavior of yielding bolts under combined loading regimes—specifically pull-out, shear, and bending—and their interaction with soft rock masses remain insufficiently characterized [11,12]. Addressing these knowledge gaps is pivotal for optimizing yielding bolt design and validating their efficacy in real-world tunneling scenarios, particularly in geotechnically challenging environments.

While experimental studies on yielding bolts have demonstrated their superior elongation capacity compared to conventional systems [13], most of the research has focused on simplified laboratory pull-out tests, leaving critical questions unanswered. For instance, the shear resistance of yielding bolts has received limited attention [14,15]. Similarly, the bending behavior of these bolts under lateral loads, such as those induced by asymmetrical ground pressures, remains poorly understood [16]. Furthermore, existing studies often neglect the holistic performance of yielding bolts in full-scale tunnel models, where complex stress paths, rock–bolt coupling effects, and time-dependent deformation interplay [17]. These limitations underscore the need for advanced numerical frameworks capable of simulating multifactorial loading conditions and providing insights into bolt–rock interactions at both local and global scales.

Numerical modeling has emerged as a powerful tool for analyzing the nonlinear behavior of support systems in soft rock tunnels [18,19,20,21]. The finite difference method, implemented in software such as FLAC3D, is particularly suited for this purpose due to its ability to handle large deformations, interface slip, and elastoplastic material responses [22,23,24]. Prior studies by Jiang et al. [25] and Bahrani and Hadjigeorgiou [26] validated the capability of FLAC3D to replicate the pull-out and shear tests of conventional bolts, establishing its credibility for simulating bolt–rock interactions. Saadat et al. [27] employed numerical simulation methods to analyze the influence of boundary conditions on the shear behavior of rock joints. Zhang et al. [28] used Flac3D to study the development law of interface for geotechnical prestressed anchorage bolt. Zhang et al. [29] revealed the shear characteristics of the bolt–grout interface under cyclic shear loading. However, these efforts largely ignored yielding mechanisms, which introduce additional complexities, such as strain-dependent stiffness degradation, frictional energy dissipation, and post-yield load redistribution [10,30]. Recent advancements in constitutive modeling—including the integration of nonlinear coupling springs, strain-softening criteria, and anisotropic damage laws—have enhanced the fidelity of bolt simulations [31,32]. For example, Tahmasebinia et al. [9] simulated yielding bolt performance under cyclic loading, highlighting the influence of bolt geometry on energy absorption. Despite these strides, comprehensive numerical investigations integrating pull-out, shear, and bending analyses for yielding bolts remain scarce [33,34]. This gap hinders the optimization of critical design parameters, such as bolt length, spacing, and yielding element configuration, which directly govern support effectiveness and economic viability [35,36,37].

To address these issues, the pull-out tests of conventional bolts and previously developed yielding bolts were simulated using FLAC3D. The detailed axial force distribution and its dynamic evolution under pull-out conditions for both bolt types were investigated. Additionally, a comprehensive analysis of the shear resistance of the bolts was conducted. Furthermore, a comparative study was undertaken to evaluate the support efficacy of conventional bolts against yielding bolts within the context of soft rock tunnel environments. The findings elucidate the mechanical behavior of yielding bolts under combined loading regimes, and this work also bridges critical gaps in the understanding of energy-absorbing support systems.

A comprehensive numerical simulation of the pull-out, shear, and bending analysis of yielding bolts was performed. A distinctive spindle-shaped axial force distribution in yielding bolts during plastic deformation was revealed, contrasting with the hourglass-shaped pattern in conventional bolts. This mechanistic insight explains the superior elongation capacity of yielding bolts.

2. Methodology

2.1. Simulation Theory

Numerical simulation of the bolt mechanical properties was conducted using the finite difference software FLAC3D 3.00. Rock bolt elements were employed to simulate the bolts, and these were defined by geometric parameters, material properties, and coupled spring parameters. A line segment between two structural nodes is represented as a single element component, with the components between two nodes sharing the same symmetric cross-section parameters. The rock bolt element effectively combines the functions of beam elements and bolt elements, making it suitable for simulating bolts subjected to both normal and axial frictional forces [22].

Each structural element component possesses its own local coordinate system, and it is used to specify inertial moments, distributed loads, and defined force and moment signs. As shown in Figure 1, the local coordinate system of the components of the simulated bolts was defined by the positions of its two nodes (1 and 2) and the vector Y with the following rules: the central axis coincides with the X axis, the X axis direction is from Node 1 to Node 2, and the Y axis lies in the cross-sectional plane.

The interaction between the bolt rod and the rock mass was simulated using coupling springs. These nonlinear, sliding coupling springs transmit forces and moments between structural element nodes and solid elements. The sign conventions for forces and moments on components are illustrated in Figure 2.

During the process of bolt pullout, the axial behavior of the bolt can be described using a one-dimensional constitutive model, as expressed in the following equation:

$$K={\displaystyle \frac{AE}{L}},$$

(1)

where $K$ represents the axial stiffness of the bolt, $A$ denotes the cross-sectional area of the bolt, $E$ stands for the elastic modulus, and $L$ signifies the length of the bolt. The tensile yield strength $F_t$ and compressive yield strength $F_c$ of the bolt were determined, and these two limits must not be exceeded during the utilization of the bolt.

When subjected to tensile loading, the bolt’s interaction with the surrounding rock is simulated through tangential coupling springs. The function of these springs is analogous to the tangential action mechanism of grouted bolts that is depicted in Figure 3; therefore, the properties of the tangential coupling springs can represent the grouting properties. The shear stress on the contact surface between the bolt and the rock mass primarily considers its cohesive force and frictional force. The influence of tangential coupling springs around the bolt is mainly reflected through parameters such as the stiffness, cohesive force, internal friction angle, outer boundary radius of the bolt, and effective stress around the bolt. The relationship between them can be expressed by the following equation [26]:

$$F_s/L=K_s\left(u_p-u_m\right),$$

(2)

where $F_s$ is the shear force, $K_s$ is the shear stiffness of the coupling spring, and $u_p$ and $u_m$ are the axial displacements of the rock bolt element and the surrounding medium, respectively.

Furthermore, we have

$$F_s^{\mathrm{m}\mathrm{a}\mathrm{x}}/L=C_s+{\sigma }_m^{l\tan {\varphi }_s},$$

(3)

where $C_s$ is the cohesive force per unit length of the shear coupling spring, ${\varphi }_s$ is the friction angle of the shear coupling spring, ${\sigma }_m$ is the pressure on the rock bolt, and $l$ is the outer circumference of the rock bolt.

When the bolt body is subjected to shear due to the sliding of the surrounding rock, it needs to resist bending moments. When there are cracks on the contact surface, sliding due to shear deformation can easily occur, where rock bolts serve to reinforce the structure. Sliding along the crack surfaces can lead to shear deformation or even failure of the bolt. By specifying the bolt’s yield strength, axial yielding can be achieved. Bolt fracture is simulated based on the tensile failure strain. The strain at various bolt positions includes the axial strain and bending plastic strain, where the axial strain is the average plastic strain along the bolt. The formula [22] for calculating plastic tensile strain is given by

$${\epsilon }_{pl}=\sum {\epsilon }_{pl}^{ax}+\sum {\displaystyle \frac{d{\theta }_{pl}}{2L}},$$

(4)

where $d$ is the bolt diameter, $L$ is the bolt length, and $\theta$ is the average twist angle of the bolt. If the strain exceeds a specified value, the forces and moments on the bolt components in the simulation will be reduced to zero, indicating the failure of the bolt.

2.2. Pull-Out Simulation of the Yielding Bolt

Jiang et al. utilized a three-dimensional finite difference method to establish a numerical simulation model for bolt pull-out tests [25]. Their research indicated that their numerical results aligned with the field test results, validating the feasibility of numerically simulating bolt pull-out behavior. In this paper, the finite difference software FLAC3D was employed to simulate the yielding bolt pull-out test and the shear test of the bolt body, aiming to demonstrate the feasibility of simulating the mechanical characteristics of the yielding bolt and the shear performance of the bolt through FLAC3D simulation.

To simulate the stress and deformation characteristics under the yielding bolt pull-out action, a computational model was established, as shown in Figure 4. The computational model of the anchoring body was a 10 m × 10 m × 10 m (X × Y × Z) cubic block. The bolt was 6 m in length, with the bolt head (that is, near the opening) positioned at (5, 5, 10) and bolt end at (5, 5, 4). The top face of the model shown in Figure 4 is a free boundary, the four sides were subject to normal constraints, and the bottom face was fully fixed. A constant velocity V was applied at the head of the bolt along the axis of the bolt. During pull-out simulation, the displacements perpendicular to the Z direction on two free surfaces were constrained, and the bolt was fixed with zero velocity in the X and Y directions.

The constitutive model of the model adopted the Mohr–Coulomb model. The material parameters of the surrounding rock were selected from the fourth category, according to the standard for engineering classification of rock masses, of surrounding rocks focused on in this study [38]. The proposed rock parameters are listed in

Table 1. The physical and mechanical parameters of the surrounding rock.

Material	Elasticity Modulus (GPa)	Poisson Ratio	Unit Weight (kN/m3)	Cohesive Force (MPa)	Internal Friction Angle (°)
Surrounding rock	4.00	0.325	21.5	0.45	33

, and the bolt calculation parameters are shown in

Table 2. The physical and mechanical parameters of the yielding bolt.

Yielding Bolt	Length (m)	Cross Sectional Area (m3)	Elasticity Modulus (GPa)	Poisson Ratio	Tensile Failure Strain
Regular part	5.5	5 × 10−4	200	0.25	0.1
Yielding element	0.5	5 × 10−4	20	0.25	0.2

.

2.3. Shear Performance Simulation of the Yielding Bolt

Considering the shear performance of the yielding bolt, a model—as shown in Figure 5—was established for simulation verification. The computational model of the anchoring body was a 10 m × 10 m × 10 m (X × Y × Z) cubic block, and it was divided into two large rock blocks above and below, with an interface function between them to allow for rock sliding. The mesh division used 8-node hexahedral elements, totaling 1000 elements and 1452 nodes. The model shown in Figure 5 was divided into two parts. The boundary conditions of the upper part were as follows: the top and bottom faces were subject to normal constraints, and all of the sides were constrained along the Z-direction. The boundary conditions of the lower part were that all faces are fully fixed. A 6 m long yielding bolt was driven vertically into the center of the model at the interface, with the bolt head positioned at (5, 5, 8) and the bolt end at (5, 5, 2). The parameters of the surrounding rock and the bolt are presented in Table 1 and

Table 3. The physical and mechanical parameters of the conventional bolt.

Material	Length (m)	Cross Sectional Area (m3)	Elasticity Modulus (GPa)	Poisson Ratio	Tensile Failure Strain
Conventional bolt	6	5 × 10−4	200	0.25	0.1

, respectively.

Bending tests also reflect the bolt body’s shear performance. To further demonstrate the feasibility of simulating the shear performance, a bolt bending test was conducted. Part of the bolt was anchored in the rock, and a force perpendicular to the bolt body was applied at the exposed head.

A model, as depicted in Figure 6, was established, with the computational model of the anchoring body being a 10 m × 10 m × 10 m (X × Y × Z) cubic block. The mesh division used 8-node hexahedral elements, totaling 1000 elements and 1331 nodes. The boundary conditions of the model were that all faces are fully fixed. The bolt length was 6 m, with the bolt end at (5, 5, 7), where half of the bolt was anchored inside the rock and the head was outside. Just like the model shown in Figure 5, the parameters of the surrounding rock and the bolt are presented in Table 1 and Table 3, respectively.

2.4. Simulation of the Soft Rock Tunnel Supported by Bolt

As shown in Figure 7, a model of 100 m × 100 m (X × Z) was established, with the Y axis representing unit length. The excavation of the tunnel was circular with a diameter of 10 m. The tunnel axis corresponded to the Y axis, with the vertical upward direction defined as the positive Z axis. The mesh utilized 8-node hexahedral elements, resulting in a total of 1584 elements and 3170 nodes. The model is a two-dimensional model. The boundary conditions of the model were as follows: the top face was a free boundary, the two sides were subject to normal constraints, and the bottom face was fully fixed. The vertical stress was set at 10.75 MPa and the horizontal stress along the tunnel axis and the horizontal stress perpendicular to the tunnel axis were 12.90 MPa, with a lateral pressure coefficient of 1.2. Considering the focus on soft rock as the main research subject, the surrounding rock parameters were selected from the fourth category of surrounding rocks based on the standard for engineering classification of rock masses [38], as detailed in Table 1. The constitutive model and yielding criteria for the surrounding rock adopted the Mohr–Coulomb model. In this study, the term “yielding bolt” refers to the bolt used for anchoring purposes.

To investigate the support effects of the two bolt types in soft rock tunnel excavation, a planar strain model of a tunnel was established. The rock mass was first brought to the initial equilibrium under the original rock stress state, with initial displacements set to zero. The excavated portion was replaced by a null model, and the support was then implemented. For comparison, simulations were conducted using three approaches: Approach One involved no bolt application, Approach Two involved the use of standard bolts, and Approach Three involved the use of yielding bolts. (Refer to Table 2 and Table 3 for the parameters.).

To more intuitively and precisely reflect the influence of the bolts on the deformation distribution of the surrounding rock, reinforcement using bolts alone was implemented after partial stress release following tunnel excavation, and this was conducted without other combined support methods. The bolt support following tunnel excavation was radially and uniformly arranged, with radial spacing along the tunnel wall approximately 1 m apart, as illustrated in Figure 8. To provide a more intuitive representation of the surrounding rock deformation within the tunnel, 16 monitoring points were evenly distributed at positions such as the arch crown, arch shoulder, sidewalls, arch foot, and arch base during simulation, as depicted in Figure 9 and

Table 4. The monitoring point locations on the surrounding rock.

Monitoring Point	Location
A	Arch crown
B	Arch shoulder (67.5° to horizontal)
C	Arch shoulder (45° to horizontal)
D	Arch shoulder (22.5° to horizontal)
E	Sidewall (left)
F	Arch foot (22.5° to horizontal)
G	Arch foot (45° to horizontal)
H	Arch foot (67.5° to horizontal)
I	Arch base
J	Arch foot (67.5°to horizontal)
K	Arch foot (45° to horizontal)
L	Arch foot (22.5° to horizontal)
M	Sidewall (right)
N	Arch shoulder (22.5° to horizontal)
O	Arch shoulder (45° to horizontal)
P	Arch shoulder (67.5° to horizontal)

.

3. Results and Discussion

3.1. Feasibility Study of Pull-Out Simulations of Yielding Bolt

Guo et al. conducted indoor experimental studies using three sets of yielding bolts, with essentially identical results [13]. The bolt initially deforms elastically under loading, and when the pressure reaches 150 kN, the yielding device begins to function, causing displacement. Experimental results demonstrate that the loading remains relatively stable. A comparison between one set of indoor experiments and simulation results is shown in Figure 10. After the bolt is subjected to tension, elastic deformation occurs. In the simulation, due to the differences in the parameters of the yielding bolt, the pressure required to reach the yield deformation was approximately 160 kN. As the pull-out test progressed, the loading increased, reaching the yield strength, after which the bolt entered a stage of plastic deformation and could be sustained for a period.

Furthermore, Guo et al. conducted pull-out tests on three sets of 35 m long bolts at an engineering site, with yielding components positioned at 29 m, 32 m, and 35 m from the bolt head. As shown in Figure 11, a comparison of the numerical simulation tests under the action of yielding bolt pull-out, indoor experiments, and field tests [13] revealed that, although there are slight differences in the yield strength and yielding amount of the bolt among the three research methods, the loading characteristics, deformation trend, and support effect were consistent. Hence, it is feasible to use the finite difference software FLAC3D to simulate pull-out tests and support effects of yielding bolts, and the simulation results align with the operating mechanism and loading characteristics of yielding bolts.

3.2. Comparative Analysis of the Bolts’ Mechanical Behavior During Pull-Out Simulation

To analyze and compare the support effects and load characteristics of the yielding bolts and conventional bolts, models, as depicted in Figure 4, were established for the pull-out simulations. The axial forces experienced at various locations along the bolt reflected the propagation of tensile forces within the bolt, providing a direct response to the loading conditions of the bolt. The distribution of the axial forces at different positions along the two types of bolts is illustrated in Figure 12.

It is evident that, under the same pull-out force, the tension borne by the bolt at different locations varies, with the axial forces experienced by the bolt gradually decreasing with increasing depth. The curve exhibits a concave shape, indicating a decreasing rate of axial force reduction along the bolt, thus approaching a plateau. As the depth increased, the yielding bolts experienced a faster decrease in axial forces compared to conventional bolts, particularly at the bolt head where higher axial forces are applied and are closer to reaching the yield strength, indicating a stronger adaptability of yielding bolts to resist larger pull-out forces. Furthermore, under the same pull-out load, the yielding bolts exhibited lower axial forces than conventional bolts at equivalent positions, suggesting that yielding bolts can withstand greater tensile forces.

The simulated pull-out involved applying a constant velocity at the bolt head, causing the pull-out force to increase incrementally until yielding. Figure 13 illustrates the variation of axial forces at different positions along the conventional and yielding bolts as the axial strain increased during the simulation process. Positions at 0.125 m, 1.125 m, 2.125 m, 3.125 m, 4.125 m, and 5.125 m from the bolt head were chosen for clarity.

Figure 13 shows that—before reaching yield, as the pull-out force and axial strain increase—the axial tensile forces increased at all sections, with the greatest increase being recorded near the head and there was also decreasing toward the inner end. Once the maximum axial force reached yield strength, its forces ceased, and this increased with further strain. Continued loading beyond yield led to brittle failure, analogous to necking in steel, and this caused rapid anchorage force decline and eventual failure.

Figure 14 depicts the axial forces at 0.125 m (head), 3.125 m (middle), and 5.125 m (end) for both bolt types. Prior to reaching the yield strength of the bolt, the bolt underwent elastic deformation, with the curve showing a significantly higher rate of axial force increase for conventional bolts compared to yielding bolts, highlighting the high elongation capacity of yielding bolts. According to the axial strain variation laws at the three positions of the two types of bolts, as shown in Figure 14, it is obvious that the elastic strains at which the conventional and yielding bolts reach yield strength were 0.08% and 1.26%, respectively, with the corresponding allowable elastic deformations of 4.92 mm and 75.46 mm, indicating elastic yielding distance of 70.54 mm for the yielding bolts.

After reaching yield strength, the bolt underwent plastic deformation. The plastic phase duration was longer for yielding bolts, again demonstrating their high elongation capacity. The plastic strains at which the conventional and yielding bolts reached ultimate strength were 0.3025% and 2.48%, respectively, corresponding to plastic deformations of 148.8 mm and 18.15 mm. This resulted in a plastic yielding distance of 130.65 mm for the yielding bolt.

At a distance of 0.125 m from the bolt head, the axial force acting on the yielding bolt was equivalent to that of a conventional bolt, while, at the 3.125 m and 5.125 m distances, the axial forces were notably lower for the yielding bolt compared to the conventional bolt. Under similar material conditions, this characteristic of yielding bolts in non-head sections determines their superior support effectiveness.

3.3. Feasibility Study of Shear Performance Simulations of Bolt

3.3.1. Shear Performance Simulation of Bolt

The upper rock block in Figure 5 was pushed to slide while the lower block remained fixed. As relative sliding progressed, the bolt’s shear resistance was demonstrated. As illustrated in Figure 15, the bending moments at several positions along the bolt varied with calculation steps. With increasing relative sliding distance, the bending moment near the interface position gradually increased. This confirms that FLAC3D’s rock–bolt element can capture the bolt’s shear resistance, enabling precise analysis of its support effect. Since the yielding device minimally influences shear resistance, the shear resistance of conventional and yielding bolts was essentially identical and is not discussed further.

3.3.2. Bending Simulation of Bolt

A force perpendicular to the bolt body was applied at a constant velocity to the exposed bolt head, as shown in Figure 6, and this was equivalent to incrementally increasing the force during calculation. The bending moments at several positions of the bolt body varied with the calculation steps, as shown in Figure 16. With increasing force magnitude and duration, the moment in the bolt section exposed outside the rock gradually increased. This illustrated the tension bending under head loading, demonstrating the bolt body’s shear resistance.

3.4. Study on the Support Effect of Bolts in Soft Rock Tunnels

3.4.1. Comparative Analysis of Surrounding Rock Deformation

To investigate the support effects of the two types of bolts after the excavation of a soft rock tunnel, a plane strain model of the tunnel, as shown in Figure 7, was established for excavation and support simulation. For comparative analysis, three scenarios were simulated: no support, conventional bolt support, and yielding bolt support. In all three scenarios, the surrounding rock mass deformed toward the tunnel, and the displacement cloud maps in the vertical and horizontal directions, along with the resultant displacement vectors, are depicted in Figure 17, Figure 18 and Figure 19.

After excavation, maximum deformation occurred at the arch crown. In Scenario One, which was without any form of support, the stabilized settlement of the arch crown reached 11.40 cm, with a deformation of 8.89 cm on the right sidewall. In Scenario Two, which was with conventional bolt support, the stabilized settlement of the arch crown was 9.79 cm, and the sidewall deformation was 8.16 cm. For Scenario Three, which was with yielding bolt support, the arch crown settlement was 9.42 cm, and the sidewall deformation amounted to 7.40 cm. Compared to no support, the yielding bolt support increased the control of crown settlement and sidewall deformation by 17.37% and 16.76%, respectively.

Resultant displacements at the monitoring points are shown in Figure 20, clearly reflecting the rock surface deformation under the three scenarios. It can be inferred that employing the bolt support after tunnel excavation effectively constrained the deformation of the surrounding rock mass. Furthermore, when yielding bolt support was used, the deformation of the rock mass was less compared to using conventional bolt support, indicating that yielding bolts are more effective in controlling rock mass deformation during tunnel excavation.

As shown in Figure 21, it can be seen that, regardless of the support scenario, the positions where the surrounding rock deformation was relatively large were the arch crown and arch base. It can be known from the curve with the error bar that the standard deviations of the surrounding rock deformation at the arch crown and arch base were larger, which indicates that the supporting effect of the yielding bolts was the most obvious at these two locations. This indicates that, within a certain deformation range, the greater the deformation of the surrounding rock, the more suitable it is to adopt yielding bolt support. On the other hand, even after the use of yielding bolt support, the deformation of the surrounding rock at the arch crown and arch base was still greater than that at the other monitoring positions without support. This indicates that reducing or avoiding the deformation of the surrounding rock in soft rock tunnels cannot rely solely on bolt support, it is also necessary to combine it with other technologies.

3.4.2. Comparative Analysis of the Mechanical Characteristics of Bolt

After the stabilization of the tunnel, the axial force distributions of the conventional and yielding bolts were recorded, as shown in Figure 22. The thickness of the black shadow in the figure represents the magnitude of the axial force. The thicker the black shadow is, the greater the axial force at this position of the bolt. Compared to other locations in the tunnel, the deformation of the surrounding rock mass was greater at the arch crown, and the stress characteristics of the bolts at different positions at the arch crown are depicted in Figure 22. It can be observed from the stress characteristics of the bolts shown in Figure 23 that the position of maximum load-bearing capacity of the bolt was 2.63 m away from the bolt head, and the variation of axial force at this position is illustrated in Figure 24.

During support, the midsection of the bolt bears significantly more load than the two ends, and this is attributed to the stress relaxation that occurs in the rock mass at the tunnel edge, leading to a slightly inward stress concentration. Both bolt types reached yield strength during support. The conventional bolts reached yield strength faster than the yielding bolts. Furthermore, the conventional bolts maintained yield strength for a shorter duration, with the central position of the maximum load-bearing capacity quickly breaking under tension, resulting in an hourglass-shaped axial force distribution. In contrast, as shown in Figure 24, after reaching yield strength, the yielding bolts, due to the good extension effect of the yielding component, were able to sustain yield strength for a longer period, allowing for a greater strain, with the axial force exhibiting a spindle-shaped distribution.

Quantitatively comparing the speed of reaching the yield strength and the strain maintained at yield validates the yielding effect. As shown in Figure 18 and Figure 19, in the initial stages of support, yielding bolts effectively release the deformation energy of the surrounding rock mass and improve the stress distribution in the support structure, fully embodying the support concept of the New Austrian Tunneling Method [39,40]. When the surrounding rock mass undergoes significant deformation, as shown in Figure 22, conventional bolts are prone to failure by breaking under tension, while yielding bolts accommodate larger deformations, continuing to reinforce the rock mass.

3.5. The Influence of the Yielding Bolt Length on Its Support Effect

To investigate the influence of the length of yielding bolts on the support effectiveness, eight different lengths of bolts, namely 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, and 9 m, were selected. A plane strain model of the tunnel, as shown in Figure 7, was employed for excavation and support simulation. The proportion of the yielding component of the bolt remained constant, relative to the total length of the bolt. During the simulation process, monitoring was conducted at the 16 positions shown in Figure 9. After the simulation calculations, the deformation of the surrounding rock at the four key locations, such as the arch crown, sidewalls, and arch base, were recorded and are presented in

Table 5. Key point deformation of the surrounding rock with different lengths of bolt support.

Bolt Length (m)	Deformation (cm)
	Arch Crown	Arch Base	Left Sidewall	Right Sidewall
2	9.99	10.23	7.67	7.64
3	9.68	9.83	7.49	7.48
4	9.45	9.65	7.42	7.38
5	9.39	9.58	7.39	7.33
6	9.42	9.58	7.40	7.36
7	9.41	9.60	7.42	7.37
8	9.43	9.63	7.44	7.43
9	9.49	9.68	7.48	7.47

, and the total deformation of the surrounding rock at the 16 monitoring points under different lengths of bolt support is illustrated in Figure 25.

With increasing the yielding bolts’ length, the deformation at the 16 key positions of the surrounding rock and the total deformation at all points exhibited a trend of initially decreasing, then stabilizing, and then slightly increasing. As the length of the yielding bolts increased from 2 m to 5 m, the deformation of the surrounding rock decreased rapidly. For every additional meter in bolt length, the total deformation at the arch crown, left and right sidewalls, and arch base decreased by 6.13 mm. During this stage, the impact of the bolt length on support effectiveness is significant. As the length of the yielding bolts increased from 5 m to 7 m, the deformation remained relatively stable with minimal changes. Further increasing the length from 7 m to 9 m resulted in a slight increase in the deformation of the surrounding rock, but the magnitude of increase was minimal. Therefore, it can be inferred that the yielding bolts with lengths of 5 m, 6 m, and 7 m exhibited better support effectiveness. Considering utilization and cost, a preliminary determination suggests that 5 m is the optimal length for yielding bolts under these conditions.

3.6. The Influence of the Yielding Bolt Spacing on Its Support Effect

To study the impact of the spacing of yielding bolts on support effectiveness, the optimal length of 5 m was selected. Utilizing the same plane strain model of the tunnel, as shown in Figure 7, excavation and support simulations were conducted. Along the circumferential direction of the tunnel, the spacing of the yielding bolt support was set at 0.75 m, 1 m, 1.25 m, and 1.5 m. After calculations, the deformation of the surrounding rock at the arch crown, sidewalls, and arch base, as well as the total displacement change at the location of maximum deformation in the tunnel under different spacing of bolt support, were obtained, as shown in

Table 6. Key point deformation of the surrounding rock with different spacings of bolt support.

Bolt Spacing (m)	Deformation (cm)
	Arch Crown	Arch Base	Left Sidewall	Right Sidewall
0.75	9.34	9.53	7.51	7.49
1	9.39	9.57	7.39	7.33
1.25	9.94	10.16	7.79	7.77
1.5	9.92	10.13	7.77	7.74

. The variation pattern of the resultant displacement at the location of maximum deformation in the tunnel is illustrated in Figure 26.

As the bolt spacing decreases, the deformation of the surrounding rock at the arch crown, left and right sidewalls, and the arch base followed a pattern of initially stabilizing, then decreasing, and then finally stabilizing again. This pattern was similarly reflected in the resultant displacement at the location of the maximum deformation in the surrounding rock. When the bolt spacing decreased from 1.5 m to 1.25 m, the deformation of the surrounding rock did not decrease but slightly increased. As the bolt spacing decreased from 1.25 m to 1 m, there was a significant reduction in the deformation of the surrounding rock, with the resultant displacement at the location of the maximum deformation decreasing from 10.16 cm to 9.57 cm and the resultant displacement at the location of minimum deformation decreasing from 7.77 cm to 7.33 cm. Further decreasing the bolt spacing from 1 m to 0.75 m resulted in a minimal reduction in the deformation of the surrounding rock, with the resultant displacement at the location of maximum deformation decreasing from 9.57 cm to 9.54 cm. Therefore, bolt spacings of 0.75 m and 1 m demonstrated better support effects, and the difference in support effects between the two was extremely small. Under the same support area, a spacing of 1 m will save a great deal of yielding bolts compared to a spacing of 0.75 m. Considering utilization and cost, a preliminary determination suggests that 1 m is the optimal spacing for yielding bolts under these conditions.

Comparing Table 5 and Table 6 and focusing on the two groups of data with the greatest change in arch crown settlement, for yielding bolt lengths of 2 m and 5 m, the difference in deformation at the arch crown was 0.69 cm, and the difference in the arch base uplift was 0.65 cm. With support spacings of 1 m and 1.25 m, the difference in arch crown deformation was 0.55 cm, and the difference in the arch base uplift was 0.59 cm. This indicates that support spacing has a greater impact on the support effectiveness than bolt length.

4. Conclusions

This study employed FLAC3D to meticulously simulate the pull-out tests of conventional bolts and previously developed yielding bolts. The axial force distribution and its dynamic evolution under pull-out conditions for both bolt types was investigated. Additionally, a comprehensive analysis of the shear resistance of the bolts was conducted. A comparative study, within the context of a soft rock tunnel environment, was undertaken to evaluate the support efficacy of conventional bolts against yielding bolts. The findings elucidate the mechanical behavior of yielding bolts under combined loading regimes. In addition, this work bridges critical gaps in the understanding of energy-absorbing support systems. Furthermore, the developed FLAC3D modeling approach provides a versatile tool for future research on advanced reinforcement strategies in geotechnically complex environments. The key conclusions are the following:

(1)

The feasibility of simulating the pull-out tests, shear tests, and support effects of yielding bolts using FLAC3D was verified. The simulation results align with the working mechanism and stress characteristics of yielding bolts, establishing a basis for subsequent numerical simulation research.

(2)

When subjected to pull-out forces, the maximum axial force at the bolt head decreased gradually toward the inner end of the bolt, with the reduction in the yielding bolts being greater than that of the conventional bolts. As the pull-out force increased, the axial force of the bolt correspondingly increased, with the rate of increase in yielding bolts significantly slower than that of conventional bolts, confirming the high extension rate of yielding bolts.

(3)

In the support of a soft rock tunnel, compared to conventional bolts, yielding bolts exhibit a slower rate of reaching yield strength and a longer duration of maintaining yield strength. The central position of the maximum load-bearing capacity of the conventional bolts breaks under tension, resulting in an hourglass-shaped distribution of axial force, whereas yielding bolts can sustain yield strength for an extended period after reaching yield strength, with the axial force displaying a spindle-shaped distribution. The elastic and plastic yielding distance of the yielding bolts are 70.54 mm and 130.65 mm, respectively. The axial force distribution law and yielding distances can provide a theoretical reference basis for engineering applications.

(4)

Bolt spacing exerts a greater influence on support effectiveness in soft rock tunnels than bolt length. It was determined that a yielding bolt length of 5 m and a spacing of 1 m × 1 m are the optimal support configuration under the geological conditions discussed in this paper. This optimal support configuration and its determination method can provide a theoretical basis for more complex engineering applications.

Overall, this study addresses critical knowledge gaps regarding yielding bolt behavior under combined loading and provides a validated framework for optimizing energy-absorbing support systems in soft rock tunnels.