Analysis of deformation and failure characteristics of rock anchor foundations in limestone strata of southern Anhui mountainous areas

This study investigates the deformation and failure mechanisms of rock anchor foundations within the limestone strata characteristic of southern Anhui’s mountainous regions. The research aims to establish quantitative design thresholds to enhance the safety and efficiency of engineering projects in these geologically complex areas.A multi-scale methodology was adopted, integrating field pull-out tests, laboratory experiments on rock-grout interfaces, and detailed numerical simulations. This comprehensive approach enabled a systematic analysis of the anchorage system’s behavior under various loading conditions and geometric configurations.The core findings provide critical quantitative benchmarks. A critical anchorage depth of approximately 3,500 mm was identified, which delineates a shift in failure mode. At depths shallower than 3,000 mm, failure occurs primarily by pull-out along the rock-grout interface. Beyond 4,000 mm, the failure mode transitions to tendon extraction from the grout column, with the peak pull-out capacity observed at the 3,000 mm depth. For group anchor systems in a 2 × 2 configuration, the ultimate pull-out capacity reached 3600 kN, representing a four-fold increase over a single anchor, attributable to stress superposition effects. An embedded rock depth of 400 mm was determined to be the optimal threshold for maximizing structural ductility. Furthermore, parameter sensitivity analysis established an optimal diameter ratio criterion (D/d > 3.2) as the most significant parameter for optimizing pull-out performance.The results demonstrate that anchorage depth, group effect, and the diameter ratio are pivotal factors controlling the performance of rock anchor foundations in limestone. The identified thresholds and failure mode transitions provide crucial insights for design. These quantitative findings offer direct theoretical support and a practical basis for the design optimization of rock anchor foundations under similar geological conditions.

1 Introduction

Rock anchor foundations are widely used as a typical foundation type for transmission lines in mountainous areas owing to their excellent pull-out resistance (Zheng et al., 2025; Wang et al., 2024; Hu et al., 2025; Shi et al., 2022; Wu et al., 2025a). These foundations are constructed by placing steel reinforcement into mechanically drilled rock sockets, which are then grouted with fine aggregate concrete or cement mortar (Wu J. et al., 2024; Wu et al., 2025b), forming primarily straight-socketed or cap-supported structural systems. Cap-supported foundations are of particular engineering significance due to their role in effectively transferring superstructure loads to the rock mass (Xue et al., 2025). However, under complex geological conditions—such as fractured rock, soil-rock composites, soft rock, and mudstone—the deformation mechanisms and failure modes of rock anchor foundations remain inadequately investigated, which constrains their safe design and practical applicability.

Research on the mechanical behavior of rock anchorage systems has been explored from various perspectives. For instance, Liu et al. (2025) employed refined numerical simulations to demonstrate that the tensile-shear composite failure of rock bolts in high in-situ stress layered rock masses is governed by the orientation of bedding planes relative to the in-situ stress field, leading to the proposal of a differentiated support design approach. Bai et al. (2024) conducted pull-out tests comparing the interfacial bond behavior of GFRP and steel anti-floating anchors, revealing that although GFRP anchors exhibit slightly lower bond strength, they display a linear bond-slip response and superior reinforcement-grout-rock synergy compared to the bilinear response of steel anchors. Grindheim et al. (2024) emphasized that joint orientation and in-situ stress—rather than overburden weight alone—are critical factors controlling the bearing capacity of rock anchors, thereby refining conventional design assumptions. Ren et al. (2010) developed a trilinear bond-slip model that captures the full-range tensile behavior of grouted rock bolts and derived closed-form analytical solutions for each of the five loading stages, providing a valuable theoretical tool for bolt analysis. Tayeh et al. (2019), Brown (2015) systematically summarized four primary failure modes of rock anchors: tensile failure of the bolt, debonding at the bolt-grout interface, failure at the grout-rock interface, and overall rock mass uplift. Grindheim et al. (2023a), Shin et al. (2023a) performed field tests on load-distributed compression anchors (LDCAS) in soft rock, noting that multi-anchor interference effects can induce tensile failure within the grout itself, significantly reducing load capacity. Liu et al. (2017), Shin et al. (2023b) highlighted that the ultimate capacity of ground anchors is highly dependent on grout quality and that stress distribution is highly nonlinear along the anchor length, necessitating design considerations beyond current codes for very short or long anchors. Grindheim et al. (2023b) experimentally determined that the bond strengths at the bolt-grout and grout-rock interfaces are approximately 20% and 5% of the uniaxial compressive strength of the grout, respectively, with the latter governed by the weaker material component. Li and Høien (2023), Li et al. (2016) introduced a critical anchor length-based stress transfer model and a non-uniform layout strategy for rock wedges based on parabolic driving force distribution, respectively advancing the prediction of load transfer and support design optimization.

Despite these advances, three major research gaps remain. First, most numerical models do not adequately account for the cohesive-frictional behavior of the grout-rock transition zone. Second, existing studies predominantly focus on single-anchor behavior, with limited analysis of stress superposition and interaction mechanisms in group anchor systems. Finally, key design parameters lack systematic optimization and clearly defined thresholds, hindering both design efficiency and economy. In the limestone formations of southern Anhui, group anchor foundations are commonly used, yet their collaborative load-bearing mechanisms and failure patterns under complex loading conditions are still not well understood.

Aiming at the aforementioned issues, this study investigates the mechanical properties of group rock anchor foundations in limestone strata in southern Anhui. Rock mass parameters were determined through field sampling and laboratory triaxial tests, and multiple sets of field pull-out tests on group anchors were conducted to characterize load-displacement responses and failure modes. Finite element models considering grout-rock interface characteristics were developed to reveal the deformation evolution laws and load distribution mechanisms of anchor groups. Based on the research findings, a critical depth (3M) for significant transformation of failure modes in single anchor systems was proposed, along with the synergistic mechanism of group anchor systems and an optimal embedded depth threshold (400 mm). The influence patterns of key design parameters on pull-out performance were quantified. Parameter sensitivity analysis established an optimal rod diameter ratio criterion (D/d ≥ 3.2), providing theoretical support for the parameter design of transmission line anchor foundations.

2 Field experiment

2.1 Laboratory experiment

The sampling site is consistent with the location of the in-situ rock bolt tests, situated alongside Provincial Highway S103 (Meilandi) in Sankou Town, Huangshan District, Huangshan City. Samples were collected from depths ranging between 1 m and 6 m; the sampling site is illustrated in Figure 1. The sampling methodology involved retrieving rock blocks from the field followed by laboratory drilling. For each representative rock type, five groups of cylindrical specimens measuring 50 mm in diameter and 100 mm in height were prepared using a drilling machine. This process ensured that the rock samples exhibited satisfactory surface finish and flatness, with no significant cracks or signs of disintegration.

Figure 1

Figure 1. Sampling site dingram.

Using triaxial compression tests (Figure 1), four groups of limestone specimens were tested under confining pressures of 10 MPa, 14 MPa, 15 MPa, and 25 MPa. Each test condition included five specimens. The loading was applied using a single continuous loading method, with an axial load increasing at a rate of 0.05 MPa per second until specimen failure occurred. The strength envelope obtained from the tests is shown in Figure 2, and the complete stress-strain curves from the triaxial compression tests on limestone are presented in Figure 3. The results indicate that the limestone specimens have an internal friction angle of 47.3° and a cohesion of 6.58 MPa.

Figure 2

Figure 2. Triaxial test apparatus diagram.

Figure 3

Figure 3. Mohr-Coulomb strength envelope.

2.2 In-situ experiment

For each typical rock type, cylindrical specimens measuring 50 mm in diameter and 100 mm in height were prepared using a drilling machine, with five sample groups processed per rock type. The rock samples were meticulously machined to ensure high surface finish and flatness, exhibiting no significant cracks or signs of fragmentation.

The field sampling campaign encompassed three representative sites, covering the mobilization and demobilization of sampling equipment. Detailed specifications regarding borehole depths and dimensions, which adhered to sampling requirements, are illustrated in Figure 4.

Figure 4

Figure 4. Strain stress anve of linestone.

2.2.1 Pre-test preparations

The area surrounding the test rock bolts was cleared to ensure sufficient and stable operating space. A safety was established, and necessary safety protection equipment was deployed.

It was confirmed that the grout for the anchor bolts had cured for the required duration and achieved the design strength. The exposed length of the anchor bolts met the test requirements, typically sufficient to facilitate the installation of hydraulic jacks and displacement measurement devices. All equipment, including the loading system, measurement system, and data acquisition system, underwent functional checks and verification of calibration certificates.

2.2.2 Assembly of the loading system

An appropriate reaction device was selected based on the design load and site conditions (Figure 5). The reaction frame was installed steadily and securely directly above the test anchor bolt, ensuring their centerlines were aligned. The supporting points of the reaction frame possessed adequate strength and stiffness to prevent excessive deformation or displacement under the maximum test load.

Figure 5

Figure 5. Reaction frame for fidd testing.

A hydraulic jack was positioned between the reaction frame and the test anchor bolt, with careful alignment to ensure the jack’s center coincided precisely with the anchor bolt’s axis, thus avoiding eccentric loading. A specialized spherical hinge-bearing load transfer block was placed between the anchor bolt head and the jack piston for leveling adjustment. The bearing plate beneath the jack was required to be flat and fully cover the load transfer block. High-pressure hoses were used to connect the electric hydraulic pump to the jack, with all connections checked for tightness and leaks. A precision pressure gauge was installed at the pump outlet to accurately measure the applied load. The load measurement system utilized an RS-JYE fully automatic static load analyzer (Figure 6), which was integrated in series within the hydraulic circuit or incorporated into an intelligent hydraulic pump, and connected to the data acquisition unit via data cables.

Figure 6

Figure 6. RS-JYE fully automatic static load analyzer.

2.3 On-site test results

This experiment investigated the depth effect and the group anchor mechanism in the rock bolt system, and the results are presented in Table 1. As shown in Figure 7, under conditions of a fixed borehole diameter (120 mm) and the use of high-strength steel bars (HRB500, 40 mm diameter), the ultimate pullout capacity of the rock bolts increased significantly with embedment depth up to 4m (rising from 440 KN to 800 KN). A clear critical point of failure mode transition was observed in the single bolt specimens.

Table 1

Table 1. Bolt parameters and failure modes.

Figure 7

Figure 7. Load-displacement curve of a single anchor rod.

When the embedment depth was ≤3,000 mm (Specimens DM1-DM3), failure manifested as the overall pullout of the bolt along the rock-grout interface (Figure 8), characterized by shear-dominated failure in the shallow rock mass. In contrast, when the embedment depth reached ≥4,000 mm (Specimens DM4-DM6), the failure mode abruptly changed to the progressive extraction of the steel bar from the grout (Figure 9), indicating that the capacity bottleneck for deeper anchors had shifted to the bond performance between the steel bar and the grout (Shi et al., 2023; Hao et al., 2025).

Figure 8

Figure 8. Failure Diagram of the hterface Between a Single Anchor and the Surrounding Rock.

Figure 9

Figure 9. Failure Diagram of the hdarfare Between a Single Anchor Bolt and the Anchorage Body.

The identification of this critical depth (approximately 3,500 mm), along with the observed strength degradation (a 21% reduction in peak load for specimen DM6 compared to DM3) occurring at embedment depths exceeding 5,000 mm due to stress concentration in the steel bar, energy dissipation in the ineffective anchorage segment, and bond stress transfer failure, confirms the existence of a saturation point in the depth effect of the anchorage system.

The load-displacement curves of group anchor specimens (QM series) in Figure 10 further reveal the coupling failure mechanism under composite loading. Quantitative analysis demonstrates that: the unembedded specimen (QM-0) exhibits an ultimate bearing capacity of 3600 kN, undergoing brittle failure at a displacement of 43.35 mm; the specimen with a 400 mm embedded depth (QM-400) maintains the same ultimate bearing capacity of 3600 KN but shows improved ductility with failure displacement extended to 51.78 mm; while the specimen with 800 mm embedding (QM-800) also maintains 3600 KN bearing capacity with a failure displacement of 51.79 mm. This indicates that appropriate rock embedding does not enhance the ultimate bearing capacity but effectively improves structural ductility at failure. Comparative analysis of Figure 11 further corroborates that increased embedding depth enhances the lateral resistance capacity parallel to the vertical load-bearing capacity. Specifically, the ultimate lateral load increases from 640 KN (QM-0) to 720 KN (both QM-400 and QM-800), representing a 12.5% enhancement. Therefore, the 400 mm embedding depth achieves an optimal balance between maintaining bearing capacity and controlling rock mass failure, providing quantitative evidence for optimizing embedded depth.

Figure 10

Figure 10. Vertical Load-D isp la cement Curve of Group Anchors.

Figure 11

Figure 11. Load -D isp la cem ent Curve of Group Anchors und er Horizontal Load.

3 Numerical calculation and analysis

3.1 Model construction

Transmission line rock anchor foundations are constructed by first drilling mechanically formed holes into the bedrock. Anchor tendons are then inserted directly into these rock boreholes and bonded into a single unit with the bedrock using fine-aggregate concrete, thereby fully utilizing the inherent strength of the rock mass. A rock anchor foundation constitutes a system composed of three materials (anchor tendon, grout, and rock mass) and two interfaces (the tendon-grout interface and the grout-rock interface). According to the State Grid Corporation of China Typical Design for Transmission Line Projects: Rock Anchor Foundation Volume (2017 Edition), the primary failure modes considered for rock anchors are as follows: tensile fracture of the anchor tendon itself; bond failure at the interface between the anchor tendon and the cement mortar/fine-aggregate concrete; bond failure at the interface between the anchor assembly and the rock mass; and shear failure of the rock mass itself.

Regarding the mechanical behavior of tension-type anchor bolts under pull-out forces, their key load-bearing mechanism is characterized in Figure 12. Under pull-out loading, the anchor tendon is subjected to axial tension. Given the presence of bond stress at the interface between the anchor tendon and the anchor body, the anchor body exerts a downward shear resistance on the tendon to maintain displacement compatibility. The stress state of the anchor tendon is governed by the static equilibrium between the upward pull-out force and the downward shear resistance.

Figure 12

Figure 12. Force Diagram of a tension-type anchor bolt.

Similarly, for the anchor body, it interacts with the surrounding soil through shear stress at their interface. The soil exerts a downward shear resistance on the anchor body. The mechanical equilibrium of the anchor body itself is dominated by the combined action of the upward shear resistance from the anchor tendon and the downward shear resistance from the soil. Thus, the pull-out load applied at the top of the anchor tendon is ultimately transferred to the surrounding soil through the complex bond-shear mechanism at the anchor tendon–anchor body interface and the anchor body–soil interface, forming an effective load transfer path.

However, when the pull-out load continuously increases and exceeds a specific critical value, the shear stress at either the anchor tendon–anchor body interface or the anchor body–soil interface may surpass the bond strength limit of the interfacial material. Once this limit is exceeded, significant relative slip occurs at the interface, causing the interfacial mechanical behavior to transition from being bond-dominated to friction-dominated.

Engineering practice and research indicate that in conventional tension-type anchor bolts, compared to the bond strength at the anchor tendon–anchor body interface, the bond at the anchor body–soil interface is more prone to softening or failure first, rapidly transitioning to a working state dominated by sliding friction. This phenomenon is attributed to the inherent characteristics of stress distribution within the anchorage zone of traditional tension anchors: the bond stress at the anchor body–soil interface exhibits a highly non-uniform distribution, significantly concentrated at the front end of the anchorage zone. As the anchor load increases incrementally, the peak bond stress is not fixed but demonstrates a stress redistribution behavior, gradually migrating from the front of the anchorage zone toward the deeper part. This stress migration process is associated with local exceedance of the bond stress limit and the propagation of bond failure zones, ultimately exhibiting a significant progressive failure mode.

3.2 Selection of calculation parameters

This study develops an ABAQUS numerical model to simulate the contact interactions between various components of the foundation. The contact interactions are established using the ABAQUS interaction module. The normal behavior of all contact surfaces is defined as hard contact. For tangential behavior, different approaches are adopted based on the characteristics of the contact interfaces: cohesive behavior models are applied to the interfaces between anchor tendons and mortar, as well as between mortar and soil, aiming to accurately capture the mechanical response of the interfacial bonding effects in practical engineering (Tables 2, 3).

Table 2

Table 2. Reinforcement model parameters.

Table 3

Table 3. Rock model parameters.

Based on pull-out test results, which indicate that the relative displacement between the anchor solid and soil is significantly greater than the minor deformation between the anchor solid and the anchor rod, an embedded model is employed in this study to simulate the interface between the anchor solid and the anchor rod. For the critical interface between the anchor solid and the soil, the characteristics of the material transition zone from cement grout to soil, as observed in practical engineering, are quantitatively analyzed. This transition zone exhibits mechanical properties intermediate between those of cement and soil, demonstrating notable and non-negligible cohesion, which bears certain similarities to the mechanical behavior of cement-soil. Accordingly, a cohesive contact constitutive relationship is established between the anchor solid and the soil in the model to thoroughly investigate the influence of this cohesion on the overall load-bearing mechanism. The stress-strain curve of this cohesive contact constitutive relationship is illustrated in Figure 12.

The adopted cohesive modeling technique is widely applicable to the analysis of adhesive interfaces, composite material connections, and rock fracture problems. There are two common implementation strategies: one involves creating cohesive elements, and the other involves defining cohesive contact. Both strategies share the same theoretical basis but differ in implementation details and application scenarios. The core principle lies in describing the mechanical behavior of the interfacial layer in the normal and two tangential directions. Figure 12 reveals that the bilinear constitutive model accurately represents the evolution of the stress-strain characteristics of the bonding layer: as strain increases, stress initially increases linearly to a peak point, then enters a damage phase with stiffness degradation. To simplify calculations, a linear softening assumption is adopted for the constitutive model during the damage phase. Ultimately, three key parameters of this cohesive contact model are determined: the damage initiation stress σ₁, the damage initiation strain ε₁, and the complete failure strain ε₂. Here, σ₁ represents the ultimate tensile strength of the adhesive; ε₁ corresponds to the damage initiation point; and ε₂ is set as the strain value at which the material completely loses its load-bearing capacity.

3.3 Verification and analysis of single rock bolt model

Taking specimen DM-1 as an example, the numerical model was validated against field in-situ test data. A vertically upward displacement of 60 mm was imposed on the top section of the rock bolt in the simulation. The resulting pull-out stress contour is shown in Figure 13a.

Figure 13

Figure 13. Load-displacement curve. (a) Comparison curve between DM-1 field data and simulation results. (b) Comparison of QM-0 test and simulation. (c) Comparison of CM-400 test and simulation.

The load-displacement curve illustrated in Figure 13 can be broadly divided into three stages: elastic-plastic deformation, plastic deformation, and residual deformation.

Elastic-plastic stage (0–20 mm): The simulated curve shows high agreement with the field data (error <5%). Stress contours indicate that the load is progressively transferred downward from the anchored end.

Plastic deformation stage (20–50 mm): The simulated peak load is approximately 8% higher than the field-measured value. Despite this slight overestimation, the overall simulation closely matches the experimental trend. Within this stage, deformations of both the bolt and the grout increase significantly, with partial cracking occurring in the grout body.

Residual deformation stage: The bolt and grout undergo further deformation, showing a pronounced increasing trend. The bearing capacity increases slowly while displacement continues to grow markedly until the anchor is pulled out, resulting in structural failure. Moreover, the failure surface inferred from the stress contour agrees well with the actual failure mode observed in the field.

3.4 Group anchor model verification and analysis

The finite element model is constructed using the QM-0 model as an example. The geotechnical mass is a homogeneous body measuring 2.4 × 2.4 × 5.5m. The interface between the anchor rod and the grout is defined as a tied contact, while the interface between the grout and the geotechnical mass is modeled as a cohesive contact. The full-section contact is treated as a general contact, primarily employing hard contact with a friction coefficient of 0.45. Specific details are shown in Figures 14, 15.

Figure 14

Figure 14. QM-0 fidd test and cross-sectional stress contour.

Figure 15

Figure 15. CM-400 field test and cross-sectional stress contour.

The boundary conditions of the model are determined as follows: displacement boundary conditions are applied around the model to restrict horizontal displacement in the corresponding directions; the bottom boundary condition is typically set as a fixed support, preventing both vertical and horizontal displacements. A vertical displacement boundary condition is applied at the coupling point of the anchor rod to simulate the uplift stress state of the foundation. Simultaneously, a horizontal displacement boundary condition is applied at the coupling point on the X-direction side of the anchor pile to simulate the stress state of the foundation under horizontal loading.

The materials included in the foundation model mainly consist of steel, concrete, mortar, and rock. Appropriate constitutive relationships are selected based on the actual stress conditions. In this model, the steel is represented using a bilinear model, where the stress-strain curve in both the elastic and plastic phases is linear. ABAQUS provides two constitutive models for concrete: the concrete smeared cracking model and the concrete damaged plasticity model. The concrete damaged plasticity model employs isotropic elastic damage combined with isotropic tensile or compressive plasticity to simulate the inelastic behavior of concrete and other quasi-brittle materials. It is suitable for this unidirectional loading scenario. Additionally, it effectively accounts for the plastic strain generated during the testing process and the changes in elastic stiffness under load, enabling accurate simulation of tensile cracking and compressive crushing of materials such as concrete. Therefore, the concrete damaged plasticity model is adopted for both the concrete and mortar constitutive models.

Compared to a single pile, the group pile configuration exhibits an expanded influence zone of displacement at the anchor head and the surrounding soil, with more pronounced interaction with the adjacent rock and soil mass. The overall load-bearing performance is superior to that of the single pile (DM-4), and the maximum load reaches 3200 kN, representing a 4-fold increase in tensile strength compared to the single pile. This demonstrates that the group pile effect can achieve a synergistic outcome where the whole is greater than the sum of its parts, enabling adaptation to more complex and challenging construction conditions.

As illustrated in the load-displacement curves of rock bolts at different locations in Figure 16, the spatial heterogeneity of force distribution within the group anchor system under coupled horizontal and uplift loads is clearly revealed. The two curves correspond to rock bolts near and far from the horizontal load application point, with the key differences lying in the divergence of peak load and displacement ductility. The rock bolt closer to the horizontal load exhibits a higher load level under the same displacement, a smaller displacement at peak load (approximately 30–40 mm), and a steeper curve slope (indicating higher initial stiffness). In contrast, the rock bolt farther from the horizontal load shows a lower load growth rate, with peak displacement delayed to 50–60 mm and superior ductility.

Figure 16

Figure 16. Load-displacement curves of anchor rods at different positions.

This discrepancy stems from the additional bending moment effect induced by the horizontal load: the horizontal force causes deflection deformation of the group anchor system around the rotation center, resulting in the rock bolt on the load-proximal side bearing a stronger coupling of axial tension and shear forces. This leads to earlier attainment of critical bond stress at the tendon-grout and grout-rock interfaces, manifesting as a mechanical response with higher stiffness but reduced ductility. This observation aligns with the von Mises stress contour results in Figures 17, 18, where stress concentration is more pronounced in the rock bolt near the horizontal load, validating the stress redistribution characteristics under composite loading.

Figure 17

Figure 17. Stress nephogram of anchor rods near the horizontal load.

Figure 18

Figure 18. Stress nephogram of anchor rods away from the horizontal load area.

Analysis based on the bond contact constitutive model indicates that, due to coupled loading, the tendon-grout interface of the load-proximal rock bolt is subjected not only to axial bond shear stress but also to additional radial shear caused by horizontal displacement. This results in an earlier attainment of the initial damage strain (ε₁), with a distinct inflection point appearing in the curve during the elastoplastic stage (20–30 mm), corresponding to the degradation of interfacial bond performance. Conversely, the rock bolt farther from the horizontal load primarily experiences uplift force, with the interface stress dominated by axial shear. The damage evolution is more gradual, leading to a larger peak displacement, which echoes the earlier conclusion that “the embedded rock effect delays failure by inducing plastic flow in the rock mass.”

4 Discussion

4.1 Analysis of shear slip failure mechanism in single rock bolt

As shown in Figure 19 the macroscopic shear slip surface is clearly defined at the rock-grout interface. This phenomenon is not random but governed by the “shear strength compatibility relationship” under shallow limestone geological conditions. According to the experimental design in Chapter 2, the single anchor specimens were constructed with a constant borehole diameter (120 mm) and HRB500 reinforcement bars (40 mm diameter), eliminating the influence of geometric parameters on interfacial weak points. The mechanical parameters of limestone obtained from laboratory triaxial tests further indicate that under low in-situ stress conditions, the shear strength of shallow rock mass is significantly lower than the bond strength at the reinforcement-grout interface. When the anchorage depth is ≤ 3,000 mm, the rock-grout interface becomes the “weak link” in the system’s pull-out capacity. As the load increases to the critical shear strength of this interface, macroscopic shear slip is inevitably triggered, manifesting as the overall pull-out of the anchor along this interface. This failure mechanism is entirely consistent with the failure patterns observed in specimens DM1–DM3 in Chapter 2.

Figure 19

Figure 19. Overall shear ship surface of a single author rod.

Based on the load-displacement curve of a single anchor in Figure 13, the evolution process of overall shear slip at the rock-grout interface can be divided into three stages, each highly coupled with the development state of the slip surface: Elastic Deformation Stage (Displacement 0–20 mm): In this stage, the pull-out load is transferred from the anchor-grout interface bond shear stress to the grout, which then acts on the rock-grout interface as uniformly distributed shear stress. Since the load does not exceed the shear strength threshold of the rock mass, only elastic deformation occurs at the interface without significant micro-cracking. The slip surface remains in a “latent state,” and the load-displacement relationship is linear (with a simulation error of less than 5% compared to field data).Micro-Crack Initiation and Propagation Stage (Displacement 20–40 mm): When the load approaches the shear strength of the shallow rock mass (approximately 900 KN for specimen DM3), the local normal stress (σ_n) and shear stress (τ) at the rock-grout interface reach a critical balance. Micro-cracks begin to appear on the limestone surface at the interface due to shear action, serving as a precursor to the formation of a macroscopic slip surface. At this point, the load-displacement curve starts to deviate from linearity, and the slope (stiffness) gradually decreases, indicating the onset of localized bond degradation at the interface, though no (through-going failure) has yet formed.Macroscopic Slip Surface Formation and Sudden Instability Stage (Displacement 40–50 mm): As the load increases to its peak (950 KN for specimen DM3), micro-cracks at the interface rapidly connect under shear stress, forming a continuous shear slip surface as shown in Figure 19. Due to the brittle shear failure characteristics of the shallow rock mass under low confining pressure, the shear resistance of the rock mass to the grout is lost almost instantaneously after the slip surface forms. This is reflected in a “precipitous drop” in the load-displacement curve (e.g., the load for specimen DM3 plummeted from 950 KN to 350 kN, a decrease of over 60%). The final failure mode manifests as the overall pull-out of the anchor along the slip surface.

In summary, the critical depth of 3,500 mm revealed in this test essentially reflects the direct influence of the “rock mass confining pressure effect controlling the transfer of the weak link at the interface.” This also corroborates, from an opposite perspective, the mechanism of shallow shear slip. When the anchorage depth is ≤3,000 mm, the low confining pressure in the shallow rock mass results in its shear strength being lower than the bond strength at the anchor-grout interface. Consequently, the shear slip surface inevitably forms preferentially at the rock-grout interface. Conversely, when the anchorage depth is ≥4,000 mm, the increased confining pressure at greater depths significantly enhances the shear strength of the rock mass. In this case, the bond strength at the anchor-grout interface becomes the system’s weak link, and the failure mode shifts to the progressive pull-out of the anchor from the grout, with the slip surface transferring to the anchor-grout interface. This correlation between depth and slip surface location further confirms that the shear slip at the rock-grout interface, illustrated in Figure 19, is an inevitable outcome dominated by “insufficient shear strength of the shallow rock mass” under shallow anchorage conditions, rather than being caused by incidental interface defects.

4.2 Analysis of group anchor shear slip failure mechanism

4.2.1 Theoretical basis and assumptions

Single Anchor Bearing Capacity Model: Based on the Mohr-Coulomb criterion, the ultimate pull-out resistance of a single anchor, denoted as Qu, is governed by the shear strength at the rock-grout interface:

$Q_u=\pi DL·{\tau }_{\max }$

${\tau }_{\max }=c+{\sigma }_n\tan \varphi$

Among them, D stands for the anchor hole diameter, L for the anchorage depth, τ_max for the maximum interface shear stress, and σ_n for the normal stress.

4.2.2 Derivation of the stress field superposition formula

Considering the superposition of stress fields, the stress field interaction of group piles can be simplified to the superposition of Boussinesq solutions. Assuming that the load on each pile is uniformly distributed, the group pile efficiency coefficient η is defined as:

$\eta =\frac{Q_{\text{group}}}{n·Q_u}$

${\tau }_{\text{group}}={\tau }_{\max }\left(1+\alpha {\displaystyle \sum _{i=1}^{n-1}}{e}^{-\beta \left(r_i/D\right)}\right)$

Considering the radial stress interference of adjacent piles in the group piles. Let the pile center distance s = 600 mm and the hole diameter D = 120mm; then the dimensionless spacing ratio s/D = 5. Based on the Mindlin solution, the interface shear stress is enhanced after the superposition of group piles: among them, rᵢ is the distance from the ith pile to the target pile, and α and β are empirical coefficients (α = 0.3, β = 0.5) (Wu W. et al., 2024; Zhang et al., 2025).

${\tau }_{\text{group}}\approx {\tau }_{\max }\left(1+0.3\times 2e^{-0.5\times 5}\right)=1.12{\tau }_{\max }$

4.2.3 Mechanical model of rock-socketed mass

Increasing the rock-socketed depth H enlarges the contact area between the column and rock mass, thus changing the failure mode. QM-0 (no rock-socketing) undergoes brittle cracking, while QM-400 transforms into uplift failure of the rock-socketed mass. Based on the limit analysis theory, the shear bearing capacity of the rock-socketed mass is:

$Q_c=A_b·c{N}_c+\gamma h{A}_b{N}_q$

Among them, A_β is the rock-socketed base area, c is the rock mass cohesion, and and are bearing capacity factors.

4.2.4 Quantitative derivation of failure delay

The rock-socketed depth delays failure by enhancing the energy dissipation capacity of the rock mass. Define the displacement ductility coefficient μ_δ:

${\mu }_{\delta }=\frac{{\delta }_{\text{peak},\mathrm{r}\text{ock}-\text{socketing}}}{{\delta }_{\text{peak},\text{no}\text{rock}-\text{socketing}}}$

$W_p={\int }_0^{{\delta }_f}{\sigma }_{y}d\delta$

${\delta }_{yield}\propto h·\tan \varphi$

4.2.5 Interface damage evolution of rock-socketed mass and rock

Laboratory triaxial tests show that limestone exhibits brittle softening under low confining pressure (10 MPa) and enhanced plasticity under high confining pressure (25 MPa). The rock-socketed depth (h) alters the stress state of the rock mass through the lateral confinement effect:

${\sigma }_v=\gamma h+\frac{K_0\gamma H}{1+\nu }$

$D=1-e^{-\alpha {\left(\epsilon /{\epsilon }_0\right)}^{\beta }}$

${\tau }_{\text{cohesive}}=c_{\text{trans}}\left[1-{\left(\frac{\delta }{{\delta }_{\max }}\right)}^m\right] \left(c_{\text{trans}}=0.35c_{\text{rock}}\right)$

Among them, α and β are limestone damage parameters, c_trans is the cohesion of the transition zone, and δ_max is the critical slip displacement.

Comparing Figure 3 with Figure 4, the variation of confining pressure simulates the in-situ stress states of rock masses at different depths. Shallow rock masses have lower confining pressure (corresponding to the 10 MPa working condition), with a significant difference between their peak strength and residual strength, an obvious stress drop, and characteristics of brittle shear failure. This is highly consistent with the phenomenon of overall pull-out of shallow anchored bodies due to insufficient shear strength of rock masses in in-situ tests. As confining pressure increases (e.g., the 25 MPa working condition), the peak strength of rock increases and the plastic deformation stage is prolonged, indicating that deep rock masses have higher shear resistance. At this point, the anchored body-rock mass interface is no longer a weak link, and the failure mode naturally shifts to the steel bar-grout interface, confirming the mechanical origin of the 3500 mm critical depth in in-situ tests.

Further, from the perspective of parameter correlation, the internal friction angle (47.3°) and cohesion (6.58 MPa) of limestone obtained from triaxial tests provide a basis for the quantitative analysis of interface shear strength. Based on the Mohr-Coulomb criterion, the shear strength of shallow rock masses under low confining pressure is relatively low, leading to the anchored bodies being prone to pull-out along the interface. However, in the deep high-confining-pressure environment, the shear strength of rock masses is significantly improved, making the bond strength between steel bars and grout the bearing capacity bottleneck. This traceability from macroscopic failure phenomena to microscopic mechanical parameters is precisely the core value of the collaborative verification of the two sets of tests.

5 Conclusion

This study focuses on the bearing mechanism and deformation-failure characteristics of rock bolt foundations under the limestone geological conditions in the mountainous area of southern Anhui Province. By adopting a multi-scale research method combining field in-situ tests, laboratory triaxial tests, and refined numerical simulations, the force mechanisms of single piles and group piles are systematically revealed. The main conclusions are as follows:

1. The failure modes and depth effect laws of rock bolt foundations are clarified. Field in-situ tests show that there exists a critical depth for the significant transformation of the failure mode of a single anchor system: when the anchorage depth is ≤3000mm, the failure manifests as the overall pull-out of the anchor solid along the rock-grout interface, controlled by the shear strength of the shallow rock mass; when the anchorage depth is ≥4000mm, the failure mode transforms into the progressive extraction of the anchor bar from the grout, and the bearing capacity bottleneck shifts to the bond performance between the steel bar and the grout, with the critical depth being approximately 3500 mm. Further analysis reveals that 3000 mm is the extreme point of the bearing capacity of a single anchor (the peak load of DM3 reaches 950 KN). However, when the anchorage depth exceeds 5000mm, due to steel bar stress concentration, energy dissipation in the ineffective anchorage section, and failure of bond stress transmission, a strength retrogression phenomenon occurs (the peak load of DM6 is 21% lower than that of DM3). Therefore, ultra-deep anchorage exceeding 5000 mm should be avoided in engineering design.

2. The synergistic mechanism of the group pile system and the optimal threshold of the rock-socketed depth are revealed. Group anchor tests confirm that the 2 × 2 arranged group pile system (QM series) significantly improves the bearing capacity through the stress superposition effect, with the ultimate uplift bearing capacity reaching 3600KN, which is 4 times higher than that of a single anchor (DM3), verifying the enhancement effect of the group pile synergistic effect on the load transfer efficiency. Regarding the optimization of rock-socketed parameters, it is found that when the rock-socketed depth increases to 4000mm, although the ultimate bearing capacity is not significantly improved (stabilizing in the 3600 KN peak range), the failure can be delayed by inducing plastic flow of the rock mass (the peak displacement increases from 40mm to 50–60 mm); when the rock-socketed depth is further increased to 800mm, the residual strain is only slightly improved (<5%). Thus, 400 mm is the optimal rock-socketed depth for anti-displacement, and this threshold provides a quantitative basis for the rock-socketed design of group pile foundations.

3. The influence of key design parameters on the pull-out performance was quantified. Parameter sensitivity analysis revealed that the anchor diameter affects the pull-out capacity following a quadratic function. When the embedment depth increases from 1M to 6M, the bearing capacity initially rises and then declines, demonstrating that under a constant anchor diameter, the embedment depth is the core parameter for optimizing pull-out performance. Accordingly, an optimal rod-to-diameter ratio criterion (D/d ≥ 3.2) was proposed, providing theoretical support for the parameter design of transmission line anchor foundations.

4. The reliability of the refined numerical model considering the cohesion of the transition zone is verified. Based on the limestone mechanical parameters obtained from laboratory triaxial tests (internal friction angle 47.3°, cohesion 6.58 MPa), a finite element model including the bond effect of the anchor bar-mortar-rock interface is established. The concrete damage plasticity model is used to characterize the behavior of the mortar-rock interface, and the bond contact constitutive model is introduced to simulate the cohesion of the transition zone. The model has a high degree of agreement with the field test data (error <8.3%), accurately reproducing the three-stage characteristics of “elastic-plastic deformation-plastic failure-residual strain” of single piles and the stress superposition process of group piles. It confirms the key role of the cohesion of the transition zone in the interface slip resistance, providing a reliable method for the numerical analysis of anchor foundations under complex geological conditions.

Given this context, this study is undertaken to address the limitations of existing research. It concentrates on the limestone strata in the mountainous areas of Southern Anhui, utilizing an integrated methodology of field testing, laboratory experiments, and numerical simulation. The research will specifically probe the collaborative mechanism of group anchor foundations, the critical thresholds for failure mode shifts, and the systematic optimization of crucial design parameters, including anchorage and rock-socketed depth. The outcomes are expected to furnish a definitive theoretical foundation and quantitative design criteria for engineering practice.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

JJ: Data curation, Formal Analysis, Writing – original draft. CW: Funding acquisition, Methodology, Validation, Writing – review and editing. CX: Supervision, Validation, Writing – review and editing. HQ: Funding acquisition, Methodology, Validation, Writing – review and editing. YZ: Supervision, Investigation, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. Research on the Rational Bearing Capacity and Calculation Theory of the Compression Arch Formed by Prestressed Anchor Cables in the Sidewalls of Deep Coal Roadways Project No. 51674005 Key Natural Science Research Project of Anhui Universities (2023AH050186) Anhui Provincial Key Research and Development Reserve Project in Transportation (2023-2027-67) Anhui Provincial Key Science and Technology Project in Transportation (2024-KJQD-028).

Conflict of interest

Author JJ was employed by Anhui Jiangnan Blasting Engineering Co., Ltd. Huohuai Branch.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The authors declare that no Generative AI was used in the creation of this manuscript.

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