Vertical Bearing Behavior and Capacity Calculation Method of Rock-Socketed Self-Drilling Hollow Bar Micropiles

Abstract

Self-drilling hollow bar micropiles (HBMPs), which integrate drilling, grouting, and reinforcement into a single process, have broad application prospects in mountainous transmission lines and offshore wind power projects. However, existing research has focused mainly on friction piles in soil layers, and there is a lack of systematic understanding of the load-transfer mechanism and bearing capacity calculation method for rock-socketed HBMPs. Based on field static load tests of rock-socketed HBMPs, this study systematically investigates the vertical bearing behavior and capacity calculation method of single rock-socketed HBMPs through a combination of test data analysis, finite element numerical simulation, and theoretical analysis. The field test results show that the load-settlement curves of rock-socketed HBMPs are of a slowly varying type, exhibiting mixed friction-end-bearing characteristics. After data screening, the average Q-s curve of Pile No. 1 and Pile No. 5 was taken as the benchmark, and the representative ultimate bearing capacity of a single pile determined by the 40 mm settlement criterion is 5860 kN. The test data of Pile No. 3 and Pile No. 4 were retained as independent validation data. A three-dimensional finite element model considering the cohesive contact behavior at the pile–rock/soil interface was established using ABAQUS. After calibration with the test results, the error between the simulated and measured bearing capacity is −3.4%, demonstrating good model reliability. Parametric analysis indicates that the bearing capacity increases linearly with the grouting volume increase rate $V_{\mathrm{inc}}$, with the expansion effect being the main enhancement mechanism; the improvement amplitude under hard rock conditions is significantly smaller than that in cohesive soils. The effect of uniaxial compressive strength $q_u$ of hard rock on bearing capacity is negligible because the capacity is controlled by the pile–rock interface shear strength. The bearing capacity increases approximately linearly with the rock-socketed depth $L_r$, and a minimum rock-socketed depth of 1.0 m is recommended. Analysis of the load-transfer mechanism shows that rock-socketed HBMPs rely mainly on shaft resistance (accounting for 90.6%), and the axial force decays significantly along the pile length. Elastic compression of the pile accounts for 78% of the pile head settlement, and the limited displacement at the pile tip leads to insufficient mobilization of end bearing. A modified bearing capacity formula considering the grouting expansion effect is established with shaft resistance as the core. A hierarchical validation strategy is adopted to test its predictive ability: for the finite element cases not participating in parameter calibration, the prediction error is within ±2%; for the field test piles, the prediction error is +7.9%; and for Pile No. 3 and Pile No. 4, the errors are +1.7% and −2.1%, respectively. These values are significantly better than those of existing methods (errors ranging from −72.1% to +54.5%). The research results can provide a theoretical basis for the design of single HBMP bearing capacity under rock-socketed conditions.

1. Introduction

A micropile is generally defined as a small-diameter (≤300 mm) drilled and grouted pile with a slenderness ratio not less than 30. Since its first being proposed, micropile technology has been widely used. It has evolved from early gravity grouting (Type A) to pressure grouting (Types B–D) and more recently to self-drilling hollow bar micropiles (HBMPs) [1,2,3,4,5]. HBMPs use a continuously threaded hollow steel bar that serves simultaneously as a drill rod, grouting conduit, and reinforcement, enabling an integrated “drilling-while-grouting” construction process [6,7]. The high-pressure grouting effectively increases the pile diameter through fracturing, compaction, and permeation of the surrounding rock and soil, thereby improving the interface bond strength [7]. Because of its unique construction characteristics, the American Association of State Highway and Transportation Officials (AASHTO) has classified HBMP as Type E micropile [8].

Abd Elaziz and El Naggar [6,9,10] conducted tests and numerical simulations in cohesive soils, revealing that HBMPs behave as friction piles dominated by shaft resistance, and proposed a modified bearing capacity formula that considers the grouting volume increase rate $V_{\mathrm{inc}}$. Abdlrahem and El Naggar [4,10] performed tests in sand and showed that increasing the drill bit-to-hollow bar diameter ratio $D_b/D_h$ from 2.25 to 3 can increase the compressive capacity by about 17%, with shaft resistance carrying more than 90% of the total load. Gomez et al. [7] summarized test data from 260 HBMPs and found that the bond strength in sandy soils can be 20–50% higher than the values recommended by FHWA. In China, Zong et al. [11,12] reported that when the grouting volume reaches three times the steel pipe volume, the bearing capacity can increase by 75–150%. Guo [13] revealed the strengthening mechanism of the “soil–cement” improved zone around the pile after grouting through microstructure analysis. For rock-socketed piles, Seo et al. [14] conducted tests on rock-socketed micropiles and observed that shaft resistance gradually becomes the dominant bearing mechanism as the slenderness ratio increases, and the applicability of the existing relative-settlement-based ultimate capacity criterion to rock-socketed micropiles requires further scrutiny. Serrano and Olalla [15] established a method for calculating the pile–rock interface shear strength based on the Hoek–Brown criterion. O’Neill and Hassan [16] highlighted the controlling effects of borehole wall roughness, rock mass discontinuity, and construction disturbance on shaft resistance. Luo et al. [17] conducted tests on steel pipe micropiles in red-bed soft rock and found that the measured bearing capacity was only 62.9–76.5% of the code-calculated value. The above studies mainly targeted conventional pile types, and it remains insufficiently verified whether their conclusions can be directly extended to HBMPs, which have an integrated construction process.

In summary, existing research on HBMPs has focused primarily on friction piles in soil layers, and there is a lack of systematic understanding of the load-transfer mechanism and bearing capacity calculation method for rock-socketed conditions. When the bedrock depth is shallow, socketing the micropile into bedrock can greatly increase its bearing capacity, which is urgently needed in projects such as mountainous transmission lines and offshore wind power. Therefore, investigating the vertical bearing characteristics of rock-socketed HBMPs, revealing their load-transfer mechanism, and establishing a corresponding bearing capacity calculation method have important theoretical significance and engineering application value.

Based on field static load test data of rock-socketed HBMPs at a test site, this study systematically investigates the vertical bearing behavior and capacity calculation method of single rock-socketed HBMPs using a combination of field test data analysis, finite element numerical simulation, and theoretical analysis. The main work includes: (1) sorting and analyzing the field static load test data of the rock-socketed HBMP test piles to determine the representative ultimate bearing capacity of a single pile; (2) establishing a three-dimensional numerical model of a single rock-socketed HBMP using ABAQUS, calibrating it with the test results, and systematically analyzing the influence of the grouting volume increase rate, uniaxial compressive strength of rock and rock-socketed depth on the bearing capacity; (3) revealing the load-ransfer mechanism dominated by shaft resistance, evaluating the applicability of existing bearing capacity calculation methods and proposing a modified formula; (4) verifying the prediction accuracy of the modified formula and providing recommendations for engineering design.

2. Experimental Overview

2.1. Engineering Geological Conditions of the Test Site

The foundation soils within a depth of approximately 30 m at the test site can be divided into six engineering geological layers. The physical and mechanical properties of each layer, determined from field investigation and laboratory tests, are summarized in

Table 1. Physical and mechanical properties of soil layers.

Layer No.	Soil Type	Depth Range (m)	Density ρ (t/m3)	Compression Modulus E (MPa)	Cohesion c (kPa)	Internal Friction Angle φ (°)
①	Fill	0–6.1	1.93	2.7	21.6	15
②	Organic soil	6.1–10.7	1.55	1.8	4.0	5.3
③	Liquefiable sand	10.7–13.7	1.87	31	0	32
④	Varved clay	13.7–19.8	1.73	2.9	33.5	18
⑤	Glacial till	19.8–29.0	2.19	50	10	38
⑥	Gneiss bedrock	>29.0	2.65	--	--	--

.

The bedrock at the site is gneiss, a metamorphic rock composed mainly of quartz, feldspar, and mica, with a granoblastic texture and gneissose structure. Its saturated uniaxial compressive strength is approximately 50 MPa, elastic modulus 1.0 × 10⁴ MPa, and Poisson’s ratio 0.25. The degree of weathering decreases from top to bottom. The rock quality designation (RQD) of the rock mass ranges from 75% to 90%, indicating good to very good rock mass; the rock mass integrity coefficient $K_v$ ranges from 0.65 to 0.80, indicating a relatively complete to complete rock mass. The equivalent cohesion of the rock mass is 1000 kPa, and the equivalent internal friction angle is 45°.

2.2. Design and Construction Process of Test Piles

The test piles were composed of TB130/60 hollow threaded bars (outer diameter 130 mm, inner diameter 60 mm, yield strength 550 MPa) and surrounding grout. The drill bit diameter was approximately 390 mm, and a pile body with a diameter of about 400 mm was formed after high-pressure grouting. A schematic diagram of the self-drilling micropile structure is shown in Figure 1.

During construction, the drilling rig advanced the hollow bar to the design depth while pumping cement slurry (water–cement ratio of 0.49, 28-day strength of about 45 MPa). The slurry exited from nozzles in the drill bit, forming a soil–slurry mixed zone. After reaching the predetermined depth, neat cement slurry was continuously injected under a pressure of 5–10 MPa, with a drilling rate of 0.5 m/min. After the hole was completed, pressure grouting was maintained for 3–5 min. A total of six test piles were completed, with a pile length of approximately 30 m, rock-socketed depth ranging from 1.0 to 2.1 m, and an average grouting volume coefficient of 1.4 (corresponding to $V_{\mathrm{inc}}=0.4$). The piles were cured for 28 days before static load testing.

The bar used was a self-drilling rod with a T-type thread produced by Luoyang Hengnuo Anchoring Technology Co., Ltd. (Luoyang, China), and the T-type thread design refers to that of prestressed threaded steel bars. The physical and mechanical parameters of the bar are given in

Table 2. Mechanical parameters of the bar.

Model	Tensile Capacity (kN)	Yield Strength (MPa)	Elongation (%)
SET130/60	7940	528	8

. The parameters of the test piles are summarized in

Table 3. Summary of test micropile parameters.

Pile No.	Effective Length (m)	Rock-Socketed Depth (m)	Grouting Volume Coefficient
1	29.38	1.12	1.75
2	28.54	1.44	1.83
3	24.41	1.83	1.23
4	29.91	2.10	1.01
5	32.15	1.12	1.95
6	25.84	0.99	1.95

Note: The grouting volume coefficient is the ratio of the actual grouting volume to the theoretical hole volume.

.

2.3. Loading and Measurement Scheme

The field static load test was performed using a loading system consisting of a hydraulic jack and a reaction frame, as shown in Figure 2. The reaction force was provided by helical piles anchored into the ground. The distance between the helical piles and the center of the test pile satisfied the code requirements to avoid interference of the reaction system with the load-bearing state of the test pile. During loading, the hydraulic jack was placed on the top of the test pile and jacked upward against the main beam of the reaction frame, thereby transferring the vertical load to the pile head. The loading capacity of the jack met the requirements of the maximum estimated load in this test.

The field static load test was carried out according to the slow maintained load method specified in the Technical Code for Testing of Building Foundation Piles. The estimated ultimate bearing capacity of a single pile was about 6000 kN. The load was applied in 10 increments, each of 600 kN (approximately 10% of the estimated ultimate capacity). After each load increment was applied, the load was maintained for no less than 2 h. During the load-holding period, the pile head settlement was measured at 5 min, 15 min, 30 min, 45 min, 60 min, and then every 30 min thereafter. When the settlement increment per hour did not exceed 0.1 mm, and this condition occurred twice consecutively, the pile head settlement under that load level was considered to have reached a relatively stable state, and the next load increment could then be applied.

The termination criteria for loading followed the provisions of Article 4.4.2 of JGJ 106-2014 [18]: when under a certain load level, the pile head settlement exceeded five times the settlement under the previous load level, or when the total pile head settlement reached 40 mm, or when the load-settlement curve exhibited a distinct steep drop, loading was terminated, and stepwise unloading began. For slowly varying load-settlement curves, the code recommends taking the load corresponding to a total pile head settlement of 40 mm as the vertical compressive ultimate bearing capacity of a single pile (JGJ 106-2014 [18]). This criterion will be adopted in the subsequent analysis to determine the ultimate bearing capacity of each test pile.

2.4. Test Results and Analysis

2.4.1. Characteristics of Load-Settlement Curves

A total of six single-pile vertical compressive static load tests were conducted in this field test program. After data screening, four valid test piles (Pile No. 1, No. 3, No. 4, and No. 5) were selected for analysis. Pile No. 2 and No. 6 encountered boulders during the drilling process, resulting in anomalous load-settlement data; therefore, the data of these two piles were excluded. The load-settlement data of the four test piles are listed in

Table 4. Summary of compression test results.

Load (kN)	Settlement (mm), Pile 1	Settlement (mm), Pile 3	Settlement (mm), Pile 4	Settlement (mm), Pile 5
0	0.00	0.00	0.00	0.00
500	2.71	1.69	1.95	1.67
1000	4.77	3.08	4.97	4.45
1500	7.21	4.87	8.52	7.05
2000	10.39	6.65	11.88	10.39
2500	13.76	8.24	16.85	13.73
3000	17.12	10.23	21.28	17.07
3500	20.68	12.21	24.82	20.79
4000	24.80	14.40	29.08	23.94
4500	28.35	16.78	33.86	27.65
5000	32.47	20.34	37.76	32.29
5500	36.78	24.50	42.02	36.56
6000	41.46	29.65	50.17	40.27

, and their load-settlement (Q-s) curves are shown in Figure 3.

It can be seen from Figure 3 that the load-settlement curves of all four test piles are slowly varying without an obvious steep drop point, exhibiting typical mixed friction-end-bearing characteristics. In the initial loading stage (0–1500 kN), the curves are approximately linear, and the pile head settlement increases uniformly with the load, indicating that the pile is in an elastic working state. During this stage, the pile head load is carried mainly by the shaft resistance, and no significant plastic deformation has yet occurred at the pile–soil interface.

As the load increases to 1500–4000 kN, the curve slope gradually decreases, and the shaft resistance is progressively mobilized. Beyond 4000 kN, the curve becomes even flatter, and the end bearing begins to participate in load sharing, which is consistent with the “shaft resistance precedes end bearing” rule observed by Luo et al. [17].

In the initial stage (0–1500 kN), the load is carried predominantly by shaft resistance along the overburden soil, while the rock–socket interface remains largely elastic. As the load increases to 1500–4000 kN, the rock–socket interface is progressively mobilized, leading to a gradual reduction in curve slope. Beyond 4000 kN, the rock-socket shaft resistance approaches its ultimate value, and further load increments are increasingly taken by end bearing.

Among the four test piles, the load-settlement curves of Pile No. 1 and Pile No. 5 essentially coincide, exhibiting consistent shapes and good repeatability, indicating that the construction quality of these two piles was well controlled and the ground conditions were relatively uniform. The curves of Pile No. 3 and Pile No. 4 show greater scatter in the later loading stage (>4000 kN). In particular, the settlement of Pile No. 4 reached 50.17 mm at 6000 kN, significantly higher than that of the other piles, which may be related to boulders encountered during construction or differences in rock-socketed conditions [4]. For the subsequent needs of numerical model calibration and formula verification, the data usage of the four valid test piles is defined as follows: the stable and highly repeatable data of Pile No. 1 and Pile No. 5, and their average load-settlement curve, serve as the benchmark for finite element model calibration and for parameter calibration of the modified bearing capacity formula. The data of Pile No. 3 and Pile No. 4 are not involved in model calibration or formula calibration but are retained as independent validation data to test the capability of the model and formula to account for engineering variability (differences in rock-socketed depth, differences in grouting volume coefficient, construction anomalies, etc.).

2.4.2. Determination of Ultimate Bearing Capacity

According to the Technical Code for Testing of Building Foundation Piles (JGJ 106-2014) [18], for a slowly varying load-settlement curve, the load corresponding to a total pile head settlement of 40 mm may be taken as the vertical compressive ultimate bearing capacity of a single pile. Using linear interpolation, the ultimate bearing capacities of the four test piles were determined as follows: Pile No. 1, 5830 kN; Pile No. 3, 6050 kN; Pile No. 4, 5700 kN; Pile No. 5, 5890 kN. The results are summarized in

Table 5. Summary of ultimate compressive loads.

Pile No.	Maximum Applied Load (kN)	Maximum Settlement (mm)	Load Corresponding to 40 mm Settlement (kN)
1	6000	41.46	5830
3	6000	29.65	6050
4	6000	50.17	5700
5	6000	40.27	5890

.

As shown in Table 5, the ultimate bearing capacities of Pile No. 1 and Pile No. 5 are very close to each other (5830 kN and 5890 kN, respectively), with a difference of only about 1%. Their arithmetic mean value is 5860 kN. The ultimate bearing capacity of Pile No. 3 is slightly higher (6050 kN), while that of Pile No. 4 is the lowest (5700 kN). Considering that boulders were encountered during the construction of both Pile No. 3 and Pile No. 4, and that the test data of Pile No. 1 and Pile No. 5 exhibit good repeatability, this paper recommends taking the average ultimate bearing capacity of Pile No. 1 and Pile No. 5, i.e., 5860 kN, as the representative value of the vertical compressive ultimate bearing capacity of a single rock-socketed HBMP at this site. This value will be used for subsequent calibration of the finite element model and verification of the bearing capacity formula.

2.4.3. Analysis of Test Result Discrepancies

The test results show that the load-settlement responses of the four test piles exhibit some discrepancies. The load-settlement curves of Pile No. 1 and Pile No. 5 essentially coincide, with ultimate bearing capacities of 5830 kN and 5890 kN, respectively, a difference of only about 1%. This indicates that the bearing performance of rock-socketed HBMPs has good repeatability under normal construction conditions. In contrast, Pile No. 3 and Pile No. 4 deviate significantly from the above two piles: the ultimate bearing capacity of Pile No. 3 (6050 kN) is about 3.2% higher, while that of Pile No. 4 (5700 kN) is about 2.7% lower. The main reasons for these discrepancies include: (1) geological condition differences—boulders encountered during construction of Pile No. 4 increased the degree of rock fragmentation near the pile tip, thereby weakening the end bearing contribution; (2) rock-socketed depth differences—the rock-socketed depth of Pile No. 3 reached 1.83 m, significantly larger than those of the other piles, increasing the shaft resistance area in the socketed section; (3) grouting volume coefficient differences—the grouting volume coefficients of Pile No. 1 and Pile No. 5 were 1.75 and 1.95, respectively, significantly higher than that of Pile No. 4 (1.01). The difference in the expansion effect directly affects the effective pile diameter and the interface bond strength. In summary, the average curve of Pile No. 1 and Pile No. 5 is taken as the benchmark for model calibration, while the data of Pile No. 3 and Pile No. 4 reflect construction variability and are retained for subsequent independent validation. The inclusion of Pile No. 4 in the independent validation dataset, despite its settlement exceeding the 40 mm criterion, is intentional: it represents a challenging case with minimal grouting expansion ($V_{\mathrm{inc}}\approx 0.01$) and construction anomalies (boulders), providing a stringent test of the modified formula’s robustness under non-ideal conditions.

3. Finite Element Numerical Simulation

3.1. Model Establishment

3.1.1. Geometry and Basic Assumptions

The pile body was modeled using a “three-layer concentric cylinder” model [7,12], consisting from inside to outside of: the hollow bar (radius 0–65 mm), the pure grout body (radius 65–150 mm), and the soil–slurry mixed zone (radius 150–200 mm). The pile diameter was 0.4 m, and the pile length was 30 m. The ground was simplified into three layers: a soft overburden layer, a medium-strength layer, and the gneiss bedrock. Under standard conditions, the rock-socketed depth $L_r$ was taken as 1.0 m. The value $L_r=1.0 \mathrm{m}$ was selected as the standard condition because it represents the average rock-socketed depth of Piles No. 1 and No. 5, which were used for model calibration. This depth is subsequently varied in the parametric study (Section 3.3.3) to investigate its influence on bearing capacity. The radial width of the soil domain was taken as 30 times the pile diameter to eliminate boundary effects. The bedrock is treated as a single homogeneous layer based on the site investigation results (RQD 75–90%, no major fractures within the socketed depth). Therefore, no additional interaction between sub-layers within the bedrock was required.

The overall model geometry is shown in Figure 4, where the three layers (soft overburden, medium-strength layer, and gneiss bedrock) are distinguished. The pile is modeled as a three-layer concentric cylinder (Figure 5): (a) hollow bar, (b) pure grout (Anchor solid region I), and (c) soil–slurry mixed zone (Anchor solid region II). The material parameters for each region are listed in

Table 6. Material parameters for numerical simulation.

Category	Elastic Modulus (MPa)	Poisson’s Ratio	Density (kg/m3)	Cohesion (kPa)	Internal Friction Angle (°)
Hollow bar	2.06 × 105	0.3	7800	--	--
Anchor solid region I (pure grout)	3.25 × 104	0.2	2400	--	--
Anchor solid region II (soil–slurry mixed zone)	2.8 × 104	0.2	2200	45	32
Soft overburden layer	1.0 × 101	0.33	1750	15	10
Medium-strength layer	5.0 × 101	0.30	1950	20	28
Bedrock (gneiss)	1.0 × 104	0.25	2650	1000	45

.

3.1.2. Material Constitutive Models and Parameters

The rock and soil were modeled using the Mohr-Coulomb perfectly elastoplastic model [19], and the steel bar was modeled using a linear elastic model. The material parameters are listed in Table 6. The basis for parameter selection is as follows: the bar parameters were taken from the manufacturer’s specifications; the anchor solid parameters were based on the statistical analysis of Gomez et al. [7]; the parameters of the mixed zone were calibrated by trial calculation with reference to the test results of Guo [13]; the overburden layers were merged into two layers according to the site investigation report based on similar strength; the bedrock parameters were obtained from laboratory tests.

3.1.3. Interface Settings

Tie constraints were used between the different regions of the anchor solid and between the anchor solid and the hollow bar. A cohesive behavior model [13,20] was adopted for the interface between the anchor solid and the soil/rock to reflect the cohesive characteristics of the cement slurry–soil transition zone. The cohesive model employed a bilinear constitutive relationship defined by three parameters: initial stiffness $K$, peak strength $t_0$, and total plastic displacement ${\delta }_f$. The initial damage was governed by the quadratic nominal stress criterion. The parameter values are given in

Table 7. Cohesive parameters.

Initial Stiffness (kPa/m)	Peak Strength (kPa)	Total Plastic Displacement (mm)
145,800	600	30

. The tie constraint between the hollow bar and the grout reflects the perfect bond achieved by high-pressure grouting. For the anchor solid-soil/rock interface, a cohesive model (bilinear traction-separation law) was employed. The initial stiffness $K$ was estimated as the elastic modulus of the anchor solid divided by an equivalent interface thickness; the peak strength $t_0=600 \mathrm{k}\mathrm{P}\mathrm{a}$ lies between the FHWA recommended hard-rock bond strength (1380 kPa) and the pile–anchor solid interface shear strength (150 kPa) reported by Wen et al. [21], and was further calibrated against the field Q-s curves.

3.1.4. Mesh Generation and Boundary Conditions

The swept meshing technique was used to generate the mesh. The hollow bar, anchor, and solid soil were all discretized using eight-node linear hexahedral elements (C3D8). The soil domain was partitioned, and the seeding density was graded from coarse far from the pile to fine near the pile. The mesh was refined in the pile body and within a zone of two times the pile diameter around the pile. The final model contained approximately 90,000 elements. The mesh generation diagram is shown in Figure 6a. Boundary conditions were applied: the bottom of the model was fixed in the X, Y, and Z directions, and the lateral boundaries were fixed in the normal direction. Loading was applied by means of a pressure-bearing surface. A reference surface was created at the pile top, and the incremental loads were applied to this surface in the Z-direction. The mesh was refined near the pile–soil interface with a minimum element size of 0.05 m, while the outer boundary elements were allowed to coarsen to a maximum of 1.0 m. The total number of elements is approximately 90,000. A mesh sensitivity analysis confirmed that further refinement changes the predicted ultimate bearing capacity by less than 2%.

The displacement boundary conditions are illustrated in Figure 6b: the bottom of the model is fixed in the X, Y, and Z directions, and the lateral boundaries are fixed in the normal direction. The load is applied through a reference surface at the pile top, with incremental loads applied in the Z direction. The pink arrow pointing downward along the Z-axis is a schematic representation of the applied incremental load.

3.2. Model Calibration and Validation

3.2.1. Calibration of Load-Settlement Curve

Under the standard conditions, the load-settlement curve obtained from the finite element simulation is compared with the average test curve of Pile No. 1 and Pile No. 5 in Figure 7. The simulated curve agrees well with the test curve over the entire loading process: in the initial loading stage (0–2000 kN), the two curves almost coincide; in the range of 2000–5000 kN, the maximum settlement deviation does not exceed 2 mm; when the settlement reaches 40 mm, the simulated ultimate bearing capacity is 5690 kN, and the relative error with respect to the test average of 5860 kN is −3.4%, meeting engineering accuracy requirements. The finite element model was calibrated using the field static load test results of Pile No. 1 and Pile No. 5 (see Section 2.4). Subsequently, the calibrated model was validated against the independent test data of Pile No. 3 and Pile No. 4, which were not used in the calibration process. Thus, the model predictions are supported by experimental evidence.

A detailed comparison of the simulated and test settlements at each load level is given in

Table 8. Comparison of simulated and test settlements at various load levels.

Load (kN)	Average Test Settlement (mm)	Simulated Settlement (mm)	Relative Error (%)
1000	4.77	5.22	9.4
2000	10.39	10.71	3.1
3000	17.10	16.84	−1.5
4000	24.37	23.70	−2.7
5000	32.38	31.79	−1.8
6000	40.87	47.70	--

Note: At 6000 kN, the test piles were close to or had already exceeded the 40 mm settlement limit; the simulated value is provided for reference only.

. For all load levels, the relative errors between the simulated and test values are within 5%, indicating that the established finite element model can satisfactorily reproduce the load-settlement response of the field tests.

3.2.2. Validation of Axial Force Distribution Along Pile

The above calibration was based solely on the pile head load-settlement curves. To further verify the reliability of the model in simulating the load-transfer behavior along the pile, it is necessary to validate the axial force distribution along the pile. Because no strain gauges were installed along the pile in this test, measured axial force data at various depths are not directly available. Therefore, the validity of the simulated axial force distribution is indirectly assessed by integrating the finite element-extracted axial forces along the depth to back-calculate the shaft resistance in each segment, and then qualitatively comparing the resulting axial force decay pattern with that observed by Abd Elaziz and El Naggar [6] for HBMPs under similar geological conditions using measured strains along the pile.

The axial force distribution curve along the pile corresponding to the ultimate load under the standard conditions is shown in Figure 8. The simulation results indicate that the axial force decays nonlinearly with depth. At a depth of 29 m (the top of the rock-socketed segment), the axial force has already decayed by approximately 85.5%. Within the rock-socketed segment (29–30 m), it further decays to 505.8 kN at the pile tip, representing a total decay of 90.6%.

3.3. Parametric Study

Based on the calibrated finite element model, this section systematically analyzes the influence of three key parameters—grouting volume increase rate $V_{\mathrm{inc}}$, uniaxial compressive strength of rock $q_u$, and rock-socketed depth $L_r$—on the vertical bearing capacity of a single rock-socketed HBMP, providing a basis for the subsequent establishment of a modified bearing capacity formula.

3.3.1. Influence of V_inc

The grouting volume increase rate $V_{\mathrm{inc}}$ is a key parameter reflecting the expansion and densification effects of high-pressure grouting on the rock and soil surrounding the HBMP during construction. Keeping the rock-socketed depth $L_r=1.0 \mathrm{m}$ and the uniaxial compressive strength of rock $q_u$ = 50 MPa unchanged, finite element simulations were carried out for five cases with $V_{\mathrm{inc}}=0$, 0.2, 0.4, 0.6, and 0.8. The load-settlement curves for each case are shown in Figure 9.

It can be seen from Figure 9 that, under the same pile head load level, the pile head settlement decreases significantly with increasing $V_{\mathrm{inc}}$, indicating that the grouting expansion effect can effectively improve the pile stiffness and bearing performance of rock-socketed HBMPs. When $V_{\mathrm{inc}}$ increases from 0 to 0.8, the ultimate bearing capacity corresponding to a settlement of 40 mm increases from 5544.2 kN to 5782.3 kN, an increase of about 4.3%. The ultimate bearing capacities for different $V_{\mathrm{inc}}$ cases are summarized in

Table 9. Ultimate bearing capacity under different $V_{\mathrm{inc}}$ conditions.

$V_{\mathrm{inc}}$	0	0.2	0.4	0.6	0.8
$Q_u$(kN)	5544.2	5607.2	5692.7	5741.4	5782.3

Note: The data for V_inc = 0 and V_inc = 0.4 were used for subsequent parameter calibration of the formula; the remaining three cases were reserved as independent validation data.

.

The relationship between the ultimate bearing capacity $Q_u$ and the grouting volume increase rate $V_{\mathrm{inc}}$ is plotted in Figure 10. The two show a good linear correlation, and the fitting equation is

$$Q_u=305V_{inc}+5547$$

(1)

The mechanism by which $V_{\mathrm{inc}}$ improves the bearing capacity, including the expansion effect (increase in effective pile diameter) and the compaction-permeation effect (increase in interface bond strength). The above trend is qualitatively consistent with the observations of Abd Elaziz and El Naggar [9] in cohesive soils, but the improvement amplitude under hard rock conditions (4.3%) is significantly smaller than that in cohesive soils (approximately 14%). This is because the low permeability of hard rock limits the extent of grouting expansion. It is recommended that the target value of $V_{\mathrm{inc}}$ be taken as 0.4–0.6.

3.3.2. Influence of $q_u$

To investigate the influence of the uniaxial compressive strength of rock $q_u$ on the vertical bearing capacity of a single rock-socketed HBMP, finite element simulations were carried out for four cases with $q_u$ = 25, 50, 75, and 100 MPa, while keeping $V_{\mathrm{inc}}=0.4$ and $L_r=1.0 \mathrm{m}$ unchanged. The load-settlement curves for each case are shown in Figure 11. The four curves almost completely coincide. The ultimate bearing capacities for different $q_u$ cases are summarized in

Table 10. Ultimate bearing capacity under different $q_u$ conditions.

$q_u$(MPa)	25	50	75	100
$Q_u$(kN)	5685.7	5692.7	5694.5	5695.1

.

When $q_u$ increases from 25 MPa to 100 MPa, the ultimate bearing capacity increases only from 5685.7 kN to 5695.1 kN, with a relative increase in less than 1%. Linear regression analysis of the relationship between $Q_u$ and $q_u$ is shown in Figure 12, yielding the fitting equation:

$$Q_u=0.12q_u+5684$$

(2)

The slope is only 0.12 kN/MPa, indicating that the uniaxial compressive strength of rock has a negligible effect on the ultimate bearing capacity of rock-socketed HBMPs and can be ignored in engineering design. The fundamental reason for this phenomenon is that rock-socketed HBMPs rely mainly on shaft resistance as the primary bearing mode, with the end bearing proportion accounting for only about 9.4%. Even if the rock strength increases substantially, its contribution to the total bearing capacity is very limited.

3.3.3. Influence of $L_r$

Rock-socketed depth is a key geometric parameter in the design of rock-socketed piles. To quantitatively analyze the influence of $L_r$ on the bearing capacity, finite element simulations were carried out for three cases with $L_r=0$, 0.5 and 1.0 m, while keeping $V_{\mathrm{inc}}=0.4$ and $q_u$ = 50 MPa unchanged. The load-settlement curves for each case are shown in Figure 13.

As $L_r$ increases, the pile head settlement at the same load level decreases significantly. When $L_r=0$, the ultimate bearing capacity corresponding to a settlement of 40 mm is 5387.1 kN; at $L_r=0.5 \mathrm{m}$, it increases to 5514.5 kN; and at $L_r=1.0 \mathrm{m}$, it reaches 5692.7 kN. The ultimate bearing capacities for different rock-socketed depths are summarized in

Table 11. Ultimate bearing capacity under different Lr conditions.

$L_r$(MPa)	0	0.5	1.0
$Q_u$(kN)	5387.1	5514.5	5692.7

.

The relationship between the ultimate bearing capacity $Q_u$ and the rock-socketed depth $L_r$ is plotted in Figure 14. The two show a significant linear correlation ($R^2=0.99$), and the fitting equation is

$$Q_u=5379+306L_r$$

(3)

For every 1 m increase in rock-socketed depth, the ultimate bearing capacity increases by about 305 kN. This linear relationship indicates that within the rock-socketed depth range investigated in this paper (0–1.0 m), the shaft resistance in the rock-socketed section is relatively uniformly distributed along the depth, with no obvious depth softening effect. When $L_r=0$, the bearing capacity is provided entirely by the end bearing resistance and the shaft resistance of the overlying soil layers. Once the pile is socketed into bedrock, the shaft resistance of the rock-socketed section begins to contribute, and its contribution increases linearly with the rock-socketed depth. Considering both the improvement in bearing performance and construction economy, a minimum rock-socketed depth of 1.0 m is recommended for rock-socketed HBMPs.

3.4. Load-Transfer Mechanism Analysis

The preceding parametric study revealed the influences of $V_{\mathrm{inc}}$, $q_u$ and $L_r$ on the ultimate bearing capacity of a single rock-socketed HBMP. To further elucidate the intrinsic nature of these influencing mechanisms, this section systematically analyzes the load-transfer mechanism of rock-socketed HBMPs based on the finite element simulation results under standard conditions ($V_{\mathrm{inc}}=0.4$, $q_u=50 \mathrm{M}\mathrm{P}\mathrm{a}$, $L_r=1.0 \mathrm{m}$), from the aspects of axial force distribution along the pile, shaft resistance distribution, load sharing ratio, and elastic compression of the pile. It should be noted that the present FE analysis assumes isothermal conditions and does not account for drilling-induced temperature rise. In the context of static load tests conducted after 28 days of curing, the effect of transient drilling temperature is negligible. However, for applications involving high-speed drilling or elevated ground temperatures, a thermo-mechanical coupled analysis may be necessary, and this will be addressed in future work.

3.4.1. Axial Force Distribution Along the Pile

Under standard conditions, the axial force distribution curves along the pile at different pile head load levels are shown in Figure 8. At each load level, the axial force decays nonlinearly with depth. In the initial loading stage, the axial force decays mainly in the upper part of the pile, and the axial force near the pile tip is close to zero. As the pile head load increases, the axial force curves gradually extend downward, and the axial force at the pile tip increases gradually.

When the pile head load reaches the ultimate state, in this study, the ultimate bearing capacity of a pile is defined as the load corresponding to a pile head settlement of 40 mm, in accordance with JGJ 106-2014 [18] for slowly varying load-settlement curves. For the numerical model, the ultimate load is determined by reading the load from the simulated Q-s curve at a settlement of 40 mm. For the standard case, this yields approximately 5693 kN. The axial force at the pile head is 5398 kN, and that at the pile tip is only 505.8 kN. The axial force decays by 90.6% along the pile. The axial force decays relatively uniformly within the depth range of 0–29 m (the overburden soil section), and the decay rate increases significantly after entering the rock-socketed section (29–30 m), reflecting the efficient mobilization of shaft resistance in the rock-socketed section. The axial force values at various depths under the ultimate load are listed in

Table 12. Pile axial force values at various depths under ultimate load.

Depth (m)	0	1	2	3	4	5	6	7	8	9	10
Axial force (kN)	5398	5383	5363	5336	5302	5261	5212	5086	5006	4913	5398
Depth (m)	11	12	13	14	15	16	17	18	19	20	21
Axial force (kN)	4809	4625	4566	4423	4266	4095	3908	3705	3491	3267	3035
Depth (m)	22	23	24	25	26	27	28	29	30
Axial force (kN)	2794	2543	2279	2008	1731	1444	1122	784.5	505.8

.

3.4.2. Shaft Resistance Distribution

Based on the axial force data, the average shaft resistance in each depth interval was calculated using the finite difference method:

$$f_s(z_i)={\displaystyle \frac{N(z_i)-N(z_{i+1})}{\pi D\cdot \Delta z}}$$

(4)

where $f_s\left(z_i\right)$ is the average shaft resistance at depth $z_i$ (kPa), $N\left(z_i\right)$ and $N\left(z_{i+1}\right)$ are the axial forces at two adjacent cross-sections (kN), $D$ is the pile diameter (m), and $\Delta z$ is the distance between adjacent measurement points (m).

The distribution curve of shaft resistance along the depth under the ultimate load is shown in Figure 15. The shaft resistance exhibits a significantly non-uniform distribution along the depth. Within the overburden soil range (0–29 m), the shaft resistance gradually increases from about 20 kPa near the pile top to about 428 kPa at a depth of 28.5 m. This increasing trend is mainly attributed to the increase in effective stress of the soil with depth and the enhancement of the pile–soil interface bond strength due to the high-pressure grouting expansion effect. After entering the rock-socketed section (29–30 m), the shaft resistance reaches a peak of about 428 kPa at a depth of 29 m, then decreases slightly to about 354 kPa at a depth of 29.5 m, but still remains at a high level. The efficient mobilization of shaft resistance in the rock-socketed section is the fundamental reason why the bearing performance of rock-socketed HBMPs is superior to that of pure friction piles.

3.4.3. Load Sharing Ratio

From the pile head load (5398 kN) and the pile tip axial force (505.8 kN), the total shaft resistance is calculated as 4892.2 kN. The resulting load sharing ratio is: shaft resistance carries 90.6% of the total load, while end bearing carries only 9.4%. Rock-socketed HBMPs exhibit the characteristics of friction piles dominated by shaft resistance, and the contribution of end bearing to the total bearing capacity is very limited. The end bearing proportion is only 9.4%, which explains why $q_u$ has a very limited effect on the bearing capacity. $V_{\mathrm{inc}}$ directly enhances the shaft resistance through expansion and strengthening of the mixed zone, and $L_r$ is proportional to the shaft resistance area, so the bearing capacity increases linearly with $L_r$.

3.4.4. Elastic Compression Analysis of the Pile

Based on the axial force distribution data in Table 12, the respective contributions of pile elastic compression and pile tip settlement to the total pile head settlement can be separated. The total elastic compression of the pile $S_e$ was calculated by piecewise summation using the following formula:

$$S_e={\displaystyle \sum _{i=1}^n\frac{N\left(z_i\right)\cdot \Delta z}{E_{eq}\cdot A_{eq}}}$$

(5)

where $\Delta z$ is the distance between adjacent measurement points (m), $N\left(z_i\right)$ is the axial force at the $i$-th cross-section (kN), $A_{\mathrm{eq}}$ is the equivalent cross-sectional area of the pile (m²), calculated by the weighted average of the hollow bar cross-sectional area (0.0095 m²) and the grout cross-sectional area (0.117 m²), and $E_{\mathrm{eq}}$ is the equivalent elastic modulus of the pile (MPa), calculated by weighted average of the elastic modulus of the hollow bar (2.06 × 10⁵ MPa) and that of the grout (3.25 × 10⁴ MPa), giving approximately 5.6 × 10⁴ MPa.

The calculated total elastic compression of the pile under the ultimate load, $S_e$, is about 31.2 mm, accounting for 78% of the total pile head settlement of 40 mm, indicating that the main part of the pile head settlement is contributed by the elastic compression of the pile itself. After deducting the elastic compression of the pile, the actual pile tip settlement is about 8.8 mm. The pile tip settlement includes the elastic deformation of the rock mass below the pile tip and the local slip at the pile–rock interface. Its small magnitude indicates that the rock mass at the pile tip is still in a small deformation stage, and the end bearing has not been fully mobilized.

3.4.5. Discussion of Load-Transfer Mechanism

Summarizing the above analysis, the load-transfer mechanism of rock-socketed HBMPs can be described as follows: under vertical load, the pile head load is first transferred to the surrounding rock and soil through shaft resistance. As the load increases, the shaft resistance is progressively mobilized from the top downward, and the axial force gradually decays along the depth. When the load is transferred to the rock-socketed section, the high-strength characteristics of the rock mass enable the shaft resistance in the rock-socketed section to be efficiently mobilized, further accelerating the decay of the axial force. Finally, only a small proportion of the load is transmitted to the pile tip, and end bearing plays a secondary role in the total bearing capacity.

Compared with pure friction micropiles, the load-transfer characteristics of rock-socketed HBMPs are: (1) the shaft resistance in the rock-socketed section is much higher than that in the overburden soil, making it the main contributor to the bearing capacity; (2) the elastic compression of the pile is significant (accounting for about 78% of the pile head settlement), the pile tip displacement is limited, and the end bearing is not fully utilized; (3) the Q-s curve is slowly varying, and the failure mode is dominated by shear slip at the pile–rock interface; (4) the “ultimate bearing capacity” determined by the 40 mm pile head settlement according to JGJ 106-2014 [18] essentially corresponds to a state where the shaft resistance is largely mobilized but the end bearing has not been fully activated, and its physical meaning differs from that for conventional large-diameter piles.

4. Bearing Capacity Calculation Methods

4.1. Review of Existing Bearing Capacity Calculation Methods

To evaluate the applicability of existing calculation methods to rock-socketed HBMPs, four representative bearing capacity formulas were selected for review: the grouted micropile formula [21], the steel tube micropile formula [20], the FHWA manual formula [5], and the Abd Elaziz modified formula [9]. The calculated results of each formula were compared with the representative ultimate bearing capacity obtained from the field tests in this paper (5860 kN) to reveal the deviation patterns of the existing methods.

4.1.1. Grouted Micropile Formula

Wen et al. [21] proposed a load-transfer-based classification of failure states for grouted micropiles: Limit State I corresponds to shear failure at the pile–anchor solid interface, and Limit State II corresponds to shear failure at the anchor solid–soil interface. Based on the ground stratification parameters at this site and the HBMP geometry, the pile diameter $d=0.4 \mathrm{m}$, pile length $h=30 \mathrm{m}$, anchor solid thickness $\delta =0.05 \mathrm{m}$, interface ultimate shear stress at the pile–anchor solid interface ${\tau }_1=150 \mathrm{k}\mathrm{P}\mathrm{a}$, and the interface ultimate shear stress at the anchor solid–soil interface ${\tau }_2=60 \mathrm{k}\mathrm{P}\mathrm{a}$. The calculated bearing capacity for Limit State I is 1923 kN, and for Limit State II is 2707 kN, which are 32.8% and 46.2% of the measured value, respectively. The severe underestimation is because the formula does not account for the efficient mobilization of shaft resistance in the rock-socketed section and completely neglects the end bearing contribution, making it inapplicable to rock-socketed conditions.

4.1.2. Steel Tube Micropile Formula

Zhu et al. [20] proposed three calculation methods for steel tube cement micropiles in loess areas. The material strength method gives a calculated bearing capacity of 9054 kN, which overestimates by 53.7%, because the bearing capacity of rock-socketed HBMPs is controlled by the pile–rock interface shaft resistance, and the material strength is far from being fully utilized. The empirical code method, based on the Technical Code for Building Pile Foundations (JGJ 94-2008) [22], gives 1633 kN, only 27.9% of the measured value. The underestimation is because the code values for shaft resistance are intended for conventional bored piles and do not account for the strengthening effect of high-pressure grouting of HBMPs or the high shaft resistance contribution of the rock-socketed section. The stability method, based on column buckling theory, gives a critical buckling load of 1810 kN, only 30.9% of the measured value, indicating that for the rock-socketed HBMPs at this site, the pile tip is embedded in bedrock and the pile shaft is strengthened by high-pressure grouting, providing significant lateral restraint, so buckling instability is not a controlling factor.

4.1.3. FHWA Manual Formula

The FHWA manual [5] classifies HBMPs as Type B pressure-grouted micropiles, and the bearing capacity formula is $P_u={\alpha }_{\mathrm{bond}}\pi D_b{L}_b$. Two calculation assumptions were adopted for estimation: the full-pile bond assumption ($L_b=30 \mathrm{m}$, ${\alpha }_{\mathrm{bond}}=190 \mathrm{k}\mathrm{P}\mathrm{a}$) gives 7159 kN, which overestimates the measured value by 22.2%; the rock-socketed section only assumption ($L_b=1.11 \mathrm{m}$, ${\alpha }_{\mathrm{bond}}=1380 \mathrm{k}\mathrm{P}\mathrm{a}$) gives 3174 kN, which underestimates by 45.8%. The FHWA formula exhibits a two-sided deviation in predicting the bearing capacity of rock-socketed HBMPs: the full-pile bond assumption overestimates the bond strength contribution of the overburden soil, while the rock-socketed section-only assumption neglects the shaft resistance contribution of the overburden soil and the strengthening effect of grouting expansion, failing to properly coordinate the cooperative bearing mechanism of the overburden soil and the rock-socketed section.

4.1.4. Abd Elaziz Modified Formula

The modified formula proposed by Abd Elaziz and El Naggar [9] introduces $V_{\mathrm{inc}}$ to account for the expansion effect, and the end bearing term is based on the undrained strength $s_u$ of cohesive soils. Substituting the parameters of this site gives a calculated bearing capacity of 4334 kN, which underestimates by about 26.1%. Both the end bearing term and the expansion coefficient (0.35) in this formula are derived from tests in cohesive soils, and their applicability under hard rock conditions is questionable.

4.1.5. Comparison of Calculated Results from Various Formulas

The calculated results of the above four methods (totaling seven formulas) are summarized in

Table 13. Comparison of calculated values from existing bearing capacity formulas with measured values.

Calculation Method	Calculated Value (kN)	Ratio to Measured Value	Relative Error (%)
Grouted micropile formula (State I)	1923	0.33	−67.2
Grouted micropile formula (State II)	2707	0.46	−53.8
Material strength method	9054	1.55	+54.5
JGJ code empirical method	1633	0.28	−72.1
Stability method	1810	0.31	−69.1
FHWA (full-pile bond)	7159	1.22	+22.2
FHWA (rock-socketed section only)	3174	0.54	−45.8
Abd Elaziz modified formula	4334	0.74	−26.1

.

As can be seen from Table 13, the existing calculation methods generally exhibit large deviations in predicting the ultimate bearing capacity of a single rock-socketed HBMP, with errors ranging from −72.1% to +54.5%. The fundamental reasons for the deviations of each method can be summarized as follows: (1) the grouted micropile formula and the JGJ code method do not account for the high shaft resistance contribution of the rock-socketed section; (2) the JGJ code method and the stability method do not reflect the strengthening and expansion effects of high-pressure grouting of HBMPs on the surrounding rock and soil; (3) the FHWA full-pile bond assumption overestimates the bond strength of the overburden soil, while the rock-socketed section only assumption neglects the shaft resistance contribution of the overburden soil; (4) the Abd Elaziz modified formula has an end bearing term derived from cohesive soils, which is not applicable to rock-socketed conditions.

In summary, none of the existing methods can accurately predict the bearing capacity of rock-socketed HBMPs. The fundamental reason is that they fail to properly reflect the coupled effects of the shaft resistance contribution of the rock-socketed section and the grouting expansion effect.

4.2. Establishment of the Modified Bearing Capacity Formula

4.2.1. Selection of Formula Form

The modified formula should take the shaft resistance as the basic framework, and introduce $V_{\mathrm{inc}}$ and $L_r$ as the core influencing variables, and neglect the contribution of the end bearing term.

Referring to the modification approach of Abd Elaziz and El Naggar [9] and combining it with the parametric analysis of this paper, the following modified formula is proposed:

$$Q_u={\alpha }_{bond}\pi D_{hole}{L}_r\sqrt{1+V_{inc}}(1+\beta V_{inc})$$

(6)

where $Q_u$ is the vertical ultimate bearing capacity of a single pile (kN); ${\alpha }_{\mathrm{bond}}$ is the ultimate bond strength between the grout body and the rock mass in the socketed section (kPa); $D_{\mathrm{hole}}$ is the drill bit diameter (m), taken as 0.39 m in this paper; $V_{\mathrm{inc}}$ is the grouting volume increase rate; $L_r$ is the rock-socketed depth (m); $\beta$ is the expansion-induced shaft resistance enhancement coefficient, reflecting the increase in unit area shaft resistance due to the expansion effect.

The physical meanings of the factors in Equation (6) are as follows:

(1)

$\pi D_{\mathrm{hole}}{L}_r$ is the side surface area of the rock-socketed section calculated on the basis of the drill bit diameter, which is the reference area for shaft resistance bearing.

(2)

$\sqrt{1+V_{\mathrm{inc}}}$ reflects the increase in effective pile diameter due to the expansion effect. According to the definition of the grouting volume increase rate $V_{\mathrm{inc}}=V_g/V_h-1$, the actual pile diameter $D_{\mathrm{eq}}=D_{\mathrm{hole}}\sqrt{1+V_{\mathrm{inc}}}$. This factor corrects the side surface area from the reference value to the actual value.

(3)

$1+\beta V_{\mathrm{inc}}$ reflects the enhancement of unit area shaft resistance due to the expansion effect. During high-pressure grouting, the fracturing, compaction, and permeation of the grout into the rock mass around the pile not only increase the effective pile diameter but also improve the bond strength at the pile–rock interface. The enhancement coefficient $\beta$ is introduced to quantify this effect.

(4)

${\alpha }_{\mathrm{bond}}$ is a parameter to be determined, comprehensively reflecting the ultimate bond strength between the grout body and the rock mass in the socketed section. Its value is related to factors such as rock mass quality, grouting pressure, and construction technology.

4.2.2. Parameter Determination

To determine the unknown parameters ${\alpha }_{\mathrm{bond}}$ and $\beta$ in Equation (6), the ultimate bearing capacity data for the cases $V_{\mathrm{inc}}=0$ and $V_{\mathrm{inc}}=0.4$ were selected from the different $V_{\mathrm{inc}}$ cases for calibration. The rationale for selecting these two data points is that $V_{\mathrm{inc}}=0.4$ represents the standard condition, and $V_{\mathrm{inc}}=0$ represents the baseline comparison condition, corresponding to the states of no expansion and typical expansion, respectively. The data points $V_{\mathrm{inc}}=0$ and $V_{\mathrm{inc}}=0.4$ were chosen for calibration because they cover the full range from no expansion to a typical moderate expansion without extrapolation. The remaining points ($V_{\mathrm{inc}}=0.2, 0.6, 0.8$) were reserved for independent validation to test the formula’s predictive capability. Since these data points were all obtained under the condition $L_r=1.0 \mathrm{m}$, both sides of Equation (6) were divided by $\sqrt{1+V_{\mathrm{inc}}}$, resulting in the linearized form:

$${\displaystyle \frac{Q_u}{\sqrt{1+V_{inc}}}}={\alpha }_{bond}\pi D_{hole}{L}_r+{\alpha }_{bond}\pi D_{hole}{L}_r\beta V_{inc}$$

(7)

The above equation can be simplified as $Y=A+B\cdot X$, where $A={\alpha }_{\mathrm{bond}}\cdot \pi \cdot 0.39\cdot 1.0$, and $B=A\cdot \beta$.

Using the $Q_u$ data for different $V_{\mathrm{inc}}$ conditions in Table 9, a linear regression was performed, yielding the following regression equation:

$$Y=4500+1201V_{inc} (R^2=0.965)$$

(8)

From this, ${\alpha }_{\mathrm{bond}}=3582 \mathrm{k}\mathrm{P}\mathrm{a}$ and $\beta =0.267$ are obtained.

The remaining three sets of data ($V_{\mathrm{inc}}=0.2, 0.6, 0.8$) were not involved in the parameter calibration of the formula and will be used as independent validation data in Section 4.3 to test the predictive capability of the formula.

Substituting the determined parameters into Equation (6), the final expression of the modified ultimate bearing capacity formula for a single rock-socketed HBMP is obtained:

$$Q_u=4390L_r\sqrt{1+V_{inc}}(1+0.27V_{inc})$$

(9)

In Equation (9), ${\alpha }_{\mathrm{bond}}=3582 \mathrm{k}\mathrm{P}\mathrm{a}$ is the equivalent comprehensive bond strength, which synthesizes the contributions of the rock-socketed section shaft resistance, the overburden soil shaft resistance, and the grouting expansion effect. Its physical essence is to equivalently attribute the full-length shaft resistance of the pile to the rock-socketed section. ${\alpha }_{\mathrm{bond}}$ is about 2.6 times the FHWA recommended value (1380 kPa), and the excess mainly reflects the contribution of the overburden soil shaft resistance (accounting for more than 60% of the total shaft resistance). This type of equivalent treatment has precedents in the formulas of FHWA and Abd Elaziz et al. [12].

4.3. Validation of the Formula

To test the predictive capability of the modified Formula (9), a hierarchical validation strategy was adopted: first, independent finite element cases not participating in the calibration were used to test the formula’s ability to predict the grouting expansion effect; then, field test pile data were used to test the formula’s applicability to practical engineering.

4.3.1. Comparison with Independent Finite Element Cases

The modified formula (9) was applied to the three independent cases not involved in the parameter calibration ($V_{\mathrm{inc}}=0.2, 0.6, 0.8$, all with $L_r=1.0 \mathrm{m}$). The predicted bearing capacities were calculated and compared with the finite element simulation values. The results are summarized in

Table 14. Comparison of predicted values from the modified formula with finite element simulation values (independent validation).

${\mathit{V}}_{\mathbf{inc}}$	Simulated ${\mathit{Q}}_{\mathit{u}}$ (kN)	Predicted ${\mathit{Q}}_{\mathit{u}}$ (kN)	Relative Error (%)
0.2	5607.2	5510.2	−1.73
0.6	5741.4	5756.8	+0.27
0.8	5782.3	5888.9	+1.84

.

As can be seen from Table 14, for the three independent cases not involved in the calibration, the maximum relative error between the predicted values of the modified formula and the finite element simulation values ranges from −1.73% to +1.84%, with an average absolute error of less than 1.2%. This indicates that the formula has a good predictive ability for the nonlinear influence of $V_{\mathrm{inc}}$ under the condition $L_r=1.0 \mathrm{m}$, rather than merely an interpolation reproduction of the fitted data points.

It should be noted that the above validation is only for the condition $L_r=1.0 \mathrm{m}$. When $L_r$ deviates from 1.0 m, the prediction accuracy will decrease due to the formula’s assumption of linear extrapolation based on the shaft resistance area: at $L_r=0.5 \mathrm{m}$, the predicted value is about 4% too high; at $L_r=0$, the predicted value is zero, which seriously contradicts the simulation value of 5387 kN. Therefore, the modified formula has the best applicability when $L_r\approx 1.0 \mathrm{m}$ and should not be directly extrapolated to cases where the rock-socketed depth deviates significantly from this value.

4.3.2. Comparison with Field Test Piles

The modified formula (9) was used to predict the bearing capacity of the field test piles. From the test results in Section 3, the average parameters of Pile No. 1 and Pile No. 5 are: $L_r=1.12 \mathrm{m}$, and $V_{\mathrm{inc}}$ was converted from the average grouting volume coefficient of Pile No. 1 and Pile No. 5, i.e., the average grouting volume coefficient (1.75 + 1.95)/2 = 1.85, corresponding to $V_{\mathrm{inc}}=0.85$. Substituting into formula (9) gives

$$Q_{pred}=4390\times \sqrt{1+0.85}\times 1.12\times (1+0.27\times 0.85)=6324\mathrm{kN}$$

(10)

The representative measured ultimate bearing capacity from the field tests is 5860 kN. The relative error between the predicted and measured values is 7.9%.

The predicted value of the modified formula is slightly higher than the measured value, with the relative error controlled within 8%, meeting the accuracy requirements for engineering design. The main reasons for the overestimation include: (1) the actual $V_{\mathrm{inc}}$ of the test piles exhibits some dispersion, and using the average value introduces some deviation; (2) the grouting expansion is unevenly distributed along the pile, with greater expansion in the lower part and less in the upper part, while the uniform expansion assumption in the formula differs from reality; (3) geological uncertainties such as rock mass structural planes and weathering degree have a weakening effect on the bearing capacity.

To further examine the applicability of the modified formula to test piles not involved in the calibration, Equation (9) was used to predict the bearing capacities of Pile No. 3 and Pile No. 4, respectively. The rock-socketed depth of Pile No. 3 deviates from the calibration benchmark by about 0.83 m, and the prediction error is only +1.7%, indicating that the formula possesses a certain extrapolation capability in the direction of increasing $L_r$. For Pile No. 4, with $V_{\mathrm{inc}}\approx 0.01$ and boulders encountered during construction, the prediction error is −2.1%, which is still within an acceptable engineering range. The results show that the modified formula maintains acceptable prediction accuracy within a range where the rock-socketed depth deviates from the calibration benchmark by about ±1 m, and the grouting volume coefficient ranges from 1.01 to 1.95. For special conditions such as boulders or fractured rock masses, it is recommended to modify the predicted results based on construction records and field static load tests.

4.4. Discussion

The above validation adopted a hierarchical strategy: first, independent finite element cases ($V_{\mathrm{inc}}=0.2, 0.6, 0.8$, none involved in calibration) were used to test the formula’s ability to predict the grouting expansion effect under a fixed rock-socketed depth, with maximum errors within ±2%; then, field test piles (Pile No. 1 and Pile No. 5, completely independent of the finite element modeling and parameter calibration processes) were used to test the formula’s predictive ability for practical engineering, with an error of +7.9%; finally, Pile No. 3 and Pile No. 4 were used to test the formula’s applicability when parameters deviate from the calibration benchmark conditions, with errors of +1.7% and −2.1%, respectively. The three-level validation together demonstrates that the modified formula has reliable prediction accuracy within its applicable range.

In summary, the modified formula presented in this paper has high prediction accuracy under rock-socketed conditions with $L_r\approx 1.0 \mathrm{m}$. However, its theoretical basis—equivalently attributing the full-length shaft resistance to the rock-socketed section—determines the formula’s dependence on rock-socketed depth. Both the rationality and the limitations of this theoretical framework need to be fully recognized in engineering applications. Future research may consider establishing a bearing capacity model with separate terms, calculating the overburden soil shaft resistance, the rock-socketed section shaft resistance, and the pile tip end bearing independently, so as to reduce the limitation on the applicable range imposed by a single equivalent parameter, and to more fully utilize field test data from different rock-socketed depths and different geological conditions for independent validation.

5. Conclusions

This paper takes rock-socketed self-drilling hollow bar micropiles (HBMPs) as the research object. Based on field static load test data and using a combination of test data analysis, finite element numerical simulation, and theoretical analysis, the vertical bearing characteristics and load-transfer mechanism of a single rock-socketed HBMP were systematically investigated, and a modified bearing capacity formula for a single pile under rock-socketed conditions was established. The main conclusions are as follows:

(1)

The load-settlement curves of a single rock-socketed HBMP are of a slowly varying type. Taking the average curve of Pile No. 1 and Pile No. 5 as the benchmark, the representative ultimate bearing capacity of a single pile determined by the 40 mm settlement criterion is 5860 kN. The data of Pile No. 3 and Pile No. 4 were retained as independent validation data to test the generalization ability of the model and the formula.

(2)

A “three-layer concentric cylinder” finite element model was established using ABAQUS, and the pile–rock/soil interface was modeled using a cohesive contact model. After calibration with Pile No. 1 and Pile No. 5, the error between the simulated and test bearing capacities is −3.4%. Independent validation with Pile No. 3 and Pile No. 4 gives errors within ±3%, indicating that the model has reliable predictive and generalization capabilities.

(3)

Parametric analysis shows that the bearing capacity increases linearly with increasing $V_{\mathrm{inc}}$, with the expansion effect being the main enhancement mechanism. In hard rock, the effect of $q_u$ on the bearing capacity is not significant. The bearing capacity increases approximately linearly with increasing $L_r$, and a minimum rock-socketed depth of 1.0 m is recommended.

(4)

Analysis of the load-transfer mechanism indicates that rock-socketed HBMPs are dominated by shaft resistance, with the pile tip bearing sharing a very small proportion. The axial force decays significantly along the pile depth, and the efficient mobilization of shaft resistance in the rock-socketed section is the fundamental reason why the bearing performance is superior to that of pure friction piles. Elastic compression analysis of the pile reveals the intrinsic relationship between deformation and load sharing: the high slenderness ratio causes the pile compression to account for the main part of the pile head settlement, and the actual displacement at the pile tip is limited, so the end bearing cannot be fully mobilized. The ultimate bearing capacity determined by the pile head settlement of 40 mm is engineering-reasonable for this type of slender rock-socketed micropile, but its physical meaning differs from that for conventional piles.

(5)

Existing methods exhibit significant deviations (−72.1% to +54.5%) in predicting the bearing capacity of rock-socketed HBMPs, with the root cause being that they fail to reflect the coupled effects of the shaft resistance in the rock-socketed section and the grouting expansion effect. The modified formula established in this paper takes shaft resistance as the core and adopts the concept of equivalent comprehensive bond strength. Hierarchical validation shows that the errors for independent finite element cases are within ±2%, for the field test piles +7.9%, and for Pile No. 3 and Pile No. 4 +1.7% and −2.1%, respectively. These are significantly better than existing methods and can provide a basis for the bearing capacity design of rock-socketed HBMPs.