Numerical study on bearing performance of rock socketed pile anchored by inclined anchors under uplift and horizontal combined load

Abstract

Rock-socketed pile anchored by inclined anchors (RPIA) is a novel pile-anchor composite foundation developed to address the challenges of transmission tower installations in mountainous areas. This study investigates the bearing performance of RPIA under uplift-horizontal combined load using a nonlinear numerical model validated through full-scale field tests. Parametric analyses were conducted to examine the effects of inclined anchor length, angle and diameter, as well as connection node depth, on RPIA performance. Analysis reveals that inclined anchors reconstruct the path of potential slip surfaces within soil and rock mass, thereby effectively expanding the participation range of geotechnical bodies. Parametric analysis demonstrates that ultimate load of RPIA decreases with increasing angle, but increases with larger length and diameter of inclined anchor. The trends are further modulated by the relative position of the connection node within the geotechnical body. Longer inclined anchors shift the failure mode of the RPIA from the pull-out of short pile to the yielding of longitudinal rebar, and are suitable for shallower connection nodes to improve horizontal resistance. While shorter inclined anchors require the increment in anchor diameter and depth of connection point. The results of this study provide guidance for the design and promotion of RPIA.

Introduction

As a crucial component of transmission lines, tower foundations serve as the fundamental safety guarantee for power grid engineering^1,2. China, being a mountainous country, faces rock foundations as the primary engineering condition for the foundations of transmission line towers³. Currently, rock anchors, rock-embedded foundations, and rock-socketed piles are the three most commonly used foundation forms in transmission tower engineering, among which rock anchors and rock embedded piles are more widely used^4,5,6,7,8.However, influenced by terrain and transportation conditions, large-scale construction equipment is difficult to deploy in mountainous tower foundation engineering, making hole formation for rock-embedded piles a challenging aspect of infrastructure construction and limiting the size of these piles^9,10,11. In contrast, rock anchors offer the advantage of hole formation using small-scale equipment^{12,13,14,15,16,17}. Nonetheless, owing to the unique characteristics of transmission lines engineering, tower foundations must not only withstand alternating tensile and compressive loads but also possess a certain degree of resistance to horizontal loads^{18,19,20,21,22}, which is an area where rock anchors fall short.

Consequently, composite foundations utilizing short piles in the overlying soil and straight rock anchors in the underlying rock strata have been proposed. Studies have shown that this type of foundation can reduce excavation volumes while enhancing uplift load-bearing capacity^{23,24,25,26,27,28,29}. However, for rock foundations with thinner overlying soil layers, issues such as difficult hole formation and low horizontal resistance still persist. In recent years, the application and research on pile-anchor composite structures have primarily focused on slope support^{30,31,32,33,34,35}. Wang et al.³⁶ employed numerical simulation to investigate the uplift bearing capacity of inclined anchor-short pile composite foundation. The findings suggested that an increase in the inclination angle and the number of rock anchors could markedly elevate the uplift bearing capacity of the foundation. Nevertheless, this type of foundation, with inclined anchors placed at the bottom of short piles, resembles traditional pile-anchor composite foundations, potentially limiting the effectiveness of the bottom-located anchors in resisting horizontal loads.

Rock-socketed short pile anchored by inclined anchors (RPIA) is a novel foundation form that has been presented specifically for thin overburden rock foundations in mountainous transmission line engineering, consisting of a rock-socketed short pile and three inclined rock anchors³⁷. Under the action of combined uplift and horizontal loads, RPIA transfers loads into the rock mass effectively through the side resistance of the short pile and the inclined anchors³⁸. Structurally, the RPIA incorporates three inclined rock anchors connected via anchor rebars to the short pile shaft. In contrast to conventional pile-anchor composite foundations employing vertical anchors installed at the bottom, this configuration enables load transfer to the surrounding geological bodies through the inclined anchors even before the short pile undergoes significant deformation relative to the ground. Furthermore, the anchors, inclined at a defined angle to the vertical, simultaneously enhance uplift resistance and restrain pile rotation, thereby providing substantial horizontal resistance. Regarding construction, the elimination of deep anchor boreholes with RPIA significantly reduces excavation volumes in hard rock formations. This translates to a fundamental shift in equipment requirements: construction no longer necessitates heavy-duty drilling rigs equipped with high-strength drill bits. Instead, the foundation can be installed using manually excavated shafts combined with light equipment. This approach partially resolves the persistent challenge of equipment access for tower foundation construction in mountainous areas. Consequently, RPIA offers superior cost-effectiveness and enhanced construction safety compared to conventional pile-anchor composite foundations. Peng et al.³⁹ and Xiang et al.⁴⁰ conducted numerical simulations to study the uplift and horizontal bearing capacities of RPIA. The results showed that compared with short piles of the same size, RPIA had significantly higher resistance. Due to the effect of inclined anchors, the stress concentration inside the short piles of RPIA was obviously improved, and the stress range in the rock mass was also greatly increased. Tian et al.^41,42 and Gao et al.⁴³ conducted model tests and simulation studies focusing on the performance of the pile-anchor connection in RPIA. The analysis, which examined the stress in short piles and crack distribution during failure using two connection methods—bent hooks and anchor plates—revealed that the anchor plate connection offered more pronounced advantages for design and bearing capacity of RPIA.

Currently, researches on RPIA predominantly focus on single uplift and horizontal loading conditions. Although some insights into the mechanical performance of RPIA connection nodes and force transmission mechanisms have been offered, the loads borne by RPIA are mostly composite loads in both vertical and horizontal directions in practical applications, with uplift-horizontal combined loading serving as the controlling condition.

This study has developed a nonlinear numerical model for RPIA and verified its accuracy using data from full-scale field tests. Based on the validated model, the axial force distribution of short piles and inclined anchors, as well as the resultant force and displacement distribution within the geological bodies were investigated to study the bearing performance and failure mode of RPIA under uplift-horizontal combined load. A parametric analysis considering the size of inclined anchors was conducted, and design suggestions were proposed according to analyzed results.

Finite element modeling

Establishment of model

The finite element (FE) software ABAQUS was utilized for the establishment and analysis of the numerical model for the RPIA. Considering the significant size differences between the short piles and inclined anchors, primarily in terms of cross-sectional dimensions and aspect ratios, 20-node three-dimensional solid elements and 2-node one-dimensional beam elements were employed to simulate the short pile and concrete of inclined anchors, respectively. Among them, the solid elements with up to 20 nodes not only simulate the elastic–plastic deformation of concrete but also prevent inaccuracies arising from shear locking due to the inclusion of additional nodes. Meanwhile, the beam elements based on Timoshenko beam theory can sustain tensile, compressive, and bending stresses simultaneously. The geological bodies, including soil and rock mass were modeled using 8-node three-dimensional solid elements, while the rebars in the short pile were represented by one-dimensional beam elements. The anchor rebars were modeled using reinforced fiber elements. These elements can share nodes with beam elements. Fundamentally, it operates by directly enhancing the stiffness matrix of the beam element through its assigned material properties and cross-sectional dimensions, thereby achieving an equivalent representation of the anchor reinforcement.

Since the inclined anchors were modeled using beam elements combined with reinforced fiber elements, their nodes must coincide with the nodes of the geological bodies elements to establish contact interactions. Simultaneously, the nodes at the other end of anchor rebars must also coincide with the nodes of the short pile elements to ensure adequate anchorage performance. Consequently, during mesh generation, the size of the solid elements was controlled based on the inclination angle of the anchors. Specifically, the ratio of the element dimensions in the transverse direction to those in the axial direction was set equal to the tangent of the anchor inclination angle. Considering the balance between computational efficiency and accuracy, the mesh size along the axial direction was determined to be 200 mm. Trial calculations confirmed that this level of mesh refinement was sufficient to achieve effective convergence in the finite element analysis. Further increasing the mesh density would significantly reduce computational speed without substantially altering the results. This occurs because each halving of the mesh size approximately doubles the total number of elements in the model, while the number of nodes requiring computation increases by a factor of 4 to 8 (depending on the number of nodes per solid element), which significantly increases the computational load per iteration, severely degrading computational efficiency. Additionally, since the 200 mm element size already adequately captures the model’s nonlinear characteristics and avoids element distortion and stress concentrations that would severely compromise the results, further reducing the element size yields only marginal benefits: while it may slightly smooth stress variations, it cannot significantly impact the overall analysis outcomes.

The established numerical model is illustrated in Fig. 1.

Fig. 1

Numerical model of RPIA (a) Whole model. (b) Model of each part in RPIA.

Material property

The material property is of paramount importance to the accuracy of FE analysis. In the FE model, the nonlinear deformation of geological bodies, including soil and rock mass, was considered using the Mohr-Coulomb (M-C) model. Concrete for both the short pile and inclined anchors were Drucker-Prager (D-P) yield criterion combined with the hardening-softening-dilation (HSD) function, respectively^44,45.

For the M-C model, the yield surface expression is shown in Eq. (1), with the required input parameters being cohesion c and internal friction angle φ of rock mass and soil, adopted based on mechanical parameters obtained from field tests; for D-P model, the yield surface expression is given by Eq. (2), requiring parameters of concrete compressive strength f_c and internal friction angle φ_c, where these values were determined according to the Chinese Code for Design of Concrete Structures (GB 50,010-2010)⁴⁶ and experimental results from Pül et al.⁴⁷, respectively, and since C25 is the most commonly used concrete strength grade in mountain transmission line engineering, f_c is taken as 16.7 MPa and φ_c as 33.6° in this study.

$$f_{MCi} = \frac{{I_{1} }}{3}\sin \varphi + \sqrt {J_{2} } \left( {\cos \theta - \frac{\sin \theta \sin \varphi }{{\sqrt 3 }}} \right) - c\,\cos \varphi$$

(1)

$$f_{DP} = f(K)\sqrt {J_{2} } + \tan \varphi_{c} \frac{{I_{1} }}{3} - f_{c} \omega_{c}$$

(2)

where I₁ is the mean principal stress (MPa); J₂ is the second invariant of the deviatoric Cauchy stress tensor; θ stands for Lode’s angle (°); f (K) is an indirect expression of Lode’s angle that combine the second and third invariants of deviatoric stress; ω_c is the HSD softening function.

The HSD function employed is given by Eqs. (3) to (5):

$$\omega_{c} = 0.2 + 0.8\sqrt {2\frac{\varepsilon }{{\varepsilon_{cm} }} - \frac{{\varepsilon^{2} }}{{\varepsilon_{cm}^{2} }},\varepsilon_{f} < \varepsilon \le \varepsilon_{cm} }$$

(3)

$$\omega_{c} = 1 - 1.05\left( {\frac{{\varepsilon - \varepsilon_{cm} }}{{\varepsilon_{cu} - \varepsilon_{cm} }}} \right)^{2} ,\;\varepsilon_{cm} < \varepsilon \le \varepsilon_{cu}$$

(4)

$$\omega_{c} = 0.25[1 + \exp ( - 4\frac{{\varepsilon - \varepsilon_{cu} }}{{\varepsilon_{cu} - \varepsilon_{cm} }})], \, \varepsilon_{cu} < \varepsilon$$

(5)

where ε_f, ε_cm, ε_cu represent the strains corresponding to concrete reaching the elastic limit, peak stress, and residual strength, respectively, which are taken as 0.59 × 10^–3, 1.56 × 10^–3, and 4.05 × 10^–3 according to GB 50,010-2010⁴⁶.

For all steel within the model, including the longitudinal and transverse rebars in the short piles, as well as the anchor rebars in the inclined anchors, a bilinear elastoplastic model was employed to represent their behavior. This model assumed that steel entered a state of perfect plasticity once the tensile stress exceeded its yield strength, exhibiting zero stiffness beyond this point. All steel had a yield strength of 400 MPa.

Contact and Interaction

During the loading process, intricate interactions occur among the geological bodies, short piles, and inclined anchors of RPIA, which were meticulously addressed in the FE model through the implementation of several techniques.

The most critical contact interactions in the model are between the concrete outside surface of the short pile and the geological bodies, and between the concrete outer surface of the inclined anchors and the geological bodies. For the pile- geological bodies interface (including pile- rock mass and pile- soil), normal behavior was governed by surface-to-surface contact elements with augmented Lagrangian formulation to prevent penetration and accommodate compression.

The normal contact stiffness coefficient was used to control the normal penetration at the contact surface, essentially serving as a multiplier for the stiffness determined by material parameters and element thickness. An excessively small value of normal contact stiffness coefficient may cause significant penetration between finite element models, severely compromising the rationality of analysis results; while an overly large value leads to an ill-conditioned stiffness matrix, making iterative convergence difficult. Therefore, multiple trial calculations are required to obtain the most reasonable value. This study employed the following procedure to adjust the contact stiffness coefficient: first performing a trial calculation using the default value; if excessive penetration occurs between models (exceeding 10% of the solid element thickness), the stiffness coefficient was increased by 50% of its current value; whereas if iterative convergence becomes difficult (failing to converge after over 50 iterations in a single step) with low penetration, the stiffness coefficient was reduced by 50%. This trial procedure was performed multiple times, ultimately determining the normal stiffness coefficient for both pile-rock mass and pile-soil contact elements to be 0.36.

Tangential interaction of interfaces between pile- geological bodies and inclined anchors- geological bodies were simulated via nonlinear spring elements connecting coincident nodes while both tangential stiffness and friction coefficient of the connect elements were specified as 0 to prevent the overestimation of foundation bearing capacity that would arise from double-counting the tangential resistance of the interface. The spring elements employed a hyperbolic stiffness model described by Eq. (6):

$$\tau_{s} = \frac{s}{a + bs}$$

(6)

where τ_s is stress of spring element; a and b are hyperbolic parameters representing reciprocal of initial stiffness and ultimate stress, respectively; s means relative displacement of contact surface. The a values for pile-soil and pile-rock mass springs were determined using algorithms proposed by Randolph et al.⁴⁸ and Carter et al.⁴⁹ based on field-tested geotechnical parameters. The b values were calculated according to the methodology prescribed in the Code for Design of Overhead Transmission Lines (DL/T 5219–2023)⁵⁰ for both soil and rock mass.

Furthermore, since the inclined anchors were modeled using beam elements, their interaction with the geological bodies was simulated via nodal constraints between corresponding nodes. For the normal interaction, displacement coupling was employed to fully constrain the two nodes, simulating the compressive effect. The tangential interaction was similarly modeled using spring elements incorporating a hyperbolic model. The stiffness model parameters for the hyperbolic model were consistent with those used for the pile- geological bodies interaction.

Boundary conditions and convergence criterion

Both uplift and horizontal loads were applied in the form of prescribed displacements. Specifically, the uplift load (T) was applied directly to the exposed longitudinal reinforcement at the top of the short pile, while the horizontal load (H) was applied to the side face of the pile protrusion. All foundations were simultaneously subjected to both uplift and horizontal loads. The bottom of the geological body model was subjected to fully fixed displacement constraints (restrained in all directions), while its external side surfaces were constrained with radial displacement constraints.

The resultant uplift force acting on the foundation was obtained by summing the vertical reaction forces at all nodes on the model bottom. The resultant horizontal force was determined by summing the horizontal reaction force in the direction of the applied horizontal load at all nodes in the bottom and external surfaces.

Convergence of the finite element calculations was controlled using the L₂ norm of the force residuals and the infinity norm of the displacement increments. The convergence tolerance was set to 1%. The stiffness matrix iteration during the solution process employed a combination of the bisection method and the Newton–Raphson method, with a maximum iteration count of 50. Trial testing confirmed that the selected parameters achieved an optimal balance between computational accuracy and convergence speed.

Model validation

The FE model developed was validated by field test data. The test site is located in Banan District, Chongqing City, in close proximity to Tower N21 of the Nanpeng-Gongping 110 kV transmission line project. The overlying soil layer is thin and mainly composed of silty clay, underlying mudstone and sandstone. The physical and mechanical parameters of each stratum are shown in Table 1.

Table 1 Properties of each stratum.

The test RPIAs are illustrated in Figs. 2(a) and 2(b). The RPIA consists of a short pile and three inclined anchors arranged at 120° intervals, labeled as IR-1, IR-2 and IR-3. The short pile was internally reinforced with longitudinal rebar at a reinforcement ratio of 0.6%, and protruded 200 mm above the ground surface for applying horizontal load. The inclined anchors were composed of an anchor rebar and fine-grained concrete, with a reinforcement ratio of 8.6%. The horizontal load direction was perpendicular to IR-2, rendering its side resistance ineffective against the horizontal load. Conversely, IR-1 and IR-3 were subjected to tensile and compressive stresses induced by the horizontal load, respectively.

Fig. 2

Schematic of test specimens (a) Front view. (b) Top view.

The anchor bars were inserted into short pile through connection nodes and being anchored by anchor plates. Based on the stratigraphic distribution of the test site and the mechanical characteristics of the RPIA, a total of three specimens were designed, as shown in Table 2. The specimens have dimensions for the short pile diameter D_p, inclined anchor angle θ, inclined anchor length l_b, and diameter D_b, which are 0.8 m, 30°, 2.5 m, and 110 mm respectively. The depth of the connection nodes of the inclined anchors from the ground surface h_b was fixed at 0.5 m. Among them, RP-1 had a short pile length h_p of 2 m and was embedded 0.6 m into sandstone, while RP-2 and RP-3 had short pile lengths of 1.5 m and were embedded 1 m and 1.4 m into mudstone respectively. Additionally, a short pile specimen SP was set up as a control, with a pile length and diameter of 1.5 m and 0.8 m respectively, embedded in mudstone of 0.9 m.

Table 2 Parameters of test specimens.

Table 2 also presents the calculated hyperbolic parameters for the spring elements at the pile- geological bodies and anchor- geological bodies interfaces, based on the respective foundation parameters. It should be noted that due to the complexity of geological mass distribution patterns and foundation configurations, the hyperbolic parameters for each foundation included: the a-value for pile-soil interaction (a_ps), the b-value for pile-soil interaction (b_ps), the a-value for pile-rock interaction (a_pr), and the b-value for pile-rock interaction (b_pr). Similarly, for inclined anchors, the hyperbolic parameters can be categorized into analogous values: a_as, b_as, a_ar, and b_ar. However, the adopted algorithm dictated that some of these parameters remain constant within the same soil type. Specifically, for the silty clay parameters given in Table 1, all foundations exhibited a_ps = 5380 MPa⁻¹ and b_ps = 25 MPa⁻¹, while a_as and b_as were consistently 204 MPa⁻¹and 25 MPa⁻¹, respectively. It is noteworthy that minor refinements to the hyperbolic parameters were implemented based on discrepancies between simulation and experimental results, given that the established theoretical calculation methods for these parameters may not fully capture the influence of the complex geological environment.

The construction of an RPIA followed a sequence where a pile hole was first excavated, followed by the excavation of anchor holes. After manually digging the pile hole to the depth of h_b, a drilling machine with an inclined drill bit was utilized to create the anchor holes. Subsequently, foam glue was employed to fill these holes and prevent collapse, allowing the pile hole to be further excavated to the designated depth. A rebar cage of the short pile and anchor rebars were inserted, and then concrete was poured, as illustrated in Figs. 3(a) and (b).

Fig. 3

Construction of test RPIAs. (a) Insert of rebar. (b) Pouring of concrete.

The comparison of load–displacement curves derived from field tests and FE analyses is depicted in Fig. 4. Upon comparison, it is evident that the load–displacement curves of the three RPIAs generally align well with the experimental results, albeit with some localized discrepancies. The slope of the elastic phase of the uplift load–displacement curve for RP-3 obtained through FE simulations is slightly smaller than that from field tests, potentially due to the conservative nature of the bond-slip model employed and the possibility that the rock material parameters at the test site are slightly higher than those used in the simulations. The curve for RP-2 in the nonlinear phase is smoother than that obtained from field tests, which can be attributed to the more complex geological conditions at the test site, where rock strength properties are variable and interbedding may exist. In contrast, the geological bodies in the FE model are homogeneous with clear interfaces, and the constitutive relationships adopted are uniformly varying. Table 3 presents a quantitative comparison of errors in ultimate uplift load (T_u) and horizontal load (H_u) obtained from simulations and field tests. It can be observed that the overall errors for the four simulated foundations are relatively small, with the majority being below 5.0%. Only the T_u error for SP reached 6.0%. This can still be attributed to the complexity of the on-site geological environment, which exceeds the homogeneous geological mass considered in the numerical simulation. This complexity is further evidenced by anomalous variations in the slope of the load–displacement curve during the elastic stage, as observed in field tests. Notably, the absolute error for this foundation is merely 50 kN, and the inherently low magnitude of T_u amplifies the percentage error. Despite these differences, the developed FE model is believed to be capable of capturing the deformation characteristics of RPIAs subjected to combined uplift and horizontal loads, rendering it reliable for further analysis.

Fig. 4

Comparison of load–displacement curves obtained by field test and FE analyses (a) RP-1. (b) RP-2. (c) RP-3. (d) SP.

Table 3 Quantitative comparison of simulation errors.

Numerical analysis of RPIA

Axial force of short pile and inclined anchors

The axial force distribution along the depth of the short piles and inclined anchors for tested RPIAs is depicted in Fig. 5, which was obtained by measuring the stresses on the longitudinal rebar and anchor rebar, respectively, followed by calculations. Considering that horizontal loads generate opposing stresses in the tensile and compressive region of the short pile, the axial forces in these two regions were analyzed separately. The pile cross-section was divided by its diameter perpendicular to the line of action of the horizontal load: the side closer to the horizontal load application point was termed the tension region, while the opposite side was termed the compression region. For the short pile, the stress distribution generally conforms to the characteristics of pile foundations bearing combined uplift and horizontal loads. Specifically, the short pile is tensioned overall under the action of uplift loads, while the compressive region experiences compression against the geological bodies, resulting in increased side resistance and a faster decline in axial force. Meanwhile, the axial force in the tensile region remains nearly constant within the soil layer but drops rapidly upon entering the rock mass, as exemplified by the control SP foundation.

Fig. 5

Axial force distributions of test foundations (a) RP-1. (b) RP-2. (c) RP-3. (d) SP.

However, under the influence of inclined anchors, the axial force distribution in the short piles of RPIA exhibits variations, which further depend on the relative relationship between the depth of connection node h_b and the thickness of the overlying soil layer. For RP-1, the axial force distribution in the compressive region of the short pile aligns with that of the SP, while the axial force curve in the tensile region presents an S-shape. The axial force initially decreases, reaching a minimum, then gradually recovers to a level close to that at the pile top, and decreases again upon entering the rock.

The distinctive axial force distribution profile can be explained through the analysis of the overall axial force of the short pile. First, it is noteworthy that the curve depicted in Fig. 5 actually represents the sum of axial forces on the longitudinal rebars. Under the influence of horizontal loads, the tensile region at the upper part of the short pile separates from the soil, resulting in minimal side resistance, causing the overall axial force of the short pile to remain nearly constant. The anchor rebars extending into the short pile share part of the axial force, reducing the force on the longitudinal rebar, as evidenced by a comparison with SP, which only begins to decrease upon failure. Below the connection node, the absence of anchor rebars results in gradually increasing axial force that longitudinal rebars need to bear.

Additionally, the emergence of an S-shaped axial force distribution curve is also related to the magnitude of the horizontal load. When the horizontal load is low, the S-shape curve is not pronounced for either RPIA or short piles. However, as the uplift load increases above 750 kN (corresponding to horizontal loads exceeding 150 kN), the tensile stress induced by the horizontal load in the middle of the longitudinal rebars in the tensile region becomes gradually significant, which also contributes to axial force recovery^51,52.

Furthermore, the rock-socketed segment of the short pile experiences small horizontal load effects, restricting significant lateral deflection, which ensures maintained contact at the short pile-rock interface even under relatively high horizontal loads. Consequently, substantial interface friction persists, reducing overall axial force in the pile and causing reduction in longitudinal rebars axial force.

Upon entering the rock, the side resistance experienced by the short pile increases, causing the overall axial force of the short pile to decrease, and consequently, the axial force on the longitudinal rebar also decreases again. This indicates that although inclined anchors do not significantly affect the overall axial force level of the short pile, they can work in coordination with the longitudinal rebar to share the load.

The axial force in the tension region of the short piles for RP-2 and RP-3 does not exhibit a similar S-shaped distribution. However, a significant increase in the rate of axial force reduction is observed near the inclined anchor connection nodes. This may be attributed to the fact that the connection nodes of RP-2 and RP-3 are located at the soil-rock interface and within the rock mass, respectively. The overall axial force of the short piles decreases due to the significant side resistance provided by the rock mass. Consequently, even without the sharing of anchor bars below the connection nodes, the axial force of the longitudinal rebars continues to decrease.

As illustrated in Fig. 5, the axial force distribution of the inclined anchors is primarily influenced by the uplift load, with an additional influence from the horizontal load. Specifically, all three inclined anchors are in tension, but RP-1, located in the tension region, experiences the highest tensile stress. In contrast, RP-3, situated in the compression region, experiences the lowest tensile stress, while RP-2 falls between the two. This difference is more pronounced when the connecting nodes are located in soil layers, as exemplified by RP-1. Conversely, when the connecting nodes are situated within the rock mass, as exemplified by RP-3, there is little difference in the axial force levels among the three inclined anchors. This phenomenon can be explained by the fact that short pile in the rock mass exhibits minimal horizontal deflection, making it difficult for horizontal loads to be transferred to the inclined anchors. In such cases, the axial force of the inclined anchors is primarily determined by the symmetrical uplift load. Additionally, the rapid decrease in axial force of the inclined anchors within the rock mass indicates that they can efficiently transfer loads to the rock mass through side resistance.

Stress and resultant force distribution in soil and rock mass

Figure 6 depicts the variations in stress distribution and resultant force of geological bodies, including soil and rock masses, obtained through numerical analysis of four test foundations. Specifically, the rose diagrams illustrate the vertical stress σ_z distribution at a specific cross-section at depth h_cal of the geological bodies, positioned 0.3 m away from the center of a short pile. Considering the effect of a horizontal load H applied at a 30° angle, σ_z is divided into σ_z,c and σ_z,t, representing the vertical stresses in the compression regions (120° ~ 300°) and tension regions (0° ~ 115° and 305° ~ 360°), respectively, which are denoted by blue and red in the rose diagrams. Additionally, the curves display the changes in resultant force F_gb at cross-sections of the geological material at different depths as the applied uplift load varies.

Fig. 6

Stress and resultant force distribution in geological bodies of test foundations (a) SP. (b) RP-1. (c) RP-2. (d) RP-3.

For short piles SP, as shown in Fig. 6 (a), the horizontal load increases the normal stress at the contact surface of the compression region, thereby enhancing the side resistance. Consequently, at relatively shallow depths, the vertical stress in the compression region is significantly higher than that in the tension region. However, as the depth increases, the deflection of the short pile tends to decrease, and the side resistance advantage of the compression region gradually diminishes, leading to the fact that the levels of σ_z,c and σ_z,t become comparable near the pile end. Furthermore, the vertical stresses in both compression and tension region peak at 30°, precisely where the horizontal load is applied. This is attributed to the fact that the short pile exerts maximum compressive force on the rock mass at this angle. Conversely, the resultant force at the cross-sections embedded within the rock, as well as its rate of augmentation with increasing load, is markedly higher compared to those in soil. This observation is in harmony with the trends observed in the axial force.

For RPIA, as illustrated in Figs. 6 (b) to (d), it is observable that the vertical stress distribution within the geological bodies does not exhibit a relatively uniform pattern as seen in SP, but rather demonstrates notable abrupt changes in the vicinity of the inclined anchors. Specifically, the vertical stresses at 0° and 120° increase, which correspond to IR-1 in the tension region and IR-2 at the boundary, respectively. This indicates that IR-1 and IR-2 transfer loads to the adjacent geological bodies through compression and friction, thereby enhancing resistance of the foundation. Conversely, at 240°, which corresponds to the inclined anchor IR-3 in the compression region, a decrease in stress is observed. This may be attributed to the fact that the geological bodies in the compression region already experience significant upward vertical stress due to large side resistance, and the deflection of the short pile causes IR-3 to exert a downward stress on the geological bodies, thereby offsetting a portion of the vertical stress.

Additionally, the influence of inclined anchors on the stress distribution within the geological bodies is more pronounced in soil. For instance, at the cross-section of RP-1 with h_cal = 1.4 m, σ_z,t at 0° reaches 51 kPa, which is four times higher than that at 60°. However, in rock, the enhanced vertical stress due to larger side resistance narrows this gap. For RP-1 at the cross-section with h_cal = 1.8 m, the difference in stress between 0° and 60° is reduced to 1.2 times.

Upon observing the variations in the resultant force on cross-sections of geological bodies, it is indicated that the resultant force induced by RPIA at different depths consistently exceeds the influence exerted by SP. This suggests the fact that, unlike SP, which solely relies on side resistance for load transfer, RPIA transmit a greater load to the geological bodies employing both side resistance and inclined anchors, thereby fully exploiting the bearing capacity of geological bodies.

Displacement distribution and failure mode

The displacement in z-direction u_z contours for the test foundations section are presented in Fig. 7. The plane of interest, designated as IR-1, represents a 30° clockwise rotation from the point of horizontal force application, providing a clear visualization of the influence of inclined anchors on the displacement and failure mode of geological bodies. Similar to the short pile, the cross-section of geological body was divided into tension and compression regions based on the direction and location of the horizontal load. The tension region of the geological body, coinciding with the tension region of the short pile on the same side of the diameter perpendicular to the line of action of the horizontal load, was subjected to additional tensile stresses transferred by the inclined anchor IR-1. The opposite side constitutes the compression region, bearing compressive stresses primarily induced by the deflection of the short pile.

Fig. 7

Displacement contour of test foundations (a) SP. (b) RP-1. (c) RP-2. (d) RP-3.

In the context of SP, under combined uplift-horizontal loading, the geotechnical bodies in the near-field zone adjacent to the pile exhibit characteristic vertical uplift displacements. Notably, the compression region develops a wedge-shaped slip surface under horizontal loading, displaying significantly greater vertical displacements compared to the tension region. The tension region manifests a linear slip surface tightly distributed along the pile shaft, with this constrained slip path resulting in limited ground surface heave.

In contrast, while RPIA demonstrates similar displacement characteristics in the compression region to SP, their tension region failure mechanism undergoes substantial modification: The inclined anchors reconstruct slip surface propagation paths through stress transfer, forcing the slip surface to extend outward along the anchor axis. This directional extension not only amplifies displacement magnitudes in the geotechnical bodies above the anchors (inducing pronounced surface uplift) but more critically demonstrates that inclined anchors effectively expand the participation range of geotechnical bodies within the load transfer zone by redirecting potential slip surface trajectories. From the perspective of failure mode evolution, this phenomenon confirms the optimization mechanism of inclined anchors in pile-geotechnical bodies interaction systems, providing critical insights into how engineered structural elements can strategically govern geotechnical failure development to enhance bearing capacity.

Combined with axial force analysis, despite marked differences in slip surface morphology between SP and RPIA, both structures share a fundamentally analogous failure mode: The piles and anchor remain structurally intact, while shear failure dominates in the surrounding geotechnical bodies, ultimately leading to the pull-out of the foundation.

Parameter analysis

Through the analysis of stress and displacement based on experimental foundations, an initial understanding can be gained that the angle, length, cross-sectional dimensions of inclined anchors and depth of connection nodes may potentially impact the bearing capacity of RPIA. Based on the developed numerical model, a parametric analysis of RPIA is conducted to investigate the influence of dimensional parameters of inclined anchors on the load-bearing capacity of RPIA under combined uplift and horizontal loads. The inclined anchors parameters considered and respective levels are listed in Table 4. The diameters of the short piles for all the RPIAs considered are uniform at 1 m, with a length of 2 m extending below the ground surface. The geological bodies comprise a soil overlay of 1 m thickness and underlying rock, both of which have material property parameters consistent with those observed in field tests.

Table 4 Inclined anchors parameters considered and respective levels.

Effect of inclined anchors angle θ

The analysis involved varying the inclined anchor angle to 10°, 20°, 30°, 40°, and 50°, resulting in the uplift and horizontal load–displacement curves presented in Figs. 8 (a) and (b), respectively. Herein, h_b = 1.6 m and 0.4 m correspond to the connection nodes located within the rock mass and overlying soil layers, respectively. The variations of T_u and H_u with θ are depicted in Fig. 8 (c). As θ increases from 10° to 50°, for the RPIAs with the connection node in rock, T_u decreases by 1%, 4%, 5%, and 10%, respectively, while H_u decreases by 4%, 3%, 4%, and 8%, respectively. For the RPIAs with the connection node in soil, T_u decreases by 3%, 6%, 10%, and 17%, and H_u decreases by 1%, 2%, 4%, and 7%, respectively.

Fig. 8

Variations of load–displacement and ultimate load with θ (a) Uplift (b) Horizontal (c) Variations of ultimate load.

It is evident that, regardless of the connection node location, the ultimate uplift and horizontal bearing capacities of the RPIA decrease with increasing θ, which may be attributed to the fact that the failure of the foundation is primarily controlled by the uplift load, and a smaller θ can provide greater vertical resistance. Furthermore, when θ is less than 40°, its impact on the ultimate bearing capacity of RPIA is relatively minor, while as θ increases from 40° to 50°, the bearing capacity of RPIA decreases significantly.

The axial force curves in the tension region and stress contours of pile-anchors, as well as axial force curves of inclined anchor IR-1 for RPIAs with inclined anchor angles θ of 10° and 50° are depicted in Fig. 9. An S-shaped axial force distribution was identified in both foundations. However, for the foundation with θ = 50°, as the uplift load exceeded 1000 kN (corresponding to a horizontal load exceeding 200 kN), the curve gradually transformed from an S-shape to a pattern of initial increase followed by decrease, while for the foundation with θ = 10°, the axial force curve remained S-shaped until the uplift load reached 1500 kN (corresponding to a horizontal load of 300 kN), with an overall lower level of axial force. This indicates that anchor rebars entering short pile at a small angle are able to share more axial stress, which can also be substantiated in the axial force curve of anchor rebar IR-1. When the inclined anchor angle is excessively large, as the horizontal load increases, the axial stress shared by the anchor rebars becomes less than that induced by the horizontal load, leading to the disappearance of the S-shaped curve.

Fig. 9

Comparison of axial force curves and axial stress contours of RPIA with θ of 10° and 50° (a) h_b = 0.4 m, θ = 10°. (b) h_b = 0.4 m, θ = 50°.

Furthermore, the axial stress (σ_a) contour plot of the RPIA shown in Fig. 9 indicates a discontinuity in axial stress between the short pile and the inclined anchor near the connection node, where the concrete of the short pile is under compression while the inclined anchor is in tension. This is because the anchor rebars inside the inclined anchor enter a certain distance into the short pile to ensure a stable anchorage. When the foundation is subjected to uplift and horizontal combined loads (particularly when the uplift load significantly exceeds the horizontal load), the inclined anchor mainly bears the tensile stress transmitted by the anchor rebars inside it, resulting in tensile stress in the axial direction.

On the other hand, under uplift load, the short pile displaces relative to the surrounding geological body and resists this displacement mainly through side resistance. Meanwhile, the inclined anchor, anchored in the geological body at an angle, has its vertical movement effectively restricted by the normal constraint of the soil and rock mass. This restraint exerts an opposing force on the short pile that tends to move uplift, creating a localized region of compressive stress concentration at the connection node. And since the inclined anchor is modeled using line elements, its connection to the short pile is represented as a single node. The modeling simplification amplifies the compressive stress concentration in the pile at the connection nodes.

Effect of inclined anchors length l _b

The load–displacement curves and variations in ultimate load of RPIA with different inclined anchor lengths l_b are presented in Fig. 10. It is evident that the influence of l_b on the bearing capacity of RPIA has a notable relationship with the depth of the connecting node h_b. When h_b = 1.6 m, indicating that the connecting node is located within the rock, both the tensile ultimate load T_u and horizontal ultimate load H_u of the RPIA initially increase with the increase in l_b. Specifically, as l_b increases from 2 to 4 m and 6 m, T_u increases by 26% and 35%, respectively, while H_u increases by 16% and 21%. However, further increasing l_b to 8 m and 10 m results in negligible change in both ultimate loads and the load–displacement curves.

Fig. 10

Variations of load–displacement and ultimate load with l_b. (a) Uplift. (b) Horizontal. (c) Variations of ultimate load.

Conversely, when h_b = 0.4 m, implying that the connecting node is situated in soil, a significant increase in ultimate loads is observed as l_b increases from 2 to 8 m. In particular, T_u increases by 41.6%, 81.8%, and 114.8%, respectively, while H_u increases by 32.4%, 70.2%, and 101.5%, respectively. Additionally, when the inclined anchor length is less than 6 m, RPIAs with connecting nodes in rock exhibit higher bearing capacities compared to those in soil. This trend reverses when the inclined anchor length exceeds 6 m. The aforementioned observations suggest that when relatively long inclined anchors are utilized, positioning the connecting node in soil or at a shallow depth appears to more fully exploit its bearing capacity.

The axial force distribution curves of the longitudinal rebar in the tensile region of short piles and anchor rebar, alongside the vertical stress contour of the pile-anchor structure for RPIAs featuring a 0.4 m-deep connection node and inclined anchor lengths of 2 m and 10 m, are depicted in Fig. 11. An S-shaped curve remains discernible in the axial force curves. Nevertheless, for the RPIA equipped with a 2 m inclined anchor, the S-shape disappears when the uplift load surpasses 1500 kN, with the axial force of the longitudinal rebar monotonically increasing within the soil. This signifies that the capacity of anchor rebar to assume tensile stress in the tensile region of the short pile has attained its maximum.

Fig. 11

Comparison of axial force curves and axial stress contours of RPIA with l_b of 2 m and 10 m (a)h_b = 0.4 m, l_b = 2 m. (b) h_b = 0.4 m, l_b = 10 m.

Conversely, for the foundation sporting a 10 m inclined anchor, the S-shape remains apparent in the axial force curve of the longitudinal rebar at an uplift load of 3600 kN, and the axial force distribution curve of the anchor rebar indicates that the axial force at the top of inclined anchor attains 800 kN, exceeding five times that of a 4 m inclined anchor. The comparisons underscore that the elongated anchor rebar can be more securely anchored within the rock mass, bolstering its pullout resistance and enabling it to shoulder a greater load. The axial stress distribution of the pile-anchor structure further elucidates that when the RPIA with a 10 m inclined anchor endures an uplift load of 3600 kN (corresponding to a horizontal load of 720 kN), the compressive stress in the compression region of the short pile aligns with that of an RPIA featuring a 4 m inclined anchor, which suggests the effectiveness of the longer inclined anchor in restraining the effect of horizontal load on the short pile.

Figure 12 provides a comparison of the variations in axial force of anchor rebar IR-1 within RPIA, featuring connection nodes situated at depths of 0.4 m and 1.6 m, respectively, under the uplift load of 2700 kN. Specifically, when the connection node is positioned at a depth of 1.6 m, minimal discrepancies are observed in the axial force of the anchor rebar within 4 m depth for RPIA with inclined anchors length 10 m and 6 m. Despite the extension in the length of the inclined anchors, there is hardly any alteration in the load transferred to them, indicating that the short piles are incapable of transmitting additional load to the inclined anchors under the given load and therefore augmenting the length of the inclined anchors does not aid in enhancing the bearing capacity of the foundation.

Fig. 12

Comparison of the variations in axial force of IR-1 (a) h_b = 0.4 m. (b) h_b = 1.6 m.

Conversely, when the connection node is positioned within the soil, as the length of the inclined anchors increases from 6 to 10 m, there persists a notable discrepancy in the axial force of the anchor rebar. This suggests that, when the connection node is proximate to the surface, the load is more readily transferred to the inclined anchors, rendering the increase in their length more effective at this situation. Furthermore, the length of the inclined anchors and the depth of the connection node collectively influence the failure mechanism of the RPIAs, resulting in the variations in ultimate load capacity as depicted in Fig. 12, which will be subject to further analysis in the subsequent section.

Effect of connect node depth h _b

Figure 13 demonstrates the variations in load–displacement curves and ultimate loads of RPIA as a function of the depth of the connection node h_b for inclined anchor lengths of 2 m and 10 m. When l_b = 2 m, both the uplift and horizontal load-bearing capacities of RPIA exhibit an upward trend with the increase in h_b. As h_b increases from 0.4 m to 1.6 m, the ultimate uplift load T_u improves by 8.5%, 16.5%, and 24.5%, respectively, accompanied by corresponding enhancements in ultimate horizontal load H_u of 6.7%, 10.8%, and 16.2%. However, when l_b = 10 m, the influence of h_b becomes more complex: the load-bearing capacity of RPIA initially diminishes and subsequently augments with the augmentation of h_b. When the connection node is embedded in soil and situated farther from the surface, it exerts a detrimental effect on the ultimate load-bearing capacity, whereas in rock, the opposite is observed. When h_b increases from 0.4 m to 0.8 m, T_u of the foundation decreases by 15.0% and H_u by 20.5%. Further increasing h_b to 1.2 m results in 26.2% and 33.1% reductions in T_u and H_u respectively. Conversely, when h_b increases from 1.4 m to 1.6 m, T_u and H_u experience increments of 6.6% and 4.0%, respectively.

Fig. 13

Variations of load–displacement curve and ultimate load with h_b (a) Uplift. (b) Horizontal. (c) Variations of ultimate load.

The variation of depth-horizontal displacement curves for the short piles and the vertical stress contours of longitudinal rebar with h_b under the same load level for RPIAs when l_b are 2 m and 10 m are shown in Figs. 14 (a) and (b) respectively, with uplift loads of 1750 kN and 3000 kN. The maximum stresses in the longitudinal rebar are marked with red arrows, accompanied by their depths from the ground surface. When l_b = 2 m, short pile of the foundation with h_b = 0.4 m exhibits the largest horizontal displacement, approaching 9 mm, which gradually decreases as h_b increases, with nearly identical deflections observed at h_b = 1.2 m and 1.6 m. The maximum stresses in the longitudinal rebar primarily occur near the soil-rock mass interface, i.e., at depths of 0.9 m to 1.1 m from the ground surface. Both the magnitude and depth of the maximum stress initially decrease and then increase with increment of h_b.

Fig. 14

Variations in the horizontal displacement distribution of short piles and the stress contour of longitudinal rebar with h_b for RPIAs (a) l_b = 2 m. (b) l_b = 10 m.

In conjunction with Fig. 11, it can be observed that when h_b is 0.4 m, the length of the inclined anchor socketed into the rock is minimal, resulting in smaller horizontal resistance and weaker restraint on the horizontal deflection of the short pile, thereby causing larger horizontal displacement of the short pile. As h_b increases to 0.8 m, both the depth of the inclined anchor socketing into the rock and the restraint on the short pile increase, reducing the lateral deflection of the short pile while also sharing more stress in the longitudinal rebar. However, when h_b further increases to 1.2 m and 1.6 m, the lateral displacement of the short pile decreases further, but this reduction is primarily indirectly caused by the restraint on the horizontal displacement due to the uplift load, because the significant increase in tensile stress in the longitudinal rebar indicates a substantial reduction in the contribution of the inclined anchor to the horizontal resistance of the RPIA.

For RPIAs with l_b = 10 m, even the inclined anchors with connecting nodes near the ground surface possess considerable socket depth into the rock. The side resistance of the inclined anchors provides sufficient resistance to constrain the horizontal displacement of the short piles. When h_b = 0.4 m, horizontal displacement at the short pile top is only 5.3 mm, with the inclined anchors achieving the best effect in reducing the horizontal deflection. This is because the inclined anchors are farthest from the rotational center of the short pile, enabling them to provide a larger counter-moment. Consequently, the maximum tensile stress level and depth of the longitudinal rebar are also relatively small. As h_b increases, the distance between the connecting node and the rotational center decreases, resulting in a smaller counter-moment provided by the inclined anchors. When h_b increases to 0.8 m and 1.6 m, the horizontal displacement at the top of the short piles reaches 20 mm to 30 mm, and the stress in the longitudinal rebar in the tension region reaches the yield strength, significantly reducing the ability of the short piles to resist and transmit loads.

The difference suggests that the foundation attains adequate uplift resistance when l_b is large, resulting in a shift of the failure mode of RPIA from pull-out failure induced by uplift loads to longitudinal rebar yielding in the short pile section above the connection node due to horizontal loads. This explains the phenomenon observed in Fig. 10(c), where the ultimate bearing capacity of the foundation with h_b = 0.4 m exceeds that of h_b = 1.6 m when l_b exceeds 6 m.

Effect of inclined anchor diameter D _b

Figure 15 compares the load–displacement curves and ultimate loads of RPIAs when the diameter of the inclined anchors D_b is incremented from 90 to 110 mm, 130 mm, 150 mm, and 170 mm, with the reinforcement ratio of the inclined anchor held constant at 16%. For foundations with h_b = 0.4 m, T_u increases by 7.4%, 14.4%, 23.5%, and 33.7%, respectively, while H_u augments by 6.0%, 10.7%, 21.5%, and 29.2%. Similarly, for foundations with h_b = 1.6 m, T_u rises by 7.4%, 18.2%, 28.4%, and 38.5%, respectively, and H_u enhances by 6.9%, 14.8%, 23.0%, and 29.6%. However, no instance is observed where the bearing capacity of the foundation with h_b = 0.4 m surpasses that of 1.6 m subsequent to D_b exceeding a particular value. This indicates that an augmentation in D_b positively contributes to the enhancement of the foundation bearing capacity, albeit its enhancement mechanism might diverge from that of the inclined anchor length l_b. Consequently, a comprehensive analysis is undertaken to delve deeper into the deflection and reinforcement stress of RPIAs featuring diverse D_b values.

Fig. 15

Variations of load–displacement curve and ultimate load with D_b (a) Uplift. (b) Horizontal. (c) Variations of ultimate load.

Figure 16 presents a comparison of the horizontal displacement and longitudinal rebar stress contour plots for short piles of RPIAs with different D_b under applied uplift load of 1500 kN. When h_b is 0.4 m, an increase in D_b from 90 to 170 mm leads to reduction in pile deflection and tensile stress in the longitudinal rebar. This observation is analogous to the changes observed in Fig. 16 when l_b increases from 2 to 10 m, as the enlargement of D_b similarly enhances the side resistance of the inclined anchors, thereby increasing their restraint on the horizontal rotation of the short pile. However, even when the influence of the inclined anchors is disregarded, the maximum tensile stress in the longitudinal reinforcement remains around 150 MPa, suggesting that foundation failure is primarily caused by the uplift load, and the increase in D_b did not change the failure mode of the RPIA. Consequently, for foundations with h_b = 1.6 m, the increase in D_b results in a higher enhancement of the lateral resistance of the inclined anchors, providing the foundation with greater resistance to uplift load. This indicates that when the inclined anchors are relatively short, locating the connection nodes within the rock can enhance the efficiency of increasing D_b in boosting ultimate load capacity.

Fig. 16

Variations in the horizontal displacement distribution of short piles and the stress contour of longitudinal rebar with D_b for RPIAs.

Discussion

Comparison of RPIA with conventional pile-anchor composite foundations

In the current body of research, resisting slope sliding represents the primary application scenario for pile-anchor composite structures. For scenarios primarily subjected to vertical loading, pile-anchor composite foundations with anchors positioned at the pile bottom constitute the more common configuration. Findings from the literature^{25,26,27,28,29} indicate that due to the significant embedment depth of anchors in this configuration, they contribute minimally to the lateral resistance of the upper pile section. Concerning uplift resistance, the side resistance at the anchor-rock interface only becomes mobilized and provides additional uplift capacity after substantial relative displacement has occurred between the upper pile and the surrounding geotechnical material.

The behavior stands in distinct contrast to RPIA investigated in this study. Results in this study demonstrate that the incorporation of inclined anchors not only provides supplementary lateral resistance to the short pile but also enables the earlier mobilization of side resistance at the anchor-rock interface to contribute to uplift resistance.

Design recommendations

The role of inclined anchors in RPIA under combined uplift-horizontal loads can be categorized into three parts, leading to several design recommendations.

Firstly, vertically, inclined anchors can provide additional uplift resistance to short piles through side friction, preventing their extraction. The unique structure of transmission towers necessitates that the uplift load acting on RPIA is five times that of the horizontal load, prioritizing the need for uplift resistance of the foundation. This makes maintaining a small angle for inclined anchors a suitable choice. The increment in anchor diameter and depth of connection point should be achieved when the length of inclined anchors is constrained, while the inclination angle should be further reduced. Secondly, in the horizontal direction, inclined anchors can restrict the deflection of short piles, thereby reducing the stress on inclined anchors in the tensile region of short piles. This effect is more pronounced when the connection node is closer to the pile top and the angle of inclined anchors is larger. Thirdly, anchor bars extending into short piles can share part of the tensile stress of longitudinal bars, enhancing the overall strength of short piles to a certain extent.

A notable contradiction arises from the above analysis, as the functional requirements for the angle and connection node position of inclined anchors in the uplift and horizontal directions are opposite. However, with appropriate design, the effectiveness in both directions can be maximized.

Firstly, inclined anchors must possess sufficient contact area with rock mass to achieve higher side friction, which can be achieved by increasing their length and diameter. According to parametric analysis results, in the stratigraphic layers considered in this study, side friction loss in soil can be neglected when the length of inclined anchors exceeds 6 m, and a larger diameter can further shorten this length. Secondly, a smaller angle of inclined anchors can enhance the vertical component of side friction and achieve greater rock-socketing depth given a fixed length, but an excessively small angle can compromise horizontal resistance. A recommended range of 20° to 30° is suggested, with adjustments needed based on actual stratigraphic conditions. Thirdly, provided that inclined anchors possess sufficient side friction to meet uplift resistance requirements, the connection node should be positioned as close to the pile top as possible to maximize restriction on short pile deflection.

Limits and future perspectives

While this study provides insights into the bearing performance of the RPIA under combined uplift and horizontal loading in shallow overburden foundations and elucidates the influence of inclined anchor design parameters on its performance, the research also reveals a significant limitation. Actual geological conditions encountered in engineering practice are vastly more complex than the homogeneous rock and soil media considered herein. The pervasive presence of joints, weathering, faults, and potential structural discontinuities within shallow rock strata may significantly influence the bearing performance of the RPIA. Investigating the impact of these geological complexities constitutes a critical focus for subsequent research.

Furthermore, the current investigation into inclined anchor design parameters was limited to an isolated single-factor approach. However, potential interactions likely exist between different parameters. Employing structured experimental designs, such as orthogonal arrays, to elucidate synergistic or antagonistic effects among key parameters represents an essential direction for future research.

Conclusion

This study presents a numerical analysis of RPIA based on field experiments. The FE model was established and validated through full-scale field tests. Based on the validated model, the bearing performance of RPIA was analyzed and compared with that of the short pile. Parametric analysis was conducted considering the length, diameter, and angle of the inclined anchors, as well as the depth of the connection node. Design recommendations for the RPIA were then proposed based on the analysis results. The main conclusions drawn are as follows:

(1) Under similar geological conditions and dimensions, the ultimate load of the RPIA is significantly higher than that of the short pile, with a higher deformation stiffness. Anchor rebars extending into short piles can share part of the tensile stress in the tension region of longitudinal rebars, resulting in an S-shaped stress distribution of longitudinal rebars when the connection node was set in the soil.

(2) Results of resultant force and displacement analysis of geological bodies show that inclined anchors can transfer more loads to the geological mass through side friction and normal compression, increasing the stress level of the geological mass and thus enhancing the bearing capacity of the foundation. The inclined anchors in RPIAs force slip surface to extend outward along the anchor axis, thereby effectively expand the participation range of geotechnical bodies.

(3) Results of parametric analysis yielded design recommendations. A smaller angle of inclined anchors helps improve the uplift resistance of RPIA. An increase in the length of the inclined anchor shifts the failure mode of the RPIA from the pull-out of short pile to the yielding of the longitudinal rebar. It is advisable to appropriately increase the yield strength of the rebar or the reinforcement ratio under these conditions. A shallower depth of the connection nodes is appropriate to enhance horizontal resistance. A deeper connection node is necessary to meet the uplift resistance requirements. The conclusions drawn from this study can provide valuable insights for the design and application of RPIA.