Coral reef limestone is brittle and porous, which makes it prone to forming jagged interfaces during grouting and hole formation. Based on the field scan results from the China-Maldives Friendship Bridge, as shown in Figures 1A, B, the interface shapes are classified into seven types: vertical pile, sloped jagged, stepped jagged, outer arc jagged, outer V-shaped jagged, inner arc jagged, and inner V-shaped jagged. Shear tests are conducted for each type. To ensure sampling quality, coral reef limestone with lower porosity is selected. Fine processing of coral reef limestone, partially prepared into blocks with dimensions of 100 mm × 100 mm × 50 mm, and partially made into irregular rock samples based on the actual dimensions of each working condition. C30 concrete with standard dimensions of 150×150×50 mm is used to simulate the pile foundation and modify the shape of the concrete according to the corresponding irregular rock sample. Three sets of tests are conducted for each interface shape under normal stresses of 100, 200, and 400 kPa. Concrete is cast on coral reef limestone with different interface shapes. The sample sizes and molding effects are illustrated in Table 1. Loading is performed using a DZJ-15 direct shear apparatus (Li et al., 2023). Considering the effect of loading rate on shear test results (Tang and Wong, 2016), the preloading rate for normal stress is set at 0.1 kN/s, and the shear rate is 0.002 mm/s.

According to the typical shear stress-shear displacement curve shown in Figure 7, the exertion of interface shear strength can be divided into five stages: initial stage, compaction stage, bearing stage (elastic deformation stage), failure stage, and friction stage. The characteristics of each stage are presented in Figure 8.

Initial Stage: The high-porosity coral reef limestone bonds with the concrete interface. The cement slurry fills the pores of the coral reef limestone near the interface to form an interface reinforcement with strength greater than that of the coral reef limestone.

Bearing stage: At the beginning of loading, the pores of the coral reef limestone near the interface reinforcement are compacted, and the shear stress increases non-linearly with shear displacement. As shear displacement increases, the compaction of the pores near the interface reinforcement is completed, and shear stress increases linearly at a rate higher than that in the compaction stage. The rock sample, interface reinforcement, and concrete are in an elastic state, in which the bonded elements of the rock sample bear the load.

Failure stage: As loading progresses, cracks start to develop in the weaker areas near the lower part of the loading point. The bonded elements begin to transform into frictional elements, while the coral reef limestone farther from the loading point remains undamaged. This asynchronous damage generates an uneven sliding surface with an upward trend. As shear displacement further increases, the interface strength is gradually controlled by the tensile strength of the coral reef limestone. Once interface strength exceeds the tensile strength, the cracks penetrate, and the shear stress drops rapidly.

Friction stage: As shear displacement continues to increase, the rock sample slid along the penetrated cracks, and the shear stress remains unchanged. At this stage, the frictional elements of the rock sample bear the load (Huang et al., 2024).

In summary, the majority of the failure zone at the pile-coral reef limestone interface was located within the coral reef limestone. Initially, the interface strength was controlled by the shear strength of the coral reef limestone, resulting in the formation of an uneven sliding surface. Later, it was controlled by the tensile strength of the coral reef limestone, leading to the penetrative failure. Overall, the interface strength of the pile-coral reef limestone is the result of the combined effects of the tensile and shear strengths of the coral reef limestone.

Figure 9A shows the physical image of cement slurry filling the pores of reef limestone (Wang et al., 2024). It can be clearly seen from the figure that the gray cement slurry penetrates into the pores of the reef limestone, filling the pores and forming a denser interface. Figure 9B shows the microscopic interface image of the bond between the reef limestone and concrete (Cheng et al., 2022). From the image, it can be seen that the concrete fills the pores of the reef limestone, forming a more tightly embedded interface reinforcement at the interface between the two. Compared with the original porous state, this interface reinforcement significantly improves the stress characteristics of the interface between the reef limestone and concrete. Therefore, most shear failure surfaces do not occur at the interface and are located inside the reef limestone with more pores (Cheng et al., 2024).

The finite-discrete element coupling method (FDEM) that integrates the finite element method based on constitutive relationships and the discrete element method based on contact rules allows for detailed analysis of the whole failure process (Gibson, 1997).

The mechanical analysis program for coral reef limestone is developed based on the constitutive model of coral reef limestone. Firstly, it is necessary to clarify the constitutive model of coral reef limestone and obtain relevant parameter values through relevant experiments. Among them, the porosity is determined as f c b through porosity testing, and the matrix parameters of the disintegration modulus and shear modulus are obtained through bonding strength and uniaxial testing to form the constitutive model of the coral reef limestone in the area where the sample is located. (Zhang et al., 2024c)

To demonstrate the reliability of the rock constitutive model, the stress-strain curves of reef limestone under uniaxial compression were compared and verified with the calculated results of the model, as shown in Figure 15. The calculation parameters are as follows: the porosity of samples I, II, and III is 0.17, 0.20, and 0.18, respectively; The values of K s , G s , K M and G M are shown in Table 3, where K s and G s are determined by the test results of the specimen during the initial loading stage, and K M and G M are determined by the test results of the specimen during the residual loading stage; The porosity value of the friction element is 1.1 times that of the bonding element. As shown in Figure 22, the calculated results are basically consistent with the experimental results, which can reflect the phenomenon of strain softening. After clarifying the key parameter values characterizing porosity and bond strength in the constitutive analysis of coral reef limestone, a mechanical analysis program for coral reef limestone can be established.

K s and G s are the bulk modulus and shear modulus of the reef limestone matrix, respectively; K M and G M are the elastic parameters of the solid soil skeleton in the constitutive model friction element, namely, the bulk modulus and shear modulus.

To validate the reliability of the FDEM-based mechanical analysis method for coral reef limestone (Zhang et al., 2024a), a numerical model is established, as displayed in Figure 14. The model parameters are taken from corresponding test values of coral reef limestone. For the main structural element parameters, the density, friction coefficient, as well as tangential and normal penalty factors are set at 2,150 kg/m³, 0.55, 45 MPa, and 450 MPa, respectively. For the joint element parameters, the tensile strength, fracture energy, cohesion, internal friction coefficient, and fracture penalty factor are at 2.30 MPa, 25 g/s², 130 kPa, 0.65, and 3.5 GPa, respectively. The viscous damping ks is set at 2.9 × 10⁷ kg/(m.s) (Wang et al., 2023; Meng et al., 2024).

The stress-strain relationships at different porosities and bonding strengths are depicted in Figure 15. Figures 15A–C correspond to the results of Sample I, Sample II, and Sample III, respectively. In the early stage of loading, the stress gradually increases and peaks with occurrence of strain softening. In contrast to the measured values and the elastoplastic model for coral reef limestone that accounts for porosity and bonding characteristics, the strain-softening phenomenon is more pronounced in the FDEM-based numerical simulation. The main reason is that the FDEM does not account for the connections between macro and micro scales and suffers from certain defects in describing the prevalent joint load-bearing behavior of bonded and frictional elements. In summary, the FDEM can effectively simulate the effects of high porosity and bonding strength on the mechanical properties of coral reef limestone.

Shear tests on the pile-coral reef limestone interface indicate that the cement slurry fills the pores of the coral reef limestone to an interface reinforcement. This results in most failure surfaces being located within the coral reef limestone, while the interface shape has minimal impact on the side friction resistance. The pile-coral reef limestone interface characteristics can be simulated using either shared nodes or interface reinforcement. Since the latter approach requires consideration of factors such as porosity and the range of the reinforcement, this study employs the shared node approach.

The FDEM for assessing the side friction resistance capacity of the pile in coral reef limestone strata requires determining three key parameters: 1) structural element parameters; 2) joint element parameters; and 3) loading block parameters. Structural and joint element parameters are obtained from laboratory test data of the coral reef limestone. The stiffness of the loading block should be maximized within permissible limits, with Lamé constants set at λ = 4.8 × 10¹² and μ = 8.5 × 10¹², and a friction coefficient of 0.1 to avoid dispersion of the computational domain. Since the loading block is not meshed, parameters such as viscous damping, normal penalty function factor, and tangential penalty function factor are not meaningful and can be selected based on the rules of the structural block. Considering factors such as peak material strength, elastic modulus, Poisson’s ratio, and time step, the fracture penalty function factor is set at five times the elastic modulus, i.e., 10 × 10¹⁰, while the maximum fracture energy is set at 20 g/s². The chosen parameters for various elements are detailed in Table 4.

The numerical simulations are compared with experimental results to verify the reliability of the simulation approach. The interface shear strength tests can be simplified into a two-dimensional problem, and then a two-dimensional model can be established, as presented in Figure 16. The concrete part is 150 mm long and 50 mm high (blue part), while the coral reef limestone measures 100 mm in length and 50 mm in height (green part), with a large stiffness loading block on the left side (red part). The meshing is performed using the meshing tool in the computational program based on solid-4 node 182 elements, with a minimum mesh size of 2 mm.

Through repeated trial calculations, the time step and loading block velocity are determined to be 5 × 10⁻⁶ s and 0.25 m/s, respectively. The analysis focuses on a shear test with a vertical load of 100 kPa to examine the interface failure process during loading. At loading time t = 0 m, the failure status of the model is illustrated in Figure 17, where the joint elements between the structural and concrete elements are undamaged, with no cracks present.

At loading time t = 30 m, the failure status of the model is depicted in Figure 18. In the early loading stage, the stress field of the elements connecting the structural and loading blocks is disturbed, while the joint elements begin to fail (green), but the cracks are not fully developed. The cracks appear, extend, and widen within the joint elements between the structural block and the concrete. The fracture surface begins to penetrate, forming long and narrow gaps that eventually propagate along the contact surface.

At the loading time t=50 m, the failure state of the model is shown in Figure 19. As the loading continues, the stress field of the components connecting the structure and the loading block is disturbed, causing the connecting components to fail and cracks to fully develop. Cracks appear, extend, and widen within the joint elements between structural blocks and concrete. The fracture surface begins to penetrate, forming narrow gaps and eventually propagating along the contact surface.The shape of the crack shows a slight upward trend, indicating that the failure location of the fracture is located inside the reef limestone.

Figure 20 compares the simulation results under various vertical stresses with the shear test results of planar concrete and coral reef limestone. Figures 20A–C correspond to the results of vertical pressures of 100kPa, 200kPa, and 400kPa, respectively. As shear displacement increases, the shear force first increases, then decreases, and finally stabilizes, and the peak shear force rises gradually as vertical stress increases. The reasons for this phenomenon are as follows. As horizontal shear displacement increases, the shear force gradually rises to a peak. However, the generation and propagation of cracks on the contact surface leads to a decrease in shear strength. With increasing vertical stress, the interlocking and embedding effects occur between contact surface elements, causing an increase in friction, which consequently enhances the shear strength. The experimental and simulation curves exhibit similar trends under various vertical stresses, suggesting that the FDEM effectively simulates the side friction resistance characteristics of the concrete-coral reef limestone interface.

The calculation Formula 5 for the allowable axial compressive bearing capacity of single rock socketed drilled piles in the “Code for Design of Highway Bridge and Culvert Foundation”. R a = c 1 A p f r k + u ∑ i = 1 m c 2 i h i f r k i + 1 2 ζ s u ∑ j = 1 n l i q s j (5)

In the formula: R a is the allowable vertical bearing capacity of a single pile (kN), c 1 is the end resistance coefficient determined based on factors such as rock integrity, A p is the cross-sectional area of the pile end (m²), f r k is the standard value of the saturated uniaxial compressive strength of the rock at the pile end (kPa), c 2 i is the lateral resistance coefficient of the i-th layer of rock determined based on factors such as rock integrity and density, u is the circumference of the pile body in each soil layer or rock layer (m), h i is the thickness of the pile embedded in each rock layer (m), m is the number of layers of rock layers, ζ s is the lateral resistance coefficient covering the coral reef soil, l i is the thickness of each coral reef soil layer (m), and q i k is the standard value of lateral resistance of the i-th layer of soil on the pile side (kPa), where n is the number of layers in the soil layer.

After studying the mechanism of lateral resistance in the injection section of coral reef limestone, combined with the FDEM calculation method of coral reef limestone, it is recommended to revise the value of the lateral resistance coefficient c 2 of the rock layer.