Abstract
The presence of karst formations significantly impacts the load-bearing capacity of pile foundations in karst geological environments, posing a challenge to their design. This study investigated the bearing characteristics of karst pile foundations using the physical model test and numerical analysis. First, the influence of cave height and span on the bearing capacity of pile foundations is examined using model tests. The results demonstrate that the height of karst caves greatly affects the bearing capacity of karst pile foundations. Subsequently, numerical analysis further explores the bearing characteristics of these foundations. It reveals that as the top load on pile increases, an arch-shaped tensile damage zone forms at the top of karst cave and gradually expands. The rock failure in this area leads to a decrease in adhesion between rock strata and pile foundation, consequently reducing its load-bearing capacity. Finally, experimental results are compared with numerical results to validate consistency and mutual verifiability between physical model tests and numerical analyses. The outcomes of the research provide valuable insights for designing rock-socketed pile foundations in similar karst areas.
1 Introduction
Karst is a geological and geomorphic phenomenon resulting from the dissolution of soluble carbonate rock, gypsum, and rock salt by groundwater and surface water. This process forms cracks and cavities within rock layers. Karst areas are widely distributed in China, with more in Guangxi, Sichuan, Yunnan, Guizhou, and Hunan and also in Guangdong, Zhejiang, Jiangxi, Jiangsu, Shandong, Shanxi, and Anhui on various scales [1,2,3,4,5].
Underground karst is generally recognized as a geological feature with potential hazards [6,7,8,9]. By identifying possible rock deformation types and analyzing expected structural loads, it is possible to determine the characteristics that may lead to karst formation. It is crucial to assess the hazards associated with identified underground karst features to evaluate their potential impact during the building’s lifespan. For areas covered by rock strata susceptible to karst disasters, comprehensive consideration should be given to the specific characteristics and mechanisms of karst that may arise during facility operation to assess the potential risks involved. Figure 1 illustrates how karst features can be categorized into six distinct groups based on their potential risk level for most buildings and facilities, ranging from high to low risk: A1: landslides or rockfalls; A2: local settlements near buildings; A3: proximity to old sinkholes; A4: foundation differential settlement caused by karstification; A5: gradual soil settlement; and A6: landslides triggered by karst (karst flooding). Karst can also be classified into three subtypes based on specific disaster types: subtype B1 signifies the threat posed by the additional load of deep foundations on karst roofs; subtype B2 entails typical underground karst deformations such as karst caves and fracture zones, within compressible overlying rock. Lastly, subtype B3 involves increased karst water flow toward underground facilities during construction or service life.
Distribution of karst in China and types and subtypes of karst danger: subtype A1 – collapse sinkholes; subtype A2 – the karst hazard resulting from local subsidence; subtype A3 – a karst hazard caused by an old sinkhole on construction; subtype A4 – threat of differential foundation settlement; subtype A5 – karst hazard created by slow subsidence; and subtype A6 – hazards related to karst (karst-suffosion) soil slumps.
Construction of highway bridges in karst areas is complicated due to various geological factors. For instance, the load on the foundation’s upper part damages the cavern’s roof, leading to ground deformation and other problems. The pile foundation can reduce the uneven settlement of buildings and the adverse effects of karst on the foundation, making it widely used in construction in karst areas [10,11,12]. Karst cavity often affects the construction of bridges, leading to insufficient bearing capacity, reduced stability, and buckling failure of piles, which can cause the uneven settlement of bridge piers and surface collapse [13,14,15,16].
When the thickness of the cavity roof cannot meet the minimum safety requirements under ultimate loads, the pile foundation needs to penetrate the cavity to form a “cavity pile foundation” to meet the bearing requirements. Based on the generalized Hoek–Brown criterion, limit analysis method of critical thickness-to-diameter ratio n of karst cave roof, and physical model test, Nie et al. [17] analyzed the influence of karst cave roof thickness on the vertical bearing capacity of rock-socketed piles in karst areas. In the study by Jiang et al. [18], by simplifying the boundary of karst cave roof to support and calculating the simplified model based on plate and shell theory, the theoretical calculation formula for determining the safety thickness of karst cave roof under pile tip under the condition of simply supporting the side was obtained. Currently, research on the bearing mechanism of hole-cross pile foundations is scattered, and determining the vertical bearing capacity of pile foundations is still in the initial stage. Therefore, understanding the vertical bearing mechanism of bridge pile foundations and clarifying the load transfer law of bridge pile foundations in karst areas has become a major focus of scholars’ discussion. Chen et al. [19] studied the vertical bearing characteristics of perforated piles by centrifugal model test and sensitivity theoretical model and found that increasing the height, span, and number of holes of piles all had adverse effects on the vertical ultimate bearing capacity of piles. Kang et al. [20] established a three-dimensional analysis model of cave–rock pile foundation by finite element method and systematically analyzed the influence of different thickness of roof and different size and position of cave on the bearing capacity of bridge foundation under different working conditions. Dong et al. [21] conducted static load tests on the bearing capacity of bridge pile foundations in karst areas, obtained the bearing characteristics of change mechanism under different vertical loads, and proposed an optimization method of rock-socketed pile depth. Huang et al. [22] used numerical methods to study the bearing characteristics of karst cave pile foundation and the stability of beaded karst cave pile foundation.
Many scholars have studied the vertical load transfer behavior of pile foundations in karst areas using numerical analysis and model tests. Liang et al. [23] studied the influence of karst caves in front and below oval piles on the stability of rock-socketed ends of anti-slide piles using horizontal load tests. Song et al. [24,25] used the orthogonal test method and range analysis method to study the influence of material content in model test on the main parameters of simulation demonstration, conducted model test research on the failure characteristics of karst roof under rock-socketed pile with the change of roof thickness and cave diameter, and provided a calculation formula. Huang et al. [26] derived the calculation formula of the minimum safe thickness to prevent rock pillar collapse based on the variation law of cave collapse aspect with cave distance. However, when determining the vertical bearing capacity, most scholars concentrate on studying the bearing capacity or stability of the cavity roof at a specific thickness or calculating the safe thickness based on the design loads. Only a few researchers have considered the presence of pile foundations passing through the cavity, which requires a different approach to determine its carrying capacity. Therefore, current research cannot meet the needs of bridge pile foundation engineering in karst areas [14,27,28].
Hence, it is necessary to carry out experimental research on karst cave pile foundation models. Based on the pile foundation project of the G318 Chizhou section in the karst area, a reasonable model test is designed based on the similarity principle to study the bearing capacity of pile foundations passing through karst caves. The relationship curves between pile top load and pile top settlement under different hole heights and hole spans and the relationship curves between pile end load and pile settlement are obtained through two groups of six single piles passing through the karst cave pile foundation model tests. The experimental results help analyze the bearing characteristics of single piles with various hole heights and spans. Based on the test model, numerical model analysis was carried out to study the bearing capacity transfer mechanism through the karst cave, and the experimental results were verified. The two results are consistent, indicating that the test results are accurate, and the research can provide a reference for similar karst geological pile foundation designs.
2 Model experiment
2.1 Similarity ratio
The model experiments were carried out according to the principle of similarity. The physical quantities considered in the design mainly include geometric size, material strength, elastic modulus, stress, and strain. The pile foundation project of G318 Chizhou Section in karst area is a part of the whole Qiupu River Bridge project. The mechanical parameters of actual limestone materials and piles were obtained through on-site geological exploration and are summarized in Table 1.
Mechanical parameters of the prototype limestone material and pile
| Material | ρ (g/cm3) | σ c (MPa) | E (MPa) | c (kPa) | φ (°) |
|---|---|---|---|---|---|
| Strongly weathered limestone | 2.48 | 28.3 | 3,000 | 800 | 32 |
| Moderately weathered limestone | 2.7 | 88 | 14,000 | 1,200 | 38 |
| Pile | 2.48 | — | 24,000 | 2,060 | — |
Note: γ = unit weight; E = modulus of elasticity; c = cohesion; φ = angle of internal friction; and λ = dilatancy angle.
To ensure the authenticity of the model experiment, the similarity coefficient was determined based on both geometry and volume. The preliminary work of this study included an investigation of the similarity ratio of model tests. Based on research findings, the geometric similarity ratio α D was set to 30 for this experiment, representing the basic geometric size and volumetric weight before the experiment. The density similarity ratio α ρ of similar materials was set to 1.5. When strain and Poisson’s ratio are both 1, the similarity ratios of φ, ε, σ, and c were set at α φ = 1, α ε = 1, α σ = 45, and α c = 45, respectively. These similarity ratios serve as a basis for deriving target parameters for similar materials, particularly limestone and piles. Table 2 lists the parameters of these materials.
Targeted parameters of the similar materials obtained using the corresponding ratio of similitude
| Similar material | ρ (g/cm3) | σ c (MPa) | E (MPa) | c (kPa) | φ (°) |
|---|---|---|---|---|---|
| Strongly weathered limestone | 1.67 | 0.63 | 66.67 | 17.77 | 32 |
| Moderately weathered limestone | 1.8 | 1.95 | 311.11 | 26.67 | 38 |
| Pile | 1.67 | — | 533.33 | 45.7 | — |
Quartz sand is chosen as the aggregate, while cement and gypsum serve as gelling agents. Mineral additives such as red clay, diatomaceous earth, and limestone powder are included. To ensure the physical parameters of similar materials in the model test, extensive matching tests were conducted [29]. Based on these tests, the parameter-fitting ratios for the required similar materials were determined and are summarized in Table 3.
Mechanical parameters of similar materials
| Mix proportion | ρ (g/cm3) | σ c (MPa) | E (MPa) | c (kPa) | φ (°) |
|---|---|---|---|---|---|
| Q:G:C:R:D:L = 66:10:10:4:4:6 | 1.788 | 1.93 | 328.93 | 32.38 | 35.50 |
| Q:G:C:R:D:L = 80:3:3:6:6:2 | 1.763 | 0.62 | 42.07 | 19.91 | 32.03 |
| Q:G:C:R:D:L = 66:12:12:4:4:2 | 1.727 | 2.57 | 448.53 | 38.21 | 36.01 |
2.2 Design of the model experiment scheme
2.2.1 Simplification of the model
In the model experiments, all components of the prototype are scaled down based on the geometric similarity ratio (α D = 30) to account for size effects. It is crucial to consider the influence of size effects when conducting physical model experiments. It has been recognized that pile foundations, when subjected to upper loads in physical model tests, exhibit boundary effects. Typically, the impact range of pile foundations on the rock and soil body under the action of upper load is 5–10 times the pile diameter. Therefore, the affected area is approximately 5–10 times the pile diameter. To address potential engineering issues, the following reasonable assumptions were made:
The karst cavity is simplified to a regular rectangle, with the pile foundation passing through the center of the karst cavity.
The rock layer’s interior is assumed to be uniform and intact, without complex phenomena such as cracks or rock joints, and clear differences between rock layers.
The pile foundation is considered an ideal uniform cylindrical composite structure, with the load at the end of the pile being similar to the load at the top of the pile.
2.2.2 Prefabrication of the model test system
The model test system for the embedded rock pile foundation in karst areas mainly consists of a model box, loading system, and data acquisition system. The experimental system is depicted in Figure 2 and includes a model test device, observation system, data acquisition system, and loading system. The model test device comprises a model test bench, while the data acquisition system consists of displacement gauges, strain gauges, soil pressure boxes, static strain gauges, and post-processing software. The loading system incorporates a jacking reaction loading device and a reaction frame that applies pressure to the model pile.
Experimental system: (a) model system, (b) monitoring system; (c) data processing system, and (d) device connection field diagram.
According to the survey report, the physical and mechanical parameters of the rock and soil mass along the Qiupu River Bridge have been obtained. Based on this information, a physical model of an embedded rock pile foundation in a karst area is established, and the geotechnical parameters of the model are determined. These parameters include an 8 m layer of strongly weathered rocks, an 18 m layer of moderately weathered rocks, and an approximately 20 m layer of weakly weathered rocks below. The proposed model involves using a bored pile with a diameter of 1 m and a length of 18 m, which is not embedded in the weakly weathered rock strata. By conducting similar theoretical and model experiments, the dimensions of the model box can be determined accordingly. The model box is constructed using 20 mm homogeneous acrylic sheets and measures 1.2 m in length, 0.9 m in width, and 0.9 m in height. To ensure the long-term usability of the model box, a partition is placed in the middle of the box, with its width adjustable according to the experimental requirements. The pile foundation model is created using simulated materials that possess physical properties similar to the defined parameters. It has dimensions of 3.3 cm × 60 cm. In simulating the karst rock strata, the optimal mixing ratio of each rock layer is used for stratification and filling. The moderately weathered rock layer, which represents the load-bearing layer of the prototype rock, is positioned at the bottom of the model box and has a thickness of 60 cm. Conversely, the heavily weathered rock layer at the top of the model box is 27 cm thick. The experimental arrangement is depicted in Figure 3.
Experimental arrangement: (a) model of variable height of cave and (b) model of variable span of cave.
2.2.3 Layout and installation of monitoring elements and loading systems
The key parameters measured in the test include pile tip settlement, pile tip resistance, and pile section axial force. As illustrated in Figure 4, the primary equipment utilized in the test comprises strain-measuring instrument, strain gauge, dial indicator, soil pressure box, model box, and a jack reaction loading device. The model box is employed to replicate the karst geological environment, while the strain gauges are used to measure the strain of the pile body. The dial indicator is utilized to measure the settlement at the top of the pile, the earth pressure box serves to quantify the resistance at the pile end, and the jack reaction loading device is responsible for applying the reaction force on the pile top to simulate the applied load from above. The equipment and measurement methods for each parameter are summarized as follows:
Laboratory equipment: (a) strain-measuring instrument, (b) strain gauge, (c) dial indicators, (d) soil pressure box, (e) model box, and (f) jack reaction loading device.
2.2.3.1 Pile top settlement
Pile top settlement is measured directly by a dial indicator (measuring range 30 mm), which is calibrated before the experiment begins. The dial indicator arrangement is shown in Figure 3.
2.2.3.2 Axial force of pile section
The axial force of the pile section is determined through the measurement of strain and the flexural stiffness of the pile. Once the load for each stage reaches stability, the strain data at each point along the pile body are measured using both dynamic and static resistance strain gauges. The strain gauges are symmetrically placed on the front and back sides of each model pile, with a total of 12 gauges evenly distributed, 6 on each side. Each strain gauge has dimensions of 3 mm × 5 mm and is securely attached to the surface of the pile body using AB glue to ensure high durability and waterproofing. All strain gauges are connected via the 1/4 bridge method. During the experiment, after the load for each stage stabilizes, the strain at each measuring point is recorded using the dynamic and static resistance strain gauges. Subsequently, the stress and axial force of the corresponding section are calculated. By measuring the strain values on both sides symmetrically at each section, the average strain and axial force of section i can be obtained using the following equations:
where ε i is the corresponding strain value of section i, ε i1 and ε i2 are the strain values on both symmetric sides of section i, E p is the elastic modulus of pile body, A is the section area of pile, and N i is the axial force of pile body in section i.
2.2.3.3 Pile side resistance
Figure 5 shows the calculation principle diagram of pile side resistance. The axial forces at both ends of pile body elements can be calculated according to equation (2), and the average side resistance of pile body elements can be deduced according to static equilibrium conditions, and the average side resistance of pile side elements can be deduced according to the following equation:
where q si is the side friction of section i pile unit, N i is the axial force at the upper end of section i pile body unit, N i + 1 is the axial force at lower end of section i pile unit, l i is the section i pile unit length, and d is the diameter of the model pile.
Pile lateral resistance calculation diagram.
2.2.3.4 Pile tip resistance
An earth pressure box is installed at the bottom of the pile, allowing for the measurement of the pile tip resistance using the sensor within the earth pressure box. Prior to conducting the model experiment, the earth pressure box is carefully calibrated, ensuring accurate measurements. The stress value obtained through the dynamic and static strain gauge represents the pile tip resistance of the model pile (Figure 6).
Arrangement of strain gauges and earth pressure box.
2.3 Pile top vertical loading device
As shown in Figure 7, weight loading is employed in this experiment. In order to simulate the actual working conditions of concrete bored cast-in-place piles and obtain more accurate and reliable results, the following three requirements must be met during the load test: (1) the pile top surface, which simulates the concrete bored cast-in-place pile, should be leveled, and the centerline of the pile top should align with the centerline of the upper portion of the pile shaft. (2) A square acrylic plate with a thickness of 3 mm should be placed as a cushion on the pile top to ensure a more even distribution of the pressure exerted by the weights, thus avoiding the occurrence of concentrated stress. (3) During the model casting process, the simulated pile should be placed at the designated position and fixed, and the fixing device should be removed once the model material reaches a certain strength.
Weight loading.
Subsequently, following the experimental method of the “slow loading test” as outlined in the “Technical Specification for Building Foundation Pile Testing” (2014-JGJ106) (referred to as the specification hereinafter), a compressive test on the pile foundation is conducted in the model experiment. The test involves gradually applying loads, waiting for the corresponding settlement requirements to be met at each load level before proceeding to the next level, until the model is destroyed. Considering the proportional relationship between the elastic modulus, compressive strength, and other physical quantities of the simulated material used in this experiment, and those obtained from actual engineering measurements, the loading design for this experiment consists of 10 levels of incremental loading, starting from 0 kN with an increment of 2 kN per level. When the settlement at the pile top does not exceed 0.1 mm within every half-hour interval, and the settlement rate reaches a relatively stable criterion, the next level of load is applied. Measurements of the pile top settlement are taken at 5, 15, and 30 min after each load level, and strain data are collected using the strain testing system (Table 4).
Loading load-level design
| Tag (times) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Load class (kN) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
2.4 Model experiment procedures
The experiment was performed according to the following procedure:
Weigh the materials needed for the simulated rock layer according to the weight proportions shown in the material ratio table earlier. Then, add water with a weight of 3/14 of the mixed material weight for mixing.
During the filling process of the simulated rock layer, maintain a thickness of approximately 10 cm per fill. Subsequently, perform compacting and leveling to increase the tightness between the filling materials, ensuring that there are no empty spaces or voids.
Position the cave model in the designated location with holes drilled at the top and bottom of the cave, allowing the pile foundation model to pass through the cave, forming a pile foundation cave model.
Secure the pile foundation model in the correct position using auxiliary fixtures during the model’s pouring. Distribute the filling materials evenly around the pile foundation, and compact them to maintain the relative position of the pile foundation model in a vertical state.
After pouring the model, maintain the same temperature and humidity as the initial material selection experiment and use the same maintenance methods for 14 days before conducting subsequent loading experiments.
After reaching the specified curing time, disassemble the model. Install and set up measurement equipment, including a dial gauge, strain gauge, earth pressure box, and connect them to the measuring instrument, computer, and software.
The slow sustained loading method is used in this experiment. After the experiment is completed, clean up the experimental site and organize the experiment data.
2.5 Data processing and analysis of model experiments
Two groups of six single-pile borehole-type pile foundation models were tested, and the bearing characteristics of different bore heights and under-hole spans of single piles were analyzed. The Q–s curve, P b –s curve, and axial force distribution curve were generated using data from these two experiments. The simulation schemes for these experiments are shown in Table 5.
Program of physical model test simulation
| Height of cave (h/cm) | Cave span (D/cm) | Pile diameter (d/cm) | Distance from the bottom of the cavity to the bottom of the pile/cm | |
|---|---|---|---|---|
| 1 | 10 | 10, 13.3, 20 | 3.3 | 10 |
| 2 | 6.7, 1, 13.3 | 20 | 3.3 | 10 |
2.5.1 Relationship between single pile top load and settlement
From Figure 8, it can be observed that under different cavity heights, the pile top settlement increases almost linearly when the load is relatively small. For instance, at a load of Q = 20 kN, when the cavity height increases from h = 2d to h = 4d, the pile top settlement increases by approximately 13.3%. Similarly, under the same load of Q = 20 kN, when the cavity span increases from three times the pile diameter (D = 3d) to six times the pile diameter (D = 6d), the top settlement increases by approximately 5.0%. Therefore, based on the Q–s experimental curve, it can be deduced that both cavity height and cavity span have an impact on the bearing characteristics of the pile foundation, but the cavity height has a more significant influence.
Q–s curve of monopile pile top: (a) different cave heights and (b) different cave spans.
2.5.2 Relationship between pile end load and settlement
From Figure 9, it can be observed that under relatively small loads, the relationship curve between pile end resistance and pile settlement gradually changes. Under different cavity height conditions, while keeping the load constant, the secondary bearing capacity at the pile end gradually increases with an increase in cavity height. Taking Q = 20 kN pile top load as an example, when the cavity height increases from h = 2d to h = 4d, the secondary bearing capacity at the pile end increases by approximately 12.3%. Similarly, under different cavity span conditions, when the cavity height increases from D = 3d to D = 6d, the load-carrying capacity of the pile components at the pile end increases by approximately 6.5%. Therefore, the P b –s test curve indicates that both cavity height and cavity span have an impact on the secondary bearing capacity of the pile foundation, but the cavity height has a greater influence on both the load-carrying capacity and secondary load-carrying capacity.
P b –s curve of monopile pile tip: (a) different cave heights and (b) different cave spans.
2.5.3 Distribution of axial forces in single pile
Based on the strain values of each section of the pile foundation under single horizontal cavity conditions, the axial forces of each section of the pile are calculated to construct the axial force diagram of the pile foundation under single horizontal cavity conditions. The distribution pattern of pile axial forces is similar under different cavity spans and cavity heights. For analysis, the axial force curve corresponding to the cavity dimensions h × D = 3 m × 6 m is selected, as shown in Figure 10. When the top load of the pile is low, the top load is mainly borne by the pile side resistance, and the top load is reduced faster at the interface between the strongly weathered rock and the moderately weathered rock, but in the karst cave section, the axial force of the pile body is almost not reduced. The pile end resistance is greater than the pile side resistance, and the pile foundation shows the characteristic of friction end-bearing pile in the model test.
Axial force distribution curve of pile body (h × D = 3 m × 6 m).
3 Numerical simulation
3.1 Model size and boundary conditions
Based on the experimental data, numerical simulations were conducted using ABAQUS software to analyze the bearing characteristics of bored piles. The established finite element model is illustrated in Figure 8. The overall dimensions of the model were determined as 15 m in both the X and Y directions and 36 m in the Z direction, effectively eliminating boundary effects. According to the geological conditions of the actual project and the experimental model, the simulated strata in the model consisted of two layers, as shown in Figure 11: an upper layer of heavily weathered tuff with a thickness of 8 m and a middle layer of moderately weathered tuff with a thickness of 28 m. The 8-node hexahedral linear reduction integral element (C3D8R) was used to grid the model. In order to ensure the calculation accuracy, the grid was encrypted in the area where stress concentration might occur, and the area near the cave and the pile side was encrypted. The cavities were simplified into regular rectangles based on the experimental model. Although this model may not fully capture the complexity of the actual cavity shape, it provides a useful approximation for studying the behavior of pile foundations in karst areas. It is worth noting that compared to the curved roof typically found in karst caves, the flat roof of the cubic model is more susceptible to the influence of the overlying load because it lacks the natural reinforcement provided by the arched shape of the cavity roof. Therefore, this model represents a conservative and potentially more hazardous scenario for evaluating the performance of pile foundations in karst areas. These dimensions are consistent with the results of geological investigations of typical karst caves in the study area. To ensure the accuracy of the simulation results, the mesh around the cavities and piles was refined.
Finite element analysis model: (a) meshing and (b) central profile.
The use of appropriate boundary conditions is a necessary prerequisite to ensure the reliability of numerical analysis results. In this study, the model’s top surface allows for free displacement, while the four vertical edges of the model are fixed in their normal directions but can move freely in the vertical direction. The bottom of the model is fixed in all three directions. No restrictions are imposed on the free surface of the cavity.
3.2 Simulation of pile–soil interaction
The interaction between the pile and soil can be divided into two components: normal interaction and tangential interaction. The normal interaction is simulated using a “hard” contact model, while the tangential interaction is simulated using a penalty method. The Coulomb friction model is utilized to describe the relative sliding between the contact surfaces of the pile and soil, with the friction coefficient μ characterizing the friction behavior between these surfaces. Tangential motion remains zero until the surface traction reaches the critical shear stress value τ crit . The critical shear stress depends on the normal contact pressure between the two surfaces, according to the following equation:
3.3 Constitutive model and parameters
The numerical analysis involves three materials: gravel, limestone, and piles. The axial load response of the pile is simulated using a linear elastic model. The Mohr–Coulomb (MC) model has been widely used in geotechnical simulations due to its effectiveness and ease of parameter acquisition, and it is employed to describe the constitutive behavior of gravel and limestone. The MC model is defined as follows:
where τ n is the ultimate shearing strength, c is the cohesion, φ is the friction angle, and σ n is the normal stress on the shearing surface (which is negative in compression).
In the state of plane stress, the MC yield criterion can be expressed as:
where σ 1 and σ 3 are, respectively, the maximum and the minimum principal stresses.
Using the stress invariants, the MC yield criterion is written as:
where I 1 is the first invariant of stress tensor and J 2 is the second invariant of deviatoric stress tensor.
In the process of ABAQUS calculation, the finite element model adopts the continuous and smooth elliptic function as the plastic potential surface to avoid the cusp singularity problem, which may occur at the corners of the calculation model.
The rock layer around the pile and the rock layer at the end of the pile were analyzed using the MC model, and the material of the pile body was analyzed using the linear elastic model. The specific material parameters of the model are shown in Table 6.
Material parameters
| Similar material | γ (kN/m3) | E (MPa) | υ | c (kPa) | φ (°) |
|---|---|---|---|---|---|
| Strongly weathered limestone | 24.8 | 2,000 | 0.3 | 30 | 32 |
| Moderately weathered limestone | 27 | 14,000 | 0.25 | 300 | 38 |
| Pile | 25 | 24,000 | 0.2 | — | — |
Note: υ = Poisson’s ratio.
In order to simulate the influence of pile side friction resistance on the roof stability, the pile–rock contact interface is defined as the contact surface calculated by the model. In the pile–rock contact surface, the rock-socketed pile is the main control surface, and the rock-socketed section’s side rock mass is the subordinate surface, and the interface sliding friction coefficient is 0.4.
3.4 Parameterized analysis plan
Based on physical model tests, two sets of six single pile numerical analysis models are established by considering different cavity heights and spans and analyzing the bearing characteristics of single piles under different heights and spans. The numerical analysis simulation plan can be found in Table 6, which is based on the simulation plan established in the model test (Table 7).
Program of numerical simulation
| Height of cave (h/m) | Cave span (D/m) | Pile diameter (d/m) | Distance from the bottom of the cavity to the bottom of the pile (m) | |
|---|---|---|---|---|
| 1 | 3 | 3, 4, 6 | 1.0 | 3 |
| 2 | 2, 3, 4 | 6 | 1.0 | 3 |
3.5 Load-bearing characteristics of hollow cross pile foundations
Figure 12 shows the stress contour plot of the progressive damage process of the pile foundation under vertical loading when the cavity dimensions are h × D = 3 m × 6 m. From the stress contour plot, it can be observed that for a single-layer cavity pile foundation, when the pile head load is 2,500 kN, a small area of tension damage appears on the top plate of the cavity. With further increase in the applied load (5,000–7,500 kN), the tension in the top plate of the cavity expands, gradually forming an arch-shaped tension damage zone. When the vertical load at the pile head continues to increase to 10,000 kN, the surrounding rock layer of the cavity’s top part fails, and the top of the cavity collapses, forming a detached arch collapse zone. Therefore, in the tension damage zone on the top of the cavity in the pile foundation, the lateral friction resistance generated by the contact between the pile and the rock layer through the single-layer cavity can be considered zero.
Gradual damage process of pile foundation under vertical graded loading (Unit: Pa, h × D = 3 m × 6 m): (a) 2,500 kN, (b) 5,000 kN, (c) 7,500 kN, and (d) 10,000 kN.
Figure 13 illustrates the distribution curve of axial force in the pile for different loads applied at the top and cavity dimensions of h × D = 3 m × 6 m. It can be observed that the axial force along the length of the pile does not change significantly when different loads are applied at the pile top above the highly weathered tuff layer. However, as it enters the moderately weathered tuff layer, the lateral friction resistance experienced by the pile increases, leading to a noticeable decrease in the axial force. Within the cavity of the karst cave, the pile is not affected by the lateral friction resistance from the surrounding rock layers, resulting in almost no change in the axial force within the cavity. The ultimate load transmitted from the pile top to the pile base is relatively small. The axial forces at the pile base under four different pile top loads are 29.2, 24.8, 28.2, and 31.95% of the pile top load, respectively. This is because the pile behaves as end-bearing friction pile. With the increase of pile top load, the pile side friction resistance and the pile end-bearing capacity both increase, but the pile side friction resistance increases faster. When the load continues to increase, the bearing capacity provided by the side friction resistance becomes smaller and the pile end bears greater load.
Pile axial force distribution curve along the pile length.
Figure 14 illustrates the lateral friction resistance of the pile for different pile top load conditions and a cavity size of h × D = 3 m × 6 m. From the graph, it can be observed that the lateral friction resistance of the pile is relatively small at the pile top when the cavity size is h × D = 3 m × 6 m. At a distance of 8 m from the pile top, the lateral friction resistance significantly increases, but decreases to zero when passing through the cavity. As the pile approaches the karst cave, there is a small abrupt change in the lateral friction resistance along the pile circumference. This is attributed to the damaged zone above the cave, where the rock layers are disrupted and the bond between the rock and the pile is weakened, resulting in a sudden decrease in friction resistance.
Distribution of pile lateral frictional resistance along the pile length (h × D = 3 m × 6 m).
In combination with Figures 13 and 14, as well as the model stratum information parameters, it can be observed that the side friction resistance between the upper strongly weathered tuff layer (with a thickness of 8 m) and the pile foundation is relatively low. Consequently, the corresponding section of the pile experiences a higher axial force, which aligns closely with the pile top load. However, as the pile penetrates deeper into the moderately weathered tuff stratum (at depths exceeding 8 m), the lateral friction resistance at the pile foundation significantly increases. This results in a rapid reduction of the axial force in the pile body, while simultaneously amplifying the vertical load sustained by the lateral friction resistance of the pile.
4 Comparison and discussion
The experimental method of “slow method” in the JGJ106-2014 Technical Code for Testing of Building Foundation Piles JGJ106-2014 is used for loading experiments of model experimental pile foundations, i.e., the load is applied first, and after the settlement of the pile foundation under this level of load meets the corresponding requirements, the next level of load is applied until the model is destroyed. However, because of the limitation of loading in this indoor model experiment, the loading did not reach the model destruction. Zhang [30] established the load transfer model of pile foundation in karst area through the load transfer method, which is a theoretical analysis method for the load transfer law of pile foundation. The basic principle is to divide pile into several elastic elements and use nonlinear spring connection between each pile element and soil to simulate the load transfer relationship between pile and soil. The basic equation of load transfer method can be deduced as follows:
where U is the circumference of pile section, and A p and E p are the section area and elastic modulus of pile body, respectively.
Zhang [30] counted and analyzed the p–s curve of pile end in karst area, and the experiment showed that S b of sudden failure ranged from 2 to 5 mm and concentrated in 3–4 mm. The measured pile top settlement and pile top load were normalized, and 182(s/s b , q/q b ) data points were obtained. The data points were plotted in the coordinate system with s/s b as the horizontal axis and q/q b as the vertical axis, and curve fitting was performed on the logarithm data points. The mathematical expression of the model to simulate the relationship between pile top load and pile top settlement of pile foundation in karst area is as follows:
This physical model experiment can fit the Q–s curve of monopile load bearing through cavity type pile foundation based on its mathematical model expression. The technical specification for construction pile testing 2014-JGJ106 section 4.3.7–5 terminates the loading: when the load–settlement curve is slowly varying, it can be loaded to a total settlement of 60–80 mm at the top of the pile, so the load corresponding to a settlement of 80 mm at the top of the pile can be taken as the ultimate bearing capacity of the pile.
The Q–s curves of the pile foundations for different cavity heights and different cavity spans are shown in Figure 12. With the increase of the pile top load, the Q–s curves of pile foundation under different cavity height and cavity span conditions have the same pattern of change, all of them are “slow.” Under the variable cavity height condition, the pile top settlement increases almost linearly when the load is small, and it shows different degrees of non-linearity as the load applied to the pile top increases step by step. Under variable cavity span conditions, the pile top settlement increases almost linearly when the load is small, and it shows the same degree of non-linearity as the load applied to the top of the pile increases in steps, but the variation is small.
As can be seen from Figures 15 and 16 in the numerical model analysis, when the length of the cavity floor from the pile bottom and the span of the cavity are certain, with the increase of the pile top load, the Q–s curve of the pile foundation at each height of the cavity does not vary greatly, and all appear to be “slow.” When the load is small, the settlement of the pile top is approximately linear with the increase of the load; with the increase of the load to a certain limit value, the Q–s curve is successively non-linear to varying degrees, the settlement increases gradually, but the increase is slow. When the length of the cavity floor from the pile bottom and the height of the cavity are certain, with the increase of the pile top load, the Q–s curve of the pile foundation under each cavity span is also not very different, but also presents a “slow type”; when the height of the cavity increases from D = 3d to D = 6d, the ultimate bearing capacity of the pile foundation decreases by 2.0%.
Q–s curves of monopile tops at different cavity heights.
Q–s curves of monopile tops at different cavity spans.
Figure 17 illustrates the comparison between the experimentally obtained Q–s curve-fitting line and the Q–s curve obtained through numerical simulation. The figure clearly demonstrates that the numerical simulation results align well with the test results, thereby validating the reasonableness and scientific nature of the testing scheme and its outcomes. During the numerical model analysis, while keeping the length of the cave floor from the pile bottom and the cave span constant, the Q–s curve of the pile foundation exhibits a “slow-type” behavior as the pile top load increases. Initially, when the load is relatively small, the pile top settlement shows an approximately linear increase. However, as the load reaches a certain threshold, the Q–s curve progressively presents varying degrees of nonlinearity, with settlement increment gradually growing, albeit at a slow pace. Increasing the cave height from h = 2d to h = 4d results in a 10.7% decrease in the ultimate bearing capacity of the pile foundation. Conversely, when maintaining a constant length of the karst cave floor from the pile bottom and the height of the karst cave, the Q–s curve of the pile foundation displays minor differences with increasing pile top load for each karst cave span, all exhibiting a “slow-type” trend. Extending the cave height from D = 3d to D = 6d leads to a 2.0% decrease in the ultimate bearing capacity of the pile foundation.
Ultimate bearing capacity of pile foundation for different: (a) cave height; (b) cave span.
The figure depicts the comparison between the experimental-fitting results and numerical calculation results for the ultimate bearing capacity of a pile foundation under varying cave heights and cave spans. It is evident from the figure that the numerical analysis results closely align with the experimental findings. Any disparities between the two can be attributed to errors arising from differences in experimental conditions, numerical model parameters, and test parameters. Consequently, the accuracy of the test results and conclusions can be verified through the numerical calculations, offering valuable references for real-world project construction.
5 Conclusions
Karst caves affect both the bearing characteristics and the calculation methods of pile foundations, resulting in unique considerations compared to those of standard embedded piles. This research uses the G318 National Highway Qiupuhe Bridge karst pile foundation construction project as the study object to systematically explore the load-bearing characteristics of embedded pile foundations in karst areas through model testing and numerical simulation. The following conclusions are drawn:
The Q–s curve for the pile foundation passing through a single-layer karst cave is obtained using the model test. The pile height and span have an impact on the pile’s bearing characteristics, with the cavity height having a significant influence. Under the maximum load, when the cavity height increases from 2 to 4 times the pile diameter, the pile top settlement increases by 13.3%, and the pile end-bearing capacity increases by 12.3%. When the cavity span increases from three times pile diameter to six times pile diameter, the settlement of pile top and bearing capacity of pile end only increase by 5.0 and 6.5%, respectively.
A finite element model further studies the bearing characteristics of the pile foundation passing through a single-layer karst cave, revealing that with the increase of pile top load, the top of the cave will gradually form a tension damage zone, and eventually, the rock layer at the top of the cave will fail, forming a separate arch collapse zone.
The Q–s curve calculated by the finite element model aligns with the ultimate bearing capacity value of the pile foundation, mutually affirming the results of the model test and numerical analysis. These findings support the conclusion’s validity, providing reference value for designing and researching similar pile foundations.
Acknowledgements
The authors thank the help and guidance from Mr. Jiye Ouyang, Mr. Shi Chen, Mr. Wenyu Shu, and Mr. Shuang Liu.
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Funding information: This research was jointly supported by the National Natural Science Foundation of China (Nos. 52308344, 52374085, 42077249), the Anhui Provincial Natural Science Foundation (2308085QE190), the Hefei Rail Transit Group Co., Ltd. Funded Project (2021BFFBZ02689), the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) (No. KFJJ23-05M), and the Fundamental Research Funds for the Central Universities (JZ2023HGTA0193, JZ2023HGQA0094).
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Author contributions: Conceptualization: P.G. and F.L.; methodology: N.J. and H.Z.; software: P.G. and F.L.; validation: J.L.; investigation: N.J. All authors have read and agreed to the published version of the manuscript.
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Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.
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Ethical approval: The conducted research is not related to either human or animal use.
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Data availability statement: All data generated or analyzed during this study are included in this published article and its supplementary information files.
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