Recycling Waste Soils for Stability Enhancement in Bored Pile Construction

Abstract

Instability in the hole wall of bored pile may cause serious environmental problems. Therefore, using the small hole expansion theory and elastic–plastic theory, we studied the instability mechanism of the hole wall of bored pile, determined the stress expansion solution of the soil layer after the excavation of pile holes in the semi-infinite elastic soil layer, and established a mechanical model. Then, the stability of the hole wall of bored pile in the cohesive soil layer and sandy soil layer was analyzed, and a formula for calculating pile hole wall stability was obtained. Finally, the stability of the hole wall of bored pile under the action of mud slurry was calculated, and the stress on the pile hole wall was analyzed when local instability and overall instability occurred, respectively. The results show that in a sandy soil layer, the safety factor of the hole wall of bored pile has no connection with the depth of the pile hole but is related to the density of mud slurry in the pile hole. In unstable soil layers, the pile hole wall was prone to instability, and the stability of the hole wall could be improved by appropriately increasing the gravity of mud slurry. With the increase in pile diameter, the lateral displacement and deformation of the hole wall increase, and the displacement of the soil layer increases when the hole wall is unstable, increasing the possibility of forming variable cross-section piles correspondingly.

1. Introduction

Disturbance of the soil layer on the side wall of a pile borehole and disruption of the original stress balance in the soil layer may cause instability and damage to the pile borehole wall during drilling. This will cause cement, mud, oil, and other construction waste to be included in the soil leakage, which may pollute the groundwater and surrounding soil, causing serious environmental problems.

To avoid these negative effects, scholars worldwide have explored the hole wall stability of bored pile. Based on the SMP criterion, Li et al. [1] gave an elastic–plastic solution for the unloading shrinkage of column holes and an analytical expression of pore wall shrinkage, and they proposed a method for calculating correlations for the minimum mud slurry weight and the pore wall stability coefficient when the pore wall is stable. Li et al. [2] analyzed the stability of borehole walls in different soil layers, established a mathematical analysis model of borehole stability, analyzed the effect of mud slurry borehole walls, and proposed corresponding measures to prevent borehole collapse. Meng et al. [3] established an analysis model of the stability of the side wall of a pile borehole based on the gray correlation theory. Gong et al. [4] studied the stability conditions of the borehole wall, analyzing the influences of parameters such as the density of mud slurry in the borehole, friction angle of the soil layer, and so on. Zhao et al. [5] derived a formula for borehole diameter change and borehole wall stability based on the Kelvin model, considering the spatial–temporal variation rule of the borehole diameter of cast-in-place piles during unloading. Chen et al. [6] studied the influence of the internal friction angle of sand, borehole diameter, borehole depth, groundwater level, and mud protection on the borehole wall stability. Zhang et al. [7] established a stress model for the pile borehole wall, analyzed its collapse mechanism, and deduced its ultimate shear stress formula. Westergard [8] and Hubbert et al. [9] analyzed the stress distribution around the borehole wall and gave a corresponding algorithm. Based on numerical simulation and the strength reduction method, Wang et al. [10] analyzed the influences of the soil layer properties, density of mud slurry, depth of the pile hole, and radius of the pile hole on the stability of the borehole wall of a pile, and they proposed that increasing the density of mud slurry in clay can reduce the shrinkage neck of the pile hole, and that the lateral pressure provided by mud slurry in the sandy soil layer can effectively ensure the stability of a pile borehole wall when it is greater than or equal to the lateral pressure of the soil layer. Hu et al. [11] used the principle of pore-forming mechanics to analyze the stress in the pore-forming process, obtained an expression related to the stress at the edge of the pile borehole wall, and analyzed the damage mechanics during the pore-forming process. Ajmi et al. [12] established an analysis model of borehole wall stability, analyzed its instability and failure, and obtained the critical gravity of mud slurry and the critical borehole depth of a pile borehole when the bore wall was stable. Xu et al. [13] used the limit analysis method to calculate the ultimate borehole depth of drilling with mud protection, and the results showed that the ultimate hole depth of a pile increased with the increase in the specific gravity of mud slurry and the internal friction angle of the soil layer, while it decreased with the increase in the bore diameter. Xiong et al. [14] established a mechanical model for analyzing the stability of a pile hole wall based on the static pressure of mud slurry, and they proposed that the friction angle, cohesion, and specific gravity of mud slurry in the soil layer are the basic elements influencing the stability of the pile borehole wall. Huang et al. [15] proposed a relevant calculation model of the maximum stable borehole depth, considering the circular arch effect of the soil layer and the influence of the aperture. Liu et al. [16] used the sliding theory related to soil mechanics to analyze the interaction mechanism between borehole wall stability and mud slurry in sandy soil. Zhang et al. [17] studied the shrinkage process of the borehole wall and derived an analytical solution for ensuring borehole wall stability based on the pore expansion theory and Mohr–Coulomb theory. Based on the elastic theory, Jiang et al. [18] considered the influence of borehole wall shrinkage and soil cohesive force on the stability of the borehole wall and obtained a relevant calculation method.

From the above studies, it can be seen that many scholars have studied the stability of the pile hole wall and achieved rich results. While much progress has been made, the existing research has focused on the effects of the water table, mud density, borehole diameter, and soil properties. Many established models and methods are limited by their specific assumptions or simplifications [19,20], such as ignoring the dynamic interaction between the soil layer and pile wall during drilling. To address those limitations, this paper aims to analyze the instability of the pile hole wall by proposing a new mechanical model combining the theory of pore expansion and the theory of elastic–plasticity. This model synthesizes the influence of local instability and stress change in different construction stages, and it considers the complex stress distribution and the spatio-temporal changes in soil properties during the drilling process, allowing us to explore the local and global instability mechanism of the pile hole wall under realistic conditions. The stability of the borehole wall is thus evaluated more comprehensively and accurately. Studying the stability of the pile hole wall during construction can effectively avoid foundation settlement or structural instability caused by instability of the pile hole wall, thus ensuring the long-term stability of the building, reducing the maintenance cost later on and thus improving the long-term benefit of the project. The mechanical model proposed in this study considers the complex interaction of mud density, soil layer characteristics, and pile hole behavior to optimize the use of resources, improve the efficiency of mud use in construction, reduce mud leakage and waste, and reduce the risk of pollutant diffusion, thus reducing the negative impact on the environment. This is of great significance for improving the environmental protection level of construction projects and reducing ecological damage in the construction process, which helps promote green construction and the environmental sustainability of projects [21,22].

2. Analysis of the Stress State in the Hole Wall of the Bored Pile

2.1. Mechanical Model of Hole Wall Stability

Mechanical stability analysis of the borehole wall of pile seeks to identify scenarios in which there is no influence of instability of the soil layer on the side wall of bored pile when it is subjected to mechanical disturbance or other factors affecting stability. The mechanical analysis of the hole formation process of bored pile is mainly based on the premise that the reverse supporting force exerted by mud slurry on the hole wall of bored pile is not enough to balance soil stress release during the excavation of the side wall of the pile hole, so the hole wall of bored pile is unstable and expands and the soil stress state changes.

Firstly, the stability of the hole wall of bored pile is regarded as the unloading expansion and contraction of the semi-infinite inner hole. According to the theory of circular hole expansion and elastoplasticity, the radial stress along the radius and tangential stress in the tangential direction are generated when the soil layer on the side wall of bored pile is disturbed during the process of hole formation. The interaction between the radial stress along the radius and the soil layer on the side wall of the pile hole benefits the stability of the latter to some extent. The tangential stress in the tangential direction makes it possible to destabilize the soil layer of the pile hole and destroy the original stress equilibrium state of the soil layer on its side wall. Before mechanical stability analysis of the side wall of bored pile is carried out, the hole and soil layer of bored pile are first regarded as infinite planes, as shown in Figure 1, and then the stress situation of the soil layer on the side wall of the bored pile hole is simplified somewhat. The basic assumptions are as follows:

(1)

The soil layer of the pile hole is within the range of elastic depth, and the soil is a homogeneous and isotropic elastoplastic material.

(2)

The pile and the pile aperture are relatively large, and the pile holes do not deviate during excavation and later pile formation.

(3)

If the yield criterion is satisfied at a certain depth of the soil layer on the side wall of the pile hole, the side wall is considered unstable in this soil layer.

2.2. Stress Analysis of the Soil Layer Around the Hole Wall

Taking the cross-section at a certain depth of the side wall of the perforated pile as the research object, a horizontal cross-section at the depth of the pile hole is taken, as shown in Figure 1, and mechanical analysis is carried out on the cross-section of the soil layer of the pile hole taken. During the process of hole formation, the lateral wall of the bored pile is subject to bore force expansion, as shown in Figure 2 and Figure 3. The force on the side wall of the pile hole comes from the soil weight, the reverse pressure of the mud in the pile hole, and other ground loads near the pile hole.

At the distance r from the central axis of the bored pile hole, the differential equation of soil microelements can be expressed as

$${\displaystyle \frac{d{\sigma }_r}{dr}}+{\displaystyle \frac{{\sigma }_r-{\sigma }_{\theta }}{r}}=0$$

(1)

The geometric equation can be expressed as the radial displacement u:

$$\left\{\begin{array}{c}{\epsilon }_r={\displaystyle \frac{du}{dr}}\\ {\epsilon }_{\theta }={\displaystyle \frac{u}{r}}\\ {\epsilon }_z=0\end{array}\right.$$

(2)

The boundary conditions when the stress balance of the pile hole wall is introduced are

$$\left\{\begin{array}{c}r=r_0,{\sigma }_r=-p_w\\ r=r_i,{\sigma }_r=-p_0\end{array}\right.$$

(3)

For elastomer materials, the stress–strain constitutive equation of pore wall expansion can be expressed as

$$\left\{\begin{array}{c}{\epsilon }_r={\displaystyle \frac{1}{E}}\left[{\sigma }_r-\mu ({\sigma }_{\theta }+{\sigma }_z)\right]\\ {\epsilon }_{\theta }={\displaystyle \frac{1}{E}}\left[{\sigma }_{\theta }-\mu ({\sigma }_r+{\sigma }_z)\right]\\ {\epsilon }_z={\displaystyle \frac{1}{E}}\left[{\sigma }_z-\mu ({\sigma }_r+{\sigma }_{\theta })\right]\end{array}\right.$$

(4)

We introduce the elastic Lamme solution under uniform pressure:

$$\left\{\begin{array}{c}{\sigma }_r=-{\displaystyle \frac{\frac{R^2}{{\rho }^2}-1}{\frac{R^2}{r^2}-1}}{P}_w-{\displaystyle \frac{1-\frac{r^2}{{\rho }^2}}{1-\frac{r^2}{R^2}}}{P}_0\\ {\sigma }_{\theta }={\displaystyle \frac{\frac{R^2}{{\rho }^2}+1}{\frac{R^2}{r^2}-1}}{P}_w-{\displaystyle \frac{1+\frac{r^2}{{\rho }^2}}{1-\frac{r^2}{R^2}}}{P}_0\end{array}\right.$$

(5)

Under the above boundary conditions, when r = r₀, the radial stress ${\sigma }_r=p_w$. When r approaches infinity, the radial stress ${\sigma }_r=p_0=k_0{\sigma }_z$. At this time, the expression of radial, tangential, and axial stresses can be obtained as follows:

$$\left\{\begin{array}{c}{\sigma }_r=p\left(1-{\displaystyle \frac{b^2}{r^2}}\right)\\ {\sigma }_{\theta }=p\left(1+{\displaystyle \frac{b^2}{r^2}}\right)\\ {\sigma }_z=0\end{array}\right.$$

(6)

where ${\sigma }_r,{\sigma }_{\theta },{\sigma }_z$ are radial, tangential, and axial stresses, respectively; ${\epsilon }_r,{\epsilon }_{\theta },{\epsilon }_z$ are the radial, tangential, and axial strains, respectively, r is the section radius from the central axis of the pile hole to the pile body, E and μ are the elastic modulus and Poisson’s ratio, respectively, p_w and p₀ are the mud slurry pressure and soil original stress, respectively, r₀ and r_i are the inner and outer radii of the selected soil layer units, respectively, and u is the radius of the plastic zone.

Based on the expressions of the radial, circumferential, and axial stresses of bored pile, the influences of the soil weight above the pile hole, reverse supporting force of mud slurry in the pile hole, and ground load near the pile hole on the stability of the pile hole wall are considered, respectively.

(1) Hole wall stress caused by the weight of the soil layer above the pile hole.

Based on the basic stress principle in advanced soil mechanics, it is deduced that z hole wall stress caused by the gravity of the soil layer above the hole of a pile is as follows:

$$\left\{\begin{array}{c}{\sigma }_r={\sigma }_{rp}{\displaystyle \frac{r_0^2}{r_i^2}}+k_0\gamma z\\ {\sigma }_{\theta }=-{\sigma }_{rp}{\displaystyle \frac{r_0^2}{r_i^2}}+k_0\gamma z\\ {\sigma }_z=\gamma z\end{array}\right.$$

(7)

where k₀ is the coefficient of static earth pressure; and ${\sigma }_{rp}$ is the radial stress at the elastic–plastic junction of soil around the pile hole.

k₀ is

$$k_0={\displaystyle \frac{\mu }{1-\mu }}$$

(8)

where μ is the Poisson’s ratio of soil mass.

The soil layer above the pile hole is usually not a single uniform soil layer, but is composed of multi-layered soil, showing the typical characteristics of a large-thickness layered foundation, and the properties of each soil layer are very different. Combined with the principle of soil mechanics and the solution to the stress state of the soil layer on the side wall of the bored pile, the hole wall stress generated at the depth of the hole wall of the soil layer above the pile hole is

$$\left\{\begin{array}{c}{\sigma }_r=k_i{\sigma }_z\left(1-{\displaystyle \frac{{r_0}^2}{{r_i}^2}}\right)\\ {\sigma }_{\theta }=k_i{\sigma }_z\left(1+{\displaystyle \frac{{r_0}^2}{{r_i}^2}}\right)=2k_i{\sigma }_z\\ {\sigma }_z={\displaystyle \sum _{i=0}^{n-1}{\gamma }_i{h}_i+{\gamma }_i\left(z-{\displaystyle \sum _{i=0}^{n-1}{h}_i}\right)}\end{array}\right.$$

(9)

where n is the number of soil layers from the ground to the pile hole depth z; h_i is the thickness of layer i soil; ${\gamma }_i$ is the weight of layer i soil; and k_i is the lateral pressure coefficient of layer i soil.

(2) The stress of the ground load on the hole wall of the bored pile.

In the process of engineering construction, various engineering materials are often stacked on the working surface, such as steel, construction machinery, brick, stone, etc. At the same time, in the process of hole formation, moving construction machinery, running vehicles, and tamping with the foundations will also affect the stability of the hole wall of the bored pile. Due to the ground load of the pile hole, the stress of the soil layer on its side wall will be redistributed, which will cause the soil layers on the pile hole to slide, especially the fine sand, sand, and breccia layers with large porosity. This effect is more serious for bored piles with large hole depths. To facilitate mechanical analysis, the ground load of a pile hole is regarded as subject to uniform load action. Assuming that the ground load at the pile hole depth z is f, the stress on the hole wall of the bored pile under a uniformly distributed load f is

$$\left\{\begin{array}{c}{\sigma }_r=k_{i}f\left(1-{\displaystyle \frac{{r_0}^2}{{r_i}^2}}\right)\\ {\sigma }_{\theta }=k_{i}f\left(1+{\displaystyle \frac{{r_0}^2}{{r_i}^2}}\right)=2k_{i}f\\ {\sigma }_z=f\end{array}\right.$$

(10)

(3) The hole wall stress caused by the reverse action of mud slurry in the pile hole.

To avoid soil stress loss and local collapse of the soil layer during the formation of the soil layer on the side wall of the bored pile, the optimal construction technology is generally adopted. If the buried depth of the pile foundation is shallow and the site soil quality is relatively good, manual digging can be used. If the buried depth of the pile foundation is great and the site geology is complex, drilling machinery is more suitable. For a mechanically drilled pile, mud slurry should be drilled in the process of hole formation to prevent the occurrence of hole expansion or necking. The analysis of the effect of pile hole slurry on the stability of the pile hole wall is related to whether the presence of mud slurry in the pile hole will affect the strength and deformation of the soil layer on its side wall. Therefore, the mechanical analysis of the reverse force of the mud slurry in the pile hole on the pile hole wall is also related to the stability of the soil layer on the side wall and the vertical bearing characteristics of the pile after perfusion.

The static pressure of mud slurry at a given point in the pile hole [23] is

$$p_c=\Delta p_n=-{\displaystyle \frac{1}{2}}m{v_n}^2$$

(11)

$$p_n={\rho }_{n}g(h_n-h_j)-p_c$$

(12)

where ρ_n is the mud slurry density; h_n is the depth of mud slurry somewhere in the side wall of the pile; h_j refers to the height at which the slurry level drops when a drilling machine is drilling; p_c is the suction force generated by mud slurry on the hole wall during drilling; and ν_n is the speed of lifting the drill.

Then, the hole wall stress caused by the reverse force of mud slurry in the pile hole on the side wall is as follows:

$$\left\{\begin{array}{c}{\sigma }_r=p_n{\displaystyle \frac{{r_0}^2}{{r_i}^2}}=\left[{\rho }_{n}g\left(h_n-h_j\right)+{\displaystyle \frac{1}{2}}m{v_n}^2\right]\cdot {\displaystyle \frac{{r_0}^2}{{r_i}^2}}=p_w\\ {\sigma }_{\theta }=-p_n{\displaystyle \frac{{r_0}^2}{{r_i}^2}}=\left[-{\rho }_{n}g\left(h_n-h_j\right)+{\displaystyle \frac{1}{2}}m{v_n}^2\right]\cdot {\displaystyle \frac{{r_0}^2}{{r_i}^2}}=-p_w\\ {\sigma }_z=0\end{array}\right.$$

(13)

In a comprehensive analysis of the above solution to the hole wall stress of bored pile, the hole wall stresses caused by the soil weight, reverse action of mud slurry, and ground load near the pile hole are superimposed to obtain the hole wall stress at a certain depth z of the bored pile hole. Formulas (9), (10), and (13) are superimposed to obtain

$$\left\{\begin{array}{c}{\sigma }_r=k_i(1-{\displaystyle \frac{{r_0}^2}{{r_i}^2}})({\sigma }_z+f)+p_w\\ {\sigma }_{\theta }=2k_i({\sigma }_z+f)-p_w\\ {\sigma }_z={\displaystyle \sum _{i=0}^{n-1}{\gamma }_i{h}_i+{\gamma }_i(z-{\displaystyle \sum _{i=0}^{n-1}{h}_i})+f}\end{array}\right.$$

(14)

Formula (14) is the hole wall stress at a certain depth z of the pile hole.

2.3. Analysis of the Stable State of the Hole Wall

In determining whether a certain soil layer in the side wall of bored pile will be unstable, the analysis of this problem is related to the quality of pile formation and the bearing capacity of the pile foundation after pile formation. Here, the Mohr–Coulomb strength theory is introduced. Coulomb’s shear test based on sand in 1776 obtained the following expression related to the shear strength of sand:

$${\tau }_f=\sigma \tan \phi$$

(15)

where ${\tau }_f$ is the shear strength of soil (kPa); $\sigma$ is the normal stress on the shear surface (kPa); and $\phi$ is the internal friction angle of sand (º).

Later, following testing, the expression of the shear strength of viscous soil was put forward:

$${\tau }_f=c+{\sigma }_{\mathrm{n}}\tan \phi$$

(16)

When the soil layer is in the ultimate equilibrium state, its shear stress is equal to its shear strength. At this time, there is a linear relationship between the soil layer and the shear stress acting on the soil layer elements, as shown in Figure 4. Based on the static equilibrium condition, the normal stress σ and shear stress τ acting on the soil elements at this time are obtained:

$$\sigma ={\displaystyle \frac{1}{2}}({\sigma }_1+{\sigma }_3)+{\displaystyle \frac{1}{2}}({\sigma }_1-{\sigma }_3)\cos 2\alpha$$

(17)

$$\tau ={\displaystyle \frac{1}{2}}({\sigma }_1-{\sigma }_3)\sin 2\alpha$$

(18)

α is

$$\alpha ={\displaystyle \frac{1}{2}}({90}^{\circ }+\phi )$$

(19)

Bringing Formulas (18) and (19) into Formula (17) gives

$${\displaystyle \frac{1}{2}}({\sigma }_1-{\sigma }_3)\cos \phi =c+[{\displaystyle \frac{1}{2}}({\sigma }_1+{\sigma }_3)+{\displaystyle \frac{1}{2}}({\sigma }_1-{\sigma }_3)\cos 2\alpha ]\tan \phi$$

(20)

When the soil layer around the hole of the bored pile is stable, and when the equilibrium condition is $\tau <{\tau }_f$, the simplified Formula (20) is

$${\tau }_f={\displaystyle \frac{{\sigma }_1-{\sigma }_3}{2}}-{\displaystyle \frac{{\sigma }_1+{\sigma }_3}{2}}\sin \phi -c\cos \phi \le 0$$

(21)

It is considered that the more unstable the soil layer in the pile hole, the greater the pressure difference between the inside and outside of the side wall, and the more obvious the stress concentration of the soil layer in the side wall. When the radius of the hole wall of bored pile reaches r = r₀, the soil layer around the side wall of the pile hole will be in a state of stress equilibrium. Under this condition, Formula (16) is transferred to Formula (21) to obtain the stress of the hole wall at the depth of the soil layer z:

$${\sigma }_z\le {\displaystyle \frac{{\rho }_{n}g(h_n-h_j)-p_c+c\cos \phi }{k(1-\sin \phi )}}$$

(22)

and

$${\sigma }_z={\displaystyle \sum _{i=0}^{n-1}{\gamma }_i}{h}_i+{\gamma }_i(z-{\displaystyle \sum _{i=0}^{n-1}{h}_i})+f$$

When Formula (22) is satisfied, the soil layer on the side wall of the cast-in-place pile is stable, and the hole wall of the pile is safe even when there is a change in soil layer. If Formula (22) is not satisfied, the hole wall stress of the cast-in-place pile is too large, and the hole wall may be damaged and become unstable.

3. Calculation of the Hole Wall Stability of Piles with Mud Slurry

Mud slurry in the bored pile hole has a reverse supporting effect on the soil layer on the side wall, and it can resist the soil pressure exerted by the soil layer around the side wall; the two phases are balanced to prevent local or overall instability of the soil layer on the side wall. Through theoretical analysis and field testing, it was found that within a certain range, the greater the proportion of mud slurry in the pile hole, the better the stability of the soil layer on the side wall of the pile hole, and conversely, the smaller the proportion of mud slurry in the pile hole, the greater the possibility of soil layer instability on the side wall.

(1) Stability analysis of the pile hole wall in a cohesive soil layer.

When there is no mud slurry action in the pile hole, the soil layer on the side wall is mainly affected by the main dynamic earth pressure p_a. According to the ultimate equilibrium condition equation of the soil,

$$\left\{\begin{array}{c}{\sigma }_1={\sigma }_3{\mathrm{tg}}^2({45}^{\circ }+{\displaystyle \frac{\phi }{2}})+2c\cdot tg({45}^{\circ }+{\displaystyle \frac{\phi }{2}})\\ {\sigma }_3={\sigma }_{1}t{g}^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})-2c\cdot tg({45}^{\circ }-{\displaystyle \frac{\phi }{2}})\end{array}\right.$$

(23)

When the soil layer around the side wall of the pile hole is in the active limit equilibrium state,

$$\left\{\begin{array}{c}{\sigma }_1={\sigma }_2=\gamma \cdot z\\ {\sigma }_3={\sigma }_x=p_a\end{array}\right.$$

(24)

Taking into account the shape of the pile hole and the pressure of the soil layer on its side wall, it can be found that the active earth pressure acting on the side wall is different from the main dynamic earth pressure p_a received by the general soil retaining structure.

$$p_a={\displaystyle \frac{1}{2}}\gamma z^{2}t{g}^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})-2c\cdot ztg({45}^{\circ }-{\displaystyle \frac{\phi }{2}})$$

(25)

where γ is the weight of the soil layer; z is the depth of the pile hole; c is the cohesive force of the soil layer; and φ is the internal friction angle of the soil layer.

If the active earth pressure on the side wall of the pile hole p_a is zero, the critical hole depth of the pile Z_cr and the safety factor of the pile hole wall F_s can be solved, namely,

$${\displaystyle \frac{1}{2}}\gamma z^{2}t{g}^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})-2c\cdot ztg({45}^{\circ }-{\displaystyle \frac{\phi }{2}})=0$$

(26)

The answer is

$$Z_{cr}={\displaystyle \frac{4c}{\gamma t{g}^2({45}^{\circ }-\frac{\phi }{2})}}$$

(27)

$$F_S={\displaystyle \frac{4c}{Z\gamma tg({45}^{\circ }-\frac{\phi }{2})}}$$

(28)

For saturated cohesive soil, its shear strength ${\phi }_u=0$, according to Equation (26), can be solved as follows:

$$Z_{cr}={\displaystyle \frac{4c}{\gamma }}$$

(29)

$$F_s={\displaystyle \frac{4c}{\gamma \cdot Z}}$$

(30)

When there is mud slurry in the pile hole, the soil layer on the side wall will also be affected by the water head pressure p of the mud slurry on the hole wall and the active soil pressure p_a generated by the soil layer around the pile hole wall. In this case, the critical hole depth Z_cr and safety stability coefficient F_s of the pile hole wall in a stable state in viscous soil can be solved as follows:

$$Z_{cr}={\displaystyle \frac{4c}{\gamma -{\gamma }_m}}$$

(31)

$$F_s={\displaystyle \frac{4c}{z(\gamma -{\gamma }_m)}}$$

(32)

where the hydrostatic pressure of mud slurry in the pile hole on the pile hole wall $p=0.5{\gamma }_m{z}^2$; and ${\gamma }_m$ is the weight of mud slurry in the pile hole.

From the analysis of Formulas (23)–(32), it can be seen that due to the presence of mud slurry in the hole, the critical hole depth Z_cr and safety factor F_s of the soil layer on the side wall of the pile hole, when it is stable, will increase with the increase in the mud slurry in the hole. If the weight of mud slurry in the pile hole is equal to the weight of the soil layer on the side wall of the pile hole, the side wall will be stable and without external disturbance; here, the safety factor F_s is independent of the depth of the pile hole. However, mud slurry in the pile hole should not be too heavy. Too much mud slurry will produce the mud skin effect, which is not conducive to the vertical bearing capacity of the pile later on after pile formation.

(2) Analysis of the hole wall stability of a cast-in-place pile in sandy soil.

For a sandy soil layer, such as breccia, fine sand, gravel sand, or gravel, because of the cohesion c = 0, if the mud wall is not used to form holes, the hole wall of bored pile will always be unstable. According to the limit equilibrium equation and the classical sliding theory in soil mechanics, the safety factor F_s of the hole wall of the cast-in pile in the sandy soil layer can be solved when the hole wall is stable, namely

$$F_s={\displaystyle \frac{\gamma +{\gamma }_m}{\gamma -{\gamma }_m}}tg\phi$$

(33)

As can be seen from Equation (33), for the sandy soil layer, the safety factor F_s when the hole wall of the cast-in-place pile is stable has nothing to do with the depth of the hole but is related to the density of mud slurry in the hole of the cast-in-place pile. Generally, the stability of the side wall of the hole of the cast-in-place pile increases with the increase in mud slurry gravity. However, the properties of the sandy soil layer are weak, the bulk density is small, and too much mud slurry will affect the stability of the pile hole wall. Therefore, the stability of the pile hole wall can be improved by selecting a suitable mud slurry weight.

4. Calculation and Analysis of the Local and Global Instability of the Hole Wall

4.1. Local Instability Calculation of a Pile Hole Wall

Local wall instability of a cast-in-place pile is mainly caused by factors such as low soil strength, large construction disturbance, and an unstable groundwater level. For example, the soft part of a soft and hard interlayer, sand lens in cohesive soil, and other soil layers are more prone to local instability. Here, according to Filz and Daviddson (2004), the local stability of the hole wall safety factor K_L is

$$K_L={\displaystyle \frac{i_0{\gamma }_w\tan {\phi }_s}{{\gamma }_1-{\gamma }_2}}$$

(34)

where ${\phi }_s$ is the internal friction angle where mud penetrates the soil layer; ${\gamma }_w$ is the groundwater gravity; ${\gamma }_1$ is the gravity of soil when there is mud infiltration; ${\gamma }_2$ is the mud weight; and i₀ is the viscosity gradient of mud slurry.

Among them, the viscosity gradient of slurry i₀ can be calculated using the following formula:

$$i_0={\displaystyle \frac{h_s(\frac{{\gamma }_2}{{\gamma }_w})-h_w}{L_s}}$$

(35)

In the formula, h_s and h_w are the height (m) from the calculation point to the slurry level and the height (m) from the groundwater level, respectively, and L_s is the horizontal length (m) of the slurry in the soil.

When pile hole mud slurry enters a soil layer with relatively large porosity, such as sand, fine sand, or sand gravel, the permeability of mud slurry in the pile hole will destroy the original earth stress around the hole wall and the stress balance of the soil layer around the pile hole, and local instability of the pile hole wall will be generated. At this time, when a certain safety and stability factor of the hole wall of bored pile is given, only the weight of mud slurry in the hole needs to be reasonably configured, and the calculation can be carried out according to Equations (34) and (35) to analyze whether the hole wall of bored pile is safe and stable in this soil layer.

4.2. Overall Instability Calculation of a Pile Hole Wall

Overall instability of the borehole wall of bored pile generally occurs in shallow soil with a soft upper and hard lower layer, or instability of the upper soil layer is caused by a weak interlayer in the soil layer. Judging that there is overall instability of the pile hole wall is half theory and half experience. Many scholars at home and abroad have carried out related analyses and research, and they have summarized the main parabolic column method, wedge method, empirical formula method, and standard method. The following will be combined with the wedge method to analyze the overall instability of the hole wall of bored pile under the action of pile hole mud slurry.

The wedge method is a classical instability analysis method in the analysis of pile hole wall stability. Its basic idea is to consider the soil around the pile hole wall as consisting of several wedges, and judge whether these wedges will slide or fail under the action of an external or internal force. The wedge method provides a simplified mechanical model for the analysis of pile hole stability, which has strong theoretical support and practical significance, but its application and accuracy are still limited by certain assumptions.

In the wedge method, the following assumptions are also required to simplify the analysis process:

(1)

It is assumed that the soil around the pile hole wall can be simplified into multiple wedges, which are regarded as rigid bodies. Each wedge is in contact with the pile hole wall at a certain angle, forming an unstable area. When the external load or the internal force of the soil exceeds a certain limit, the wedge will slide or break.

(2)

It is assumed that sliding between the wedge and the surrounding soil mass is controlled by the internal friction force of the soil mass, and that the magnitude of the friction force is related to that internal friction angle. The friction force is evenly distributed, and the shear strength of the soil is linearly related to the normal stress.

(3)

The sliding failure surface is a plane.

(4)

The borehole wall instability model as shown in Figure 5 is established by using the classical sliding theory of soil mechanics. The unstable soil layer is a trapezoidal unstable body moving toward the central axis of the pile hole, the slip plane of the soil layer is at an angle a to the horizontal direction, the height of the mud and slurry surface in the pile hole is Z_n, the height of the crack zone at the top of the pile hole is Z_c, the length of the damaged soil layer is L₀, the width of the soil layer is B₀, $B_0=(H-Z_c)\cot \alpha$, and H is the height of the unstable soil layer. The geometric calculation analysis model is shown in Figure 5 and Figure 6.

The safety stability coefficient of the sliding surface of the soil layer when wedge instability occurs in the soil layer of the side wall of the bored pile under the action of pile hole mud slurry will be deduced below. First of all, according to the geometric relationship, the calculation formula for the forces on the unstable soil when the soil layer of the pile hole wall is unstable is obtained as follows:

Soil layer weight G when the lateral wall of the pile hole is unstable.

When the influence of groundwater in the construction area is considered, the volume V_a of the soil layer above the groundwater level at the time of pile hole instability is calculated based on the geometric relationship:

$$\begin{array}{l}{V}_a=Z_c(H-Z_c)\cdot L_0\cot \alpha +H(Z_w-Z_c)\cdot L_0\cot \alpha \\ +{\displaystyle \frac{1}{2}}({Z_c}^2-{Z_w}^2)\cdot L_0\cot \alpha \end{array}$$

(36)

Volume above the water table V_b is

$$V_b={\displaystyle \frac{1}{2}}{(H-Z_w)}^2{L}_0\cot \alpha$$

(37)

Then, the dead weight G of the unstable soil layer of the pile hole is

$$G=A{L}_0{H}^2\cot \alpha ={\gamma }_1{V}_a+{{\gamma }^{\prime }}_1$$

(38)

When the soil layer on the side wall of the pile hole is unstable, the ground load near the soil layer Q is

$$Q=q{B}_0{L}_0=q(H-Z_c)L_0\cot \alpha$$

(39)

The resultant P of the mud slurry pressure P_m in the pile hole and the groundwater pressure P_w in the soil layer of the pile hole wall are

$$\begin{array}{l}P=P_{\mathrm{m}}-p_w={\displaystyle \frac{1}{2}}{\gamma }_2{Z_n}^2-{\displaystyle \frac{1}{2}}{\gamma }_w(H-Z_w{)}^2\\ ={\displaystyle \frac{1}{2}}\left[{\gamma }_2{Z_n}^2-{\gamma }_w{(H-Z_w)}^2\right]\end{array}$$

(40)

The friction force between the unstable soil layer and the pile hole side wall P_c is

$$\begin{array}{l}{P}_{\mathrm{c}}=\cot \alpha \cdot {\displaystyle \underset{Z_c}{\overset{Z_w}{\int }}(c+{\sigma }_{h1}\tan \phi )(H-z)dz}\\ +\cot \alpha \cdot {\displaystyle \underset{Z_w}{\overset{H}{\int }}(c+{\sigma }_{h2}\tan \phi )(H-z)dz}\\ =2A\cot \alpha \end{array}$$

(41)

and

$$\left\{\begin{array}{c}{\sigma }_{h1}=K_0(q+{\gamma }_{1}z)\\ {\sigma }_{h2}=K_0\left[q+{\gamma }_1{Z}_w+{{\gamma }^{\prime }}_1(z-Z_w)\right]\\ Z_c={\displaystyle \frac{(2c\sqrt{K_a}-\mathrm{q})}{{\gamma }_1}}\\ \sqrt{K_a}=\tan ({45}^{\circ }-{\displaystyle \frac{\phi }{2}})\end{array}\right.$$

(42)

The reverse supporting force N of the soil layer on the sliding surface where the pile hole wall is unstable is

$$N=(G+Q)\cos \alpha +P\sin \alpha$$

(43)

The shear stress of the soil layer on sliding surface T is

$$T=N\tan \phi +c(H-Z_c)\csc \alpha$$

(44)

Combined with Formulas (36) to (44), the safety factor F_s when the soil layer on the side wall of the bored pile is stable is finally obtained:

$$F_s={\displaystyle \frac{T+2P_c+P\cos \alpha }{(G+Q)\sin \alpha }}$$

(45)

5. Project Example Verification

The above derivation takes into account the stratification characteristics of the soil layer and its different properties, and it assumes the slip form when the soil layer on the side wall of the pile hole is unstable under the action of pile hole mud and slurry. On this basis, it deduces and solves the safety stability factor when the soil layer on the side wall of the pile hole is stable. Verification will be presented through specific engineering examples below.

In a pile foundation project in Lanzhou City, Gansu Province, China, a test pile hole was selected for analysis. During the construction of the pile hole, mud slurry inside extended to about 1 m away from the top surface of the pile hole, and the underground water level was about 23 m away from the ground. Due to the large burial depth of the groundwater level, the impact of groundwater is not considered for short piles. According to the analysis of the field investigation report, sand layers such as gravel and fine sand, which were distributed within about 10 m from the top of the pile hole, were prone to instability, while the soil layers below the pile hole comprised silty clay and would not collapse easily. During construction, the ground load was 40 kPa, and the mud slurry in the pile hole was 10.8 kN/m³. The parameters of each soil layer are shown in

Table 1. Parameters of each soil layer.

Soil Layer Name	Weight (kN/m3)	Cohesion (kPa)	Internal Friction Angle (°)	Thickness (m)
Gravel	18.5	0	30	2.1
Fine sand	18.0	0	22	3.6
Silty clay	18.8	20	13	1.6

.

According to the above parameters, it could be determined that γ₁ = 18.3 kN/m³, γ’₁ = 8.3 kN/m³, γ₂ = 10.8 kN/m³, γ’₂ = 0.8 kN/m³, H = 10 m, Z_n = 13.4 m, Z_m = 12.9 m, q = 40 kPa, c = 0, φ = 26.8°, α = 30°, and L = 0.8 m. The above parameters were entered into Formulas (36)–(44), and the safety stability coefficient was finally obtained as 0.47, indicating that there was a high probability of unstable borehole walls during the drilling process, which was consistent with the observed pile hole expansion rates of 18.01% in the gravel layer, 15.32% in the fine sand layer, and 5.2% in the silty clay layer during actual construction.

At the same time, parameters such as pile hole diameter and mud slurry specific gravity, affecting the stability of the borehole wall of bored pile, were analyzed via finite element simulation based on two pile holes in the same background project. The hole size, specific gravity of mud slurry, and soil layer distribution of the selected bored pile are shown in

Table 2. Site pile test situation.

Pile Hole Number	Aperture D (mm)	Pile Depth (m)	Specific Gravity of Mud Slurry
1 #	800	25	1.08–1.13
2 #	1200	25

and

Table 3. Soil layer condition on-site.

Soil Layer	Depth (m)	Elastic Modulus (MPa)	Poisson’s Ratio	Unit Weight (kN·m−3)	Internal Friction Angle (°)	Cohesion (kPa)
Plain fill	0~2	20	0.17	15.72	23.4	24.3
Fine sand	2~6	60	0.32	16.8	33	0
Gravel	6~16	150	0.18	21.3	40	0
Silt	16~20	30	0.35	17.444	23.9	22.8
Silty clay	20~38	70	0.33	19.208	21.2	22.6

, respectively.

A three-dimensional axisymmetric finite element model of the effect of pile hole slurry on the pile hole wall was established in numerical simulation software to analyze the influence of mud slurry in the pile hole on the soil layer within the depth range of the pile hole during the formation of the borehole. The finite element model for the stability analysis of the pile hole wall is shown in Figure 7. The model is 100 m long, 25 m wide, and 50 m high, and it consists of 75,472 units. To simulate the boundary conditions of the bored pile in actual engineering, the nodes at its bottom were constrained in six directions (angle and displacement), and the normal direction of its side was constrained [24]. For the soil, we adopted the molar Coulomb constitutive model, and the basic mechanical parameters are shown in Table 3.

The specific gravity of the mud slurry was selected based on the designed specific gravity of the mud slurry 1.08~1.13, and the selected values of specific gravity of the mud slurry were 1.08, 1.10, and 1.13. The lateral displacement of the hole wall under drilling unloading and mud slurry protection are shown in Figure 8.

As can be seen from Figure 8, when the specific gravity of mud slurry in the pile hole decreases, the lateral displacement and deformation of the soil layer of the hole wall of the bored pile increase. When the depth of the pile hole is within the range of 10~15 m, the soil layer of the side wall has a large lateral deformation. The lateral deformation of the soil layer on the side wall of the pile hole decreases when the mud slurry weight in the pile hole increases. It can be seen that soil instability easily occurs on the side wall of the pile hole in the unstable soil layer, and properly increasing the proportion of mud slurry in the pile hole can improve the stability of the hole wall of the bored pile.

The diameter of the pile hole also has a great influence on the stability of the soil layer on the side wall. The larger the diameter of the pile hole, the greater the gravity redistribution of the soil layer on the side wall, and the soil layer on the side wall will be more prone to instability and damage, resulting in a large lateral displacement and deformation of that soil layer. When the depth of the pile hole was 25 m and the specific gravity of the mud slurry was 1.13, two pile hole diameters (800 and 1200 mm) were selected to analyze their influence on the stability of the pile hole. The lateral displacements under different pile hole diameters are shown in Figure 9.

As can be seen from Figure 9, when the diameter of the pile hole increases, the displacement and deformation of the side wall increase. Similarly, within the range of 10~15 m pile hole depth, the lateral displacement and deformation of the soil layer of the side wall are relatively large, and the soil layer mainly comprises fine sand and gravel. When the depth of the pile hole is 15–22 m, the lateral displacement and deformation of the soil layer in the pile hole clearly decrease. After checking the strata, it was found that the soil layers at a depth of 15–22 m were mainly silt and silty clay, and the lateral displacement and deformation of the latter soil layer gradually increased, while those of the former soil layer gradually decreased.

When the lateral displacement and deformation of the soil layer on the side wall of the bored pile are too large, if no effective measures are taken to restrain deformation, the soil layer on the side wall of the bored pile will become partially or as a whole unstable, thus forming a variable or irregular section, rather than the smooth section originally designed or an equal section with specific shape specifications. Due to the instability of the soil layer on the side wall of the bored pile, the load-bearing performance, safety index, and reliability of the pile are paid increasing attention.

Similarly, the lateral displacement and deformation of the soil layer on the side wall of the pile hole differ with different diameters, and the state of either local or overall instability of the soil layer on the side wall also differs. When a variable or irregular section is formed in the direction of pile hole depth, the instability mechanism of the pile hole wall and the vertical bearing characteristics of the pile foundation after the formation of a variable-section pile should be comprehensively analyzed and considered.

6. Conclusions

Based on a pile foundation project in Lanzhou City, Gansu Province, China, this paper studied and discussed the stability of the hole wall of bored pile. The main conclusions are as follows:

(1)

In viscous and sandy soils, the safety factor when the hole wall of the cast-in-place pile is stable has nothing to do with the depth of the pile hole but increases with the increase in the mud slurry weight in the hole of the cast-in-place pile. However, too large a mud slurry weight is not conducive to the bearing capacity of the pile. Therefore, the mud slurry weight should be strictly controlled in practical engineering to achieve the optimal value.

(2)

The critical safety stability coefficient of hole wall instability of bored pile in this project was 0.47. When the safety stability coefficient falls lower than 0.47, the soil layer of the hole wall of the drilled pile will become unstable and the hole wall will collapse.

(3)

In unstable soil layers, the pile hole wall was prone to instability, and the stability of the hole wall could be improved by appropriately increasing the specific gravity of mud slurry. With the increase in pile diameter, the lateral displacement and deformation of the hole wall increase, and the displacement of the soil layer increases when the hole wall is unstable, increasing the possibility of forming variable cross-section piles correspondingly.

(4)

In this paper, the analysis of hole wall stability of bored pile mainly considers the influence of mud slurry in the hole on the stability of the pile hole wall, while the influences of other factors on the stability of the pile hole wall have not been deeply analyzed and discussed. This needs to be addressed and related tests performed for comparative analyses in the future.