Research On Soil Displacement Prediction Method Caused By High-pressure Jet Grouting Pile Group Construction Under Adjacent Pile Response

This paper analyzes the soil displacement effect during high-pressure jet grouting pile construction by simplifying the construction process as a series of pressure-controlled spherical cavity expansions in a semi-infinite soil medium. Based on the elastic-plastic solution of pressure-controlled spherical cavity expansion theory, a calculation method is proposed to estimate the soil displacement caused by high-pressure jet grouting pile construction. Non-linear contact theory between the pile and soil is introduced, and the finite difference method is employed to determine the internal forces and deformations of the pile. The proposed method is applied to field construction cases, and its validity is verified by comparing the results with on-site monitoring data. The research findings reveal that the soil displacement effect decreases with depth and primarily occurs in shallow soil layers during the construction of individual high-pressure jet grouting piles. The lateral displacement at the ground surface increases first and then decreases with the increase in horizontal distance, while the ground uplift exhibits an exponential decay trend.


Introduction
With the continuous construction of many infrastructures in China, a significant increase of foundation reinforcement project has emerged.As an effective measure for foundation reinforcement, highpressure jet grouting pile has gained widespread application due to its advantages of convenient construction, simplified processes, and readily available raw materials [1].The construction of highpressure jet grouting pile involves injecting a substantial volume of high-pressure cement slurry into the soil, leading to significant deformations in the soil structure [2][3], which+h will cause adverse effects on existing piles and structures.
Wang et al. [4] and Sui [5] employed the circular cavity expansion theory to introduce a method for computing the soil squeezing effect in displacement-controlled jet grouting pile construction.Liu et al. [6] utilized the spherical hole expansion theory to deduce an approximate solution for the displacement-controlled spherical cavity expansion problem.Rao et al. [7] applied coordinate transformation to adjust the boundary conditions of spherical hole expansion and derived an elastic analytical solution for spherical hole expansion under non-axisymmetric boundary conditions.Zhu et al. [8] put forward an analytical solution for the internal force and deformation of piles based on the nonlinear Pasternak two-parameter foundation model, amalgamating the nonlinear contact theory of piles and soil.
In this paper, the construction of high-pressure jet grouting piles is simplified as a sequence of spherical cavity expansions.A predictive method which builds upon the theory of semi-infinite medium spherical cavity expansion is introduced for calculating both vertical displacement and lateral displacement within the soil resulting from high-pressure jet grouting pile construction.Then, this paper integrates nonlinear pile-soil contact theory with the finite difference method to address the response issues of existing piles.This leads to the development of analytical solutions for lateral deformation in existing piles and soil displacements caused by pile group construction.Finally, the method is applied in practical engineering contexts to evaluate its feasibility and practicality.

Modelling of High-Pressure Jet Grouting Pile Construction
Throughout the high-pressure jet grouting pile construction process, an intense expulsion of highpressure cement slurry from the nozzle relentlessly erodes the surrounding soil, leading to shear failure of the soil adjacent to the nozzle.This intricate process culminates in the amalgamation of the expelled cement slurry with the native soil, forming a composite known as soil-cement slurry.Subsequently, a high-strength pile emerges through the process of solidification.Currently, there exist numerous analytical and semi-analytical solutions for the spherical cavity expansion problem based on an infinite medium.However, the construction process of high-pressure jet grouting piles actually takes place within a semi-infinite soil medium, which imposes certain limitations on the applicability of the spherical cavity expansion theory in an infinite medium.In this study, the construction process is postulated to be replicable through a pressure-controlled spherical cavity expansion process.As shown in figure 1, the deformation of the soil induced by construction is conceptually partitioned into elastic zone and plastic zone for computational analysis.

Theory of Spherical Hole Expansion in Infinite Soil
In 1986, Carter [9] proposed an elastoplastic solution for spherical cavity expansion under pressurecontrolled boundary conditions.This theory provides a mathematical expression for the relationship between pressure and expansion at the conclusion of expansion.Theoretical insights into Carter's spherical cavity expansion: An initial spherical cavity with a radius of r0 expands under the action of uniform pressure p, and the surrounding soil gradually transitions from an elastic state to a plastic state.The plastic zone continuously enlarges until the circular cavity pressure brings the soil to a limit state.Carter provided an analytical solution for the displacement of soil at a radial distance r from the center of the spherical cavity within an infinite medium: Plastic Zone: Plastic Zone: Where εR is the displacement at the interface between the elastic zone and the plastic zone, R is the radius of the plastic zone, A, B, M, and N are all introduced spherical cavity expansion coefficients.
The displacement εR at the interface between the elastic zone and the plastic zone can be calculated: where c is the cohesion, φ is the internal friction angle, G is the shear modulus of the soil, and p0 is the initial overburden stress.
Plastic Zone Radius: Where r0 is the initial radius of the spherical cavity, σR is the stress at the interface between the elastic zone and the plastic zone, and p is the applied cavity expansion pressure, which can be determined by the following formula: 0.5 0 0 / n p Kd P x  (5) Where P0 is the pressure at the nozzle, d0 is the nozzle diameter, x is the distance from any point on the central axis of the jet to the center axis of the nozzle, K and n are coefficients related to the medium.In this paper, K=0.01, n=2.
The stress σR at the interface between the elastic zone and the plastic zone is: The spherical cavity expansion coefficients A, B, M, and N can be determined according to the following equation: (2 ) Where ψ is the soil dilation angle.Thus, the soil displacement caused by the expansion of single spherical cavity within an infinite soil mass can be calculated by the anaiytical solution of Carter's spherical cavity expansion.

Theory of Spherical Hole Expansion in Semi-Infinite Soils
The Luo [10] found that the correction of ground stress can be ignored when selecting the sourcemirror method reasonably to calculate the soil displacement caused by ball hole expansion.The problem of ball hole expansion in semi-infinite soil can be transformed into the problem of ball hole expansion in infinite soil for solution.As shown in figure 2, a spherical cavity is considered as the primary source, and symmetric mirror-image sources are created in relation to the semi-infinite soil boundary.the lateral displacement δr and the vertical displacement δz induced by spherical cavity expansion in a semi-infinite soil could be obtained.
Where δre is the lateral displacement of the soil induced by the source spherical cavity expansion, and δim is the lateral displacement of the soil induced by the expansion of the mirrored spherical cavity.
Equations ( 11) and ( 12) are the analytical solutions for the expansion of a single spherical cavity under the influence of pressure.The construction process of a high-pressure jet grouting pile involves a series of spherical cavity expansions.As shown in figure 3, the soil displacement caused during the construction of high-pressure jet grouting piles can be obtained by superimposing the solutions for the expansion of multiple spherical cavities.

Real spherical cavity expansion
Imaginary spherical cavity expansion Through the equivalent volume conversion, the spacing between spherical cavities can be determined:

The Soil Displacement Caused by the Construction of Piles in Response to Adjacent Piles
The squeezing effect of pile group on soil needs to consider the influence of the already constructed piles and the surrounding soil.The analysis involves the response of the constructed piles and the surrounding soil to soil displacement loads.It is assumed that the pile group construction satisfies the following assumptions:  During the subsequent construction of high-pressure jet grout piles, there is no relative sliding at the pile-soil interface of the already constructed piles.
 The high-pressure jet grouting pile is a linear elastomer that satisfies Hooke's law.
 The soil displacement in different directions is independent and has no mutual influence.By introducing nonlinear foundation beam theory, the lateral displacement of the soil in the free field is used as the basis to obtain the lateral displacement of the already constructed pile body using the p-y curve method.Then, the theory combines with the results from the first phase to analyze the shielding effect of the already constructed pile body on the free field soil displacement.This leads to the development of a response model for existing piles affected by high-pressure jet grouting pile construction and a method for calculating the soil squeezing effect in pile group.Figure 4 shows the nonlinear foundation-beam model.This model simplifies the lateral interaction between pile sand the soil as a series of lateral nonlinear springs, and it incorporates the iterative solution of the pile-soil interaction using the hyperbolic p-y curves proposed by Kondner [11].The nonlinear hyperbolic p-y relationship for pile-soil interaction proposed is as follows: (17) Where δ is the relative displacement between pile and soil, k0 is the initial subgrade reaction coefficient, Pu is the ultimate lateral soil pressure on the pile, Ei is the soil's elastic modulus, and su is the undrained shear strength.
The deformation-deflection equation for the pile subjected to a concentrated force P based on the nonlinear foundation beam is as follows: The pile-soil deformation coordination equation: The simultaneous combination of the two equations yields: In the equation: P is the lateral soil pressure on the pile; k is the subgrade reaction coefficient; rs is the computed lateral displacement of the free-field soil; rp is the lateral displacement of the pile shaft; D is the pile diameter; EpIp is the pile's flexural stiffness.
By discretizing the pile into n+4 linear elastic elements using the finite difference method, the solution to Equation (20) can be obtained as follows: The bending moment, shear force, and rotation of the pile shaft can be obtained from the following equations, respectively: 2 Formulation of a nonlinear system of equations using equation ( 21): Involving n known equations and n+4 unknowns, facilitated by the inclusion of four pile-head and pile-toe boundary conditions for solving.0 0 0, 0 The pile top is free 0, 0 The pile end is free The comprehensive nonlinear system of equations is as follows： In the context of the response of the soil between piles under the influence of shielding effects, as indicated by Equations ( 1) and ( 2), for the fixed points A and B: Where rs,A and rs,B are the displacements of the free-field soil at fixed point A and point B, In order to quantify the shielding effect of the constructed pile on the soil's lateral extrusion, a dimensionless shielding coefficient η is introduced.This coefficient establishes a relationship between the free-field soil displacement at point A, rs,A, and the soil displacement at point A under the influence of shielding effects, rp,A, as follows: The soil displacement at point B, rp,B, under the influence of shielding effects, is given by:

Engineering examples
A project involving the expansion of an existing intercity railway embankment requires on-site testing of high-pressure jet grouting piles.As shown in figure 5 and figure 6, a total of four high-pressure jet grouting piles were selected for on-site testing, designated as 1X through 4X.The construction sequence was 1X, 2X, 3X, and 4X.Monitoring of the inclinometer boreholes was conducted all after the completion of pile 1X and after the completion of pile 4X.The soil parameters were determined based on field measurements and in conjunction with the engineering geological manual.The Poisson ratio ν=0.32, the dilation angle ψ=0, and other parameters are listed in table 1.As shown in figure 7, comparative analysis of theoretical and measured lateral soil displacement at the inclinometer location following the construction of single and multiple piles.It can be observed that the maximum lateral soil displacement occurs at the ground surface and decreases with depth.Furthermore, within the depth range of 0-5m, lateral displacement decreases rapidly.This is due to the lower shear strength of shallow soil, which experiences significant deformation under the impact of high-pressure cement slurry.As the depth progresses, there is an increase in the shear strength of the soil, leading to a concurrent reduction in deformation.Additionally, theoretical predictions display more consistent trends in contrast to the measured values, which largely attributed to non-uniform flow pulsations and fluctuating lifting speeds experienced during high-pressure jet grouting pile operations.These dynamics potentially result in cement slurry congestion.The presence of this irregularity can lead to fluctuations in grouting pressure and localized stress concentration.Consequently, these factors give rise to the manifestation of partial water wedge effects and fractures within the densely grouted area of the high-pressure jet grouting pile and introduces a degree of variability in the acquired measurement data As shown in figure 8, the normalized soil displacement δ at the ground surface caused by the construction of a single high-pressure jet grouting pile is plotted against the radial distance r from the pile center, with normalization done using the pile radius r0.This analysis offers valuable insights into the variations of normalized soil displacement at the ground surface concerning the normalized horizontal distance from the pile center.The vertical soil displacement at the ground surface decreases continuously as the radial distance from the pile center increases, and its rate of decrease varies from fast to slow.In contrast, the lateral soil displacement at the ground surface initially increases and then decreases as the radial distance from the pile center increases, with its peak occurring at a distance equal to one pile radius.
In the primary compaction zone of soil (at depths ranging from 0-5m), a comparative analysis of soil displacement induced by the construction of multiple piles reveals a slight discrepancy between theoretical and measured values.Wang [12] reports that soil undergoes rebound after the completion of high-pressure jet grouting pile construction, primarily occurring within the depth range of 0-5m.The maximum rebound value is around 1mm.The theoretical calculations in this paper do not account for soil rebound.Furthermore, in cases where high-pressure jet grouting piles are initiated from the base, a bottom grouting process is undertaken, resulting in a marginal increase in grout volume at the base relative to other segments of the pile.This variation leads to a notable discrepancy, with the measured soil displacement in the localized area at the pile's base surpassing the corresponding theoretical values.Considering the mechanism of soil compaction, the impact of the construction of a single highpressure jet grouting pile on the surface soil can be divided into three regions: the primary disturbance zone, the secondary disturbance zone and the undisturbed zone.The primary disturbance zone is within one pile radius, where the soil experiences direct effects of high-pressure jet grout injection.The interaction between high-pressure cement slurry and the soil in this specific region is defined by a complex interplay of mixing, compaction, fracturing, and permeation phenomena.The secondary disturbance zone lies between the primary disturbance zone and six times the pile radius.Due to the swift attenuation of mixing, fracturing, and permeation influences within the primary disturbance zone, compaction effects persist with partial attenuation and come into play within the secondary disturbance zone, thereby culminating in soil displacement.The undisturbed zone is located beyond six times the pile radius, where the compaction effects transmitted to this area are extremely weak, and the resulting soil deformation can be neglected.

Conclusion
(1) In this paper, a theoretical prediction and computation method is introduced for assessing the response of adjacent piles and soil displacement induced by the constriction effects during highpressure jet grouting pile construction.This method is primarily built in the application of Carter's infinite soil ball hole expansion theory, supplemented by the source-sink theory.It addresses the construction dynamics of high-pressure jet grouting piles, conceptualizing it as a sequence of pressurecontrolled ball hole expansion processes.The analytical solutions for soil displacement under single and multiple pile conditions in semi-infinite soil are derived, and then the finite difference method is used.Based on the nonlinear contact theory of piles, a calculation method for soil displacement caused by the squeezing effect of pile group under the response of adjacent piles was derived.
(2) This proposed method was applied to on-site tests of high-pressure rotary jet grouting piles in a wide-section rail embankment of an existing intercity railway, which has been integrated into an ongoing high-speed railway construction project.Comparative analysis was conducted between the predicted and measured values of soil compaction effects during both single-pile and multiple-pile construction scenarios.This validation process serves to confirm the suitability of the method for practical engineering applications.
(3) The scope of soil influence within high-pressure jet grouting piles is systematically classified into three discrete regions: the primary disturbance zone, the secondary disturbance zone and the undisturbed zone.This paper delves into the intricacies of compaction mechanisms encompassing mixing, separation, permeation, and densification, operating within these delineated areas.

Figure 1 .
Figure 1.The diagram of construction model simplification

Figure 2 .Figure 3 .
Figure 2. The diagram of spherical cavity expansion in semi-infinite soil

Figure 4 .
Figure 4.The diagram of nonlinear transverse foundation beam model

Figure 5 .Figure 6 .
Figure 5.The diagram of inclination hole and multi-pile layout

Figure 7 .Figure 8 .
Figure 7.The diagram of comparison between the theoretical value of lateral displacement of soil and the measured value

Table 1 .
Soil layer parameters