# Abstract

In this paper, a model of the composite foundation reinforced with geosynthetic encased stone columns was established using the discrete element method, and the characteristics of its action under cyclic loading were studied. The influence of the length and radius of the pile on the settlement of the composite foundation is analyzed. The deformation characteristics of the pile and the stress ratio of pile-soil are studied under different pile lengths and radius. Then, based on this, the analysis of the lateral deformation characteristics of the piles under cyclic loading, the calculation model of the geosynthetic encased pile composite foundation is established. The settlement calculation formula of the composite foundation is solved according to the deformation coordination relationship between the pile and soil, the equilibrium condition, and the boundary condition. The results show that the theoretical value is in good agreement with the simulation value, which verifies the rationality of the theoretical calculation formula.

Keywords

# 1 INTRODUCTION

The expressway and railway in China are developed rapidly in recent twenty years. However, soft soil foundations will inevitably be encountered in expressway and railway construction because of the limitation of geographical conditions such as low shear strength, a large void ratio, high water content, strong compressibility, and rheological properties. These characteristics are easy to induce uneven settlement and lateral sliding failure of the foundation under the repeated action of fixed load and traffic load on the pavement, resulting in the failure of the soft foundation to achieve the expected goal. Therefore, the soft soil foundation must be strengthened in order to meet the bearing capacity and deformation requirements.

This paper introduces the basic principle of encased stone column to reinforce the soft soil, and establishes the settlement calculation model of encased stone column composite foundation based on the integral expression of the displacement of the rigid circular support plate in the semi-infinite elastic solid and deformation coordination theory. The settlement of the encased stone columns with different pile lengths and pile radius under dynamic loads was calculated by using the formula. By establishing a discrete element numerical model, the influence of different pile lengths and pile radius on the bearing capacity and deformation of the foundation was further analyzed, and the results were compared with the formula calculation results. The results show that the formula used in this paper can accurately calculate the settlement of the encased stone column under different working conditions.

# 2 NUMERICAL SIMULATION OF COMPOSITE FOUNDATION

## 2.1 Establishment of particle flow model

When the model is established, the vertical force is applied to the ‘clump’ for loading. The cyclic loading waveform applied in the simulation is a half sine wave, the 110 peak load is 1500 N, the frequency is 1 Hz, and the number of cycles is 1500.

## 2.2 Macro and micro parameters of simulated materials

In order to provide satisfactory DEM simulation results and refer to the existing experimental data (Wang et al.,2016Wang, Z., Jacobs, F., & Ziegler, M. (2016). Experimental and DEM investigation of geogrid–soil interaction under pullout loads. Geotextiles and Geomembranes. 44(3): 230-246.), the linear modulus of the wrapping material of the stone column is set to 2e7 MPa, the linear stiffness ratio is 1.0 and the bond modulus is 1e12. The micro input parameters in the composite foundation are calibrated. The physical and mechanical properties of sand and aggregates are obtained by establishing the numerical model of biaxial test. The physical and mechanical properties of the simulated sand are as follows: the Poisson's ratio of the sand is μs = 0.3, the Young’s modulus of the sand is Es = 10 MPa. The physical and mechanical properties of the aggregates are as follows: the Poisson's ratio of the aggregates μa = 0.25, the elastic modulus of the aggregates Ea = 30 MPa, and the internal friction angle of the aggregates φa = 35°.

## 2.3 Simulation working conditions

In the simulation process, the settlement change of composite foundation is analyzed by changing the pile radius and pile length of the stone column in the composite foundation. The simulation condition in this paper is shown in Table 1.

Table 1
Working conditions of numerical simulation

# 3 THEORETICAL ANALYSIS OF SETTLEMENT OF COMPOSITE FOUNDATION

The total settlement of the composite foundation with geosynthetic encased stone columns under cyclic loading consists of three parts: pile settlement w1, settlement of upper cushion w2, settlement of underlying layer w3 (Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ), which can be expressed as:

$w = m w 1 + w 2 + w 3$ (1)

## 3.1 Basic assumptions and calculation models

When calculating the settlement of the composite foundation with geosynthetic encased stone columns, due to the interaction of the pile and soil, the stress situation is complicated. Several simplifying assumptions are usually made to make the problem simpler with reference to major factor:

1. 1

The stone columns and the surrounding soil are linear elastic materials.

2. 2

The effect of the self-weight stress of the piles and soil on the compression deformation of the composite foundation is not considered.

3. 3

Ignoring the influence of the pile group’s effect on the mechanical properties of the element, the settlement and deformation of the composite foundation are approximately analysed by the "pile-soil element".

4. 4

The settlement of the pile and the soil around the pile at the same depth is equal.

Brauns (Brauns,1978) considered that the stone column will produce bulging deformation under the vertical load as shown in Fig. 2. Therefore, the bulging failure length can be given by:

$l 0 = 2 r tan ⁡ 4 5 ° + φ a 2$ (2)

Where l0 is the depth of bulging failure along the pile direction (mm), r is the initial radius of the stone column (mm), and φa is the internal friction angle (°) of the pile material, which are presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) .

In this paper, the pile is divided into a bulging area and non-bulging area. Due to the bulging deformation of the upper half of the pile, it is difficult to determine the pile radius, which makes the calculation difficult. Therefore, an equivalent radius re is introduced into the calculation in this paper. re can be calculated from Eqs. (3)-(4):

$S 1 = S 0 + ∫ 0 l 0 y x d x$ (3)
$r e = S 1 S 0 r$ (4)

Where: S0 is the initial cross-sectional area of the stone column (mm2), and S1 is the cross-sectional area (mm2) of the pile in the bulging area, see Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) , S0=2rl0.

## 3.2 Settlement calculation

### 3.2.1 Settlement of any point in soil surrounding pile

Fig. 2 (b) presented calculation model of single pile after equivalent analysis of bulging area. Take the pile-soil interface for analysis, as shown in Fig. 3. The settlement of the soil around the pile consists of three parts: the settlement wPR caused by the pile end counterforce; the settlement wuu and wtt caused by the skin friction of the pile, and the settlement wPs caused by the load on the upper part of the soil around the pile, which are described in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) .

The composite foundation load is transferred to the soil at the pile end, PR is the pile end counterforce (presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ), and the distribution of the pressure stress of the soil at the pile end under the pile end counterforce is as follows:

$f = 2 P R π r 2$ (5)

Where f is the compressive stress value of the lower edge of the circular bearing plate, PR is the pile end counter force (N), see Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) .

The settlement of soil under the pile end counterforce is wPR, which is presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) . Through the integral expression of the displacement of the rigid circular support plate in the semi-infinite elastic body, the settlement expression of the soil under the pile end can be obtained:

$w y = 1 + μ s f r 2 E s α 1$ (6)

Where μs is the Poisson's ratio of sand and Es is the Young’s modulus of sand, which are shown in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) . δ and α1 are parameters demonstrated as:

$δ = 2 1 - μ s$ (7)
$α 1 = 2 1 - μ s π 2 - arctan ⁡ y r + y r y 2 + r 2$ (8)

Substituting Eq. (5) into Eq. (6) gives:

$w P R = β R 1 - μ s 2 P R E s r$ (9)

Where βR is the correction factor for settlement calculation (Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ), which is determined by the physical and mechanical properties of the soil around the pile, the length and radius of the pile and other parameters. Generally, the value is 0.5 ~1.0 (Zhang et al., 2013Zhang, M.J., Luo Q., Zhan X.Q., et al. (2013). Settlement analysis of settlement calculation of CFG pile composite foundation of high-speed railway. Rock and Soil Mechanics. (02): 219-225 + 245. (in Chinese)), and it is 0.65 in this paper.

The settlement of the soil around the pile caused by the skin friction is wuu and wtt, respectively. Take a section of the bulging area of the stone column for analysis as shown in Fig. 4. In order to simplify the calculation, the pile skin friction is equivalent transformed.

The conversion formula of pile skin friction is as follows:

$∫ 0 2 π 2 1 - μ s 2 τ y 2 π E s r e r e d y d θ = 2 P x r e 1 - μ s 2 E s$ (10)

Where τy is the pile skin frictional resistance, which is presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°)

$P x = τ y r e d y$ (11)

Then, the settlement of the soil surrounding the pile caused by the skin friction of the pile in the bulging area and the non-bulging area is as follows:

$w u u = ∫ 0 l 0 β R τ u y 1 + μ s 2 E s α 2 d y$ (12a)
$w t t = ∫ l 0 l β R τ t y 1 + μ s 2 E s α 3 d y$ (12b)

Where α2 and α3 are parameters demonstrated as:

$α 2 = δ π 2 - arctan ⁡ y r e + y r e y 2 + r e 2$ (13a)
$α 3 = δ π 2 - arctan ⁡ y - l 0 r + y - l 0 r y - l 0 2 + r 2$ (13b)

The load of the soil around the pile of the composite foundation is Ps. The settlement of the soil under the load of the soil around the pile can be solved according to the formula of the displacement of the half space under the action of the concentrated load, and the settlement of the soil can be solved integrally within the long strip range:

$w P s = 4 β R P s D 1 + μ s π E s [ η + χ ]$ (14)

$η = δ 1 + y 2 D 2 - y D$ (15)
$χ = y D 1 - y D 1 + y 2 D 2 - 0 . 5$ (16)

According to the above Eqs. (9) - (14), the settlement of the soil around the pile under various loads can be obtained, and the settlement of each part can be accumulated to any point of the soil around the pile.

$w 11 = w P R + w u u + w t t + w P s$ (17)

### 3.2.2 Settlement of any point on the pile

The settlement of any point on the pile consists of three parts: the settlement wPRa of the soil at the pile end; the settlement wuua and wtta of the soil at the pile end due to the skin friction of the pile (presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ), and the compression deformation of the pile itself under the load of the pile top. It can be seen from Eq. (6) that the settlement of the soil mass at the pile end under the reaction of the pile end is as follows:

$w P R a = ζ R P R 1 - μ s 2 r E s$ (18)

Where ζR is the correction factor for settlement calculation (Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ), which is determined by the physical and mechanical properties of the soil around the pile, the length and radius of the pile and other parameters. Generally, the value is 0.5 ~ 1.0 (Zhang et al., 2013Zhang, M.J., Luo Q., Zhan X.Q., et al. (2013). Settlement analysis of settlement calculation of CFG pile composite foundation of high-speed railway. Rock and Soil Mechanics. (02): 219-225 + 245. (in Chinese)), and it is 0.65 in this paper.

The settlement of the pile bottom caused by the skin friction in the bulging area and the non-bulging area is as follows:

$w u u a = ∫ 0 l 0 ξ R τ u y 1 + μ s 2 E s α 4 d y$ (19a)
$w t t a = ∫ l 0 l ξ R τ t y 1 + μ s 2 E s α 5 d y$ (19b)

Where α4 and α5 is a parameter demonstrated as:

$α 4 = δ π 2 - arctan ⁡ l 0 - y r e + ( l 0 - y ) r e ( l 0 - y ) 2 + r e 2$ (20a)
$α 5 = δ π 2 - arctan ⁡ l - y r + ( l - y ) r ( l - y ) 2 + r 2$ (20b)

According to the internal stress distribution of the stone column, the compression deformation of the stone column in the bulging area and non-bulging area are as follows:

$w u a = ∫ 0 l 0 1 E a σ u y - 2 μ a τ u y t a n φ a d y$ (21a)
$w t a = ∫ l 0 l 1 E a σ t y - 2 μ a τ t y t a n φ a d y$ (21b)

Where Ea is the Young’s modulus of stone column, σuy and σty are the axial stress of pile body in the bulging and non-bulging area respectively, and μa is the Poisson's ratio of the pile body (Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ).

According to the above Eqs. (18) -(21b), the settlement of the pile under various loads can be obtained, and finally the final settlement of each part is settled and accumulated:

$w 12 = w P R a + w u u a + w t t a + w u a + w t a$ (22)

## 3.3 Pile soil stress balance

Take a certain segment of stone column for analysis, as shown in Fig. 5.

According to the force balance of the element segment, the balance equation can be obtained:

$σ y + d σ y π r 2 + 2 π r τ y d y = σ y π r 2$ (23)

Eliminating the term of Eq. (23) gives:

$r d σ y + 2 τ y d y = 0$ (24)

At the top of the stone column, y = 0, the axial force σy of the pile is:

$σ y = P π r 2$ (25)

Where P is the top load of the stone column (presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ).

At the bottom end of the stone column, y = l, the axial force σy of the pile is:

$σ y = P R π r e 2$ (26)

# 4 Settlement calculation formula solution

The stress parameters (σy, τy, PR, see Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ) can be determined by the force balance and deformation coordination relationship between the piles and the soil. The settlement of the composite foundation is calculated from the soil and pile directions respectively. According to the assumption of equal strain, that is, that the settlement of the pile and the soil at the same depth is equal before the pile and soil break away under the load.

$w 11 = w 12$ (27)

Combining Eq. (24) and Eq. (27) rearranged: Eq. 6, substituting them into Eqs. (25)-(26) gives:

$σ y = P π r e 2 ρ + κ 1 β R ξ R + 4 β R P s 1 + μ s π r e 1 - μ s 2 n 1$ (28)

Where ρ, κ1 and n1 are parameters demonstrated as:

$ρ = exp ⁡ B A 2 ln ⁡ A y + B B - y A$ (29)
$κ 1 = exp ⁡ B A 2 ln ⁡ A l 0 - y + r e + B B - D$ (30)
$D = l 0 - y - r e A$ (31)
$n 1 = δ ( D + y - y 2 + D 2 ) + y y y 2 + D 2 - 1$ (32)
$τ y = P 2 π r e ρ v + κ 1 β R ξ R ψ 1 + 2 β R P s 1 + μ s π 1 - μ s 2 ζ$ (33)

Where v, ψ1 and ζ are parameters demonstrated as:

$v = y A y + B$ (34)
$ψ 1 = l 0 - y + r e A l 0 - y + r e + B$ (35)
$ζ = 1 - 2 μ s y y 2 + D 2 - 1 - y D 2 y 2 + D 2 1.5$ (36)
$P R = f β R ξ R θ 1 + 4 β R P s r e 1 + μ s 1 - μ s 2 n 2$ (38)

Where θ1 and n2 are parameters demonstrated as:

$θ 1 = exp ⁡ B A 2 ln ⁡ A r e + B B - r e A$ (39)
$n 2 = δ ( D + l 0 - l 0 2 + D 2 ) + l 0 l 0 l 0 2 + D 2 - 1$ (40)

In the non-bulging area of the pile:

The upper load F1 of non-bulging area (listed in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ) is:

$P 1 = F - ∫ 0 l 0 2 π r e τ y d y$ (41)

Where τy is the skin friction of the pile in the bulging area (see Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) ).

$σ y = P 1 π r 2 ρ + β R ξ R κ 2 + 4 β R P s 1 + μ s π r 1 - μ s 2 n 1$ (42)

Where κ2 is a parameter demonstrated as:

$κ 2 = exp ⁡ B A 2 ln ⁡ A ( l - y + r ) + B B - l - y - r A$ (43)
$τ y = P 1 2 π r ρ v + β R ξ R κ 2 ψ 2 + 2 β R P s 1 + μ s π 1 - μ s 2 ζ$ (44)

Where ψ2 is a parameter demonstrated as:

$ψ 2 = l - y + r A ( l - y + r ) + B$ (45)
$P R = P 1 β R ξ R θ 2 + 4 β R P s r 1 + μ s 1 - μ s 2 n 3$ (46)

Where θ2 is a parameter demonstrated as:

$θ 2 = exp ⁡ B A 2 ln ⁡ A r + B B - r A$ (47)
$A = μ a r e t a n φ a , B = 3 π E a r e 2 1 - μ s 16 E s$ (48)

Substituting the above formulas into Eq. (17) and Eq. (19), the total settlement of the stone column can be calculated. In the calculation model of stone column composite foundation in this paper, because the interaction between geogrid and soil particles between piles can effectively reduce the settlement of soil between piles in composite foundation, considering the above reasons, a correction coefficient is introduced to correct the settlement of soil on the pile side.

Then the settlement of composite foundation is:

$w 1 = λ w 11 = λ w 12$ (49)
$w 2 = P l 2 E s$ (50)

Where: λ is the correction coefficient, which is presented in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) , and λ is taken as 0.75 through the numerical simulation results. w2 is the amount of compression of the sand cushion. l2 is the thickness of the sand cushion. w3 is the amount of settlement of the underlying layer soil, the amount of which can be calculated directly according to the code for design of building foundations, and then the amount of settlement of the foundation can be calculated.

# 5 Results and analysis

## 5.1 Calculation of equivalent radius of bulging area

Fig. 6 shows the change of lateral displacement along the pile length during the tests.

It can be known from Fig. 6 that after the loading is completed, the pile body will undergo a certain bulging deformation along the direction of the pile body. The deformation effect is different under different pile lengths and radius. When the pile length is constant, as the radius of the pile increases, the range of bulging will gradually increase, which is consistent with the research by Brauns (Brauns, 1978). When the radius of the pile is constant, the range of bulging of the pile is basically the same. As the length of the pile increases, the location where the maximum bulging deformation occurs will gradually deepen.

The values of radius re of the bulging area under different pile radius and pile lengths can be obtained by curve fitting with Eqs. (3)-(4).

## 5.2 Stress concentration ratio

In the initial loading stage of cyclic loading, the stress concentration ratio of geosynthetic encased stone column composite foundation increases rapidly with the increase of the number of loadings and then gradually tends to a certain limit value. The stress concentration ratio under different pile radius and lengths is obtained by numerical simulation, as shown in Table 2.

Table 2
Pile-soil stress ratio under various working conditions

Table 2 illustrates the stress concentration ratio under various working conditions. It shows that the stress concentration ratio increases with the increase of the pile radius and length. Therefore, the soil mass tends to be more stable, as confirmed in the load shared by the piles in the composite foundation increases gradually.

When the applied load is known, the upper load of the pile and the upper load of the soil between piles can be solved by the following formula.

$n = P P s$ (51)

Where:nis the stress concentration ratio;Pis the upper load of the pile;Ps is the upper load of the soil surrounding the pile, which are described in Appendix 1 Appendix 1 D width of the loading board (mm) Ea Young’s modulus of the gravel pile (MPa) Es Young’s modulus of the sand (MPa) F1 the upper load of non-bulging area (N) f compressive stress value of the lower edge of the circular bearing plate (MPa) H height of composite foundation model (mm) l length of pile (mm) l0 depth of bulging failure (mm) l1 thickness of the upper cushion (mm) l2 thickness of the underlying layer (mm) m ratio of cyclic displacement (-) n stress concentration ratio P the top load of the gravel pile (N) PR pile end counterforce (N) Ps load of the soil surrounding the pile (N) r initial radius of pile (mm) re equivalent radius (mm) S0 initial cross-sectional area of the gravel pile (mm2) S1 cross-sectional area of pile in the bulging area (mm2) u lateral displacement of pile (mm) W width of composite foundation model (mm) w total settlement of the composite foundation (mm) w1 pile settlement (mm) w2 settlement of upper cushion (mm) w3 settlement of underlying layer (mm) wa1 settlement value of numerical simulation wPR settlement caused by the pile end counterforce (mm) wPRa settlement of the soil at the pile end (mm) wPs settlement caused by the load Ps(mm) wta compression deformation in the non-bulging area (mm) wtt settlement caused by the skin friction of pile (mm) wtta settlement of the pile bottom caused by τty (mm) wua compression deformation in the bulging area (mm) wuu settlement caused by the skin friction of pile (mm) wuua settlement of the pile bottom caused by τuy (mm) βR correction factor for settlement calculation (-) ζR correction factor for settlement calculation (-) μa Poisson's ratio of the gravel pile (-) μs Poisson's ratio of the sand (-) σty the axial stress of pile body in non-bulging area σuy the axial stress of pile body in the bulging area τy pile skin frictional resistance (-) φa internal friction angle of the gravel pile (°) .

## 5.3 Result verification and analysis

### 5.3.3 Comparison of simulated and theoretical results

The pile settlement value can be obtained by substituting the equivalent radius of the bulging area under different pile radius and pile lengths, the upper load of the pile body and the upper load of the soil between piles into Eq. (1), as shown in Table 3.

Table 3
Comparison of theoretical and numerical results under different working conditions

It can be seen from Table 3 that the calculated value of the settlement under cyclic loading is basically consistent with the numerical simulation data, and the maximum error between them is not more than 16%, which shows that it is feasible to divide the pile into bulging and non-bulging areas for analysis when calculating this settlement.

In order to study the influence of the pile length and pile radius on the settlement of the composite foundation, the results of numerical simulation and theoretical calculation under different pile radius and lengths are compared Fig.8 shows the relationship between the settlement of composite foundation and the pile length and radius.

According to Fig. 8, when the pile length is fixed, the total settlement of the geosynthetic encased stone column composite foundation decreases with the increase of the pile radius, indicating that the settlement of the composite foundation can be significantly reduced by increasing the replacement rate of the composite foundation. When the pile radius is fixed, the total settlement of the geosynthetic encased pile composite foundation changes little with the increase of the stone column length. The results show that when the pile radius is fixed, the effect of the pile on the composite foundation increases less with the increase in the pile length.

# 6 CONCLUSIONS

In this paper, particle flow software is used to simulate the geosynthetic encased pile composite foundation under cyclic loading and to analyse the stress and deformation and, on this basis, calculation of the foundation settlement under cyclic loading is theoretically deduced. Through numerical simulation and theoretical calculation, the following conclusions are reached.

• The calculated results in this paper are consistent with the simulation results, and the maximum error is no more than 16%. The results show that it is feasible to divide the stone column into expansion zone and non-expansion zone in the settlement calculation and analysis of the model. The calculation method in this paper can accurately reflect the settlement of encased stone column composite foundation.

• The numerical simulation and theoretical calculation results show that, compared with pure sand foundation, the encased stone column can effectively reduce the settlement of the foundation by about 25%-40%. Increasing pile radius and pile length of the encased stone column can improve the replacement rate of composite foundation and increase the foundation stiffness, so as to reduce the settlement of foundation under cyclic loading.

• The simulation results show that under cyclic loading, the bulging deformation occurs in the upper part of the encased stone column, resulting in foundation settlement. When the pile length is fixed, the bulging zone will gradually increase with the increase of the pile radius. When the pile radius remains unchanged, the maximum bulging position will gradually deepen with the increase of pile length.

# Appendix 1

Ea Young’s modulus of the gravel pile (MPa)

Es Young’s modulus of the sand (MPa)

F1 the upper load of non-bulging area (N)

f compressive stress value of the lower edge of the circular bearing plate (MPa)

H height of composite foundation model (mm)

l length of pile (mm)

l0 depth of bulging failure (mm)

l1 thickness of the upper cushion (mm)

l2 thickness of the underlying layer (mm)

m ratio of cyclic displacement (-)

n stress concentration ratio

P the top load of the gravel pile (N)

PR pile end counterforce (N)

Ps load of the soil surrounding the pile (N)

r initial radius of pile (mm)

S0 initial cross-sectional area of the gravel pile (mm2)

S1 cross-sectional area of pile in the bulging area (mm2)

u lateral displacement of pile (mm)

W width of composite foundation model (mm)

w total settlement of the composite foundation (mm)

w1 pile settlement (mm)

w2 settlement of upper cushion (mm)

w3 settlement of underlying layer (mm)

wa1 settlement value of numerical simulation

wPR settlement caused by the pile end counterforce (mm)

wPRa settlement of the soil at the pile end (mm)

wPs settlement caused by the load Ps(mm)

wta compression deformation in the non-bulging area (mm)

wtt settlement caused by the skin friction of pile (mm)

wtta settlement of the pile bottom caused by τty (mm)

wua compression deformation in the bulging area (mm)

wuu settlement caused by the skin friction of pile (mm)

wuua settlement of the pile bottom caused by τuy (mm)

βR correction factor for settlement calculation (-)

ζR correction factor for settlement calculation (-)

μa Poisson's ratio of the gravel pile (-)

μs Poisson's ratio of the sand (-)

σty the axial stress of pile body in non-bulging area

σuy the axial stress of pile body in the bulging area

τy pile skin frictional resistance (-)

φa internal friction angle of the gravel pile (°)

# Acknowledgments:

The research described in this paper was financially supported by the National Natural Science Foundation of China (Grant numbers No. 41772300)

# References

• Ayadat, T., Hanna, A.M. (2005). Encapsulated stone columns as a soil improvement technique for collapsible soil. Ground Improvement. 9(4): 137–147.
• Ali, K., Shahu, J.T., Sharma, K.G. (2012). Modeltests on geosynthetic-reinforced stone columns: A comparative study. Geosynthetics International. 19(4): 292–305.
• Afshar, J. N., & Ghazavi, M. (2014). Experimental studies on bearing capacity of geosynthetic reinforced stone columns. Arabian Journal for Science and Engineering. 39(3): 1559-1571.
• Brauns, J. (1978). The initial load of stone column in the clay foundation. The construction technology. 55(8): 263-271.
• Bongiorno, D., Cantarelli, E., DI PRISCO, C. G., Galli, A. (2006). Geo-reinforced sand columns: Small scale experimental tests and theoretical modelling. In Proceedings of the 8th international conference on geosynthetics. pp. 1685-1688.
• Borges, J.L., Marques, D.O. (2011). Geosynthetic-reinforced and jet grout column-supported embankments on soft soils: Numerical analysis and parametric study. Computers and Geotechnics. 38(7): 883-896.
• Castro, J., Sagaseta, C. (2011). Deformation and consolidation around encased stone columns. Geotextiles and Geomembranes. 29(3): 268-276.
• Guo, P.F., Wang, X., Yang, l.C., & Luo, H.W. (2015). Model test study on settlement characteristics of single pile in Loess under long-term vertical cyclic load. Journal of geotechnical engineering. 37 (3): 551-558. (in Chinese).
• Hosseinpour, I., Riccio, M., Almeida, M. S. (2014). Numerical evaluation of a granular column reinforced by geosynthetics using encasement and laminated disks. Geotextiles and Geomembranes. 42(4): 363-373.
• Hosseinpour, I., Almeida, M.S.S., Riccio, M. (2015). Full-scale load test and finite-element analysis of soft ground improved by geotextile-encased granu-lar columns. Geosynthetics International. 22(6): 428-438.
• Khabbazian, M., Kaliakin, V. N., Meehan, C. L. (2015). Column supported embankments with geosynthetic encased columns: validity of the unit cell concept. Geotechnical and Geological Engineering. 33(3): 425-442.
• Kahyaoğlu, M. R., Vaníček, M. (2019). A NUMERICAL STUDY OF REINFORCED EMBANKMENT-SUPPORTED BY ENCASED FLOATING COLUMNS. Acta Geotechnica Slovenica. 2.
• Lo, S. R., Zhang, R., Mak, J. (2010). Geosynthetic-encased stone columns in soft clay: a numerical study. Geotextiles and Geomembranes. 28(3): 292-302.
• Malarvizhi, S. N., Ilamparuthi, K. (2004). Load versus settlement of clay bed stabilized with stone and reinforced stone columns. In 3rd Asian Regional Conference on Geosynthetics. pp. 322-329.
• Murugesan, S., Rajagopal, K. (2010). Studies on the behavior of single and group geosynthetic encased stone columns. Journal of Geotechnical and Geoenvironmental Engineering. 136(1): 129-139.
• Niu, T., Liu, H., Ding, X., Zheng, C. (2018). Model tests on XCC-piled embankment under dynamic train load of high-speed railways. Earthquake Engineering and Engineering Vibration. 17(3): 581-594.
• Pulko, B., Majes, B., Logar, J. (2011). Geosynthetic-encased stone columns: analytical calculation model. Geotextiles and Geomembranes. 29(1): 29-39.
• Tan, X., Zhao, M., Chen, W. (2018), Numerical Simulation of a Single Stone Column in Soft Clay Using the Discrete-Element Method. International Journal of Geomechanics. 18(12): 1-12.
• Tang, Y., Xiao, S., & Yang, Q. (2020). The behaviour of geosynthetic-reinforced pile foundation under long-term dynamic loads: model tests. Acta Geotechnica. 1-21.
• Wang, Z., Jacobs, F., & Ziegler, M. (2016). Experimental and DEM investigation of geogrid–soil interaction under pullout loads. Geotextiles and Geomembranes. 44(3): 230-246.
• Yoo, C. (2015). Settlement behavior of embankment on geosynthetic-encased stone column installed soft ground–a numerical investigation. Geotextiles and Geomembranes. 43(6): 484-492.
• Zhang, M.J., Luo Q., Zhan X.Q., et al. (2013). Settlement analysis of settlement calculation of CFG pile composite foundation of high-speed railway. Rock and Soil Mechanics. (02): 219-225 + 245. (in Chinese)
• Zhang, L., & Zhao, M. (2015). Deformation analysis of geotextile-encased stone columns. International Journal of Geomechanics. 15(3): 1-10.

# Edited by

Editor: Rogério José Marczak

# Publication Dates

• Publication in this collection
28 Mar 2022
• Date of issue
2022