^{1}

^{2}

^{1}

^{1}

^{3}

^{3}

^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

^{3}

The settlement calculation of composite foundation bidirectionally reinforced by piles and geosynthetics is always a difficult problem. The key to its accuracy lies in the determination of pile-soil stress ratio. Based on the theory of double parameters of the elastic foundation plate, the horizontal geosynthetics of composite foundation are regarded as the elastic thin plate, and the vertical piles and surrounding soil are regarded as a series of springs with different stiffness. The deflection equation of horizontal geosynthetics considering its bending and pulling action is obtained according to the static equilibrium conditions. The equation is solved by using Bessel function of complex variable, and the corresponding deflection function of horizontal geosynthetics is deduced. Then, the calculation formula of pile-soil stress ratio and settlement of composite foundation is derived by considering the deformation coordination of pile and soil. The results of engineering case analysis show that the theoretical calculation results are in good agreement with the measured values, which indicates that the proposed method is feasible and the calculation accuracy is good. Finally, the influence of composite modulus of horizontal geosynthetics, tensile force of geosynthetics, and pile-soil stiffness ratio on pile-soil stress ratio and settlement is further analyzed. The results show that the pile-soil stress ratio increases with the increase of the composite modulus of the horizontal geosynthetics, the tensile force of geosynthetics, and the pile-soil stiffness ratio, and the settlement decreases with the increase of the composite modulus of the horizontal geosynthetics, the tensile force of geosynthetics, and the pile-soil stiffness ratio. When the flexural stiffness of the horizontal geosynthetics is small, the influence of the tensile action of the geosynthetics on the pile-soil stress ratio and settlement of the composite foundation cannot be ignored.

In recent years, with the rise of highway and railway construction in China, the problems of soft soil foundation have become increasingly prominent, the bearing capacity of the foundation is insufficient, the settlement is too large, and uneven settlement is particularly serious. The composite foundation reinforced by piles and geosynthetics, a new type of soft foundation treatment consisting of vertical piles and horizontal geosynthetics, has a good effect on the treatment of the abovementioned foundation problems, and the foundation treatment method has simple construction and rapid progress. In recent years, it has been widely used in railway soft foundation treatment [

Research on the working mechanism of composite foundation has achieved many results so far [

However, the above literature all assume the deformation of the geosynthetics in advance, which cannot really reflect the stress state of the geosynthetics, which can easily lead to a large error between the calculated results and the measured results. For this reason, Tan et al. [

From the above analysis, we can know how to comprehensively consider the flexible raft effect, net effect, and pile-soil joint deformation of cushion without assuming the deformation of geosynthetics is the key to accurately solve the pile-soil stress ratio and settlement of composite foundation reinforced by piles and geosynthetics. Therefore, on the basis of the above research work, based on the Filonenko–Borodich two-parameter foundation model [

Take the typical unit within the influence range of the single pile as shown in Figure _{d}, the diameter of single pile reinforcement range is _{e}, square pile _{e} = 1.13_{d}, and piles are arranged as the shape of plum blossoms: _{e} = 1.05_{d}.

Sketch map of composite foundation.

In order to further simplify the calculation, the following assumptions are made:

As shown in Figure

where

It is assumed that the vertical supporting force _{p} of the pile to the thin plate is uniformly distributed, and according to references [

The distribution of embankment load caused by differential settlement is not considered, that is, the embankment load is uniformly distributed [

Calculation model for composite foundation.

Let _{p}, the control differential equation of the horizontal geosynthetics is as follows:

Let

The general solution of the homogeneous equation corresponding to equation (_{N} and _{N} are the first and second kinds of _{1}, _{2}, _{3}, and _{4} are undetermined constants.

Under the action of uniformly distributed load

Combining formula (

Let _{s} is the spring coefficient of soil foundation between piles.

To simplify the calculation,

The homogeneous equation corresponding to equation (

It can be seen from formula (_{21} and _{22} are divided into real and imaginary numbers according to the positive and negative of ^{2} − 4_{s}. The solution of equation (

(1) When ^{2} ≧ 4_{s}, order:

The general solution of equation (_{n} is the second kind of _{5}, _{6}, _{7}, and _{8} are undetermined constants.

Under the action of uniformly distributed load _{s}, and the combination formula (^{2} ≧ 4_{s}:

When ^{2} < 4_{s}, the general solution of equation (_{n} and _{n} and _{5}, _{6}, _{7}, and _{8} are undetermined constants.

In order to determine the bending function of the thin plate, it is also necessary to compare the unknown parameters _{1}, _{2}, …, _{8} and _{p} in equations (

As the turning angle of the thin plate at the center of the circular plate is 0, _{2} = _{3} = 0 can be obtained. At the pile-soil junction, the continuity conditions of thin plate deflection, rotation angle, radial moment, and shear force are as follows:

In addition, according to hypothesis (

The value of pile top reaction _{p} is the sum of the upper embankment load and the load transferred from the cushion to the pile, that is,

According to reference [_{e}/2 of the element are as follows:

At the same time, according to the bending functions (

Within the range of the top of the pile, that is,

Within the range of the top of the pile, that is,

(1) When ^{2} ≧ 4_{s},

(2) When T^{2} ≥ 4_{s},in order to simplify the expression, order

Combining formula (_{MN}(_{n}(_{n}(

According to the above analysis, the simultaneous equations (_{1}, _{4}, …, _{8} and _{p}. Thus, the deflection expression of the plate is obtained.

From the above analysis, if the strain of the steel bar is known before the calculation, the tension of the steel bar can be calculated directly by using formula (

According to the above research results, the maximum deflection _{e}/2) of geosynthetics is selected as the settlement

First of all, assuming an initial tension value _{0} (_{0} can be a very small value), it is substituted into equations (_{e}/2) are obtained.

Then, take the value of _{e}/2) obtained before as S, replace (_{e}/2).

Select an error value and compare the obtained _{e}/2) with the previous one. If the comparison results are less than the error value, stop the calculation; otherwise, continue the iterative calculation according to the steps of second and third.

The flowchart of the above steps is shown in Figure

Flowchart of calculation.

After obtaining the flexure function and _{p} of the geosynthetics, it can be seen from the calculation model shown in Figure _{s} at the top of the soil between piles can be obtained as follows:

Then, the pile-soil stress ratio

Reference [

In order to comprehensively consider the nonlinearity in the process of pile deformation, the deformation stiffness of pile is taken as the secant slope of the load test

In order to comprehensively consider the nonlinearity in the process of soil deformation between piles, the coefficient of soil foundation between piles _{s} takes the secant slope on the load test _{s} is the deformation modulus of soil, the weighted average of multilayer soil is depth, and _{s} is the thickness of soil layer.

From the point of view of the joint action of the geosynthetics and the bulk material pile, the thickness of the composite structure formed by the geosynthetics and its wrapped filler is taken as the thin plate thickness if it is a multilayer grille, that is, the distance between the top layer and the bottom grille. If it is a geotechnical cell, the thickness of the geotechnical cell can be taken directly.

For the grid cushion, Zheng et al. [

Technical treatment of soft soil foundation according to foundation reinforced by piles and geosynthetics in ^{3}, the filling height of roadbed is 10 m, and the crushed stone cushion is 25 kN/m^{3}. According to the static load test, the stiffness coefficient of pile is _{p} = 2000 kN/m. The thickness of the thin plate is 0.3 m, the composite elastic modulus

The parameters for calculation.

Station | Type of soils | Depth, _{s} (m) | Deformation modulus, _{s} (MPa) | Grid tension, | ^{2} − 4_{s} |
---|---|---|---|---|---|

Soft soil | 7.0 | 3.1 | 60.3 | <0 | |

Soft soil | 10.6 | 3.1 | 82.7 | <0 |

The comparison between the calculated and measured values of the central settlement of the embankment and the pile-soil stress ratio under the geogrid geosynthetics is shown in Table

Results of settlement and pile-soil stress ratio.

Station | Settlement, | Pile-soil stress ratio, | ||
---|---|---|---|---|

Calculated value | Measured value | Calculated value | Measured value | |

28.8 | 26.6 | 4.53 | 3.87 | |

33.9 | 32.0 | 6.64 | 5.82 |

The development trend of foundation settlement

Comparison between calculated and measured results of embankment settlement.

As can be seen from Tables

Results of settlement and pile-soil stress ratio.

Items | ^{2} − 4_{s} | Pile-soil stress ratio, | Settlement, |
---|---|---|---|

Measured value | — | 6.5 | 5.3 |

Method of literature [ | — | 5.8 | — |

The proposed method | <0 | 6.1 | 5.7 |

The test section of an expressway lying on soft soil foundation in Hunan is treated with geocell + mixing piles. The pile is arranged in the form of plum blossoms, and its diameter is 0.50 m with its spacing 1.2 m. The pile top is filled with a thick 30 cm sand cushion, and the center is equipped with a geotechnical cell with the thickness of 10 cm. The foundation is muddy clay, the load of the upper fill is 20 kN/m^{3}, and the filling height of the roadbed in the test section is 4 m. The measured settlement value _{p} is 2355 kN/m. After treatment, the coefficient of soil foundation between piles _{s} is 1024 kN/m^{3}, the thickness of thin plate

The pile-soil stress ratio and settlement are calculated as shown in Table

It can be seen from Table

In order to further discuss and analyze the effects of geosynthetics composite elastic modulus, grille tension and pile-soil stiffness ratio on the settlement and pile-soil stress ratio of composite foundation reinforced by piles and geosynthetics, based on the parameters in example 2, the corresponding parameters are analyzed according to the abovementioned factors.

Without considering the influence of tension of geosynthetics, _{p} = 4_{p}/(^{2}) is introduced to characterize the comprehensive influence of pile deformation stiffness and replacement ratio, and other parameters remain unchanged.

As shown in Figure

Relation between settlement and _{p}/_{s}.

As can be seen from Figure

Relation between pile-soil stress and _{p}/_{s}.

As can be seen from Figures

Relation between settlement and tension.

Relation between pile-soil stress and tension.

In addition, the effect of

Based on theory of the Filonenko–Borodich two-parameter elastic foundation model, the horizontal geosynthetics of composite foundation are regarded as the elastic thin plate, and the vertical piles and surrounding soil are regarded as a series of springs with different stiffness. The deflection equation of horizontal geosynthetics considering its bending and pulling action is obtained according to the static equilibrium conditions. The equation is solved by using Bessel function of complex variable, and the corresponding deflection function of horizontal geosynthetics is deduced. The calculation method for pile-soil stress ratio and settlement of composite foundation is derived by considering the deformation coordination of pile and soil.

The settlement of composite foundation decreases with the increase of pile-soil stiffness ratio, geosynthetics tension, and composite elastic modulus of geosynthetics. The pile-soil stress ratio of composite foundation increases with the increase of pile-soil stiffness ratio, elastic modulus of geosynthetics, and its tension. When the bending stiffness of geosynthetics is small, the influence of tensile action of reinforcement on pile-soil stress ratio and settlement cannot be ignored.

The key to calculating with this method is to accurately measure the composite elastic modulus of geosynthetics-reinforced cushion, the strain of geosynthetics, and the stiffness of pile and soil.

Although this presented method does not rely on preassumed deformation when calculating the pile-soil stress ratio and settlement of composite foundation, the net effect of geosynthetics-reinforced cushion can be considered only when the strain of geosynthetics is known, which brings defects to the convenience of calculation. The proposed method still cannot consider the coupling relationship between deformation of geosynthetics-reinforced cushion and tension of geosynthetics, which can be carried out in the next work.

The tension of geosynthetics

The average strain of geosynthetics

The tensile stiffness of geosynthetics

The bending stiffness of the thin plate

The elastic modulus of the thin plate

Poisson’s ratio of the thin plate

The thickness of the thin plate

_{N}:

The first kind of N-order Bessel functions

_{N}:

The second kind of N-order Bessel functions

The distributed load

_{p}:

The supporting force

_{s}:

The spring coefficient of surrounding soil

_{N}:

The second kind of N-order virtual variable Bessel function

The real part of the first kind of Hankel function of order N

The real part of the first kind of Bessel function of order N

_{n}:

The imaginary part of Hankel function of order N

The imaginary part of Bessel function of order N

_{p}:

The value of pile top reaction

The relevant fitting parameters

_{0}:

The initial tension value

The settlement

_{s}:

The average vertical stress at the top of the soil between piles

_{p}:

The stiffness coefficient of pile

_{s}:

The deformation modulus of soil

_{s}:

The thickness of soil layer

The calculating thickness of cushion

The composite elastic modulus of cushion

_{1},

_{2},

_{3},

_{4}:

The undetermined constants

_{5},

_{6},

_{7},

_{8}:

The undetermined constants.

The [Data Type] data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work described in this paper was fully supported by these grants from the National Natural Science Foundation of China (award nos. 51778227, 51308208, and 41372303), the Provincial Natural Science Foundation of Hunan (award nos. 2015JJ3069 and 18C0311), the Youth Talent Plan Program of Hunan (award no. 2016RS3032), and the Postgraduate Scientific Research Innovation Project of Hunan Province (award no. CX20200992).