# Analysis of Internal Forces and Deformation for a Single Pile in Layered Soil Based on the p-y Curve Method

^{1}

^{2}

^{*}

## Abstract

**:**

_{u}is taken to be constant, and the diversity of soil layers around the pile side are also not considered. Second, in the analysis of the internal forces and deformation of the pile, the influence of the vertical load from the top of the pile and the self-weight of the pile, are both ignored. Third, in the analysis of internal forces and deformation, the pile side soil is set equivalent to a homogeneous soil layer, and the layering of the soil is not considered at all. In order to study the nonlinear problem of internal forces and deformation of a single pile in layered soil in greater detail, this paper analyzes a calculation model based on Wang’s calculation model, and compares several commonly used p-y curve calculation models. An internal force and deformation analysis model for a laterally loaded single pile, that explicitly considers the second-order effect is then established, by considering the differences between p-y curves of different soil types, as well as the change in C

_{u}with depth. The differential equation of pile deflection for a single pile in layered soil is also presented, together with the corresponding finite differential solution algorithm program. This model was validated using a horizontal load test of a pile, and comparison of the calculated results with the measured results shows that the method outperforms existing p-y curve methods for deformation and internal force analysis of horizontally loaded piles.

## 1. Introduction

_{u}is taken as fixed, ignoring the influence of its change with depth on the p-y curve. Second, when the p-y curve is used to calculate the forces on a laterally loaded pile, the influence of the vertical load on the pile top is not considered. Third, the soil mass is divided into soft clay (C

_{u}< 96 kPa) and hard clay (C

_{u}≥ 96 kPa) only, according to the undrained shear strength C

_{u}value, and then analyzed assuming a uniform soil layer, which ignores the possible coexistence of soft and hard clay, as well as other cohesionless soil that is often observed in practice. Based on field tests and Wang’s method, this paper corrects for the above three problems, in order to study the nonlinear problem of internal forces and deformation of a single pile in layered soil in greater detail. A calculation method for the internal forces and displacement of the pile body in layered soil is proposed, based on the improved p-y curve method, and an engineering application analysis is also conducted.

## 2. Calculation Model and Solution Method

#### 2.1. Calculation Model and Basic Equation

_{0}, bending moment M

_{0}, and vertical load N

_{0}are assumed to be applied to the pile top. The other basic assumptions are enumerated below.

_{0}is the calculated width of the pile; and c

_{z}is the horizontal resistance coefficient of the foundation. When the p-y curve is used to describe the pile side resistance, c

_{z}is the secant modulus of the p-y curve, ${c}_{z}=p/y$, kN/m

^{3}.

#### 2.2. Establishment and Solution of the Difference Equation

#### 2.2.1. The Difference Equation

#### 2.2.2. Boundary Conditions

_{0}, bending moment M

_{0}, and axial force ${N}_{0}={\left(N\right)}_{3}$.

#### 2.2.3. Solution Ideas

_{n}

_{+2}to y

_{3}, can be obtained, and then the expressions for node displacements y

_{3}, y

_{4}, and y

_{5}obtained by the above method can be substituted into the boundary condition, Equations (5) and (6), for the pile top, to obtain the expressions of node displacements y

_{1}and y

_{2}, so that all node displacements, y

_{i}, can be inversely deduced.

#### 2.2.4. Solving Steps

_{z}

_{0}, of the soil at the node of the pile element. In the third step, the finite difference method is used to solve the displacement of each node, y

_{i}, and the fourth step is to determine the pile side soil resistance, p

_{i}, from y

_{i}, according to the p-y curve of the different soil layers. Next, step 5 is to obtain a new c

_{zi}from ${c}_{zi}={p}_{i}/{y}_{i}$. Finally, the sixth step is to use the finite difference method to calculate the new node displacements according to the new c

_{zi}. If the displacement difference of each node obtained by two iterations is within the error limit ε, then the process is stopped, otherwise steps 4 to 6 are repeated until the error limit is reached.

#### 2.2.5. Programming

#### 2.3. p-y Curve Characteristics and Selection

_{z}, varies with depth. When the soil displacement at the pile side is large, the nonlinear relationship between resistance, p, and displacement, y, is often best described by a p-y curve.

#### 2.3.1. Characteristics and Application of Common p-y Curves

_{u}is the standard value of the ultimate horizontal soil resistance per unit area at a depth below the surface, and y

_{50}is the lateral horizontal displacement of the pile when the soil around the pile reaches half of the ultimate horizontal soil resistance.

#### 2.3.2. Comparison of Applications of p-y Curves for Homogeneous Soil

^{3}, and the undrained shear strength, C

_{u}, was 17 kPa. Comparisons of the theoretical pile top displacement and the maximum pile shaft bending moment with the measured results are shown in Table 2 and Table 3.

_{50}= Aε

_{50}D, A = 0.05(1/D + 4), and ε

_{50}is the strain value when the maximum principal stress difference is half: p

_{u}= KAC

_{u}D, $K=\frac{100D}{3+8.3D}+\frac{4z/D}{1+0.4z/D}$.

## 3. Improvement of Wang’s p-y Curve

#### 3.1. The Change in Undrained Shear Strength, C_{u}, with Depth

_{u}in Table 1 is often described by C

_{u}, the undrained shear strength C

_{u}, has a great influence on the results, when a p-y curve is used to calculate the internal forces and lateral displacement of piles. The Code for Pile Foundations of Port Engineering [16] gives the p-y curve for C

_{u}< 96 kPa and C

_{u}> 96 kPa, respectively. ε

_{50}and C

_{u}is taken as a fixed value during calculation. According to the basic principle of soil mechanics [17], even for the same soil, undrained shear strength C

_{u}increases with depth, and so is not a fixed value. At present, there are three main ways to obtain the undrained shear strength, C

_{u}: (1) direct measurement by field or indoor test; (2) the effective consolidation stress method [18], which considers the failure of a soil mass to be a sudden undrained shear process, where the water content and strength of soil elements remain unchanged; here, the undrained shear strength can be measured by the effective consolidation stress on the potential fracture surface before failure; and (3) the empirical fitting method, which is based on the main factors that affect the undrained shear strength, C

_{u}, of the soil mass, and can give the C

_{u}according to several parameters that are easy to measure, hence this method’s name.

_{50}is obtained from Table 4:

#### 3.2. The Case of Layered Soil

_{u}, and puts forward the construction method of p-y curve clusters for dealing with different types of soils. However, most current studies do not consider the coexistence of the three types of soil. Thus, a p-y curve cluster in layered soil can be established by considering the differences in p-y curves between different soil layers, as follows.

^{3}); ψ is the calculation coefficient; and d is the pile diameter or pile width.

## 4. Application of the Improved p-y Curve Method in Layered Soil

#### 4.1. Overview of the Site Horizontal Load Test

^{3}. The test device is shown in Figure 4, which shows that the horizontal displacement of piles on the ground was measured with a dial indicator. Two dial indicators were installed symmetrically on the horizontal force action surface, at heights of 10 cm, 60 cm, and 95 cm above the bottom of the pile cap, with a spacing of 1.0 m. Pile A was a test pile, where a reinforcement meter was symmetrically arranged on the reinforcement cage, to measure the strain on the pile body. A total of 10 test sections were set up at a depth of 34 m below the ground, with 20 elements in total, according to the principle of “upper dense” and “lower sparse”. The strain gauge on the top layer was 50 cm from the ground, and the spacing below was 200 cm, 250 cm, 300 cm, and 400 cm.

#### 4.2. Validation

## 5. Conclusions

_{u}, in the clay layer with depth, the p-y curve of the Wang method was the most applicable to our model and was therefore used for subsequent analysis.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Comparison between the calculated and measured value of the pile body displacement with depth.

**Figure 6.**Comparison between the calculated and measured value of the pile bending moment with depth.

Literature Source | p-y Curve Expression | Features and Applicability |
---|---|---|

Matlock [1] | $p=\{\begin{array}{l}0.5{p}_{u}{\left(\frac{y}{{y}_{50}}\right)}^{1/3}\left(y\le 8{y}_{50}\right)\\ {p}_{u}\left(y>8{y}_{50}\right)\end{array}$ | The curve is in the form of a subsection function and is applicable to soft clay. y_{50} is the horizontal displacement of the pile when the ultimate resistance reaches half [16]. |

Sullivan et al. [3] | $p=\{\begin{array}{l}0.5{p}_{u}{\left(\frac{y}{{y}_{50}}\right)}^{1/3}\left(y\le 8{y}_{50}\right)\\ {p}_{u}\left(y>8{y}_{50}\right)\left(\mathrm{z}>12{b}_{0}\right)\\ {p}_{u}\left(F+\frac{\left(1-F\right)z}{12{b}_{0}}\right)\left(y>30{y}_{50}\right)\left(z<12{b}_{0}\right)\end{array}$ | The curve is in the form of a piecewise function and is applicable to any clay. F is a parameter such that, when the curve describes soft clay, F = 1, and when the curve describes hard clay, F < 1 |

Wang Method [4] | $p=\{\begin{array}{l}\frac{y/{y}_{50}}{a+by/{y}_{50}}{p}_{u}\left(y\le \beta {y}_{50}\right)\\ {p}_{u}\left(y>\beta {y}_{50}\right)\end{array}$ | The curve is hyperbolic and suitable for soft clay and hard clay. Here, a = β/(β − 1), b = (β − 2)/(β − 1), where the value of β is obtained from a geotechnical triaxial undrained test. In the absence of test data, the value used for soft clay is 9 and is 12 for hard clay. |

Zhang Method [5] | $p=\{\begin{array}{l}\left[{F}_{s}+\left(1-{F}_{s}\right)z/{z}_{r}\right]{p}_{u}\left(z\le {z}_{rs}\right)\\ {p}_{u}\left(z>{z}_{rs}\right)\\ 0.5{p}_{u}{\left(y/{y}_{50}\right)}^{1/3}\end{array}$ | The curve is in the form of a piecewise function and is suitable for soft clay, such as that in the middle and lower reaches of the Yangtze River. Fs is the reduction factor, which is related to the nature and load of the soil, and Zrs is the influence depth, taken as 1/4 of the pile length |

Load Level | Project | p-y Curve Model | Field Measurement Results | |||
---|---|---|---|---|---|---|

Sullivan | Wang Method | Matlock | Zhang Method | |||

200 kN | Pile top displacement/mm | 11.76 | 17.11 | 20.72 | 13.43 | 19.95 |

Error percentage (%) | −41.06 | −14.25 | 3.88 | −32.68 | ||

300 kN | Pile top displacement/mm | 24.90 | 37.16 | 44.31 | 25.70 | 42.34 |

Error percentage (%) | −41.19 | −12.24 | 4.65 | −39.30 | ||

350 kN | Pile top displacement/mm | 33.11 | 52.31 | 59.19 | 32.92 | 75.49 |

Error percentage (%) | −56.14 | −30.71 | −21.59 | −56.39 |

Load Level | Project | p-y Curve Model | Field Measurement Results | |||
---|---|---|---|---|---|---|

Sullivan | Wang Method | Matlock | Zhang Method | |||

200 kN | Maximum bending moment of pile shaft/kN.m^{2} | 517.7 | 517.6 | 623.7 | 685.1 | 628.2 |

Error percentage (%) | −17.6 | −17.6 | −0.7 | 9.1 | ||

300 kN | Maximum bending moment of pile shaft/kN.m^{2} | 876.1 | 924.3 | 1045.1 | 1111.3 | 873.5 |

Error percentage (%) | 0.3 | 5.8 | 19.6 | 27.2 | ||

350 kN | Maximum bending moment of pile shaft/kN.m^{2} | 1068.3 | 1172.1 | 1269.5 | 1333.9 | 1234.5 |

Error percentage (%) | −13.5 | −5.1 | 2.8 | 8.0 |

C_{u} (kPa) | 12–24 | 24–48 | 48–107 | 107–215 | 215–430 |
---|---|---|---|---|---|

ε_{50} | 0.02 | 0.01 | 0.007 | 0.005 | 0.004 |

Type of Soil Layer | Thickness/m | Mass Density/(kg/m^{3}) | Compression Modulus Es/MPa | Poisson’s Ratio | Friction Angle/(°) |
---|---|---|---|---|---|

Miscellaneous fill | 4.8 | 1845 | - | 0.38 | 12.8 |

Silt | 8.3 | 1676 | 2.09 | 0.40 | 10.2 |

Silty clay with silt | 4.8 | 1915 | 4.31 | 0.35 | 12.8 |

Silty clay | 12.6 | 1769 | 3.24 | 0.35 | 11.2 |

Silty clay | 12.7 | 1777 | 4.63 | 0.30 | 12.3 |

Silty clay | 1.8 | 1810 | 3.77 | 0.30 | 11.3 |

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**MDPI and ACS Style**

Wei, L.; Wang, J.; Zhai, S.; He, Q.
Analysis of Internal Forces and Deformation for a Single Pile in Layered Soil Based on the *p-y* Curve Method. *Appl. Sci.* **2023**, *13*, 3520.
https://doi.org/10.3390/app13063520

**AMA Style**

Wei L, Wang J, Zhai S, He Q.
Analysis of Internal Forces and Deformation for a Single Pile in Layered Soil Based on the *p-y* Curve Method. *Applied Sciences*. 2023; 13(6):3520.
https://doi.org/10.3390/app13063520

**Chicago/Turabian Style**

Wei, Limin, Junpeng Wang, Shun Zhai, and Qun He.
2023. "Analysis of Internal Forces and Deformation for a Single Pile in Layered Soil Based on the *p-y* Curve Method" *Applied Sciences* 13, no. 6: 3520.
https://doi.org/10.3390/app13063520