Building foundation piles are mainly affected by horizontal loads. Vertical bearing capacity and settlement are generally not controlled [1]. Both building and pile foundation codes require the following formula to check the horizontal bearing capacity:
_{ik}_{h}

The methods for calculating horizontally loaded piles mainly include the p-y curve method and the three-dimensional finite element method. The m method is a linear elastic method. This method is only used when the horizontal displacement is less than 10mm. Theoretically, the three-dimensional finite element method can better simulate the problem of pile-soil contact, but its modeling is more complicated [2]. This method is inconvenient to reflect the continuous change of soil parameters. The nonlinear p-y curve method is a common method for analyzing horizontally loaded piles. This method is theoretically rigorous and can reflect the gradual change of soil parameters with depth. The main forms of the p-y curve are the American API standard method, Resee method, Hohai University method, Tongji University method, ideal elastic-plastic method, sand strain wedge model, etc. The method for calculating the deflection of the pile body by the p-y curve is mainly the finite difference method, and the nonlinear finite element method and the mathematical programming method are also used. This paper introduces the Lagrangian multiplier of horizontal force and moment balance to derive the Lagrangian equation of the deflection curve array. At the same time, we use Newton's method to solve it. The method herein can be used for any form of the p-y curve. The typical stratum in my country's coastal waters is soft clay on the surface and sandy soil underneath [3]. The API specification gives detailed recommended methods for the p-y curves of these two soils. The soft soil p-y curve is shown in Figure 1. As the lateral displacement increases, the horizontal resistance first increases nonlinearly. This part will not change after reaching the ultimate resistance. The p-y curve of shallow soft clay also has a descending segment. In Figure 1, x is the depth, y is the lateral displacement, and _{R}_{50} is the lateral displacement when the soil around the pile reaches half of the ultimate resistance. _{50} = 2.5_{50}_{50} is the strain at the half of the maximum principal stress difference on the _{u}_{u}^{3}) of the sand foundation reaction force. It is obtained from a look-up table for the effective internal friction angle.

The horizontal load pile and the coordinate axis setting are shown in Figure 2. _{1} is the potential energy of the external load. _{2} is the potential energy stored by the bending of the pile body. _{3} is the reaction potential energy of the deformation of the foundation soil. When the pile foundation is not displaced, the potential energy of the external force is 0, then there is
_{1},
_{1} and

The bending potential energy of the pile body in the mechanics of materials is
_{1} and _{2} to each increase by a constant. The pile is an elongated structure [5]. The reaction potential energy of the foundation soil is equal to the work done by the foundation soil to reset from the deflection

The actual p-y curves are mostly piecewise functions. It is difficult to obtain analytical solutions for the functional extremum problems represented by equations (3) to (8). We subdivide the pile length in soil into N segments of equal length [6]. We mark the length of each segment as _{i}_{i}_{i}_{2} − _{1}) / _{i}_{i}_{i}_{i}_{1} + _{2}) / _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{50} < 0.1 segment as a straight line, and this difference cannot be reflected in Figure 1. After we obtain the array

The Newton iteration method can solve the Lagrangian equation (14) and the constraints. We denote the multiplier, _{N}_{+1} and _{N}_{+2} in the augmented deflection vector _{N}_{+1} and _{N}_{+2} in the Laplace equation vector ^{0} = (0, 0, ⋯,0)^{T} is taken as the initial deflection vector. We, using the iterative equation:
_{i}_{j}_{i}_{i}_{j}^{−6}) after 7 iterations. The process takes less than 1s.

Calculation parameters of static pile foundation

Source | Matlock | Red Steel City |
---|---|---|

EI/(kN·m^{2}) |
3.193×10^{4} |
1.332×10^{6} |

L/m | 12.81 | 35.6 |

k/m | 0 | 8.055 |

D/m | 0.324 | 1 |

_{s}^{3}) |
18 | 18 |

cu/kPa | 39.1 | 18 |

_{50} |
0.012 | 0.08 |

Clay layer thickness/m | 2 | |

26 |

We take the soft clay static load p-y curve recommended by the API specification as an example. The curve is divided into 3 segments in the actual calculation. where _{50} < 0.1 and _{50} > 8 are linear. The interval 0.1 ≤ _{50} ≤ 8 is the nonlinear segment of the ^{1/3} form. The derivative function of the p-y curve is also divided into 3 segments accordingly. Two of them are constants, and one is in the form of ^{2/3}. In the actual calculation, it is found that the method in this paper is completely unaffected by the nonlinearity and piece wiseness of the p-y curve [9]. At this point, the deflection array quickly converges. We used the static load p-y curve recommended by the API specification to obtain the soil surface displacement and the maximum bending moment curve of the pile body under different loads. The calculation results are shown in Figure 3. The red steel city pile foundation in the sand is within the load range of Figure 3. Its loading curve is linear. The loading curves of Matlock piles in soft clay are also significantly nonlinear.

This paper studies the variation of soil surface displacement and rotation angle under different load heights. We can realize a series of load height change processes by adding a loop about the parameter

The displacement ratio-height ratio curve varies with pile-soil conditions and loads. There is no fixed relationship between the two. The relationship between the angle ratio and

Matlab as mentioned above is used to study design parameters for horizontally loaded piles with large bending moments. We changed the cycle parameter to the outer diameter D of the pipe pile. The horizontal load is 2.3MN, and the bending moment at the top of the pile is 100MN·m. The parameter design needs to meet the deformation control requirements under the normal service limit state. After the parameter design is completed, we carry out the check calculation of the limit state of the bearing capacity. The soil layer is layered soil. The upper layer is soft clay with a layer thickness of 10m. Undrained shear strength _{u}_{50} = 0.01. The saturation severity is 18^{3}. The bottom layer is sandy soil. Effective internal friction angle ^{3}. The sandy soil was judged to be non-liquefiable. The design pile foundation adopts a steel pipe pile. The elastic modulus of steel is 210GPa. Note the diameter of the hollow part of the pipe pile as d. The common void ratio of steel pipe piles is F

This section calculates the pile top deformation of steel pipe piles with different diameters under normal service and bearing capacity conditions. The pile length ranges from 30 to 60 m. The wind and wave loads acting on the offshore piles are all long-term cyclic loads. Therefore, we use the p-y curve of the periodic load in the API specification. Figures 5 and 6 show pile top deformation calculation results under normal and bearing capacity conditions, respectively. In this example, the pile length L can be taken as 40m. The pile diameter is determined by the intersection of the deformation curve of L = 40m and the control line in Figures 5 and 6. The horizontal dotted line in the figure is the control line. It can be seen from Figure 5 that the minimum pile diameter is 5m under the requirement of this turning angle. Although the displacement of the pile top increases significantly under the bearing capacity condition, the minimum pile diameter required by the experiment is also 5m. When the soil surface displacement is limited to 30mm, the minimum pile diameter is 5.8m. The minimum pile diameter is 6.25m when the displacement is limited to 25mm.

The rotation angle and settlement requirements are relatively easy to meet, but the limit of horizontal displacement is not easy to meet. This indicates that more explicit horizontal displacement criteria are required for building foundation design. A horizontal displacement limit of 25mm will result in a mud surface turning angle that is half the value allowed by the specification. In this paper, the horizontal displacement limit is 30mm, and the design pile diameter is 5.8m. The soil surface rotation angles under the normal service limit and bearing capacities are 0.14° and 0.30°, respectively.

We derive the variational form of the deflection curve problem for a horizontally loaded single pile. In this paper, the energy expression is discretized into the extreme value problem of the Lagrangian function. At the same time, the article derives the corresponding Lagrange equations. We use Newton's method to solve the equation system to obtain the pile displacement and bending moment at different depths. This process can be realized by Matlab matrix operation. This method applies to any nonlinear or piecewise form of the p-y curve. The algorithm calculates quickly and can simulate continuous load and pile parameters changes. Through this algorithm, we realize the batch calculation of horizontally loaded piles. By calculating loads of two kinds of horizontally loaded piles, it is found that the deformation caused by the horizontal force of the single-pile structure with the same load height only accounts for a very small part. The bearing capacity check needs to consider horizontal load and bending moment combined.

#### Calculation parameters of static pile foundation

Source | Matlock | Red Steel City |
---|---|---|

EI/(kN·m^{2}) |
3.193×10^{4} |
1.332×10^{6} |

L/m | 12.81 | 35.6 |

k/m | 0 | 8.055 |

D/m | 0.324 | 1 |

_{s}^{3}) |
18 | 18 |

cu/kPa | 39.1 | 18 |

_{50} |
0.012 | 0.08 |

Clay layer thickness/m | 2 | |

26 |

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