Single-pile p-y model bearing characteristic analysis method considering long-term horizontal cyclic load

Abstract

The invention relates to the field of pile foundation bearing analysis, and discloses a long-term horizontal cyclic load-considered single pile p-y model bearing characteristic analysis method, which comprises the following steps of: obtaining test data of a single pile under the action of a horizontal cyclic load; on the basis of test data, a global optimization algorithm is adopted to calibrate parameters, and a foundation reaction modulus empirical formula capable of reflecting the comprehensive influence of the cycle index, the burial depth and the loading force amplitude is established; establishing a finite element model of a single pile; in the finite element numerical analysis process, the rigidity of the nonlinear spring unit is dynamically updated in real time through a user-defined subprogram according to the current cycle index; and evaluating the bearing characteristic of the single pile based on a numerical analysis result. According to the method, the cyclic evolution behavior of the soil body rigidity in pile-soil interaction can be accurately simulated, the bearing characteristic of a single pile under long-term cyclic loading is accurately evaluated, and the analysis efficiency is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of pile foundation bearing analysis, in particular to a method for analyzing bearing characteristics of a single pile PY model considering long-term horizontal cyclic loads. Background Art

[0002] As an important foundation form, single pile foundations are widely used in bridge engineering, port terminals, offshore platforms, and the rapidly developing offshore wind power generation industry in recent years. In these engineering structures, single pile foundations are often subjected to complex environmental loads, especially long-term horizontal cyclic loads caused by wind, waves, and currents. Such long-term cyclic loading can cause significant changes in the mechanical properties of the soil surrounding the pile, such as stiffness degradation and increased cumulative deformation, which in turn affects the bearing capacity and normal performance of the single pile foundation. Therefore, accurately evaluating the bearing characteristics of single piles under long-term horizontal cyclic loading is crucial to ensuring the safety and economic efficiency of engineering structures.

[0003] Currently, the main methods for analyzing the horizontal load characteristics of single piles include the py curve method and the finite element method. The py curve method simulates the pile-soil interaction through a series of nonlinear springs. It has the advantages of clear concepts and simple calculations, and has been widely used in engineering practice. However, existing standard py curve models, such as the py curve recommended by the American Petroleum Institute (API), are mostly based on static or a small number of cyclic loading tests under specific soil conditions. They do not take into account the evolution of the mechanical behavior of soil under different soil conditions and thousands or even more cyclic loads. It is difficult to accurately reflect the cumulative effects and complex changes of soil stiffness and strength under long-term cyclic loading. Directly applying these py curves to analyze the response of single piles under long-term cyclic loading often limits the reliability of the results.

[0004] Finite element methods can more accurately simulate pile-soil interactions and the complex constitutive behavior of soil. However, the accuracy of the analysis is highly dependent on the rationality of the soil constitutive model and pile-soil contact model used. For long-term cyclic loading, establishing a constitutive model that can accurately describe complex phenomena such as cyclic softening, hardening, pore pressure accumulation, and deformation accumulation in the soil is itself a huge challenge. At the same time, obtaining the parameters required for these complex constitutive models often requires a large number of special and costly indoor and outdoor tests, which to some extent limits their widespread application in conventional engineering design.

[0005] In addition, when dealing with pile-soil interaction, existing analysis methods are still not perfect in quantitatively describing the comprehensive influence of key factors such as the number of cycles, load amplitude, and soil depth on the foundation reaction characteristics. Many models either simplify the coupling of these factors or fail to provide a systematic method to extract parameterized models that can dynamically reflect these influences from test data. This means that the adaptability and prediction accuracy of existing methods need to be improved when faced with changing load histories and site conditions. In actual engineering design, it is often necessary to analyze multiple working conditions. If it relies entirely on complex manual modeling and parameter adjustments, its analysis efficiency will be difficult to meet the needs of rapid engineering iteration.

[0006] Therefore, there is an urgent need to develop a single pile bearing characteristics analysis method that can more accurately consider the effects of long-term horizontal cyclic loads, effectively combine experimental data with numerical simulations, and be conveniently applied to engineering practice. Summary of the Invention

[0007] The present invention provides a method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic loads. The method aims to address the shortcomings of existing PY curve models in simulating pile-soil interaction under long-term horizontal cyclic loads, namely, the failure to fully consider the dynamic changes of soil stiffness with the number of cycles, burial depth and load amplitude, as well as the high calculation cost and low efficiency of traditional three-dimensional finite element methods, thereby providing a more accurate and efficient means for analyzing the bearing characteristics of single pile foundations.

[0008] To achieve the above objectives, the present invention is implemented through the following technical solutions:

[0009] The present invention provides a method for analyzing the bearing characteristics of a single pile PY model considering long-term horizontal cyclic loads, comprising the following steps:

[0010] First, obtain test data on a single pile subjected to horizontal cyclic loading under specific site conditions. This test data, which forms the basis of the analysis presented in this method, should at least cover the pile displacement and bending moment response at different depths (z), as well as the corresponding loading force amplitude (F) and the cumulative number of cycles (N). This data can be obtained through in-situ field testing, indoor model testing, or centrifuge model testing.

[0011] Secondly, based on the acquired test data, a quantitative functional relationship model between the foundation reaction modulus (k) and the aforementioned key influencing factors, namely the number of cycles (N), burial depth (z) and loading force amplitude (F), was systematically established.

[0012] This step specifically includes:

[0013] 1. Based on the pile bending moment distribution measured in the test, the soil reaction force (p) at different depths, different numbers of cycles and load amplitudes is calculated using the mechanical inverse analysis method.

[0014] 2. Based on the calculated soil reaction (p) and the corresponding pile horizontal displacement (y), determine the corresponding foundation reaction modulus (k).

[0015] 3. In view of the variation of the foundation reaction modulus (k) with the number of cycles (N), a mathematical function that can reflect the cyclic softening or hardening effect (for example, a logarithmic function or its combination) is selected for fitting; in view of the variation of the foundation reaction modulus (k) with the burial depth (z), a power function is usually used for fitting; in view of the influence of the loading force amplitude (F) on the foundation reaction modulus, a polynomial function (for example, a quadratic polynomial) can be used to characterize it.

[0016] 4. Combine the functional relationships of the above-mentioned single influencing factors to form a comprehensive expression for the foundation reaction modulus (k), which includes multiple undetermined parameters. Subsequently, using all valid test data, a global optimization algorithm (e.g., differential evolution algorithm) is used to simultaneously optimize and calibrate these undetermined parameters to ensure that the established functional relationship model can optimally match the test results. In some cases, in order to more accurately describe the different mechanical behaviors of the soil in the initial loading stage (N = 0) and the long-term cyclic loading stage (N > 0), the established foundation reaction modulus (k) model can be in the form of a piecewise function.

[0017] Again, a finite element analysis model of the single pile is established. In this model, the pile body itself can be simulated by beam elements or solid elements, while the interaction between the pile and the soil is simulated by multiple groups of nonlinear spring elements arranged along the pile body. The force-displacement characteristics of these nonlinear spring elements (for example, they can be implemented by connector units), especially their stiffness, are directly defined by the foundation reaction modulus (k) model established and optimized and calibrated in the previous step. In order to improve modeling efficiency and accuracy, the generation of the pile body geometric model, the parametric arrangement of the nonlinear spring elements, and the setting of related boundary conditions can be automated through a scripting language (such as Python) that interfaces with finite element analysis software (such as ABAQUS).

[0018] Subsequently, the established finite element model is numerically analyzed under simulated long-term horizontal cyclic loads (e.g., sinusoidal loads). The core innovation of this step lies in the fact that the stiffness of the nonlinear spring element is not fixed throughout the entire numerical analysis process. Instead, it is calculated and dynamically updated in real time based on the cumulative number of cycles (N) corresponding to the current analysis step (this number of cycles can be derived from parameters such as the analysis time and time step). A user-defined subroutine (e.g., the USDFLD user subroutine) integrated with finite element analysis software (such as ABAQUS) calls the previously established functional relationship model for the foundation reaction modulus (k). This means that the spring stiffness can truly reflect the stiffness degradation or strengthening process that occurs in the soil under cyclic loading. During the numerical analysis, a static general analysis step that takes geometric nonlinearity into account is typically used, and the total analysis time and incremental step size are appropriately controlled to ensure accurate tracking of the number of cycles and convergence of the calculation.

[0019] Finally, based on the calculation results obtained from the above numerical analysis, such as the evolution of pile top displacement, pile bending moment, and foundation reaction force distribution with the number of cycles, a comprehensive analysis and evaluation of the bearing characteristics of a single pile under long-term horizontal cyclic loading was conducted.

[0020] The present invention provides a method for analyzing the bearing characteristics of a single pile PY model considering long-term horizontal cyclic loads.

[0021] It has the following beneficial effects:

[0022] 1. The present invention establishes a foundation reaction modulus model driven by experimental data and combined with a global optimization algorithm. The model comprehensively considers the effects of the number of cycles, burial depth and load amplitude on soil stiffness, significantly improving the accuracy of the py curve model and the simulation accuracy of actual working conditions.

[0023] 2. The present invention uses a user-defined subroutine to achieve real-time dynamic update of the soil spring stiffness in the finite element model, which can more realistically capture the time-varying characteristics of pile-soil interaction under long-term cyclic loading and is particularly suitable for simulating the cyclic cumulative effect of soil.

[0024] 3. By combining parametric modeling scripts with finite element analysis, the present invention improves the efficiency of complex working condition analysis and reduces dependence on high-performance computing resources. At the same time, it retains the relatively high efficiency advantage of the py model method, enabling it to better serve multiple rounds of iterations in engineering design and the analysis of large-scale pile foundation groups.

[0025] 4. The present invention provides a more refined and reliable analysis tool for pile foundations subjected to complex cyclic loads in fields such as marine engineering and bridge engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1It is a schematic diagram of the layout of the centrifuge test device of the present invention;

[0027] Figure 2 This is a schematic diagram of the changes in burial depth and loading amplitude of the present invention;

[0028] Figure 3 Schematic diagram of the polynomial fitting curve of the present invention;

[0029] Figure 4 Schematic diagram of the relationship between the stiffness k and the burial depth z of the present invention;

[0030] Figure 5 Schematic diagram of the change of spring stiffness k of the present invention;

[0031] Figure 6 Flowchart of the Python script of the present invention;

[0032] Figure 7 This is a schematic diagram of the pile spring structure of the present invention;

[0033] Figure 8 The graph of pile displacement versus cycle number obtained by actual measurement and numerical model calculation for loading force amplitudes of 6MN and 8MN in the present invention;

[0034] Figure 9 Schematic diagram of the method of the present invention. DETAILED DESCRIPTION

[0035] The following is combined with Figure 1 -Attached Figure 9 , the present invention is described in further detail.

[0036] Example 1:

[0037] Please see the attached Figure 9 The present invention provides a method for analyzing the bearing characteristics of a single pile under long-term horizontal cyclic loading using the PY model. This method aims to accurately evaluate the mechanical behavior of a single pile under long-term cyclic loading by combining experimental data with numerical simulation techniques. The method includes the following steps:

[0038] S1. Obtain test data of a single pile under horizontal cyclic load.

[0039] This step is the physical basis for subsequent model establishment and verification.

[0040] Specifically, it is necessary to first design and implement a series of single pile horizontal load tests based on the type of pile foundation being studied and the soil conditions of the site. These tests may include:

[0041] 1. On-site in-situ testing: Loading the prototype pile or test pile directly at the project site. The data obtained most directly reflects the actual working conditions.

[0042] 2. Indoor 1g model test: Under laboratory conditions, model piles and foundation soil are made according to a certain geometric similarity ratio, and loading tests are carried out under conventional gravity fields.

[0043] 3. ng centrifuge model test: In order to better simulate the stress state of the soil, the scaled-down model piles and foundation soil are placed on a centrifuge, and loading tests are carried out under a high gravity acceleration field, which can better meet the similarity law of soil mechanics.

[0044] During the test, the following key data need to be measured and recorded:

[0045] The time history of horizontal cyclic loads applied to the pile top includes the load amplitude (F), loading frequency, and the number of cycles at each load level (N). The load waveform can typically be a sine wave or other typical waveforms selected based on the actual working conditions.

[0046] Response data for piles at different depths (z) primarily includes bending moments along the pile axis and horizontal displacements. This is typically achieved by attaching strain gauges along the length of the pile's inner and outer surfaces to measure strain, which is then converted into bending moments. Horizontal displacements of specific sections of the pile are then measured using displacement sensors (such as LVDTs).

[0047] Basic physical and mechanical parameters of site soil: obtained through in-situ tests (such as standard penetration test, static penetration test) or indoor geotechnical tests to assist in the analysis and understanding of soil behavior.

[0048] The collected raw test data are preprocessed as necessary, such as signal denoising, data screening, unit conversion, etc., to ensure the accuracy and reliability of the data and provide high-quality input for the subsequent establishment of an accurate foundation reaction modulus model.

[0049] S2. Based on the test data, a functional relationship model between the foundation reaction modulus (k) and the number of cycles (N), burial depth (z) and loading force amplitude (F) is established.

[0050] This step is one of the core of the method of the present invention. Its purpose is to quantitatively describe the evolution of the mechanical properties of the soil around the pile under long-term cyclic loading. This step specifically includes the following sub-steps:

[0051] S21. Calculate the soil reaction (p) based on the measured bending moment of the pile body.

[0052] The bending moment (M) of any section of the pile body is the result of integrating the soil reaction force (p) around the pile along the pile length. In pile foundation engineering, the pile body is often regarded as an elastic beam. Based on the beam deflection differential equation, the horizontal resistance provided by the soil at each depth of the pile body, i.e., the soil reaction force (p), is calculated by performing mathematical processing such as difference or fitting derivative on the distribution data of the bending moment of the pile body measured in the test. For example, The calculation is based on the relationship between M and z, where M is the bending moment of the pile and z is the burial depth.

[0053] S22. Determine the foundation reaction modulus (k) based on the soil reaction (p) and the corresponding pile displacement (y).

[0054] The foundation reaction modulus (k), also known as the secant stiffness of the foundation, is an important parameter that describes the soil's ability to resist deformation during pile-soil interaction. Given the soil reaction (p) and the horizontal displacement (y) at the corresponding pile position (the horizontal displacement (y) can be obtained by integrating the pile bending moment or directly measuring it), the foundation reaction modulus (k) can be defined as:

[0055]

[0056] By processing the test data under different numbers of cycles (N), different burial depths (z), and different loading force amplitudes (F), a series of corresponding (p, y) data points can be obtained, and then a series of foundation reaction modulus (k) values can be calculated.

[0057] S23. Use mathematical functions to fit the relationships between the foundation reaction modulus (k) and the number of cycles (N), the foundation reaction modulus (k) and the burial depth (z), and the foundation reaction modulus (k) and the loading force amplitude (F).

[0058] In order to establish a mathematical model that can predict the foundation reaction modulus (k), it is necessary to analyze the intrinsic relationship between (k) and various influencing factors.

[0059] Relationship between foundation reaction modulus (k) and number of cycles (N):

[0060] Numerous experimental studies have shown that the stiffness of soils subjected to cyclic loading changes with the number of cycles, exhibiting cyclic softening (stiffness reduction) or cyclic hardening (stiffness increase), or more complex behaviors. Logarithmic functions and their combinations are often used to describe this relationship. For example, a binomial logarithmic function of the following form can be used for fitting:

[0061] k N (N) = a·ln 2 (N)+b·ln(N)+c

[0062] Among them, k N(N) represents the foundation reaction modulus when only the influence of the number of cycles is considered, and a, b, and c are parameters obtained by fitting the test data.

[0063] Relationship between foundation reaction modulus (k) and burial depth (z):

[0064] Generally speaking, the initial stiffness of soil increases with increasing burial depth. The power function is a common and effective form to describe this relationship:

[0065] k z (z) = A·z d

[0066] Among them, k z (z) represents the foundation reaction modulus when only the influence of burial depth is considered, and A and d are fitting parameters.

[0067] Relationship between foundation reaction modulus (k) and loading force amplitude (F):

[0068] The magnitude of the loading force significantly affects the nonlinear behavior of the soil and the rate of stiffness degradation. Generally, as the load amplitude increases, the strain level of the soil increases and the stiffness tends to decrease. Polynomial functions, especially quadratic polynomials, can be used to describe the effect of the loading force amplitude on the foundation reaction modulus:

[0069] k F (F) = e·F 2 +f·F+g

[0070] Among them, k F (F) represents the foundation reaction modulus correction coefficient or component when only the influence of the loading force amplitude is considered, and e, f, and g are fitting parameters.

[0071] S24. Construct an empirical formula for the comprehensive foundation reaction modulus k(N,z,F).

[0072] After studying the effects of N, z, and F on k separately, it is necessary to combine these influencing factors and construct a unified empirical formula that can simultaneously consider the effects of all three. A common approach is to combine the functional forms of each of the above single effects in the form of a product, as shown in the following formula:

[0073] k(N,z,F)=(a·ln 2 (N)+b·ln(N)+c)·z d ·(e·F 2 +f·F+g)

[0074] This formula is only an example. In actual construction, other reasonable combinations can be selected based on the patterns reflected by experimental data. In this case, a, b, c, d, e, f, and g in the formula are all parameters to be determined through global optimization fitting.

[0075] S25. Parameter calibration and model refinement based on global optimization algorithm.

[0076] Because comprehensive empirical formulas are typically nonlinear and contain multiple unknown parameters, traditional local optimization methods are prone to falling into local optimal solutions. Therefore, the present invention uses global optimization algorithms, such as differential evolution algorithms, genetic algorithms, and particle swarm optimization algorithms, to simultaneously optimize all unknown parameters in the empirical formula.

[0077] Taking the differential evolution algorithm as an example, its basic idea is to perturb and select through the differences between individuals in a population, guiding the search process towards a global optimal solution. The optimization objective function is usually to minimize the sum of squared errors between the foundation reaction modulus predicted by the model and the foundation reaction modulus calculated from test data.

[0078] Through the global optimization algorithm, a set of optimal parameter combinations can be obtained so that the constructed foundation reaction modulus k(N,z,F) model can best fit all the test data.

[0079] S26. A piecewise function model considering the difference between initial and cyclic loading is introduced.

[0080] The mechanical behavior of soil during the initial loading phase (e.g., a few cycles with N = 0 or N close to 0) can differ significantly from its behavior after a large number of cycles. To more accurately describe this difference, the foundation reaction modulus k(N, z, F) model can be constructed using a piecewise function. For example, different stiffness calculation expressions or correction factors can be defined for N = 0 (representing the initial state before static loading or cycling) and N > 0 (representing cyclic loading).

[0081] A specific example could be:

[0082] When N>0,

[0083] k(N,Z,F)=C1·(a·ln 2 (N)+b·ln(N)+c)·z d ·(e·F 2 +f·F+g)

[0084] When N=0,

[0085] k(0,z,F)=C0·k static (z,F)

[0086] Among them, k static(z, F) represents the initial static stiffness model, and C1 and C0 are the corresponding adjustment coefficients. The specific function form and parameter values must be determined through experimental data.

[0087] S3. Establish a finite element model of a single pile, wherein the pile-soil interaction is simulated by a plurality of nonlinear spring units, and the stiffness of the nonlinear spring units is defined by the foundation reaction modulus (k).

[0088] This step is the process of converting the physical problem into a mathematical model that can be solved by numerical methods.

[0089] Specifically, a three-dimensional or two-dimensional finite element model of a single pile is established in a commercial finite element analysis software (such as ABAQUS, ANSYS, etc., and this embodiment is described by taking ABAQUS as an example).

[0090] Pile simulation: The pile itself can be discretized using beam elements (such as B31 and PIPE31 elements) or 3D solid elements (such as C3D8R and C3D20R elements) based on its geometry and load characteristics. This gives the pile realistic material properties, such as elastic modulus, Poisson's ratio, and density.

[0091] Simulation of pile-soil interaction: The core of the present invention is to simplify the complex effect of the soil around the pile on the pile into a series of discrete nonlinear spring units distributed along the pile body. One end of these spring units is connected to the node of the pile body, and the other end is usually connected to a fixed reference point or a node representing the far-field soil. Each spring unit represents the resistance provided by the soil within a small length range of the pile body. In ABAQUS, this spring can usually be realized by a connector unit (such as CONN3D2 type unit), and its mechanical behavior (especially stiffness) will be defined by the foundation reaction modulus k (N, z, F) model established in step S2.

[0092] S4. Perform a numerical analysis of the finite element model under the simulated long-term horizontal cyclic load. During the analysis, the stiffness of the nonlinear spring element is dynamically updated in real time based on the current number of cycles (N) using the established foundation reaction modulus (k) model.

[0093] This step is the key to realizing "taking long-term effects into consideration" of the present invention.

[0094] Dynamic update mechanism of nonlinear spring stiffness:

[0095] To dynamically update the spring stiffness with the number of cycles, the finite element software's user-defined subroutine functionality is required. For example, in ABAQUS, a user-defined field (USDFLD) subroutine can be created. This subroutine, written in a language such as Fortran, is called by the ABAQUS main program at each analysis increment (or at a specific time point). Within the USDFLD subroutine, the equivalent number of cycles (N) is calculated based on the current analysis time, time increment, and other information.

[0096] The k(N,z,F) function model established in step S2 is then called to calculate the spring's stiffness in the current cyclic state. This stiffness is then passed to the material properties or connector cross-section properties associated with the spring via field variables, enabling real-time updates of the spring's stiffness.

[0097] Automated modeling process:

[0098] To improve modeling efficiency and facilitate parametric analysis, the finite element model creation (including pile geometry creation, soil spring element generation and placement, material property assignment, boundary conditions, and load definition) and its integration with the USDFLD subroutine can be automated using scripting languages such as Python. Python scripts can directly call ABAQUS kernel commands or graphical user interface commands to implement parametric modeling, batch analysis, and result post-processing.

[0099] For example, a script can automatically read the design parameters of the pile, generate nodes at preset intervals along the pile shaft, create connector elements as springs, and link the USDFLD subroutine to the analysis job.

[0100] Cyclic load application and analysis step settings:

[0101] In the finite element model, a horizontal cyclic load history consistent with the test conditions is applied to the pile top or a designated location. This is typically achieved by defining an amplitude curve. The analysis step configuration is crucial; a static general step (Static, General) is typically used with the geometric nonlinearity option enabled (NLGEOM = ON) to account for large deformation effects.

[0102] The total length of the analysis step should correspond to the total number of simulation cycles. Solution control parameters, such as the initial increment, minimum / maximum increments, and maximum number of iterations, need to be carefully set to ensure convergence and accuracy of the calculation.

[0103] S5. Analyze the bearing characteristics of the single pile based on the results of the numerical analysis.

[0104] After the numerical analysis is completed, the following results can be extracted and analyzed from the post-processing module of the finite element software or through scripts:

[0105] Cumulative variation of horizontal displacements of the pile top and key sections of the pile body with the number of cycles.

[0106] Distribution of pile bending moment and shear force along the pile length and their evolution with the number of cycles.

[0107] Corner or kicker displacement at the base of the pile.

[0108] The reaction force, deformation and stiffness of each soil spring element calculated by USDFLD vary with the number of cycles.

[0109] By comprehensively analyzing these results, we can evaluate the deformation characteristics, bearing capacity decay patterns, and pile-soil interaction evolution mechanisms of single piles under long-term horizontal cyclic loading. The numerical simulation results can be compared with the experimental data obtained in step S1 to verify the accuracy and reliability of the proposed analysis method.

[0110] Through the above steps, this embodiment systematically explains how to establish a dynamic foundation reaction modulus model that takes into account the influence of multiple factors based on experimental data, and integrate it into the finite element analysis, thereby realizing a complete process of accurately analyzing the bearing characteristics of a single pile under long-term horizontal cyclic loads.

[0111] Example 2:

[0112] This embodiment of the present invention discloses a method for determining parameters, performing numerical simulation, and verifying a single pile PY model based on centrifuge test data and considering long-term horizontal cyclic loads. This embodiment aims to illustrate the feasibility and effectiveness of the technical solution of the present invention through specific experiments and numerical analysis.

[0113] S2.1. Conduct single pile centrifuge tests.

[0114] In order to obtain the load characteristics of a single pile under different loading amplitudes, this embodiment carries out single pile centrifuge tests under horizontal static force and cyclic load.

[0115] The experimental design takes a large-diameter single pile with a prototype pile diameter of 6.3m and a burial depth of 34m as the research object.

[0116] The overall layout of the test device, loading system, and the layout of force and displacement sensors are shown in the attached Figure 1 shown.

[0117] The test preparation work mainly includes:

[0118] 1. Foundation soil preparation: Prepare soil for the model test according to the particle grading curve provided in the geological survey report of the target site as shown in the table below.

[0119]

[0120] 2. Model pile installation: Design and process the model pile according to the centrifugal acceleration and geometric similarity ratio, and accurately install it in the predetermined position in the test soil box.

[0121] 3. Sensor Placement and Calibration: Strain gauges are attached to key locations on the model pile (e.g., at different depths) to measure bending moments. Displacement sensors (e.g., laser displacement sensors, LVDTs) are placed at the pile top and other locations of interest to measure horizontal displacement. All sensors are rigorously calibrated before use.

[0122] 4. Cyclic loading equipment installation: Install an actuator system capable of applying precisely controlled horizontal cyclic loads.

[0123] This embodiment carried out four groups of centrifugal model tests, and the specific test conditions are shown in the following table:

[0124]

[0125] The four sets of centrifuge model tests include:

[0126] A set of static loading tests.

[0127] Three sets of cyclic loading tests with different load amplitudes.

[0128] All tests were conducted under a centrifugal acceleration of 100 g to simulate the stress level at prototype scale. The number of cycles for each cyclic loading test was set to 2000.

[0129] S2.2. Experimental data processing and specific construction of foundation reaction modulus k(N,z,F) model.

[0130] S2.2.1. Calculate the soil reaction (p) for different cyclic load amplitudes based on the measured pile bending moment.

[0131] After the centrifuge test, the recorded pile strain data is extracted and converted into a pile bending moment distribution based on the principles of material mechanics. Subsequently, by numerically differentiating the bending moment distribution curve (e.g., using the difference method), the soil reaction force (p) at different pile depths (z), at different numbers of cycles (N), and at different loading force amplitudes (F) is calculated.

[0132] S2.2.2. Use multiple functions to fit the relationship between foundation stiffness (k) and number of cycles (N).

[0133] The calculated soil reaction (p) is divided by the corresponding horizontal displacement of the pile (y) (the displacement can be obtained by integrating the bending moment or directly measured) to obtain the foundation reaction modulus (k).

[0134] In order to determine the most appropriate functional expression between the foundation reaction modulus (k) and the number of cycles (N), this embodiment tried to fit multiple functions and compared the determination coefficient (R 2 ) and other statistical indicators for analysis and selection.

[0135] Polynomial fitting: k = 0.57ln 2 N+0.85lnN+25.61, its determination coefficient R 2 =88.4%.

[0136] Linear fitting: k = 0.57lnN + 25.96, its determination coefficient R 2 =86.51%.

[0137] Exponential fit: k = 26.77e 0.015lnN , its coefficient of determination R 2 =85.94%.

[0138] Based on the goodness of fit, this embodiment selects a quadratic logarithmic polynomial form to describe the relationship between (k) and (N).

[0139] S2.2.3. Use the power function expression to analyze the relationship between foundation stiffness (k) and burial depth (z).

[0140] The change of foundation reaction modulus (k) with burial depth (z) usually shows a trend of increasing with increasing depth. The power function is a common form to describe this relationship, that is, k z (z) = A·z d .

[0141] S2.2.4. Use the global optimization fitting method based on the differential evolution algorithm to construct a comprehensive model.

[0142] To comprehensively consider the coupled effects of the number of cycles (N), burial depth (z), and loading force amplitude (F) on the foundation reaction modulus (k), and to determine the hyperparameter values of the nonlinear model containing multiple parameters, this example compiled a Python script and used the differential evolution algorithm for global optimization fitting.

[0143] Through this method, the expression of foundation reaction modulus (k) is finally obtained as follows:

[0144] k(N,z,F)=(0.0011ln 2 N-0.1lnN+7)·z 0.689 (-0.034F 2+0.36F+0.48)

[0145] By calculating the coefficient of determination (R 2 ) were used to evaluate the fitting effect of the comprehensive model and verify that it could better reflect the experimental data.

[0146] The attenuation characteristics of the spring stiffness (k) here can be calculated by combining the Python subroutine with the USDFLD user subroutine of ABAQUS.

[0147] S2.2.5. Distribution and adjustment of spring stiffness.

[0148] When discretizing the continuous foundation reaction modulus into spring stiffness in the finite element model, the spring arrangement needs to be considered. In this example, it is assumed that each pile has 4a nonlinear springs uniformly distributed across its cross-section, with the springs spaced at an angle α = 90° / a. When simulating piles using a solid three-dimensional model, to apply the concept of the foundation reaction modulus under two-dimensional plane strain to the three-dimensional springs, the composite foundation reaction modulus value obtained above needs to be multiplied by an adjustment factor, which in this example is set to 1 / (a + 1).

[0149] S2.2.6. Stiffness (k) calculation shall be in piecewise function form.

[0150] To account for the different mechanical responses of the soil during initial and cyclic loading, as well as the cyclic hardening and softening effects described by the logarithmic function, the stiffness (k) is calculated using a piecewise function. For a loading force amplitude of F = 8 MN, the piecewise function expression is as follows:

[0151]

[0152] Where N = 0 represents the initial static loading or pre-cycling state, and N > 0 represents the cyclic loading state. The coefficient of 1.18 and the exponent of 0.859 are adjusted based on the specific test data.

[0153] S2.3. Establishment of ABAQUS finite element model and detailed parameter setting.

[0154] S2.3.1. Model geometry and components.

[0155] According to the model pile parameters of the centrifuge test, the corresponding finite element model was established in ABAQUS.

[0156] Model pile: outer diameter 63mm, wall thickness 2mm, height 364.5mm.

[0157] Pile top flange: outer diameter 83mm, inner diameter 63mm, height 6mm.

[0158] Tower bottom flange: outer diameter 83mm, inner diameter 56mm, height 6mm.

[0159] Tower (simulating superstructure): outer diameter 56 mm, wall thickness 2 mm, height 1025.5 mm.

[0160] Tower top counterweight: a rectangular block with a length and width of 66mm and a height of 20mm.

[0161] S2.3.2. Material property definitions.

[0162] The model piles, flanges, towers and counterweights are all made of aluminum tubes, with Young's modulus set to 70 GPa and Poisson's ratio set to 0.33.

[0163] S2.3.3. Connections between components.

[0164] Surface-to-surface binding constraints (TieConstraint) are used between the tower and the top mass block and flange, and between the pile and flange to ensure deformation coordination between components.

[0165] S2.3.4. Grid division strategy.

[0166] The pile mesh element type adopts hexahedral eight-node element (C3D8R, R stands for reduced integration).

[0167] Since the pile body is equipped with nonlinear springs, the pile-soil interaction is the key area of analysis. Therefore, a refined mesh division strategy is adopted in the pile body, and the nodes are encrypted along the axial direction of the pile body with a 1mm basis.

[0168] In the tower part, the forces acting on it are relatively simple, and a 50mm grid size is used to improve the calculation efficiency of the model.

[0169] Since this model has a relatively simple geometric structure and is mainly composed of piles and springs, and the spring stiffness is calculated and converted accordingly based on the coordinates of the grid nodes to which it is connected, changes in grid size have little effect on the calculation of spring stiffness. Therefore, no special grid sensitivity verification is performed in this embodiment.

[0170] S2.3.5. Automated modeling process.

[0171] By compiling Python scripts, the pile geometry processing, soil spring connector unit generation and boundary condition definition are automatically completed. The process of the Python script can be found in the attached Figure 6 shown.

[0172] The script first initializes the ABAQUS model and creates a static analysis step with the geometric nonlinearity option enabled (NLGEOM=ON) to capture possible large deformation effects.

[0173] Then, the script obtains the node set of the pile-soil contact surface (i.e., the outer surface of the pile).

[0174] For each pile node, the script automatically creates a connector element (type CONN3D2) to act as a nonlinear spring to simulate soil reaction. The connector element geometry is generated using the WirePolyLine command to create a line object connecting the pile node to a virtual reference point at a specified distance outside the pile. In this example, this distance is set to 2 meters.

[0175] S2.4. Compilation and integration of USDFLD user subroutines.

[0176] This embodiment uses Python script embedding and integration with ABAQUS user subroutine (USDFLD) to generate and manage nonlinear springs.

[0177] S2.4.1. Write the USDFLD subroutine.

[0178] Write a USDFLD user subroutine (for example, named usdfld_cyclic.f) in Fortran. The core function of this subroutine is to calculate and update the stiffness data of each spring element in real time based on the comprehensive expression (including piecewise function form) of the foundation reaction modulus (k) determined in step S2.2.

[0179] Within this subroutine, the time increments during the ABAQUS analysis are linked to the number of load cycles N. For example, the total analysis time can be directly mapped to the total number of load cycles, or each unit of analysis time can represent one or more load cycles.

[0180] Based on the currently calculated number of cycles N, the buried depth z of the spring location (which can be obtained from the node coordinates), and the current load amplitude F (which can be passed in as a global parameter or field variable), the subroutine calls the stiffness formula to calculate the instantaneous stiffness k.

[0181] S2.4.2. Compile the USDFLD subroutine.

[0182] After configuring the Fortran compiler in ABAQUS, open a command prompt, go to the directory where the USDFLD subroutine file (usdfld_cyclic.f) is located, and run the following command to compile the subroutine:

[0183] abaqus make library=usdfld_cyclic.f;

[0184] After successful compilation, the target file will be generated in the current directory, such as usdfld_cyclic.f.obj.

[0185] S2.4.3. Integration of USDFLD subroutine and Python script.

[0186] Integrate the compiled USDFLD subroutine into the ABAQUS Python script used for automated modeling in step S2.3.5. Specifically, modify the settings related to the user subroutine in the Python script. For example, specify the userSubroutine parameter to be the full path to the generated .obj file.

[0187] S2.4.4. Field variable output.

[0188] In the ABAQUS analysis settings, add a field variable output request (for example, FV1) to track how the spring stiffness values calculated and updated by USDFLD change with the number of cycles in postprocessing. By viewing FV1, you can verify that the stiffness model works as expected.

[0189] After running the Python script, nonlinear springs are generated at each external mesh node of the pile in the ABAQUS model. The stiffness of these springs is calculated by the USDFLD subroutine according to the formula and is continuously updated as the analysis time progresses (that is, the number of cycles increases).

[0190] The arrangement of nonlinear springs in the pile body and the local refinement can be found in the attached Figure 7 Due to the overall scaling in the left image, the pile springs appear densely packed, which may give the visual illusion of vertical extension. The actual arrangement effect is shown in the partial enlarged image on the right.

[0191] S2.5. Load application and analysis step setting.

[0192] S2.5.1. Definition of cyclic load.

[0193] A horizontal cyclic load is applied in the form of a sine wave. To precisely control the load history, in this embodiment, the load amplitude curve data points are pre-edited in an Excel spreadsheet, and then imported into ABAQUS in the form of a table to create an amplitude curve.

[0194] Taking the simulation of 100 cycles as an example, the derived cyclic amplitude curve has values between 0 and 1, where the period from 0 to 0.5 represents the loading process, and the period from 0.5 to 1 represents the unloading process. When applying the load, this amplitude curve is multiplied by a constant representing the maximum load amplitude to achieve cyclic loading of a specific amplitude. For different load amplitude test conditions, specific load values are applied in the corresponding loading direction.

[0195] S2.5.2. Analysis step setting.

[0196] Use the static general analysis step (Static, General) to solve.

[0197] Taking the simulation of 100 cycles as an example, the time length of the analysis step (Total time period) is set to 100, so that the analysis time can be directly used to represent the number of cycles.

[0198] Considering the possible geometric nonlinear effects caused by pile-soil interaction, the geometric nonlinearity option (NLGEOM=ON) is turned on in the analysis step settings.

[0199] In order to ensure the convergence and efficiency of the calculation, automatic stabilization (if specified, or automatic incrementation) control is adopted.

[0200] The maximum number of increments is set to 100,000.

[0201] The initial increment size is set to 0.005.

[0202] The maximum increment size is set to 1 to avoid a single increment that is too large, which may cause sudden load changes or convergence difficulties.

[0203] The minimum allowed increment size is set to 1e-5 to help the model try to solve with smaller steps when encountering local convergence difficulties.

[0204] S2.6. Comparison and verification of numerical simulation results with experimental results.

[0205] After the numerical simulation calculations are completed, the key results are extracted and compared with the measured data from the centrifuge test.

[0206] For example, for the working conditions with loading force amplitudes of 6MN and 8MN, the changes in pile deformation with the number of cycles obtained by actual measurement and numerical simulation are compared. Figure 8 As shown in Figure 2, the displacement of the pile body obtained by actual measurement and numerical model calculation for 6MN and 8MN varies with the number of cycles.

[0207] The horizontal displacement of the pile at the mud surface and the displacement of the pile bottom both increase with the increase in the number of cycles, and the growth of the displacement is particularly significant in the first 100 cycles.

[0208] Comparison revealed that the single pile deformation morphology derived from numerical simulations exhibits greater rigidity (i.e., relatively smaller bending deformation) than the experimental results. This may be due to factors such as model simplification and material parameter selection. Nevertheless, for two key responses, displacement at the mud surface and displacement of the pile base, the numerical simulation results agree well with the measured experimental data.

[0209] This comparative verification shows that the finite element model based on the USDFLD dynamically updated spring stiffness adopted in this embodiment can better simulate the cyclic weakening (or hardening) characteristics of sand (or other types of soil) under long-term cyclic loading, providing an effective analysis method for the long-term performance evaluation of single pile foundations.

[0210] Example 3:

[0211] The embodiment of the present invention discloses a method for realizing the automated analysis of a single pile py model considering long-term horizontal cyclic load based on Python script. This method realizes parametric modeling and automated analysis process by secondary development of ABAQUS finite element analysis software, which significantly improves the analysis efficiency and flexibility. The automated process can be referred to in the attached Figure 6 (Python script flow chart) shown.

[0212] The core of the method provided in this embodiment is to use Python scripts to programmatically execute the following key steps:

[0213] S3.1. Automated model initialization and pile geometry construction.

[0214] The Python script is first responsible for initializing the ABAQUS analysis environment and automatically constructing the geometric model of the pile based on the preset pile design parameters (such as pile diameter, pile length, burial depth, material properties, etc., which can be read from external files or directly defined in the script).

[0215] Specifically include:

[0216] 1. Create a new model database through script commands and define the necessary analysis steps, such as creating a static general analysis step (Static, General) for subsequent analysis and automatically enabling the geometric nonlinear option (NLGEOM=ON) to consider large deformation effects.

[0217] 2. Based on the read parameters, the script calls the geometric modeling function of ABAQUS to automatically generate a three-dimensional solid unit model or beam unit model of the pile.

[0218] S3.2. Automated soil-spring connector unit generation and parametric layout.

[0219] This step is the core of automating the py model. The Python script is responsible for accurately and efficiently generating and arranging nonlinear spring elements for simulating pile-soil interaction.

[0220] Specifically include:

[0221] 1. Pile-soil interface node identification and selection: The script automatically traverses the generated pile model's outer surface nodes and selects the node set located in the pile-soil contact area based on node coordinates (such as burial depth range) or predefined rules.

[0222] 2. Soil Spring Layout and Geometric Support Generation: For each selected pile node, the script creates a virtual reference point radially outward by a specified distance (this distance can be parameterized). Using the ABAQUS WirePolyLine command, a straight line segment (wire feature) is automatically generated between the pile node and the corresponding virtual reference point. This line segment serves as the geometric support for the subsequent connector element. Spring placement can be automated using pre-defined algorithms, such as axially even spacing or depth gradients.

[0223] 3. Connector element definition and nonlinear behavior assignment: The script automatically defines CONN3D2 type connector elements on the generated straight line segments. Crucially, the script defines the connector section properties for these connector elements and associates their nonlinear stiffness behavior with the dynamic foundation reaction modulus k(N,z,F) subsequently defined via the USDFLD user subroutine, which varies with the number of cycles N, the embedment depth z, and the load amplitude FF.

[0224] S3.3. Automated boundary conditions, load application, and analysis step parameter settings.

[0225] The Python script further automates the definition of model constraints and loading:

[0226] Boundary condition definition: The script automatically applies pile bottom fixed constraints (for example, U1=U2=U3=0) and other necessary far-field boundary conditions according to preset conditions.

[0227] Load and amplitude curve definition: The script can automatically apply horizontal cyclic loads to the pile top or specified location, and can call the Amplitude module function of ABAQUS to automatically create the amplitude curve required for the load (such as a sine wave) based on parameters (such as the number of cycles and the total number of cycles).

[0228] Analysis step parameter setting: For the created static general analysis step, the script can automatically set its detailed solution control parameters, such as time length, maximum / minimum / initial incremental step size, etc. The setting method of these parameters can refer to the relevant discussion in Example 2, but the emphasis here is on the automation of the script.

[0229] S3.4. Integration and management of USDFLD user subroutines by Python scripts.

[0230] To achieve dynamic updates of spring stiffness, Python scripts play a key management role in collaboration with USDFLD user subroutines:

[0231] Compilation and linking of the USDFLD subroutine: The script may include logic for calling system commands to automatically compile the user-written Fortran language USDFLD subroutine (the subroutine includes the calculation logic of the foundation reaction modulus k(N, z, F), as described in Example 2) and generate the required target file.

[0232] Binding of subroutines to model properties: The script ensures that the compiled USDFLD subroutine is correctly associated with the material properties or connector cross-sectional behavior of the soil-spring connector elements defined in the ABAQUS model. In this way, the stiffness of the connector elements will be calculated and updated in real time by the USDFLD subroutine based on the current state (e.g., number of iterations) at each increment of the finite element analysis.

[0233] Coordination of field variables and parameter transfer: The script ensures that the USDFLD subroutine can obtain the necessary input information (such as recording the current cycle number through field variables) and correctly pass the calculated stiffness value to the ABAQUS main program.

[0234] Through the above-mentioned Python script-driven automated process, the present invention can efficiently and accurately establish and analyze a single pile py model considering the effects of long-term cyclic loading, which is convenient for parametric research and engineering applications.

[0235] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. The method for analyzing the bearing characteristics of a single pile py model under long-term horizontal cyclic load is characterized by: The following steps are involved: S1. Acquire test data of a single pile under horizontal cyclic loading, wherein the test data includes at least pile displacement, pile bending moment, loading force amplitude, and number of cycles at different burial depths; S2. Based on the test data, a functional relationship model between the foundation reaction modulus and the number of cycles, burial depth and loading force amplitude is established; S3. Establishing a finite element model of the single pile, wherein the pile-soil interaction is simulated by a plurality of nonlinear spring units, wherein the stiffness of the nonlinear spring units is defined by the foundation reaction modulus; S4. performing a numerical analysis on the finite element model under a simulated long-term horizontal cyclic load, wherein during the analysis, the stiffness of the nonlinear spring unit is dynamically updated in real time based on the current number of cycles using an established foundation reaction modulus model; S5. Analyze the bearing characteristics of the single pile based on the results of the numerical analysis.

2. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 1 is characterized in that: Step S2 further comprises: S21. Calculate the soil reaction force based on the measured pile bending moment; S22, determining the foundation reaction modulus according to the soil reaction and the corresponding pile displacement; S23, using a mathematical function to fit the relationship between the foundation reaction modulus and the number of cycles, the foundation reaction modulus and the burial depth, and the foundation reaction modulus and the loading force amplitude; S24. Based on the test data, a global optimization algorithm is used to determine the parameters of the foundation reaction modulus model.

3. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 2, wherein: The relationship between the foundation reaction modulus and the number of cycles is expressed by a logarithmic function, the relationship between the foundation reaction modulus and the burial depth is expressed by a power function, and the influence of the loading force amplitude is expressed by a polynomial function.

4. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 2 is characterized in that: The global optimization algorithm is a differential evolution algorithm.

5. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 1 is characterized in that: The establishment of the finite element model in step S3 and the execution of the numerical analysis in step S4 are implemented using finite element analysis software, and the dynamic update of the spring stiffness is implemented through a user-defined subroutine integrated with the finite element analysis software.

6. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 5 is characterized in that: The finite element analysis software is ABAQUS, and the user-defined subroutine is the USDFLD user subroutine.

7. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 5 is characterized in that: In step S3, the generation of the pile geometry model, the arrangement of the nonlinear spring units and the definition of the boundary conditions are automatically completed through a scripting language interfaced with the finite element analysis software.

8. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 1 is characterized in that: The nonlinear spring unit in step S3 is a connector unit, and its nonlinear stiffness behavior is defined by the dynamically updated foundation reaction modulus.

9. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 1, characterized in that: The numerical analysis performed in step S4 includes: S41, applying a cyclic loading waveform to the pile model; S42, using the static general analysis step considering geometric nonlinearity; S43. Control the analysis time and increment step so that the analysis time corresponds to the number of loading cycles, allowing real-time update of stiffness through user-defined subroutines.

10. The method for analyzing the bearing characteristics of a single pile PY model under long-term horizontal cyclic load according to claim 1, characterized in that: When establishing the foundation reaction modulus model in step S4, a piecewise function form is used to characterize the different mechanical behaviors of the soil under the initial loading state and the cyclic loading state.

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