Pile dynamic response and post-earthquake residual deformation prediction method based on shear dilation effect

Abstract

The present application belongs to the field of geotechnical engineering and disaster prevention and mitigation technology, and relates to a pile foundation dynamic response and post-earthquake residual deformation prediction method based on shear dilation effect. The method comprises the following steps: obtaining soil dilatancy angle and seismic load spectrum characteristics; constructing soil horizontal dynamic stiffness model and comprehensive dynamic damping model varying with load frequency and dilatancy angle; calculating pile-soil interface additional normal stress based on dilatancy angle, and correcting interface ultimate shear strength; establishing and solving pile body dynamic balance non-homogeneous differential equation in frequency domain to obtain frequency domain displacement response; reconstructing time domain displacement from frequency domain response through inverse Fourier transform, judging plastic flow state based on plastic potential function, and extracting post-earthquake pile body residual deformation through time domain integration. Through the mixed strategy of frequency domain analysis and time domain reconstruction, the dynamic correction of interface strength by shear dilation effect and the pile body residual deformation are accurately captured while the elastic response is efficiently solved, which is suitable for seismic design and post-earthquake evaluation of pile foundation in liquefiable sand and silt soil layers.

Description

Technical Field

[0001] This invention relates to the fields of geotechnical engineering and disaster prevention and mitigation, and in particular to a method for predicting the dynamic response of pile foundations and post-earthquake residual deformation based on the dilatation effect. Background Technology

[0002] As the primary foundation type for engineering structures such as bridges, high-rise buildings, offshore wind power, and port terminals, pile foundations directly affect the safety of the superstructure due to their dynamic response characteristics under seismic loads. Under seismic loading, pile-soil interaction is the key mechanical process determining the stress and deformation behavior of pile foundations. Accurately evaluating pile-soil dynamic interaction is of significant engineering importance for the seismic design of pile foundations.

[0003] Currently, the analysis methods for pile-soil dynamic interaction can be mainly divided into the following categories: (1) Equivalent linear analysis method based on Winkler foundation model This type of method simplifies the soil around the pile as independent springs and dampers, characterizing the dynamic properties of the soil by setting the equivalent shear modulus and damping ratio, and solving the dynamic equilibrium equations of the pile in the frequency domain. Representative works include the Py curve method recommended by the API standard and its improved dynamic form. This type of method has high computational efficiency and relatively simple parameter acquisition, making it the most widely used method in engineering practice.

[0004] (2) Numerical analysis method based on continuum mechanics This mainly includes methods such as the finite element method and the finite difference method. By establishing a global numerical model of the pile-soil system, appropriate soil constitutive models (such as the Mohr-Coulomb model, the Drucker-Prager model, and isotropic / kinematic hardening models) are used to analyze the dynamic response in the time domain. Theoretically, this type of method can comprehensively consider the nonlinear characteristics of the soil and the complex mechanical behavior of the pile-soil interface.

[0005] (3) Experiment-based empirical / semi-empirical methods The dynamic response characteristics of the pile-soil system are obtained through indoor unit tests (such as cyclic triaxial tests and resonant column tests), centrifuge model tests, and field prototype tests, and empirical prediction formulas or correction coefficient tables are established accordingly. This type of method has strong engineering practical value, but its universality is limited by test conditions and scale effects.

[0006] (II) Deficiencies and shortcomings of existing technologies After systematic analysis, the existing technology has the following main shortcomings in predicting the dynamic response of pile foundations under seismic loads: 1. The dynamic influence of soil dilatation on the shear strength of the pile-soil interface is generally ignored. Under seismic cyclic shear loading, granular materials such as dense sand and silt exhibit significant dilatation / contraction characteristics, meaning that volumetric strain changes with the development of shear deformation. This dilatation effect directly alters the effective normal stress state at the pile-soil interface, thereby dynamically adjusting the ultimate shear strength of the interface. However, existing Winkler models and most numerical analysis methods still treat soil as a volume-invariant material or use equivalent linearization methods, failing to incorporate the dilatation angle as an independent physical parameter into the interface strength correction model. This leads to a systematic bias in the calculation of pile side friction—significantly underestimating the peak shear capacity of dilatant soils and overestimating the residual strength of contractile soils, ultimately affecting the accuracy of predicting the dynamic response amplitude of the pile.

[0007] Second: Traditional frequency domain methods cannot extract post-earthquake residual deformation. Frequency domain analysis methods (such as the transfer matrix method and frequency domain impedance method) play an important role in seismic response analysis due to their high computational efficiency. However, frequency domain methods are essentially based on linear or equivalent linearization assumptions, and can only provide the elastic vibration displacement response of the pile body. They cannot reflect the plastic flow characteristics of the soil under strong earthquakes, and therefore cannot obtain the irreversible permanent displacement of the pile foundation after the earthquake. However, post-earthquake residual deformation is precisely one of the core indicators for performance-based seismic design and post-earthquake safety assessment of pile foundations.

[0008] Third: Pure time-domain nonlinear methods have excessively high computational costs. Theoretically, time-domain nonlinear numerical methods (such as explicit finite element method and implicit finite difference method) can simultaneously solve the problems of dilatation effect characterization and residual deformation extraction by introducing an elasto-plastic constitutive model. However, seismic loads are usually long-duration (tens to hundreds of seconds) dynamic inputs rich in high-frequency components. Solving large nonlinear equations iteratively step-by-step in the time domain will generate considerable computational overhead, making it difficult to meet the computational efficiency requirements of multi-condition and multi-parameter sensitivity analysis in engineering practice. Especially in the preliminary design stage, a large number of trial calculations are required for different seismic motion inputs and different pile-soil parameter combinations. The computational cost of pure time-domain methods has become the main bottleneck for their engineering promotion.

[0009] Fourth: Existing hybrid analysis methods lack a physical coupling mechanism for the dilatation effect. In recent years, although a few studies have attempted to combine frequency domain methods with nonlinear corrections (such as the equivalent linearization iterative method), their nonlinear processing remains at the equivalent level of modulus and damping. They have failed to establish a direct quantitative relationship between the dilatation angle and the additional normal stress at the level of interface shear strength, nor have they realized a dual-domain collaborative calculation framework that combines frequency domain solution of elastic vibration with time domain discrimination of plastic accumulation. This has led to the artificial separation of the intrinsic relationship between the physical effects of dilatation and the extraction of residual deformation. Summary of the Invention

[0010] The purpose of this invention is to address the following problems in existing pile-soil dynamic analysis methods: the traditional Winkler model neglects the physical mechanism of soil dilatation, leading to an underestimation of pile side friction in dense sand or expansive soil; it ignores the frequency dependence of soil dynamic parameters, making it difficult to accurately predict dynamic response under broadband loads; and existing simplified methods struggle to predict irreversible residual deformation after earthquakes while maintaining computational efficiency. This invention provides a method for predicting pile foundation dynamic response and post-earthquake residual deformation based on the dilatation effect.

[0011] This invention provides a method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on the dilatation effect, comprising the following steps: Step 1: Obtain the soil dilatation angle parameters; Step 2: Based on the dilatation angle parameter, construct a dilatation correction model for the shear strength of the pile-soil interface, calculate the additional normal stress at the interface caused by dilatation, and correct the ultimate shear strength of the interface. Step 3: Based on the modified interface ultimate shear strength, establish and solve the dynamic equilibrium equation of the pile in the frequency domain to obtain the frequency domain displacement response of the pile. Step 4: Reconstruct the frequency domain displacement response into a time domain displacement using inverse Fourier transform; Step 5: Based on the reconstructed time-domain displacement, determine whether the soil is in a plastic flow state according to the plastic potential function, and accumulate the irreversible displacement generated during the earthquake by time-domain integration, and extract the residual deformation of the pile body after the earthquake.

[0012] Preferably, in step 2, correcting the interface ultimate shear strength specifically includes: calculating the incremental normal stress at the interface caused by dilatation based on the dilatation angle parameter and the current strain state of the soil; superimposing the incremental normal stress on the initial normal stress to obtain the corrected interface normal stress; and then calculating the corrected interface ultimate shear strength based on the corrected interface normal stress.

[0013] Preferably, in step 3, the dynamic equilibrium equation of the pile body is a non-homogeneous differential equation in the frequency domain, and the stiffness term and damping term of the equation are both related to the modified interface ultimate shear strength.

[0014] Preferably, step 2 further includes obtaining a frequency influence coefficient; the frequency influence coefficient is determined by fitting cyclic triaxial test data or resonant column test data, and is related to the dominant frequency of the seismic load.

[0015] Preferably, step 1 further includes: obtaining the acceleration time history of the seismic load, and performing a Fourier transform on the acceleration time history to obtain the spectral characteristics of the seismic load; in step 3, the dynamic equilibrium equation of the pile body is established and solved in the frequency domain based on the spectral characteristics.

[0016] Preferably, in step 5, after the residual deformation accumulation is completed at each time step, the soil strain state, interface normal stress and interface ultimate shear strength at the current time are updated, and the updated parameters are used to participate in the frequency domain solution of the next time step to realize the time-varying nonlinear iterative calculation of the pile-soil system.

[0017] Preferably, after each time step, it is also necessary to determine whether the energy error between the frequency domain solution and the time domain reconstruction meets the preset tolerance requirements; if not, the stiffness parameters and damping parameters of the current time step are adjusted, and the frequency domain solution and time domain reconstruction are performed again until the tolerance requirements are met.

[0018] Preferably, the soil is liquefiable sand or silt, and the shear dilatation angle is dynamically determined based on the relative density of the soil and the effective confining pressure.

[0019] Preferably, in step 5, when determining the plastic flow state based on the plastic potential function, the accumulation of unrecoverable displacement for the current time step is only performed when the stress state of the current time step reaches the modified interface ultimate shear strength and the gradient direction of the plastic potential function is clear; otherwise, all displacements of the current time step are classified as recoverable elastic displacements.

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention introduces the dilatation angle as an independent physical parameter to establish a dilatation correction model for the shear strength of the pile-soil interface. It calculates the additional normal stress at the interface caused by dilatation and corrects the ultimate shear strength. This model can truly reflect the enhancing effect of dilatation on pile side friction in dense sand and silt during cyclic shearing. It overcomes the shortcomings of traditional methods that ignore the physical mechanism of dilatation and improves the prediction accuracy of pile dynamic response under seismic load.

[0021] 2. This invention constructs a soil horizontal dynamic stiffness model and a comprehensive dynamic damping model that vary with load frequency and dilatation angle, and correlates the frequency influence coefficient with the dominant frequency of seismic load. This can accurately characterize the frequency dependence of soil dynamic impedance under broadband seismic input, and avoid the prediction distortion of traditional static parameters when dealing with broadband loads.

[0022] 3. This invention adopts a hybrid calculation strategy of frequency domain analysis and time domain reconstruction. In the frequency domain, the dynamic equilibrium equation of the pile body is solved linearly to obtain the elastic vibration response. Then, the time domain displacement is reconstructed through inverse Fourier transform. Based on the plastic potential function, plastic flow is determined, and residual deformation is extracted through time domain integration. While ensuring calculation efficiency, the quantitative output of the permanent displacement of the pile body after the earthquake is realized.

[0023] 4. This invention establishes a dynamic updating mechanism for the dilatation angle with relative density and effective confining pressure, and introduces excess pore water pressure to correct the effective normal stress. This can reasonably reflect the coupling effect of pore pressure accumulation and dilatation effect during earthquakes, and is applicable to the seismic design and post-earthquake assessment of pile foundations in liquefiable sand and silt strata. Attached Figure Description

[0024] Figure 1 This is a flowchart of the present invention.

[0025] Figure 2 A schematic diagram of the overall model of pile-soil dynamic interaction considering frequency dependence.

[0026] Figure 3 This is a partially enlarged schematic diagram of the shear expansion stress enhancement mechanism at the pile-soil interface.

[0027] Figure label: 1- Seismic load or dynamic load input; 2- Pile-soil dynamic interaction element; 3- Pile body; 4-Soil particles; 5-Surrounding soil constraints; 6-Pile surface; 7-Direction of soil shear movement tendency; 8-Direction of interface shear resistance; 9-Additional normal stress. Detailed Implementation

[0028] To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the invention will be further described in detail below with reference to preferred embodiments and specific application scenarios. It should be understood that the specific embodiments described herein are only for explaining the invention and are not intended to limit the invention.

[0029] This invention provides a method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on the dilatation effect. The aim is to establish a modified dynamic stiffness model and interface strength criterion by introducing the dilatation angle and frequency influence coefficient, thereby achieving efficient and accurate prediction of the dynamic response and post-earthquake residual deformation of pile foundations under complex geological conditions.

[0030] In constructing the simplified analysis method described above, this invention employs a numerical-analytical joint modeling strategy. First, based on the plane strain assumption, a multi-concentric ring shear complex impedance model of the soil surrounding the pile (hereinafter referred to as the refined model) is established. Iterative calculations are used to obtain accurate solutions for the dynamic stiffness of the soil under different frequencies, dilatation angles, and shear strain amplitudes. Subsequently, based on a large amount of numerical simulation data from the refined model, the variation of soil stiffness with frequency and dilatation angle is analyzed, revealing significant logarithmic growth and tangential hardening characteristics.

[0031] The simplified formula proposed in this invention is derived from fitting and parameterizing numerous numerical simulation results of the aforementioned refined model. This approach retains the refined model's ability to accurately capture nonlinear physical mechanisms such as dilatational hardening and frequency dependence, while significantly reducing the computational threshold for engineering applications through analytical formulas, thus achieving a balance between theoretical depth and engineering practicality.

[0032] Figure 1 This is a flowchart illustrating the method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on the dilatation effect, as described in this invention. Figure 1 As shown, the method includes the following steps: Step S1, obtaining the soil dilatation angle parameter; Step S2, based on the dilatation angle parameter, constructing a dilatation correction model for the shear strength of the pile-soil interface, calculating the additional normal stress at the interface caused by dilatation, and correcting the ultimate shear strength of the interface; Step S3, based on the corrected ultimate shear strength of the interface, establishing and solving the dynamic equilibrium equation of the pile in the frequency domain to obtain the frequency domain displacement response of the pile; Step S4, reconstructing the frequency domain displacement response into a time domain displacement through inverse Fourier transform; Step S5, based on the reconstructed time domain displacement, determining whether the soil is in a plastic flow state according to the plastic potential function, and accumulating the irreversible displacement generated during the earthquake through time domain integration, extracting the residual deformation of the pile after the earthquake.

[0033] The following details the steps of this prediction method, combined with... Figures 1-3 As shown: Step 1: Obtain key calculation parameters 1.1 Obtaining parameters of the soil around the pile A geological survey of the site to be analyzed was conducted to obtain soil parameters around the piles, including at least the soil dilatation angle. Static reference stiffness k 0. Internal friction angle φ, shear dilatation influence coefficient α Frequency influence coefficient β g and reference frequency 0; The above parameters are determined through in-situ field tests, such as standard penetration tests and static cone penetration tests, combined with indoor geotechnical tests, including consolidation tests, direct shear tests, and triaxial compression tests.

[0034] As a preferred option, the frequency influence coefficient β g The preferred value range is 0.02-0.08; the reference frequency The value of 0 is 1.0 rad / s.

[0035] As the core physical parameter of the present invention, the value range of the shear dilatation angle ψ is 0°-15°, preferably 3°-10°.

[0036] For strata where the effective confining pressure changes significantly during an earthquake (such as liquefiable sand layers), the dilatation angle ψ is not taken as a constant value, but is based on the relative density D of the soil. r Dynamically update the current effective confining pressure σ′: Specifically, the effective confining pressure σ′(t) is updated by calculating the accumulated pore water pressure during the earthquake process, and then the shear dilatation angle under the current state is determined based on the updated effective confining pressure.

[0037] As a preferred technical solution, the shear dilatation influence coefficient The value range is 1.5-3.0, preferably 2.0-2.5.

[0038] Obtain the geometric and material parameters of the pile, including: pile diameter D, pile length L, pile cross-sectional area A, moment of inertia I, pile elastic modulus E, and pile material density ρ. p The above parameters are determined by the design documents or on-site testing data.

[0039] Seismic load input acquisition and spectrum analysis Obtain the acceleration time history a(t) of the design ground motion, and discretize the acceleration time history into N sampling points according to the sampling time interval Δt. The discrete acceleration sequence is represented as: For the discrete acceleration sequence Perform a Discrete Fourier Transform (DFT) to obtain the complex spectrum corresponding to each discrete frequency point: in, For the first Complex spectrum values ​​corresponding to discrete frequency points The imaginary unit, This represents the total number of sampling points in the acceleration time history. The angular frequencies corresponding to each frequency point are: Based on the spectral amplitude within the positive frequency range The maximum value determines the dominant angular frequency of the earthquake, that is: Obtain the spectral characteristics A(ω) of the seismic load and identify the dominant frequency ω of the seismic load. p This provides a basis for determining the value of the subsequent frequency influence coefficient.

[0040] Step 2: Based on the dilatation angle parameter, construct a dilatation correction model for the shear strength of the pile-soil interface, calculate the additional normal stress at the interface caused by dilatation, and correct the ultimate shear strength of the interface. 2.1 Constructing a dynamic model of soil horizontal stiffness that varies with frequency and dilatation angle To characterize the frequency dependence and dilatation hardening effect of soil under cyclic loading, a horizontal dynamic stiffness model of soil is constructed as follows: in: K h0 ω is the initial horizontal stiffness coefficient under static conditions; f(ω) is the frequency influence coefficient, which is related to the dominant frequency of the seismic load. Its value varies with the angular frequency of the load and is used within the applicable frequency range of the test or numerical calibration; g(ψ) is the dilatation hardening function, which is used to characterize the effect of dilatation on the dynamic stiffness of the soil.

[0041] The frequency correction function is: in, For the first k A discrete angular frequency; As the reference angular frequency, and ＞0 ; The frequency influence coefficient, and The value was determined by fitting data from cyclic triaxial tests, resonant column tests, or pile-soil dynamic response tests. Calculate the frequency correction function to ensure that the dynamic parameters corresponding to the positive and negative frequency components are consistent.

[0042] In the complete frequency domain calculation process, all discrete angular frequencies should be considered. Calculate the frequency correction function and dominant angular frequency respectively. It is used to identify the main energy frequency band of earthquakes, and is not used to replace all discrete frequency points.

[0043] The dilatation hardening function g(ψ) takes the form shown in equation (4): in, The dominant angular frequency is identified based on the spectral amplitude of the design ground motion acceleration time history; The frequency enhancement factor and Let be the shear dilatation hardening coefficient, and Within the scope of this invention, the shear dilatation angle The value satisfies The parameters were determined by fitting data from cyclic triaxial tests or pile-soil dynamic response tests.

[0044] To ensure the applicability of the frequency correction function, when < When the frequency range is 0 or exceeds the test calibration range, for f( Set preset upper and lower limits to avoid non-physical values ​​for dynamic stiffness.

[0045] 2.2 Constructing a comprehensive dynamic damping model that varies with the dilatation angle The dynamic damping of soil consists of two parts: material damping and radiation damping, and is affected by the dilatation effect. (Combined dynamic damping) The construction method is as follows: in: This is the radiation damping coefficient. This is the hysteresis damping coefficient; The dynamic stiffness of the soil as a function of frequency in step 2.1 is used to determine the overall damping. It varies accordingly with frequency and shear dilatation angle.

[0046] 2.3 Calculation of Additional Normal Stress at the Interface Based on Dilatation Angle This step is the key to achieving the breakthrough in the physical mechanism of this invention. Under cyclic shear loading, the soil around the pile experiences volumetric strain due to shear. This volumetric strain is constrained by the surrounding soil, generating an additional normal stress Δσ at the pile-soil interface. n .

[0047] Based on the plastic shear strain increment dγ at the current time step p Using the dilatation angle ψ, calculate the volumetric strain increment, and then calculate the additional normal stress increment at the interface. The additional normal stress is then superimposed onto the initial effective normal stress σ. n0 The corrected real-time effective normal stress σ is obtained above. n : The additional normal stress Considering depth The attenuation characteristics are given by the following formula: Corrected interfacial shear force The calculation formula is: in, For shear expansion modulus, Characteristic shear strain, This is the depth attenuation coefficient.

[0048] Figure 3 A partially enlarged schematic diagram of the shear expansion stress enhancement mechanism at the pile-soil interface is shown. Under cyclic shear loading, soil particles 4 in the surrounding soil undergo shear motion along the pile surface 6. The shear motion trend 7 drives the surrounding particles to expand in volume. This expansion trend is constrained by the surrounding soil 5, generating additional normal stress 9 at the pile-soil interface. This additional normal stress 9 acts on the pile surface 6, increasing the effective normal stress at the pile-soil contact surface, thereby enhancing the ultimate shear strength in the shear resistance direction 8. Thus, a "shear dilatation self-locking" effect is formed at the interface, that is, the volume expansion trend induced by shear deformation is transformed into additional effective confining pressure under the constraint of the surrounding soil, significantly improving the shear stiffness and ultimate skin friction of the pile-soil interface.

[0049] Figure 2 A frequency-dependent pile-soil dynamic interaction model is shown. Seismic load or dynamic load input 1 is applied to the pile body 3, and the soil around the pile is simulated by pile-soil dynamic interaction element 2. This element contains stiffness springs and integrated dynamic dampers that vary with the load frequency to characterize the frequency dependence and shear dilatation hardening effect of the soil dynamic impedance.

[0050] Step 3: Based on the modified interface ultimate shear strength, establish and solve the dynamic equilibrium equation of the pile in the frequency domain to obtain the frequency domain displacement response of the pile. 3.1 Establishing the non-homogeneous differential equations for the dynamic equilibrium of the pile body Treating the pile as an elastic beam element, based on the dynamic stiffness model established in step 2. and integrated dynamic damping model and the corrected interfacial ultimate shear strength Establish the non-homogeneous differential equations of dynamic equilibrium of the pile body in the frequency domain.

[0051] For a horizontally loaded pile, the typical governing equation is a fourth-order ordinary differential equation, in which both the stiffness and damping terms are related to the modified interface ultimate shear strength.

[0052] For a horizontally loaded pile, its frequency domain dynamic equilibrium non-homogeneous differential equation is expressed as: in, For the bending stiffness of the pile body, The mass per unit length of the pile body. This is the complex displacement response in the frequency domain. The imaginary unit; External dynamic loads are to be determined.

[0053] 3.2 Solving the frequency domain dynamic equilibrium equations The acceleration spectrum obtained in step S1 Converted into an equivalent inertial force, as a non-homogeneous loading term. Applying a load to the right-hand side of the differential equation described in step 3, a non-homogeneous dynamic equation incorporating the external load is constructed; the complex displacement response of the pile in the frequency domain is obtained analytically by using the transfer matrix method or the finite difference method. .

[0054] Step 4: Reconstruct the frequency domain displacement response into a time domain displacement using inverse Fourier transform; For the discrete frequency domain response obtained in step S3, the inverse fast Fourier transform (IFFT) is used to transform the frequency domain response. Convert to time-domain displacement time history Based on the displacement time history Using the horizontal dynamic stiffness of the soil established in step S2, calculate the time-domain shear force at the pile-soil interface. ; with interfacial shear strength limit For the yield surface threshold, when When the soil is deemed to have entered a plastic flow state, the excess stress state is substituted into the plastic potential function. The direction of plastic strain increment is determined according to the flow law, and the irreversible residual deformation of the pile body after the earthquake is extracted by time-domain integration. .

[0055] Step 5.1 Determination of Plastic Flow State Based on the reconstructed time-domain displacement in step 4 Based on the corresponding stress state, determine whether the soil is in a plastic flow state at each time step.

[0056] 5.1.1 Yield Criterion The Mohr-Coulomb yield criterion or the modified Cambridge model yield criterion is used to determine whether the soil has reached the yield state. Taking the Mohr-Coulomb criterion as an example, the yield function F is: Where τ is the shear stress at the pile-soil interface, and σ n This is the effective normal stress corrected in step 2.3.3. When F≥0, the soil reaches the yield state.

[0057] 5.1.2 Plastic Potential Function and Flow Rules Using the non-associated flow rule, the plastic potential function g is defined as follows: in, The generalized shear stress is taken as the shear force at the pile-soil interface at a certain moment. ; Generalized mean normal stress (including additional normal stress) The values ​​are the initial static earth pressure and the additional normal stress. The sum of The critical stress ratio. These are the hardening parameters; To account for the correction term for shear dilatation and compaction, we utilize... Determine the direction of plastic shear strain flow.

[0058] Dual criteria for determining plastic flow state As a key determination logic of this invention, not all time steps that reach the yield state produce irreversible residual deformation. Preferably, the current time step is determined to be in a plastic flow state and residual deformation accumulation only occurs when both of the following conditions are met simultaneously: Condition 1 is a stress condition: the stress state at the current time step reaches the interface ultimate shear strength τ after the correction in step 2.3.4. f That is, the yield function F ≥ 0; Condition two is a directional condition: the direction of the gradient of the plastic potential function is clear, that is... Q / σ ≠ 0, and the angle between the stress increment direction and the plastic potential function gradient direction is less than 90° (ensuring that plastic flow occurs in the correct direction).

[0059] When both of the above conditions are met, the time step is determined to be in a plastic flow state; otherwise, even if the stress reaches the yield surface, the displacement of the time step will be classified as a recoverable elastic displacement, and no permanent deformation accumulation will occur.

[0060] 5.3 Calculation of Irrecoverable Displacement Increment For the time step determined to be in a plastic loading state in step 5.2, obtain the shear stress, effective normal stress, and ultimate shear strength of the interface after correction for dilatation effect at that time step. Determine the direction of plastic flow based on the plastic potential function described in step 5.1.2, and determine the plastic multiplier increment for that time step according to the yield condition and consistency condition; combine the elastoplastic constitutive relationship of the pile-soil interface to determine the tangential plastic slip increment for that time step, and use the tangential plastic slip increment as the irreversible displacement increment for that time step.

[0061] For time steps that do not meet the plastic loading criteria described in step 5.2, the irreversible displacement increment for that time step is set to zero. Following the chronological order of the seismic action, the irreversible displacement increments of each plastic loading time step are accumulated to obtain the cumulative irreversible displacement at the pile-soil interface at the end of the seismic action. This cumulative irreversible displacement is used for subsequent calculations of the pile's post-earthquake residual deformation.

[0062] 5.4 Update and Iterative Feedback of Time Step Parameters After the residual deformation is accumulated at each time step, the soil strain state, interface normal stress and interface ultimate shear strength at the current moment are updated, and the updated parameters are used to participate in the frequency domain solution of the next time step to realize the time-varying nonlinear iterative calculation of the pile-soil system.

[0063] 5.5 Energy Error Convergence Criteria After each time step, it is determined whether the energy error between the frequency domain solution and the time domain reconstruction meets the preset tolerance requirements. If it does not meet the requirements, the stiffness and damping parameters of the current time step are adjusted, and the frequency domain solution and time domain reconstruction are performed again until the tolerance requirements are met.

[0064] 5.6 Extraction of residual deformation after earthquake By performing time-domain integration over the entire earthquake process and accumulating all irreversible displacement increments, the residual deformation of the pile body after the earthquake is obtained.

[0065] Case Analysis Example 1: Analysis of Residual Deformation of Pile Foundations in Liquefied Sand Recorded by El Centro Strong Earthquake This embodiment aims to verify the applicability of the simplified pile-soil dynamic interaction analysis method of the present invention, which considers soil dilatation effect and frequency dependence, in practical engineering. The calculation process is divided into two stages: simplified model construction and calibration, and specific working condition calculation.

[0066] Simplified construction and parameter calibration of dynamic stiffness model Before proceeding with the specific calculations, we will first explain the origin and physical basis of the core simplified formula of this invention. The traditional Winkler foundation beam model usually assumes that the foundation stiffness is constant and ignores the rheological and dilatational properties of the soil.

[0067] In order to construct a simplified model that is more in line with physical reality, this embodiment first establishes a concentric ring iterative benchmark model as the basis for theoretical derivation.

[0068] The soil around the pile is radially discretized into N concentric thin-layered rings.

[0069] Introducing the dilatation angle based on the generalized Hooke's law The resulting increase in volumetric strain is used to establish the dynamic equilibrium equations under plane strain conditions. The transfer matrix method is then used to solve for the dynamic equilibrium equations of each ring at different frequencies. The shear strain distribution under the given conditions is determined, and the shear modulus of each ring is iteratively updated using an equivalent linearization method. This benchmark model can accurately reflect the nonlinearity of the soil, but the computation time is relatively long. A full-parameter domain scan (dilatation angle) is then performed using the aforementioned benchmark model. ,frequency ).

[0070] Under a fixed dilatation angle, the logarithm of the dynamic stiffness of soil versus frequency. The correlation is highly linear, indicating that high-frequency loading significantly improves the apparent stiffness of the soil; at a fixed frequency, the dynamic stiffness of the soil increases with the dilatation angle. tangent value The near-linear increase reflects the confining pressure strengthening mechanism caused by dilatation.

[0071] Based on the above numerical patterns, this invention proposes the following modified dynamic stiffness analytical model, which achieves mathematical dimensionality reduction of the complex benchmark model. By fitting the model data with the least squares method, a range of recommended parameter values ​​applicable to liquefiable sandy soil is determined: , This process ensures that the simplified formulas of this invention are not simply empirical fits, but rather a precise capture of complex nonlinear mechanical behavior.

[0072] Numerical Calculation Process of Dynamic Response and Residual Deformation Step S1: Data Acquisition and Frequency Domain Conversion First, determine the physical parameters of the computational model and the external load input: Pile foundation parameters: Large-diameter reinforced concrete cast-in-place piles were selected, with a pile diameter of... Pile length Bending stiffness of pile body Mass per unit length of pile .

[0073] Soil parameters: The soil around the pile is a homogeneous liquefiable sandy soil layer with an internal friction angle of [missing information]. Shear expansion angle coefficient of earth pressure at rest Effective unit weight of soil Initial shear modulus .

[0074] Model correction coefficients: Based on geological surveys and centrifuge testing, the shear dilatation influence coefficients were determined. Frequency influence coefficient Reference frequency Depth attenuation coefficient .

[0075] Load input: The 1940 El Centro earthquake wave (north-south component) was selected as the dynamic input. The acceleration time history was converted into a spectrum using Fast Fourier Transform (FFT) to identify the dominant frequencies of energy concentration. (Approximately 1.6 Hz).

[0076] Step S2: Construct a dilatation-frequency coupled dynamic stiffness model According to the formula described in the claims of this invention, the calculation at the dominant frequency is performed. The corrected dynamic impedance parameters of the soil are given below.

[0077] Dynamic stiffness calculate: Setting static reference stiffness .

[0078] Introducing frequency influence coefficient and shear dilatation influence coefficient .

[0079] According to the corrected equation proposed in this invention, wherein, .

[0080] Substituting each parameter into the numerical calculation yields a dynamic stiffness value equivalent to the combined writing method.

[0081] Substitute the numerical values ​​into the calculation: ; ; The results show that, after considering the dilatation term (0.175) and the frequency term (0.115), the dynamic stiffness of the soil increased by about 29% compared with the static reference value, reflecting the hardening characteristics of the soil under dynamic load.

[0082] Integrated dynamic damping calculate: Take the radiation damping coefficient , Hysteresis damping coefficient .

[0083] According to the equation: , Will Convert to Substitute the numerical values ​​into the calculation: ; The contributions of radiation damping and hysteresis damping are clearly distinguished here, with hysteresis damping accounting for a significant proportion, which is consistent with the energy dissipation characteristics of low-frequency vibration.

[0084] Step S3: Calculate the interface ultimate shear dilatation potential correction according to the claims, quantifying the enhancing effect of shear dilatation on pile side friction. Based on the pile depth... Let's take this as an example for calculation.

[0085] Calculate additional normal stress : Define the characteristic shear strain of soil when it is sheared to a dilatational steady state. (i.e., 1%). It should be noted that this embodiment, for the purpose of demonstrating the calculation process, directly selects the characteristic shear strain value after iterative convergence. .

[0086] In practical engineering applications, the determination of this parameter follows the convergence criteria: initial assumptions Substituting the values ​​into the formula described in this invention, the initial stiffness matrix is ​​calculated; a full-frequency domain scan is performed to extract the maximum shear strain amplitude of the pile body. ;like Then let Repeat the above calculations. Verification shows that this example reaches the above convergence criterion after 3 iterations.

[0087] ; Substitute the values ​​( Pick ): ; Calculate the corrected interfacial shear strength Calculate the initial static earth pressure at this depth: .

[0088] The corrected total normal stress is: .

[0089] The corrected shear strength is: At a depth of 10m, the additional stress (55.3kPa) generated by the dilatation effect contributed about 38% of the normal constraint, which significantly enhanced the anti-slip ability of the pile-soil interface, thus forming a dilatation self-locking effect.

[0090] Steps S4-S5: Frequency Domain Dynamic Response Solution. Substitute the above parameters into the dynamic equilibrium equation of the pile body, and first calculate the complex wavenumber of the system. : , Substitute the values ​​(note) Unit conversion unified to Or keep Consistent): ; Calculated .

[0091] The transfer matrix method is used to analytically solve the equations and obtain the pile displacement response in the frequency domain. .

[0092] Step S6: Time-domain reconstruction and residual deformation extraction. The frequency domain response is reconstructed into a time-domain displacement time history using the inverse fast Fourier transform (IFFT).

[0093] Based on the decoupling analysis strategy: Considering the extremely time-consuming nature of direct full-time-domain nonlinear coupling calculations, this scheme adopts a stress-driven calculation mode in this step. Step S5, through frequency domain analytical methods, has actually obtained the true pile stress time history after considering the shear dilatation stiffness hardening effect. In this step S6, this stress time history is used as an independent driving variable and substituted into the plastic potential function. Integrating is performed. This is equivalent to independently calculating the cumulative plastic flow of the soil based on an accurate assessment of the impact of dilatation locking on the stress level, thus achieving a balance between computational efficiency and physical accuracy. An assessment of the residual state after the earthquake is then conducted. Nonlinearity determination and integration: Plastic potential function according to the claims The residual deformation is calculated by integrating the plastic displacement rate over the earthquake duration: Calculated .

[0094] Result comparison and verification: The method of this invention: The maximum displacement of the pile head was calculated to be 45mm, and the residual displacement after the earthquake was... .

[0095] Comparative Example 1 Traditional method (not considering dilatation, i.e.) Under the same working conditions, the calculated residual displacement is 18 mm.

[0096] Conclusion: The prediction results of the method of this invention are reduced by approximately 33% compared to traditional methods. This trend is highly consistent with the physical law observed in centrifuge shaking table tests that "dense sand shear dilatation leads to a reduction in residual deformation of pile foundations," proving the accuracy and superiority of this invention in predicting post-earthquake recoverability.

[0097] Example 2 Implementation methods for stress analysis of pile foundations in expansive soil strata The analysis steps in this embodiment are basically the same as those in Embodiment 1, except that the selected soil is an expansive soil layer.

[0098] In this embodiment, considering the varying sensitivities of expansive soils to frequency, the frequency influence coefficient should be adjusted based on the results of indoor dynamic triaxial tests. (Typically, the value ranges from 0.05 to 0.08), and a larger dilatation angle parameter is selected and substituted into the calculations in the above steps. Analysis results show that the method described in this invention has good applicability and stability under different dilatational soil conditions.

[0099] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect, characterized in that, Includes the following steps: Step 1: Obtain the soil dilatation angle parameters; Step 2: Based on the dilatation angle parameter, construct a dilatation correction model for the shear strength of the pile-soil interface, calculate the additional normal stress at the interface caused by dilatation, and correct the ultimate shear strength of the interface. Step 3: Based on the modified interface ultimate shear strength, establish and solve the dynamic equilibrium equation of the pile in the frequency domain to obtain the frequency domain displacement response of the pile. Step 4: Reconstruct the frequency domain displacement response into a time domain displacement using inverse Fourier transform; Step 5: Based on the reconstructed time-domain displacement, determine whether the soil is in a plastic flow state according to the plastic potential function, and accumulate the irreversible displacement generated during the earthquake by time-domain integration, and extract the residual deformation of the pile body after the earthquake.

2. The method for predicting pile foundation dynamic response and post-earthquake residual deformation based on dilatation effect according to claim 1, characterized in that, In step 2, correcting the ultimate shear strength of the interface specifically includes: calculating the incremental normal stress at the interface caused by dilatation based on the dilatation angle parameter and the current strain state of the soil; superimposing the incremental normal stress on the initial normal stress to obtain the corrected interface normal stress; and then calculating the corrected ultimate shear strength of the interface based on the corrected interface normal stress.

3. The method for predicting pile foundation dynamic response and post-earthquake residual deformation based on dilatation effect according to claim 1, characterized in that, In step 3, the dynamic equilibrium equation of the pile body is a non-homogeneous differential equation in the frequency domain, and the stiffness term and damping term of the equation are both related to the modified interface ultimate shear strength.

4. The method for predicting pile foundation dynamic response and post-earthquake residual deformation based on dilatation effect according to claim 1, characterized in that, Step 2 further includes obtaining the frequency influence coefficient; the frequency influence coefficient is determined by fitting cyclic triaxial test data or resonant column test data, and is related to the dominant frequency of the seismic load.

5. The method for predicting pile foundation dynamic response and post-earthquake residual deformation based on dilatation effect according to claim 4, characterized in that, Step 2 specifically includes: constructing a soil horizontal dynamic stiffness model that varies with both load frequency and dilatation angle, and a comprehensive dynamic damping model that varies with dilatation angle; and establishing the pile dynamic equilibrium equation based on the dynamic stiffness model and the comprehensive dynamic damping model.

6. The method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect according to claim 5, characterized in that, Step 1 further includes: obtaining the acceleration time history of the seismic load and performing a Fourier transform on the acceleration time history to obtain the spectral characteristics of the seismic load; in step 3, the dynamic equilibrium equation of the pile body is established and solved in the frequency domain based on the spectral characteristics.

7. The method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect according to claim 1, characterized in that, In step 5, after the residual deformation accumulation is completed at each time step, the soil strain state, interface normal stress and interface ultimate shear strength at the current time are updated, and the updated parameters are used to participate in the frequency domain solution of the next time step to realize the time-varying nonlinear iterative calculation of the pile-soil system.

8. The method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect according to claim 7, characterized in that, After each time step, it is also necessary to determine whether the energy error between the frequency domain solution and the time domain reconstruction meets the preset tolerance requirements. If it does not meet the requirements, the stiffness and damping parameters of the current time step are adjusted, and the frequency domain solution and time domain reconstruction are performed again until the tolerance requirements are met.

9. The method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect according to claim 1, characterized in that, The soil is liquefiable sand or silt, and the dilatation angle is dynamically determined based on the relative density of the soil and the effective confining pressure.

10. The method for predicting the dynamic response and post-earthquake residual deformation of pile foundations based on dilatation effect according to claim 1, characterized in that, In step 5, when determining the plastic flow state based on the plastic potential function, the accumulation of unrecoverable displacements in the current time step is only performed when the stress state of the current time step reaches the modified interface ultimate shear strength and the gradient direction of the plastic potential function is clear; otherwise, all displacements in the current time step are classified as recoverable elastic displacements.

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