Hybrid analytic-numerical method for solving soil-structure nonlinear interaction

Abstract

The invention discloses a mixed analytic-numerical method for solving soil-structure nonlinear interaction, and belongs to the technical field of geotechnical engineering and structural engineering. According to the method, a soil-structure nonlinear resistance curve is subjected to quasi-linearization, and numerical iteration is carried out by utilizing a matrix analytic solution of a multi-section elastic foundation beam theory, so that the internal force and displacement response of the structure is solved. The hybrid analytic-numerical method for solving soil-structure nonlinear interaction provided by the invention can truly simulate a mechanical behavior path of nonlinear resistance without depending on a traditional difference format. According to the method, absolute stability is kept in the solving process, better precision is achieved under the same calculation efficiency, meanwhile, precision improvement is based on fine adjustment of a small number of key variables, and the transparency and interpretability of the algorithm are guaranteed.

Description

Technical Field

[0001] The present invention belongs to the technical fields of geotechnical engineering and structural engineering, and particularly relates to engineering situations involving the interaction between structures and soil, such as anti-slide piles and underground pipelines. The present invention aims to provide a method for solving the internal force and displacement of a structure under the action of non-linear resistance, namely a hybrid analytical-numerical calculation method. Background Art

[0002] For the problem of solving the structural response under the action of non-linear resistance, the method of constructing a difference scheme is usually adopted. Based on the concept of setting spring supports by the Newmark method, a non-linear finite element analysis model for analyzing the deformation and internal force of pile structures under the p-y curve is mainly established. On this basis, there is a coupling algorithm that combines the advantages of the p-y curve method and the Poulos elastic theory method. This algorithm also adopts a finite difference form and requires the prior assumption of the initial lateral displacement distribution of the soil body. At the same time, there is a non-linear analysis method for foundation piles in multi-layered foundations using the p-y curve based on the m method. There is also a finite difference numerical analysis method for thrust piles with a horizontal trapezoidal distributed load where the foundation coefficient is expressed according to the p-y curve resistance mode. There is also a method of applying the p-y curve calculation to the deformation problem of horizontally loaded piles by establishing a mathematical programming model, which transforms the problem into determining how to distribute the load among the soil springs to minimize the total potential energy of the system, that is, finding a set of displacements that makes the total potential energy of the entire system reach an extreme value. These methods together provide a comprehensive solution for solving the structural response problem under the action of non-linear resistance.

[0003] However, when approximately solving non-linear differential equations by the difference method, a reasonable difference scheme needs to be constructed, and the selection of the difference scheme often affects the stability, convergence, accuracy, and calculation efficiency of the solution. Assuming the initial distribution will affect the result of the final solution.

[0004] This hybrid analytical-numerical method takes a unique approach. It neither requires the idea of the difference method nor the prior assumption of the initial displacement of the soil body. It fully utilizes the existing analytical solution form in the elastic foundation beam theory and combines the matrix framework of the multi-layer elastic foundation beam theory derived from matrix expansion. Through matrix transfer iteration of this matrix framework, a hybrid analytical-numerical iterative solution method is formed. This method can truly simulate the mechanical behavior path of non-linear resistance without relying on the traditional difference scheme. This method maintains absolute stability during the solution process, has better accuracy under the same calculation efficiency, and the improvement of accuracy is based on the fine adjustment of a few key variables, ensuring the transparency and interpretability of the algorithm. Summary of the Invention

[0005] The purpose of the present invention is to solve the hybrid analytical-numerical method for soil-structure non-linear interaction.

[0006] To achieve the above objectives and ideas, the present invention adopts the following technical solutions: S1. Divide the foundation beam into n sections, and perform quasi-linearization processing on the non-linear resistance curves of each section to obtain the stiffness data values with sectional accuracy. According to the method shown in the appendix Figure 2 The approximate linear stiffness characteristics (secant stiffness) of each section can be obtained. Based on the boundary conditions and the stiffness values of each section, establish the initial stiffness matrix [K], and determine the load increment value △P and the number of iterations at each level to obtain the external load matrix [P]. Apply the multi-segment elastic foundation beam theory to solve the initial parameter distribution of the beam under the load increment value at each level; S2. Analyze the obtained displacement results to check whether the stiffness data of each section fall within the accuracy interval defined by the non-linear resistance curve. If the stiffness values of all sections meet the requirements, directly superimpose the current solution; otherwise, for the inconsistent sections, use the updated stiffness series values to form the corrected stiffness matrix [K*], and re-solve using the multi-segment elastic foundation beam theory again; S3. Loop and execute the above-mentioned solution and correction processes until the predetermined number of load increments is completed. Superimpose and integrate the initial parameter solutions obtained from each iterative calculation within the entire beam range to obtain the final displacement and stress distribution diagrams. The resistance within each iterative step is equal to the product of the stiffness and the displacement, and superimposing them can obtain the resistance of the structure.

[0007] Furthermore, the quasi-linearization processing described in step S1 is specifically as follows: At different horizontal depths, perform quasi-linearization processing on the non-linear resistance curve according to the secant stiffness. As the sectional accuracy improves, the secant stiffness can approximate the non-linear resistance curve more accurately. Construct a comparison database with the secant stiffness and the corresponding displacement data, where the initial judgment displacement is set to 0, corresponding to the initial stiffness value. Establish a calculation relationship between this initial stiffness and the parameter β i β i is used to form the stiffness matrix [K] based on the elastic foundation beam theory. Perform the first elastic foundation beam calculation with the set load increment step size to obtain the displacement values at the corresponding depths. Subsequently, compare these displacement values with the data in the comparison database. If it is found that some displacements exceed the original interval, the corresponding secant stiffness needs to be updated, and re-establish the calculation relationship with β i to form the updated stiffness matrix [K*]. Repeat this process until the numerical iteration stops. In this way, by continuously iterating and updating the stiffness matrix, a high-precision solution for the structural response is achieved.

[0008] Furthermore, the specific derivation of the multi-segment elastic foundation beam theory described in step S1 is as follows: The Winkler foundation beam (i.e., elastic foundation beam) model idealizes the foundation layer as a continuous spring bed structure with spatially uniform distribution but variable local elastic response, so as to simulate the support of the foundation for the beam above. Therefore, for any cross-section position of the beam, a differential equation corresponding to deformation and external force can be established: For the homogeneous differential equation in the unloaded case, its general solution has a clear analytical expression form and can be accurately described by natural exponential functions. The Krylov functions are a special family of functions that integrate the characteristics of natural exponential functions. By using the Krylov functions, the complexity of the original problem is significantly reduced, and finally the solution of the problem can be completely characterized by a set of four initially set parameters (attached Figure 4 ), that is, the so-called "initial parameter method": where φ1~φ4 are Krylov functions: Taking the initial bending moment and shear force as known initial parameters, Equation (2) can generate a matrix form of a beam segment, as shown in Equation (3): The multi-segment elastic foundation beam theory can divide the foundation into multiple different segments (attached Figure 5 ), and different foundation resistance coefficients can be assigned to each segment according to the actual situation to better reflect the discontinuity of the stratum. At the same time, this theory retains the boundary conditions of continuous displacement, rotation, shear force, and bending moment of the beam. For each additional segment, 4 differential equations are added, and continuity conditions are added to the two bottom boundary conditions of the previous segment and the newly added two equations. Listing such continuously expanded equations in matrix form is as shown in Equation (4). In the formula: M0, M i are the bending moments at the starting section of the structure-soil interaction and the bending moment at the i-th section of the pile body, unit: kN·m. Q0, Q i are the shear forces at the starting section of the structure-soil interaction and the shear force at the i-th section of the pile body, unit: kN. y0, y i are the horizontal displacements at the starting section of the structure-soil interaction and the horizontal displacement at the i-th section of the pile body, unit: m. θ0, θ i are the beam end rotations at the starting section of the structure-soil interaction and the rotations at the i-th section of the pile body, unit: rad. EI is the flexural rigidity of the cross beam, unit: kN·m 2 . L0, L i are the lengths of the starting section of the structure-soil interaction and the lengths of the segments at the i-th section of the pile body. Β0, β i are the characteristic coefficients of the starting section of the structure-soil interaction and the characteristic coefficients at the i-th section of the pile body, Among them, k0, k i is the subgrade reaction coefficient considering the calculation width, unit: kN / m 2 .

[0009] Furthermore, the separation of the parameters of the equations and the formation of matrix equations described in step S4 are specifically described as follows: The end boundary conditions can usually be simplified to free end, hinged end or fixed end. This is a reasonable simplification method for complex geological conditions, especially in the case of lack of specific parameters of the lateral constraint and anti-rotation stiffness of the surrounding rock at the pile bottom. At the same time, with the increase of pile foundation tests and the in-depth of frontier research, especially in the papers on offshore pile foundation engineering and numerical simulation, the design analysis is gradually tending to use more refined models, such as introducing three-spring or four-spring models to more accurately consider the lateral restraint effect and anti-rotation characteristics of the surrounding rock on the pile bottom. Next, the matrix deformation and re-expansion methods for such problems are described. Equations (3) and (4) show the matrix expression forms in the case of free pile bottom. For the other two boundary conditions mentioned in the specification, the two elements in the bottom right corner of the matrix can be adjusted accordingly to adapt to different boundary constraints. The adjusted contents are shown in the following equations (5) and (6): The hinged end is the displacement and moment boundary conditions at the end: The fixed end is the displacement and rotation boundary conditions at the end: In addition to considering the lateral horizontal resistance of the soil (p-y curve), the three-spring model also takes into account the effect of the moment - center rotation of the base effect resistance (M R -M θ ), and the effect of the moment - rotation of the side wall friction at each position along the pile axis (M P -M θ ). Formula (7) shows its change idea taking the M R -M θ spring at the base as an example: Description of the Drawings

[0010] Figure 1 is the schematic diagram of the hybrid analytical-numerical method process.

[0011] Figure 2 is the schematic diagram of the initial parameter method.

[0012] Figure 3 is the schematic diagram of the multi-segment elastic foundation beam.

[0013] Figure 4It is a schematic diagram of the piecewise precision secant stiffness formed by quasi-linearizing the nonlinear resistance curve.

[0014] Figure 5 It is a schematic diagram of the two-step calculation process of the hybrid analytical-numerical method.

[0015] Figure 6 It is a schematic diagram of the parameters of the unidirectionally loaded pile and the distribution of the soil around the pile in the implementation example of the present invention.

[0016] Figure 7 It is a schematic diagram of the proposed p-y curves for each layer of soil in the implementation example of the present invention.

[0017] Figure 8 It is a diagram of the horizontal displacement of the pile body at different load levels in the implementation example of the invention.

[0018] Figure 9 It is a diagram of the bending moment of the pile body at different load levels in the implementation example of the invention.

[0019] Figure 10 It is a diagram of the shear force of the pile body at different load levels in the implementation example of the invention.

[0020] Figure 11 It is a diagram of the resistance of the pile body at different load levels in the implementation example of the invention.

[0021] Figure 12 It is a comparison diagram of the actual calculated resistance path and the proposed p-y curve in the implementation example of the invention. Specific calculation method

[0022] The following further illustrates the present invention in conjunction with the implementation example, but the present invention is not limited to this example.

[0023] The technical solution adopted by the present invention is a hybrid analytical-numerical method for solving the nonlinear interaction between soil and structure The specific steps are as follows: First, the foundation beam is divided into n sections. The nonlinear resistance curves of each section are subjected to piecewise linearization (displacement interval) to obtain the secant stiffness values at the specified piecewise accuracy (displacement interval). Based on the boundary conditions and the stiffness values of each partition section, the initial stiffness matrix [K] is established, and the load increment △P and the number of iterations for each level of load are determined to form the external load matrix [P]. The multi-segment elastic foundation beam theory is applied to calculate the initial parameter distribution of the beam under each level of load. Second, the obtained displacement results are analyzed to check whether the stiffness data (displacement interval) of each section meets the requirements of the nonlinear resistance curve within the preset accuracy interval. If all sections meet the conditions, the current solution is directly superimposed; otherwise, for the sections that do not meet the requirements, a modified stiffness matrix [K*] is constructed using the updated stiffness series values, and the multi-segment elastic foundation beam theory is applied again to solve. The above solution and correction process is repeated until the predetermined number of load applications is completed. Finally, the initial parameter solutions obtained from each iterative calculation are integrated over the entire beam range, so that the internal force and displacement distribution diagrams of the structure under any load can be obtained. In this example, a test pile is selected for analysis. As shown in Figure 6 the figure, the test pile is a steel pipe pile with a pile diameter of 0.5 m, an embedded depth of 13.28 m, a free height of 0.72 m, a flexural stiffness of 97939.67 kN·m2, and a horizontal load H0 = 60.0555 kN acting on the pile top. The geotechnical parameters at the test site are shown in Table 1. The p-y curves of the nonlinear resistance of each soil layer adopt the following empirical formula for calculating the p-y curve of clay under one of the assumptions, and the nonlinear resistance curves of each soil layer are shown in Figure 7 the figure. P < P u P / P u = 0.48(Y / Y 50 ) 1 / 3 P ≥ P u P / P u = 1 Where: P u = min(P u1 , P u2 ) P u1 = (a + b·z)σ1D a = 1.121, b = 0.6613 / D Table 1 Geotechnical parameters of the site for the implementation example

[0023] By adopting the hybrid analytical-numerical method for analysis, we calculated the distributions of the horizontal displacement, bending moment, shear force, and resistance of the pile shaft along the depth. The specific results are shown inFigure 8 to Attachment Figure 11 。The comparison diagram of the actually calculated resistance path and the proposed p-y curve is shown in Attachment Figure 12 . The internal force and displacement diagrams show the states at five different load levels of 20%, 40%, 60%, 80% and 100%, and these states are selected from typical samples of numerous analysis steps. For the comparison of the actual path, the data at depths of 0.5 m, 0.75 m, 1.40 m and 4.30 m are selected, and the absolute values of these data are processed to facilitate clear comparison in the same coordinate system. It can be seen from the results that this calculation method can achieve high-precision simulation of the stress path, verifying its effectiveness and reliability in engineering applications.

Claims

1. A hybrid analytical-numerical method for solving soil-structure nonlinear interaction, characterized in that, The method includes the following steps: S1. Divide the foundation beam into n sections, perform quasi-linearization on the nonlinear resistance curves of each section to obtain the stiffness data values with sectional precision; obtain the approximate linear stiffness characteristics (i.e., secant stiffness) of each section according to the initial parameter method; based on the boundary conditions and the stiffness values of each section, establish the initial stiffness matrix [K], determine the load increment value △P and the number of iterations at each level, obtain the external load matrix [P], and apply the multi-segment elastic foundation beam theory to solve the initial parameter distribution of the beam under the load increment value at each level; S2. Analyze the obtained displacement results to check whether the stiffness data of each section fall within the precision interval defined by the nonlinear resistance curve; if the stiffness values of all sections meet the requirements, directly superimpose the current solution; otherwise, for the inconsistent sections, form a corrected stiffness matrix [K*] using the updated stiffness series values, and re-solve using the multi-segment elastic foundation beam theory again; S3. Repeat the above solution and correction processes until the predetermined number of load increments is completed; superimpose and integrate the initial parameter solutions obtained from each iterative calculation within the entire beam range to obtain the final displacement and stress distribution diagrams. The resistance within each iterative step is equal to the product of the stiffness and the displacement, and their superposition gives the resistance of the structure.

2. The hybrid analytical-numerical method for solving the soil-structure nonlinear interaction according to claim 1, characterized in that: At different horizontal depths, the nonlinear resistance curve is quasi-linearized according to the secant stiffness; as the segmentation accuracy improves, the secant stiffness can approximate the nonlinear resistance curve more accurately; the secant stiffness and the corresponding displacement data are constructed into a comparison database, where the initial judgment displacement is set to 0, corresponding to the initial stiffness value; this initial stiffness value and the parameter β i Establish a calculation relationship, β i is used to form the stiffness matrix [K] based on the elastic foundation beam theory; the first elastic foundation beam calculation is performed with the set load increment step size to obtain the displacement values at the corresponding depths; these displacement values are compared with the data in the comparison database; If it is found that some displacements exceed the original range, the corresponding secant stiffness needs to be updated and re-established the calculation relationship with β i to form the updated stiffness matrix [K*]; repeat until the number of numerical iterations terminates; by continuously iterating and updating the stiffness matrix, a high-precision solution of the structural response is achieved.

3. The hybrid analytical-numerical method for solving soil-structure nonlinear interaction according to claim 1, characterized in that: This method makes full use of the existing analytical solution form in the elastic foundation beam theory, combines the matrix framework of the multi-layer elastic foundation beam theory derived from matrix expansion, and performs matrix transfer iteration on this matrix framework to form a hybrid analytical-numerical iterative solution method.

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