A method, system, equipment, and storage medium for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert.

Abstract

This invention discloses a method, system, equipment, and storage medium for calculating the deformation of a pipe-jacking structure considering the soil resistance between the pipe and culvert. The method includes the following steps: S1. Simplifying the pipe-jacking structure as an elastic thin plate, and further simplifying the problem to the deformation of an elastic thin plate with fixed opposite sides and simply supported opposite sides under a uniformly distributed load on a Winkler elastic foundation, obtaining the differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression for the boundary conditions of the thin plate; S2. Determining the elastic modulus when the pipe-jacking structure is simplified to an elastic thin plate, and calculating the bending stiffness D of the pipe-jacking structure; S3. Solving the differential equation of the bending surface of the thin plate on the Winkler elastic foundation using a hybrid solution method combining trigonometric series and the force method, obtaining the deflection formula of the elastic thin plate placed on the elastic foundation under a uniformly distributed load. This invention can accurately predict the deformation of pipe-jacking structures.

Description

Technical Field

[0001] This invention relates to the field of tunnel engineering technology, and in particular to a method, system, equipment and storage medium for calculating the deformation of a pipe-culvert structure that takes into account the soil resistance between the pipe and the culvert. Background Technology

[0002] The pipe-jacking box culvert method is widely used in underground excavation projects where surface control is crucial. It involves pre-inserting a row of steel pipes into the soil layer to form a pre-support system, and then jacking a box culvert beneath the pipe-jacking structure to reduce soil disturbance and control surface settlement. Considering the interaction between the steel pipes and the soil, models such as the Winkler elastic foundation beam and the Pasternak two-parameter elastic foundation beam are used. These models can accurately calculate the final deformation of the pipe-jacking structure. However, in actual construction, the deformation of the pipe-jacking structure is a gradual process, which these models cannot accurately predict. Simplifying the pipe-jacking structure as a single continuous beam for analysis does not reflect the actual bidirectional stress state of the structure and ignores the influence of the interaction between the steel pipes on the support effect, leading to conservative analysis results.

[0003] As a whole, the transverse and longitudinal deformations of a pipe-jacking structure are coupled, with a mechanical mechanism more inclined towards that of a slab. In the practical application of the pipe-jacking-box culvert method, the soil between the pipe-jacking structure and the box culvert will generate resistance, inevitably affecting the support effect and construction safety of the pipe-jacking structure. Furthermore, the selection of steel pipe diameter, wall thickness, and spacing relies heavily on engineering experience. In addition, research on the coordinated transverse and longitudinal deformation of pipe-jacking structures is still in its early stages, and there is a lack of in-depth understanding of the linkage and feedback mechanism between the soil layer and the pipe-jacking structure system.

[0004] Therefore, the above problems urgently need to be solved. Summary of the Invention

[0005] Purpose of the invention: The first purpose of this invention is to provide a method for calculating the deformation of a pipe curtain structure that takes into account the soil resistance between the pipe and the culvert. This method fully considers the soil resistance between the pipe and the culvert and can accurately predict the deformation of the pipe curtain structure.

[0006] The second objective of this invention is to provide a deformation calculation system for pipe curtain structures that takes into account the soil resistance between the pipe and the culvert.

[0007] A third objective of this invention is to provide an electronic device.

[0008] A fourth objective of this invention is to provide a computer storage medium.

[0009] Technical Solution: To achieve the above objectives, this invention discloses a method for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert, comprising the following steps:

[0010] S1. The tube curtain structure is simplified to an elastic thin plate. At the same time, according to the boundary conditions, the problem is simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on the Winkler elastic foundation. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression of the boundary conditions of the thin plate are obtained.

[0011] S2. Determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate, and calculate the bending stiffness D of the tube curtain structure;

[0012] S3. The differential equation of the bending surface of a thin plate on a Winkler elastic foundation is solved by a hybrid solution method of trigonometric series and force method, and the deflection formula of an elastic thin plate subjected to uniform load on an elastic foundation is obtained.

[0013] Optionally, the differential equation for the bending surface of the thin plate on the Winkler elastic foundation obtained in step S1 is:

[0014]

[0015] In equation (4): D is the bending stiffness of the thin plate, w is the deflection of the plate, k is the foundation coefficient, (x, y) is the coordinate of the point on the thin plate at the mid-surface, and q(x, y) is the load perpendicular to the plate surface.

[0016] Optionally, the specific steps for obtaining the boundary condition expression of the thin plate in step S1 include:

[0017] Assuming the transverse sides of the tubular curtain structure are simply supported and the longitudinal sides are fixed, the deformation problem of the tubular curtain structure can be simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The boundary condition expressions for the tubular curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows:

[0018] Since the fixed side is parallel to the x-axis, the deflection and slope of the plate on both sides at y=0 and y=b are both 0.

[0019] (w) y＝0 =(w) y＝b =0(5)

[0020]

[0021] Since the direction of the simply supported side is parallel to the y-axis, the deflection and bending moment on both sides of the plate at x = 0 and x = b are both zero.

[0022] (w) x＝0 =(w) x＝a =0(7)

[0023]

[0024] In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate.

[0025] Optionally, the specific steps of step S3 include:

[0026] The simplified boundary conditions for the thin plate are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides. At the same time, bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, and x=0 and x=a, respectively. From equation (8), it can be seen that M(y) is 0. According to the differential equation of the bending surface of the thin plate on the Winkler elastic foundation, that is, according to formula (4), the deflection w of the thin plate is taken as a double sine series solution:

[0027]

[0028] In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mn Let be a function of x and y; for the entire system of elastic thin plate and Winkler elastic foundation, when deformation occurs under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation.

[0029] The formula for calculating the deformation energy U1 of an elastic thin plate is as follows:

[0030]

[0031] The formula for calculating the deformation energy U2 of an elastic foundation is as follows:

[0032]

[0033] Total strain energy U:

[0034]

[0035] The slope of the elastic thin plate on the four sides y=0, y=b and x=0, x=a can be expressed as:

[0036]

[0037] when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is:

[0038]

[0039] Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given:

[0040]

[0041] Where E m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively:

[0042]

[0043] Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain:

[0044]

[0045] Substituting the bending stiffness D of the pipe curtain structure in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get:

[0046]

[0047] The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equations (16) and (17), we can obtain:

[0048]

[0049] Summing the series of equation (26):

[0050]

[0051] In the formula:

[0052]

[0053] From equation (27), we can obtain:

[0054]

[0055] Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

[0056] Based on the same inventive concept, this invention discloses a deformation calculation system for pipe-culvert structures that considers the soil resistance between the pipe and culvert, comprising:

[0057] The simplification module is used to simplify the tube curtain structure into an elastic thin plate. At the same time, based on the boundary conditions, the problem is simplified into the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on a Winkler elastic foundation. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression of the boundary conditions of the thin plate are obtained.

[0058] The calculation module is used to determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate, and to calculate the bending stiffness D of the tube curtain structure.

[0059] The solver module is used to solve the differential equation of the bending surface of a thin plate on a Winkler elastic foundation using a hybrid solution method of trigonometric series and force method, so as to obtain the deflection formula of an elastic thin plate placed on an elastic foundation and subjected to a uniformly distributed load.

[0060] Optionally, the differential equation for the bending surface of a thin plate on a Winkler elastic foundation obtained in the simplification module is:

[0061]

[0062] In equation (4): D is the bending stiffness of the thin plate, w is the deflection of the plate, k is the foundation coefficient, (x, y) is the coordinate of the point on the thin plate at the mid-surface, and q(x, y) is the load perpendicular to the plate surface.

[0063] Optionally, the boundary condition expression for the thin plate obtained in the simplification module is:

[0064] Assuming the transverse sides of the tubular curtain structure are simply supported and the longitudinal sides are fixed, the deformation problem of the tubular curtain structure can be simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The boundary condition expressions for the tubular curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows:

[0065] Since the fixed side is parallel to the x-axis, the deflection and slope of the plate on both sides at y=0 and y=b are both 0.

[0066] (w) y＝0 =(w) y＝b =0(5)

[0067]

[0068] Since the direction of the simply supported side is parallel to the y-axis, the deflection and bending moment on both sides of the plate at x = 0 and x = b are both zero.

[0069] (w) x＝0 =(w) x＝a =0(7)

[0070]

[0071] In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate.

[0072] Optionally, the simplified thin plate boundary conditions in the solution module are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides, and bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, and x=0 and x=a, respectively. From equation (8), it can be seen that M(y) is 0. According to the differential equation of the bending surface of the thin plate on the Winkler elastic foundation, that is, according to formula (4), the deflection w of the thin plate is taken as a double sine series solution:

[0073]

[0074] In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mn Let be a function of x and y; for the entire system of elastic thin plate and Winkler elastic foundation, when deformation occurs under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation.

[0075] The formula for calculating the deformation energy U1 of an elastic thin plate is as follows:

[0076]

[0077] The formula for calculating the deformation energy U2 of an elastic foundation is as follows:

[0078]

[0079] Total strain energy U:

[0080]

[0081] The slope of the elastic thin plate on the four sides y=0, y=b and x=0, x=a can be expressed as:

[0082]

[0083] when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is:

[0084]

[0085] Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given:

[0086]

[0087] Where E m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively:

[0088]

[0089] Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain:

[0090]

[0091] Substituting the bending stiffness D of the pipe curtain structure in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get:

[0092]

[0093] The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equations (16) and (17), we can obtain:

[0094]

[0095] Summing the series of equation (26):

[0096]

[0097] In the formula:

[0098]

[0099] From equation (27), we can obtain:

[0100]

[0101] Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

[0102] This invention discloses an electronic device, including a processor and a memory.

[0103] The memory is used to store computer programs that are executed by the processor to perform the above-mentioned method for calculating the deformation of a pipe curtain structure that considers the soil resistance between the pipe and the culvert.

[0104] The present invention discloses a computer storage medium storing a computer program, which is executed by a processor to perform the above-described method for calculating the deformation of a pipe curtain structure considering the soil resistance between the pipe and the culvert.

[0105] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: (1) The selection of boundary conditions in the present invention is more in line with reality than the existing theories. Currently, the existing theories on the deformation of pipe jacking box culverts all set the boundary as simply supported on four sides, and the boundary in the longitudinal direction does not match reality. When constructing the pipe jacking box culvert, after the pipe jacking steel pipe above the box culvert jacking area is completed, the soil in the starting and arriving ends of the box culvert will be grouted and reinforced to prevent the collapse of the soil at the tunnel entrance when the box culvert starts and arrives, and to fix the longitudinal ends of the pipe jacking structure. Therefore, the present invention selects fixed constraints as the pipe jacking structure. The boundary conditions at both ends in the longitudinal direction remain unchanged, while the boundary conditions in the transverse direction remain unchanged, which is more in line with the actual situation. At the same time, the existing theories only simplify the boundary of the pipe jacking structure to simply supported four sides. During the jacking process of the box culvert, the pipe jacking structure in the non-jacking area has a more fixed boundary because the soil below has not been excavated and the soil has a suppressive effect on the longitudinal rotation of the pipe jacking structure. Therefore, it cannot predict the deformation of the pipe jacking structure during the jacking process. The boundary conditions selected in this invention fully take into account both the situation during and after the jacking of the box culvert, and can provide an accurate reference for predicting the deformation of the pipe jacking structure during and after the jacking process of the box culvert.

[0106] (2) The present invention addresses the solution approach for the differential equation of deformation of pipe curtain structures by introducing the force method into the solution process. Compared with the traditional solution method, by introducing the strain energy generated by the deformation of both the pipe curtain structure and the soil, the present invention fully considers the coordinated deformation process of the pipe curtain structure and the soil. The traditional solution method only solves the problem by simply expanding the deflection and load into a series and substituting the boundary conditions, ignoring the coordinated deformation of the soil and the pipe curtain structure, which leads to a large error in the calculation results. The final result of the present invention is a series with fast convergence. The calculation of the first three terms can obtain a more accurate result. Compared with the existing theory, it can effectively reduce the amount of calculation and improve its wide applicability in engineering applications. Attached Figure Description

[0107] Figure 1 This is a diagram of the theoretical calculation model of the elastic thin plate in this invention;

[0108] Figure 2 This is a simplified schematic diagram of the lateral boundary of the tube curtain structure in this invention;

[0109] Figure 3 This is a simplified schematic diagram of the longitudinal boundary of the tube curtain structure in this invention;

[0110] Figure 4 This is a simplified calculation diagram of the tube curtain structure in this invention;

[0111] Figure 5 This is a schematic diagram of the coordinate system with "simply supported opposite sides and fixed opposite sides" in this invention;

[0112] Figure 6This is a simplified schematic diagram of the tube curtain structure in this invention. Detailed Implementation

[0113] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0114] Example 1: The present invention provides a method for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and the culvert, comprising the following steps:

[0115] S1. The tube curtain structure is simplified into an elastic thin plate, and the corresponding assumptions are determined. At the same time, the problem is simplified into the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on a Winkler elastic foundation, based on the boundary conditions. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the boundary condition expression of the thin plate are obtained.

[0116] The specific steps of step S1 are as follows: According to the provisions of elasticity mechanics, the feasibility of simplifying the tube curtain structure into an elastic thin plate is verified, and the necessary assumptions are determined to ensure the accuracy of the model; at the same time, according to the boundary conditions, the problem is simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides subjected to uniformly distributed load on a Winkler elastic foundation.

[0117] In the field of elasticity, a plate whose thickness to its short side length is between 1 / 80 and 1 / 5 is called a thin plate. In the pipe jacking-box culvert method, the diameter of the steel pipe is generally 800-1000 mm, which is the thickness when simplified to an elastic thin plate. The longitudinal jacking distance of the box culvert is greater than the transverse width of the pipe jacking, and the transverse width of the pipe jacking is slightly greater than the cross-sectional width of the box culvert. The transverse distance of the pipe jacking is the short side of the pipe jacking structure. The transverse width of the pipe jacking is generally 6-9 m. Therefore, the pipe jacking structure can be simplified as an elastic thin plate.

[0118] like Figure 1 As shown, the maximum deflection of the pipe jacking structure caused by the unloading of the soil beneath it during excavation by the pipe jacking machine generally does not exceed 100mm. This maximum deflection is much smaller than the thickness of the pipe jacking structure, falling within the category of small deflection problems in thin-plate bending, based on the following assumptions:

[0119] (a) Neutral surface assumption: When the elastic thin plate deforms in the z-axis direction, each point on the mid-surface has displacement only in the z-axis direction, and the displacement in the x-axis and y-axis directions is 0.

[0120] (b) Straight normal assumption: The elastic thin plate is imagined to be composed of several line segments of height h perpendicular to the mid-surface. Before the elastic thin plate deforms, the line segments perpendicular to the mid-surface of the elastic thin plate remain perpendicular to the bent mid-surface after the elastic thin plate deforms; that is, it is assumed that when the elastic thin plate deforms in the z-axis direction, there is no shear strain in the mid-surface of the elastic thin plate.

[0121] (c) Non-compression assumption: When the elastic sheet deforms, the fibers in each layer of the elastic sheet do not compress against each other, that is, the normal stress σ in each layer section parallel to the middle surface is zero. z Much smaller than the normal stress σ in the x-direction within the cross-section x σ in the y-direction y and the shear force τ in the xy plane xy Therefore, the normal stress σ z Negligible;

[0122] In actual construction, the gap between the soil layers between the pipe and the culvert is generally 10-20cm, and the foundation thickness is much smaller than the thickness of the elastic thin plate. When the planar dimensions of the plate are fixed, the deformation calculation results of the elastic thin plate on the Winkler elastic foundation and the two-parameter elastic foundation are similar. Therefore, the Winkler elastic foundation model is selected.

[0123] Winkler's elastic foundation assumes that the elastic foundation consists of a series of closely spaced, unconnected linear springs. When the foundation surface is subjected to a load, the deformation at a point on the foundation is proportional to the pressure exerted at that point, which can be expressed as:

[0124]

[0125] In equation (1): w is the foundation deformation, i.e., w is the deflection of the plate, k is the foundation coefficient, and q is the pressure per unit area;

[0126] If an elastic thin plate is located on a continuous Winkler elastic foundation and is bent by a load q(x,y) perpendicular to the plate surface; when the deflection of the thin plate is much smaller than the thickness of the thin plate, and according to the assumption of the Winkler elastic foundation, the pressure per unit area at any point on the thin plate is proportional to the deflection of the thin plate at that point, as shown in equation (2):

[0127] q=kw(2)

[0128] Therefore, the magnitude of the load acting on each point of the thin plate is the overlying load on the thin plate minus the ground reaction force generated by the deformation of the thin plate, as shown in equation (3):

[0129] q(x,y)-kw(3)

[0130] Therefore, the differential equation for the bending surface of a thin plate on a Winkler elastic foundation is:

[0131]

[0132] In equation (4): D is the bending stiffness of the thin plate, w is the deflection of the plate, k is the foundation coefficient, and (x, y) is the coordinate of the point on the thin plate at the mid-surface.

[0133] In actual construction using the pipe-jacking-box culvert method, the jacking of the box culvert causes minimal disturbance to the soil beneath the steel pipes at both ends. The transversely arranged steel pipes are connected by interlocking joints, resulting in relatively weak bending resistance. Therefore, it is assumed that the transverse sides of the pipe-jacking structure are simply supported. After the pipe-jacking structure is completed and before the box culvert is jacked, grouting reinforcement is required at the portal to prevent soil collapse when the box culvert arrives. During the box culvert jacking process, the soil beneath the pipe-jacking structure in the unjacked area experiences minimal disturbance. Therefore, it is assumed that the longitudinal sides of the pipe-jacking structure are fixed. The longitudinal calculation length is the jacking distance of the box culvert, and the simplified calculation diagram is shown below. Figure 2 and Figure 3 As shown;

[0134] Thus, the deformation problem of the tube jacking structure has been simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The x-direction is the transverse direction of the tube jacking, and the y-direction is the longitudinal direction of the tube jacking, i.e., the jacking direction of the box culvert. The simplified calculation diagram is as follows: Figure 4 As shown;

[0135] Establish a rectangular coordinate system with the upper left corner of the tube curtain structure as the origin, such as Figure 5 As shown, the boundary condition expressions for the tube curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows:

[0136] Since the fixed side is parallel to the x-axis, the deflection and slope of the plate on both sides at y=0 and y=b are both 0.

[0137] (w) y＝0 =(w) y＝b =0(5)

[0138]

[0139] Since the direction of the simply supported side is parallel to the y-axis, the deflection and bending moment on both sides of the plate at x = 0 and x = b are both zero.

[0140] (w) x＝0 =(w) x＝a =0(7)

[0141]

[0142] In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate;

[0143] S2. Based on the assumptions in step S1, determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate, and calculate the bending stiffness D of the tube curtain structure.

[0144] In the actual construction of the pipe jacking box culvert method, the displacement deviation of the steel pipe is limited by welding male and female locking buckles on both sides of the steel pipe, thus connecting the pipe jacking steel pipes into a whole.

[0145] As can be seen from equation (4), in the differential equation of the bending surface of a thin plate on a Winkler elastic foundation, the bending stiffness D of the thin plate is a core parameter in the deflection calculation. The formula for calculating the bending stiffness D of the thin plate is as follows:

[0146]

[0147] In equation (9): D is the bending stiffness of the thin plate, E eq For the elastic modulus of the thin plate, t eq Let μ be the thickness of the thin plate, and μ be the Poisson's ratio of the thin plate.

[0148] When the tube curtain structure is simplified to an elastic thin plate, the elastic modulus E of the thin plate is... eq It can be determined using the following formula:

[0149]

[0150] In equations (10) and (11), K1 and K2 are the bending stiffness of the steel pipe and the grout, D1 and D2 are the compressive stiffness of the steel pipe and the grout, and b is the bearing width of the pipe curtain. The pipe curtain structure is simplified as a schematic diagram of an elastic thin plate, as shown in the figure. Figure 6 As shown;

[0151] S3. The differential equation of the bending surface of a thin plate on a Winkler elastic foundation is solved by a hybrid solution method of trigonometric series and force method, and the deflection formula of an elastic thin plate subjected to uniform load on an elastic foundation is obtained.

[0152] The simplified boundary conditions for the thin plate are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides. At the same time, bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, and x=0 and x=a, respectively. From equation (8), it can be seen that M(y) is 0. According to the differential equation of the bending surface of the thin plate on the Winkler elastic foundation, that is, according to formula (4), the deflection w of the thin plate is taken as a double sine series solution:

[0153]

[0154] In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mn Let be a function of x and y; for the entire system of elastic thin plate and elastic foundation, when it deforms under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation.

[0155] The formula for calculating the deformation energy U1 of an elastic thin plate is as follows:

[0156]

[0157] The formula for calculating the deformation energy U2 of an elastic foundation is as follows:

[0158]

[0159] Total strain energy U:

[0160]

[0161] The slope of the elastic thin plate on the four sides y=0, y=b and x=0, x=a can be expressed as:

[0162]

[0163] when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is:

[0164]

[0165] Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given:

[0166]

[0167] Where E m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively:

[0168]

[0169] Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain:

[0170]

[0171] Substituting the flexural stiffness D of the pipe curtain structure calculated in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get:

[0172]

[0173] The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equations (16) and (17), we can obtain:

[0174]

[0175] Summing the series of equation (26):

[0176]

[0177] In the formula:

[0178]

[0179] From equation (27), we can obtain:

[0180]

[0181] Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

[0182] Example 2: This invention discloses a deformation calculation system for a pipe-culvert structure considering the soil resistance between the pipe and culvert, comprising:

[0183] The simplification module is used to simplify the tube curtain structure into an elastic thin plate and determine the corresponding assumptions. At the same time, based on the boundary conditions, the problem is simplified into the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on a Winkler elastic foundation. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression of the boundary conditions of the thin plate are obtained.

[0184] In the simplification module, the feasibility of simplifying the tube curtain structure into an elastic thin plate is verified according to the provisions of elasticity mechanics, and the necessary assumptions are determined to ensure the accuracy of the model. At the same time, based on the boundary conditions, the problem is simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides subjected to uniformly distributed load on a Winkler elastic foundation.

[0185] In the field of elasticity, a plate whose thickness to its short side length is between 1 / 80 and 1 / 5 is called a thin plate. In the pipe jacking-box culvert method, the diameter of the steel pipe is generally 800-1000 mm, which is the thickness when simplified to an elastic thin plate. The longitudinal jacking distance of the box culvert is greater than the transverse width of the pipe jacking, and the transverse width of the pipe jacking is slightly greater than the cross-sectional width of the box culvert. The transverse distance of the pipe jacking is the short side of the pipe jacking structure. The transverse width of the pipe jacking is generally 6-9 m. Therefore, the pipe jacking structure can be simplified as an elastic thin plate.

[0186] like Figure 1 As shown, the maximum deflection of the pipe jacking structure caused by the unloading of the soil beneath it during excavation by the pipe jacking machine generally does not exceed 100mm. This maximum deflection is much smaller than the thickness of the pipe jacking structure, falling within the category of small deflection problems in thin-plate bending, based on the following assumptions:

[0187] (a) Neutral surface assumption: When the elastic thin plate deforms in the z-axis direction, each point on the mid-surface has displacement only in the z-axis direction, and the displacement in the x-axis and y-axis directions is 0.

[0188] (b) Straight normal assumption: The elastic thin plate is imagined to be composed of several line segments of height h perpendicular to the mid-surface. Before the elastic thin plate deforms, the line segments perpendicular to the mid-surface of the elastic thin plate remain perpendicular to the bent mid-surface after the elastic thin plate deforms; that is, it is assumed that when the elastic thin plate deforms in the z-axis direction, there is no shear strain in the mid-surface of the elastic thin plate.

[0189] (c) Non-compression assumption: When the elastic sheet deforms, the fibers in each layer of the elastic sheet do not compress against each other, that is, the normal stress σ in each layer section parallel to the middle surface is zero. z Much smaller than the normal stress σ in the x-direction within the cross-section x σ in the y-direction y and the shear force τ in the xy plane xy Therefore, the normal stress σ z Negligible;

[0190] In actual construction, the gap between the soil layers between the pipe and the culvert is generally 10-20cm, and the foundation thickness is much smaller than the thickness of the elastic thin plate. When the planar dimensions of the plate are fixed, the deformation calculation results of the elastic thin plate on the Winkler elastic foundation and the two-parameter elastic foundation are similar. Therefore, the Winkler elastic foundation model is selected.

[0191] Winkler's elastic foundation assumes that the elastic foundation consists of a series of closely spaced, unconnected linear springs. When the foundation surface is subjected to a load, the deformation at a point on the foundation is proportional to the pressure exerted at that point, which can be expressed as:

[0192]

[0193] In equation (1): w is the foundation deformation, i.e., w is the deflection of the plate, k is the foundation coefficient, and q is the pressure per unit area;

[0194] If an elastic thin plate is located on a continuous Winkler elastic foundation and is bent by a load q(x,y) perpendicular to the plate surface; when the deflection of the thin plate is much smaller than the thickness of the thin plate, and according to the assumption of the Winkler elastic foundation, the pressure per unit area at any point on the thin plate is proportional to the deflection of the thin plate at that point, as shown in equation (2):

[0195] q=kw(2)

[0196] Therefore, the magnitude of the load acting on each point of the thin plate is the overlying load on the thin plate minus the ground reaction force generated by the deformation of the thin plate, as shown in equation (3):

[0197] q(x,y)-kw(3)

[0198] Therefore, the differential equation for the bending surface of a thin plate on a Winkler elastic foundation is:

[0199]

[0200] In equation (4): D is the bending stiffness of the thin plate, w is the deflection of the plate, k is the foundation coefficient, and (x, y) is the coordinate of the point on the thin plate at the mid-surface.

[0201] In actual construction using the pipe-jacking-box culvert method, the jacking of the box culvert causes minimal disturbance to the soil beneath the steel pipes at both ends. The transversely arranged steel pipes are connected by interlocking joints, resulting in relatively weak bending resistance. Therefore, it is assumed that the transverse sides of the pipe-jacking structure are simply supported. After the pipe-jacking structure is completed and before the box culvert is jacked, grouting reinforcement is required at the portal to prevent soil collapse when the box culvert arrives. During the box culvert jacking process, the soil beneath the pipe-jacking structure in the unjacked area experiences minimal disturbance. Therefore, it is assumed that the longitudinal sides of the pipe-jacking structure are fixed. The longitudinal calculation length is the jacking distance of the box culvert, and the simplified calculation diagram is shown below. Figure 2 and Figure 3 As shown;

[0202] Thus, the deformation problem of the tube jacking structure has been simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The x-direction is the transverse direction of the tube jacking, and the y-direction is the longitudinal direction of the tube jacking, i.e., the jacking direction of the box culvert. The simplified calculation diagram is as follows: Figure 4 As shown;

[0203] Establish a rectangular coordinate system with the upper left corner of the tube curtain structure as the origin, such as Figure 5 As shown, the boundary condition expressions for the tube curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows:

[0204] Since the fixed side is parallel to the x-axis, the deflection and slope of the plate on both sides at y=0 and y=b are both 0.

[0205] (w) y＝0 =(w) y＝b =0(5)

[0206]

[0207] Since the direction of the simply supported side is parallel to the y-axis, the deflection and bending moment on both sides of the plate at x = 0 and x = b are both zero.

[0208] (w) x＝0 =(w) x＝a =0(7)

[0209]

[0210] In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate;

[0211] The calculation module is used to determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate based on the assumptions in the simplification module, and to calculate the bending stiffness D of the tube curtain structure.

[0212] In the actual construction of the pipe jacking box culvert method, the displacement deviation of the steel pipe is limited by welding male and female locking buckles on both sides of the steel pipe, thus connecting the pipe jacking steel pipes into a whole.

[0213] As can be seen from equation (4), in the differential equation of the bending surface of a thin plate on a Winkler elastic foundation, the bending stiffness D of the thin plate is a core parameter in the deflection calculation. The formula for calculating the bending stiffness D of the thin plate is as follows:

[0214]

[0215] In equation (9): D is the bending stiffness of the thin plate, E eq For the elastic modulus of the thin plate, t eq Let μ be the thickness of the thin plate, and μ be the Poisson's ratio of the thin plate.

[0216] When the tube curtain structure is simplified to an elastic thin plate, the elastic modulus E of the thin plate is... eq It can be determined using the following formula:

[0217]

[0218] In equations (10) and (11), K1 and K2 are the bending stiffness of the steel pipe and the grout, D1 and D2 are the compressive stiffness of the steel pipe and the grout, and b is the bearing width of the pipe curtain. The pipe curtain structure is simplified as a schematic diagram of an elastic thin plate, as shown in the figure. Figure 6 As shown;

[0219] The solver module is used to solve the differential equation of the bending surface of a thin plate on a Winkler elastic foundation using a hybrid solution method of trigonometric series and force method, so as to obtain the deflection formula of an elastic thin plate placed on an elastic foundation and subjected to a uniformly distributed load.

[0220] The simplified boundary conditions for the thin plate are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides. At the same time, bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, and x=0 and x=a, respectively. From equation (8), it can be seen that M(y) is 0. According to the differential equation of the bending surface of the thin plate on the Winkler elastic foundation, that is, according to formula (4), the deflection w of the thin plate is taken as a double sine series solution:

[0221]

[0222] In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mnLet be a function of x and y; for the entire system of elastic thin plate and elastic foundation, when it deforms under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation.

[0223] The formula for calculating the deformation energy U1 of an elastic thin plate is as follows:

[0224]

[0225] The formula for calculating the deformation energy U2 of an elastic foundation is as follows:

[0226]

[0227] Total strain energy U:

[0228]

[0229] The slope of the elastic thin plate on the four sides y=0, y=b and x=0, x=a can be expressed as:

[0230]

[0231] when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is:

[0232]

[0233] Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given:

[0234]

[0235] Where E m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively:

[0236]

[0237] Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain:

[0238]

[0239] Substituting the flexural stiffness D of the pipe curtain structure calculated in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get:

[0240]

[0241] The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equations (16) and (17), we can obtain:

[0242]

[0243] Summing the series of equation (26):

[0244]

[0245] In the formula:

[0246]

[0247] From equation (27), we can obtain:

[0248]

[0249] Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

[0250] Example 3: Corresponding to the method of Example 1 of the present invention, Example 3 of the present invention also provides an electronic device. In Example 3, the electronic device includes: at least one communication bus, at least one processor, at least one memory, at least one network interface, and at least one peripheral interface. The memory contains programs and data.

[0251] A communication bus can be a communication device that transmits data between components within an electronic device, such as an internal bus (CPU and memory bus) or an external bus (Universal Serial Bus port, Peripheral Component Interconnect Fast Port, etc.).

[0252] The memory may include high-speed RAM, and may also include non-volatile memory, such as at least one disk storage device.

[0253] The processor calls the program and data stored in the memory to execute the deformation calculation method of the pipe curtain structure considering the soil resistance between the pipe and the culvert provided in Embodiment 1 of the present invention.

[0254] Peripheral interfaces are used to connect to peripherals, which are external devices. External devices may include, but are not limited to, keyboards, monitors, cursor control devices (such as mice, touchpads or touch screens), video input devices, etc.

[0255] A network interface provides wired or wireless communication with external networks (e.g., the Internet, intranets, local area networks, mobile communication networks, etc.).

[0256] Example 4: Corresponding to the method of Example 1 of the present invention, Example 4 of the present invention also provides a computer storage medium for data acquisition and reception. The computer storage medium stores a computer program, which is run by a processor to execute the deformation calculation method of the pipe curtain structure considering the soil resistance between the pipe and the culvert provided in Example 1 of the present invention.

[0257] The functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware, as a software functional unit, or in a combination of software and hardware.

[0258] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as portable hard drives, USB flash drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0259] This invention provides a method, system, device, and storage medium for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert. The above description is merely a preferred embodiment of this invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A method for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert, characterized in that, Includes the following steps: S1. The tube curtain structure is simplified to an elastic thin plate. At the same time, according to the boundary conditions, the problem is simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on the Winkler elastic foundation. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression of the boundary conditions of the thin plate are obtained. The differential equation for the bending surface of the thin plate on the Winkler elastic foundation obtained in step S1 is as follows: （4） In equation (4): D is the bending stiffness of the tube curtain structure, w is the deflection of the plate, k is the foundation coefficient, and (x, y) are the coordinates of the point on the thin plate at the mid-surface. The load is perpendicular to the plate surface; S2. Determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate, and calculate the bending stiffness D of the tube curtain structure; S3. The differential equation of the bending surface of a thin plate on a Winkler elastic foundation is solved by a hybrid solution method of trigonometric series and force method, and the deflection formula of an elastic thin plate subjected to uniform load on an elastic foundation is obtained. According to the differential equation of the bending surface of a thin plate on a Winkler elastic foundation, i.e., according to formula (4), the deflection w of the thin plate is taken as a double sinusoidal series solution: （12） In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mn Let x be a function of x and y; For the entire system of elastic thin plate and Winkler elastic foundation, when deformation occurs under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation. when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is: （20） Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given: （21） Where E m and F n These are the coefficients when M(x) and M(y) are expanded into Fourier series, respectively. Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain: （24） Substituting the bending stiffness D of the pipe curtain structure in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get: （25） （30） Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

2. The method for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert, as described in claim 1, is characterized in that: The specific steps for obtaining the boundary condition expression of the thin plate in step S1 include: Assuming the transverse sides of the tubular curtain structure are simply supported and the longitudinal sides are fixed, the deformation problem of the tubular curtain structure can be simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The boundary condition expressions for the tubular curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows: The direction of the fixed side is parallel to the x-axis, therefore in and The deflection and slope on both sides of the upper plate are both 0. （5） （6） The direction of the simply supported side is parallel to the y-axis, therefore in and The deflection and bending moment on both sides of the upper plate are both 0: （7） （8） In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate.

3. The method for calculating the deformation of a pipe-culvert structure considering the soil resistance between the pipe and culvert, as described in claim 2, is characterized in that: The specific steps of step S3 also include: The simplified boundary conditions of the thin plate are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides. At the same time, bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, and at x=0 and x=a, respectively. From equation (8), it can be seen that M(y) is 0. The formula for calculating the deformation energy U1 of an elastic thin plate is as follows: （13） The formula for calculating the deformation energy U2 of an elastic foundation is as follows: （14） Total strain energy U: （15） The slope of the elastic thin plate on its four sides at y=0, y=b and x=0, x=a can be expressed as: （16） （17） （18） （19） E in formula (21) m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively: （22） （23） The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equation (16) and equation (17), we can obtain: （26） Summing the series of equation (26): （27） In the formula: （28） （29） From equation (27), we can obtain equation (30).

4. A deformation calculation system for a pipe-culvert structure considering the soil resistance between the pipe and culvert, characterized in that, include: The simplification module is used to simplify the tube curtain structure into an elastic thin plate. At the same time, based on the boundary conditions, the problem is simplified into the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides under uniformly distributed load on a Winkler elastic foundation. The differential equation of the bending surface of the thin plate on the Winkler elastic foundation and the expression of the boundary conditions of the thin plate are obtained. The differential equation for the bending surface of a thin plate on a Winkler elastic foundation obtained from the simplified module is as follows: （4） In equation (4): D is the bending stiffness of the thin plate, w is the deflection of the plate, k is the foundation coefficient, and (x, y) are the coordinates of the point on the thin plate at the mid-surface. The load is perpendicular to the plate surface; The calculation module is used to determine the elastic modulus when the tube curtain structure is simplified to an elastic thin plate, and to calculate the bending stiffness D of the tube curtain structure. The solution module is used to solve the differential equation of the bending surface of a thin plate on a Winkler elastic foundation using a hybrid solution method of trigonometric series and force method, and obtain the deflection formula of an elastic thin plate placed on an elastic foundation and subjected to a uniformly distributed load. According to the differential equation of the bending surface of a thin plate on a Winkler elastic foundation, i.e., according to formula (4), the deflection w of the thin plate is taken as a double sinusoidal series solution: （12） In equation (12): a is the transverse length of the thin plate, b is the longitudinal length of the thin plate, and m and n are equal to 1, 2, 3, ...; mn Let be a function of x and y; for the entire system of elastic thin plate and Winkler elastic foundation, when it deforms under load, the deformation energy consists of the deformation energy U1 of the elastic thin plate itself and the deformation energy U2 of the elastic foundation. when a mn Increase to a mn +δa mn At that time, the strain energy increment δU is: （20） Given an external load q, a bending moment M(x), and the work δT done by the bending moment M(y), the following is given: （21） Where E m and F n These are the coefficients when M(x) and M(y) are expanded into Fourier series, respectively. Equations (20) and (21) are equal. Combining this with the principle of virtual displacement, we can obtain: （24） Substituting the bending stiffness D of the pipe curtain structure in step S2 into equation (24) yields a. mn , the obtained a mn Substituting into equation (12), we get: （25） （30） Substituting equation (30) into equation (25) yields the deflection formula for an elastic thin plate subjected to a uniformly distributed load on an elastic foundation.

5. The deformation calculation system for a pipe-culvert structure considering the soil resistance between the pipe and culvert as described in claim 4, characterized in that, The boundary condition expression for the thin plate obtained in the simplification module is as follows: Assuming the transverse sides of the tubular curtain structure are simply supported and the longitudinal sides are fixed, the deformation problem of the tubular curtain structure can be simplified to the deformation problem of an elastic thin plate with fixed opposite sides and simply supported opposite sides, placed on a Winkler elastic foundation and subjected to a uniformly distributed load. The boundary condition expressions for the tubular curtain structure, i.e., the boundary condition expressions for the thin plate, are as follows: The direction of the fixed side is parallel to the x-axis, therefore in and The deflection and slope on both sides of the upper plate are both 0. （5） （6） The direction of the simply supported side is parallel to the y-axis, therefore in and The deflection and bending moment on both sides of the upper plate are both 0: （7） （8） In equation (8): w is the deflection of the thin plate, and μ is the Poisson's ratio of the thin plate.

6. The deformation calculation system for a pipe-culvert structure considering the soil resistance between the pipe and culvert as described in claim 5, characterized in that: The simplified thin plate boundary conditions in the solution module are: the opposite sides in the x direction are fixed, and the opposite sides in the y direction are simply supported; the thin plate boundary is equivalent to an elastic thin plate with simply supported sides on all four sides, and bending moments M(x) and M(y) that vary with the value of x are applied at y=0 and y=b, x=0 and x=a respectively. From equation (8), it can be seen that M(y) is 0. The formula for calculating the deformation energy U1 of an elastic thin plate is as follows: （13） The formula for calculating the deformation energy U2 of an elastic foundation is as follows: （14） Total strain energy U: （15） The slope of the elastic thin plate on its four sides at y=0, y=b and x=0, x=a can be expressed as: （16） （17） （18） （19） E in formula (21) m and F n The coefficients of M(x) and M(y) when expanded into Fourier series, respectively: （22） （23） The thin plate parallel to the y-axis has simply supported constraints on both sides, therefore M(y) = 0, i.e., F n =0; The two sides of the thin plate parallel to the x-axis are fixed constraints, and the rotation angle is 0. From equation (16) and equation (17), we can obtain: （26） Summing the series of equation (26): （27） In the formula: （28） （29） From equation (27), we can obtain equation (30).

7. An electronic device, characterized in that, Including processor and memory, A memory for storing a computer program that is executed by a processor to perform the method described in any one of claims 1-3.

8. A computer storage medium, characterized in that, The computer storage medium stores a computer program that is executed by a processor to perform the method described in any one of claims 1-3.

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