Beam bridge flexural rigidity inversion method based on displacement influence line

Abstract

The invention discloses a beam bridge flexural rigidity inversion method based on a displacement influence line, and belongs to the technical field of bridge flexural rigidity inversion, and the method comprises the following steps: S1, introducing a rigidity distribution function; s2, establishing a transfer function; and S3, carrying out flexural rigidity inversion. According to the method, a displacement influence line analytical solution is deduced through a force method basic equation, a complex stiffness function is processed in combination with numerical integration, and theoretical preciseness and engineering practicability are considered; by introducing Gaussian-Legendre numerical integration, the non-linear change of the rigidity of the variable cross-section beam is directly processed, and errors caused by simplified assumption are avoided; by utilizing displacement influence line data, structural response can be obtained through quasi-static loading (such as moving vehicles), traffic interruption is not needed, and the test cost is reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge bending stiffness inversion, and in particular to a beam bridge bending stiffness inversion method based on displacement influence lines. Background Art

[0002] The flexural stiffness of a bridge structure is an important parameter for assessing its load-bearing capacity and safety. Traditional methods for identifying flexural stiffness rely primarily on static load tests, inferring stiffness values by measuring the displacement response under specific loads. However, static load tests require traffic interruption and are costly, making them difficult to implement frequently. Furthermore, existing methods are often based on the assumption of uniform cross-section beams, while actual bridges often use variable cross-section designs (such as continuous box girders), whose stiffness varies nonlinearly along the span. Traditional methods suffer from large inversion errors due to simplified assumptions.

[0003] In recent years, modal analysis methods based on dynamic testing have been increasingly used. However, these methods rely on high-order modal information, which is susceptible to interference from local damage or environmental noise, resulting in unstable identification results. Furthermore, existing methods face challenges in inverting the stiffness of variable-section beams, including complex mathematical processing and difficulty in analytical solutions. They often rely on finite element model modifications, resulting in low computational efficiency and limited versatility. Therefore, a method for inverting the flexural stiffness of beam bridges based on displacement influence lines is proposed. Summary of the Invention

[0004] The technical problem to be solved by the present invention is how to solve the problems faced by existing methods in the inversion of the stiffness of variable-section beams, such as complex mathematical processing and difficulty in analytical solution, and provide a method for inverting the bending stiffness of beam bridges based on displacement influence lines.

[0005] The present invention solves the above technical problems through the following technical solutions, which include the following steps:

[0006] S1: Introducing stiffness distribution function

[0007] The bending stiffness distribution function is introduced into the virtual work principle, and the force method equation is reconstructed to obtain the displacement influence line expression under the virtual work principle.

[0008] S2: Create a conversion function

[0009] According to the quadratic polynomial of the height change of the variable-section beam, the displacement influence line expression in step S1 is rewritten into an integral form, and then the Gauss-Legendre quadrature method is introduced to solve the integral value, thereby establishing a conversion function between the bending stiffness and the displacement influence line;

[0010] S3: Bending Stiffness Inversion

[0011] Based on the conversion function obtained in step S2, the bending stiffness of the bridge is solved through the displacement influence line, thereby realizing the bending stiffness inversion.

[0012] Furthermore, in step S1, the specific processing process is as follows:

[0013] S11: For a beam with variable cross-section, the bending stiffness EI(x) varies with the position x. Assuming the moving unit force F = 1 and the action position is x0, the expression of the influence line of the measuring point displacement needs to be modified to:

[0014]

[0015] in, M is the bending moment function expression of the bridge structure when a unit moving force acts on the structure. p (x, x0) is the functional expression of the bending moment generated in the structure by the fictitious unit force at the measuring point;

[0016] S12: Based on the basic equation of the force method, the redundant constraints of the system are released, and redundant forces X1, X2, and X3 are introduced. The redundant forces X1, X2, and X3 are solved by numerical integration, and then the functional expression of the bending moment of each span is obtained to achieve the reconstruction of the force method equation;

[0017] S13: Based on the functional expression of the bending moment of each span, the displacement influence line expression under the virtual work principle is obtained:

[0018]

[0019] Among them, M k (x) is the bending moment function expression when the moving unit force acts on the kth span, M p,k (x, x0) is the bending moment function expression generated by the fictitious unit force at the measuring point in the structure, EI k (x) is the bending stiffness distribution function of the kth span of the bridge.

[0020] Furthermore, in step S12, the basic equation of the force method is as follows:

[0021]

[0022] Among them, the displacement of the j-th unit force point along its action direction caused by the unit force applied at the i-th point is δ ij , also known as the flexibility coefficient, the vertical displacement of the i-support position generated by the structure under load is Δ ip , that is, the free term, the values of i and j are 1, 2 or 3.

[0023] Furthermore, in step S12, when reconstructing the force method equation, the flexibility coefficient δ in the basic force method equation is ij and the free term Δ ip It needs to be calculated by segmented integration, and the calculation formula is as follows:

[0024]

[0025] Furthermore, in step S2, the integral form of the displacement influence line expression is as follows:

[0026]

[0027] Where m and u are constants related to the bridge structure; D(x) is a sixth-order polynomial about x, which originates from the nonlinear variation of the section moment of inertia with the beam height.

[0028] Furthermore, assuming that the change of each span height with position x h(x) conforms to a quadratic parabola, the moment of inertia I(x) is proportional to the cube of h(x):

[0029]

[0030] Where B is the cross-sectional width of the bridge;

[0031] After substituting h(x) into the expansion, I(x) becomes a sixth-degree polynomial, which can be factored and extracted to obtain D(x).

[0032] Furthermore, in step S2, the conversion function between the bending stiffness and the displacement influence line is as follows:

[0033] EI=Δ H δ(x)

[0034] Among them, δ(x) is the bending stiffness EI and the displacement influence line Δ H The conversion function of .

[0035] Furthermore, in step S2, the Gauss-Legendre quadrature method can dynamically adjust the distribution density of the integration nodes according to the severity of the change of the integrand in the integration interval. The formula is as follows:

[0036]

[0037] When the integration interval is not [-1,1], the mapping relationship between the standard interval and the physical interval [a,b] is established through affine transformation:

[0038]

[0039] Use the Gauss-Redjean quadrature formula to integrate the right side of the integral form of the displacement influence line expression to obtain the integral value. The Gauss-Redjean quadrature formula is as follows:

[0040]

[0041] Among them, A k is the weighting coefficient.

[0042] Compared with the prior art, the present invention has the following advantages:

[0043] 1. Using displacement influence line data, structural response can be obtained through quasi-static loading (such as moving vehicles) without interrupting traffic and reducing testing costs.

[0044] 2. By introducing the Gauss-Legendre numerical integration, the nonlinear stiffness change of the variable cross-section beam is directly processed to avoid the errors caused by simplified assumptions.

[0045] 3. The analytical solution of the displacement influence line is derived from the basic equation of the force method, and the complex stiffness function is processed by numerical integration, taking into account both theoretical rigor and engineering practicality.

[0046] 4. Combining the displacement influence line and modal frequency information, the displacement influence line is used to invert the overall stiffness distribution, and the rationality of the result is verified by the modal frequency to improve the inversion accuracy and reliability. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 1 is a flow chart of the design and verification process of the inverse method for the bending stiffness of a beam bridge based on displacement influence lines in an embodiment of the present invention;

[0048] Figure 2 is a schematic diagram of a continuous beam model in an embodiment of the present invention;

[0049] Figure 3 is a bending moment diagram of a unit force acting in the second span in an embodiment of the present invention;

[0050] Figure 4 is a finite element model of a four-span continuous beam of uniform cross-section according to an embodiment of the present invention;

[0051] Figure 5 is the stiffness fitting curve in the embodiment of the present invention;

[0052] Figure 6 is a bending stiffness diagram obtained by inversion in an embodiment of the present invention;

[0053] Figure 7 This is a diagram of a vehicle for an influence line loading test according to an embodiment of the present invention;

[0054] Figure 8 1 is a comparison diagram of the model influence line and the measured influence line in an embodiment of the present invention;

[0055] Figure 9 3 is a diagram of the measured influence line fitting result in an embodiment of the present invention. DETAILED DESCRIPTION

[0056] The following is a detailed description of an embodiment of the present invention. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process. However, the protection scope of the present invention is not limited to the following embodiment.

[0057] like Figure 1 As shown, this embodiment describes the design and verification process of the beam bridge bending stiffness inversion method based on the displacement influence line of the present invention, which specifically includes the following contents:

[0058] 1. Establishment of the inverse theory of continuous beam bending stiffness

[0059] 1.1 Theoretical Basis of Continuous Beam Displacement Influence Line Analysis

[0060] According to the displacement reciprocity theorem: F p1 =F p2 , let the displacement caused by unit force be δ, and the load F p1 The corresponding displacement influence coefficient δ 21 Equal to the load F p2 Caused by F p1 The corresponding displacement influence coefficient δ 12 .

[0061] According to the principle of virtual work, when a deformable body is in equilibrium, at any infinitesimal virtual displacement, the sum of the virtual work done by the external forces is equal to the virtual deformation work received by the deformable body. The elastic displacement Δ of any point under load is expressed as follows:

[0062]

[0063] The displacement of the beam is mainly caused by the bending deformation of the rod. The axial deformation and shear deformation have little effect on the structure and can be ignored.

[0064] Simplify formula (1) to formula (2).

[0065]

[0066] 1.2. Establishment of the inverse method for the bending stiffness of continuous beams based on displacement influence lines

[0067] Taking a four-span continuous beam as an example, a continuous beam model is established. Figure 2 , the analytical solution of the displacement influence line at the mid-span of the second span is derived. When the unit displacement force F acts on different spans, assuming that the distance between F and end A of the bridge is x0, the bending moment M(x) at any section is given by Equation (3).

[0068] Apply a unit force F in the second span P = 1, remove the redundant constraints of the system and replace them with unknown forces X1, X2, and X3. According to the boundary conditions of the continuous beam, the basic equation of the force method is shown in formula (4). In formula (4), δ 11It represents the vertical displacement at support B when a unit force acts on the structure; Δ 1p It represents the vertical displacement of the B support position generated by the structure under the action of load, and so on.

[0069] According to the principle of virtual work, after solving the coefficients in the basic equation of force method, we can substitute them into the basic equation of force method to get equation (5). Substituting X1, X2, and X3 into the basic equation of force method, we can get the bending moment diagram, see Figure 3 .

[0070]

[0071] Depend on Figure 3 The bending moment function of each section is derived as shown in formula (6):

[0072]

[0073] After integrating Equations (3) and (6), the displacement influence line expression of the second span is shown in Equation (7), and the bending stiffness and displacement influence line Δ are easily obtained. H The mathematical conversion relationship.

[0074]

[0075] In this embodiment, the analytical solution of the displacement influence line derived from the force method equation (7) provides a theoretical basis for stiffness inversion, avoiding the errors of pure numerical methods.

[0076] 1.3. Establishment of the inverse analysis method for the bending stiffness of variable-section continuous beam bridges

[0077] Continuous beam bridges often use variable cross-section designs to optimize structural performance. For variable cross-section beams, the bending stiffness EI(x) varies with position x, requiring the introduction of a stiffness distribution function within the principle of virtual work. Assuming the moving unit force F = 1 and the applied position x0, the expression for the displacement influence line at the measuring point needs to be modified to:

[0078]

[0079] Taking the four-span variable cross-section continuous box girder as an example, let the change of each span height h(x) with x conform to the quadratic parabola. For example, h of span i is i (x) = a i (x) 2 +b i x+c i , the parabola parameters can be solved by the height of the beam section at the support and the mid-span, and its moment of inertia function is a sixth-order polynomial about x. Based on formula (4), the redundant constraints of the system are released and redundant forces X1, X2, and X3 are introduced, but the flexibility coefficient δ ij and the free term Δ ip It requires piecewise integral calculation, see formula (9):

[0080]

[0081] By solving the excess forces X1, X2, and X3 through numerical integration, we can obtain the functional expression of the bending moment of each span. Then, the displacement influence line expression under the virtual work principle can be written as formula (10):

[0082]

[0083] Among them, M k (x) is the bending moment function expression when the moving unit force acts on the kth span, M p,k (x, x0) is the bending moment function expression generated by the fictitious unit force at the measuring point in the structure, EI k (x) is the bending stiffness distribution function of the kth span of the bridge.

[0084] When EI(x) is a piecewise constant function or a linear function, the integral can be expressed analytically as a polynomial function. However, the stiffness of actual variable-section beams often varies continuously and does not conform to linear continuity. If simplified assumptions are made for variable-section beams, the deviation from reality will be large. Taking a common variable-section box beam as an example, its height variation conforms to a quadratic polynomial, and the integral is written as Equation (11):

[0085]

[0086] Where m and u are constants related to the bridge structure; let D(x) be a sixth-order polynomial about x.

[0087] It should be noted that the sixth-order polynomial of D(x) is derived from the nonlinear variation of the section inertia moment with the beam height. Assume that the variation of each span height with x conforms to a quadratic parabola. For example, h(x) of span i i (x) = a i x 2 +b i x+c i , taking a rectangular cross section or box beam as an example, the moment of inertia I(x) is proportional to the cube of h(x):

[0088]

[0089] Where B is the cross-sectional width of the bridge;

[0090] Substituting h(x) into the expansion, I(x) becomes a sixth-order polynomial. Factoring and extracting the parameters yields D(x).

[0091] D(x) can be explicitly expressed as a sixth-order polynomial integral:

[0092]

[0093] Substitute I(x) into δ ij With Δ ip We can get D(x).

[0094] In this embodiment, the quadratic parabola height-varying box girder is directly processed without discretizing the girder into equal-section segments, which is more in line with actual bridge design.

[0095] By moving the EI parameter contained in D(x) to the left side of the equation, equation (11) can be transformed into:

[0096] EI=Δ H ·δ(x) (12)

[0097] Among them, δ(x) is the bending stiffness EI and the displacement influence line Δ H The conversion function of .

[0098] The integral can be solved by partial fraction decomposition or complex function, but the result usually includes natural logarithm and inverse tangent terms, and the expression is complex and difficult to be directly used for redundant force solution. If EI(x) is a high-order polynomial or nonlinear polynomial, the integral result cannot be explicitly expressed by a finite number of elementary functions, so Gauss-Legendre integral is used for approximate numerical solution. Gauss-Legendre integral can dynamically adjust the distribution density of the integral nodes according to the drastic change of the integrand on the integral interval, see formula (13), where: A k is the weighting coefficient.

[0099]

[0100] When the integration interval is not [-1,1], the mapping relationship between the standard interval and the physical interval [a,b] is established through affine transformation:

[0101]

[0102] The Gauss-Legend quadrature formula can be used to integrate the right side of equation (11), as shown in equation (15):

[0103]

[0104] 1.4. Establishment of the inverse method for the flexural stiffness of continuous beam bridges based on modal information

[0105] Based on the differential equation of free vibration of a beam, there is a specific relationship between the modal frequency and bending stiffness of a beam structure:

[0106]

[0107] where w(x,t) is the displacement of the beam at position x at time t, and ρA is the mass per unit length.

[0108] Assuming that the free vibration of the beam can be regarded as a superposition of modes, let:

[0109] w(x,t)=φ i (x)*sin(ωt) (17)

[0110] Among them, φ i (x) is the vibration shape of each mode.

[0111] Substituting the above formula into (16) we can obtain:

[0112]

[0113] In order to obtain the modal vibration shape φ i (x) and angular frequency ω, the above formula can be rewritten as:

[0114]

[0115] Defining β i is a frequency parameter, which is directly related to the bending stiffness of the beam. The natural frequency is:

[0116]

[0117] For continuous beam bridges, the frequency parameter β i Often the length of each span L i It is related to the boundary conditions of the support. For hinged support bridges, the frequency parameter β i It can be expressed as:

[0118] β i L i =λ i (twenty one)

[0119] Among them, λ i is the modal eigenvalue, which is related to the modal order n and boundary conditions.

[0120] For the i-th order mode, by considering the relationship between modal frequency and bending stiffness, and introducing boundary conditions and span, we can obtain equation (22):

[0121]

[0122] Among them, f i is the natural frequency of the i-th order; L i is the span of each span.

[0123] According to each mode, the bending stiffness value of the continuous beam bridge can be calculated, and then the rationality can be verified by multi-order modal solution results.

[0124] 2. Influence line verification

[0125] 2.1 Verification of the influence line of the theoretical model

[0126] To prove the accuracy of the derivation of the transfer function, we use Figure 4 The example shown is a four-span continuous beam with uniform cross-section. The continuous beam span is 50m+80m+80m+50m, and 260 beam elements are evenly divided along the longitudinal direction of the beam using finite element software.

[0127] The box girder adopts a single-box, single-cell cross-section. The beam height at mid-span is 2.0m, the web thickness is 0.5m, the top plate thickness is 0.28m, and the bottom plate thickness is 0.3m. A unit concentrated force (F = 1000N) is applied sequentially to the beam element nodes. The deflection at mid-span of the second span is calculated and output for different loading positions. This allows the quasi-static displacement influence line to be obtained, with the measurement point located at the mid-span of the second span.

[0128] Now let α = 1, β = 1.6, γ = 1.6, and δ = 1 and substitute them into formula (5), and we can get X1, X2, and X3:

[0129]

[0130] Substituting X1, X2, and X3 into formula (7), we can obtain the displacement influence line expression when the measuring point is located in the middle of the second span:

[0131]

[0132] The modal information of each order of the continuous beam bridge is calculated by finite element software, and the bending stiffness is inverted by the modal information, as shown in formula (25):

[0133]

[0134] Considering that the high-order modal frequencies are more affected by the local characteristics of the structure, the first five modal frequencies are taken for calculation. The bending stiffness values obtained by inversion of formula (25) are discrete points. The analytical solution is now used to perform cubic polynomial fitting. The fitting results are shown in Figure 5 .

[0135] In this embodiment, the stiffness formula (25) is inverted by the first five modes and the fitting curve ( Figure 5 ), avoiding single-mode errors and ensuring the stability of the results.

[0136] In bridge stiffness assessment, the support area will cause significant bending stiffness enhancement due to the boundary constraint effect. This non-uniform stiffness distribution will cause the stiffness identification results of the overall analytical solution inversion to be artificially high near the support. To eliminate this interference, the bending stiffness value at the support is truncated, and the bending stiffness obtained by inversion is shown in Figure 2. Figure 6 .

[0137] 2.2. Real bridge influence line test verification

[0138] In order to verify the reliability of identifying the flexural stiffness of a bridge through displacement influence lines, a four-span variable cross-section continuous beam bridge was used as the experimental object for research. The bridge was selected for influence line loading. The loading process used a four-axle loading vehicle with a total weight of 34.50t. The axle loads were 1.34t, 9.58t, and 23.58t (the total weight of the last two axles), and the axle spacing was 1.35m, 3.8m, and 1.8m. Figure 7 To measure the displacement response of the bridge structure, the test used a quasi-static vehicle influence line loading, with the loading path selected along the bridge centerline. A dynamic deflection measurement point was located at the mid-span of the second span, and the vehicle traveled across the bridge at a uniform speed of 10 km / h.

[0139] The same finite element model is established and the influence line is obtained through the finite element model for comparative analysis.

[0140] Plot the influence lines extracted from the model and the influence lines identified from the real bridge. Figure 8 The processed influence line is fitted with spline curve interpolation according to formula (11), see Figure 9 In this embodiment, the inversion result ( Figure 9 ) clearly reflects the stiffness change trend of the variable-section beam, and the stiffness difference between the mid-span and support areas is in line with engineering expectations.

[0141] Due to the limitation of the influence line identification accuracy, the solution of the bending stiffness of the interval span is not ideal. Therefore, the bending stiffness values of the target measuring point span and the adjacent span can be accurately solved.

[0142] In summary, the inversion method for the bending stiffness of a beam bridge based on displacement influence lines in the above embodiment derives an analytical solution of the displacement influence line through the basic equation of the force method, combines numerical integration to process complex stiffness functions, and takes into account both theoretical rigor and engineering practicality; by introducing the Gauss-Legendre numerical integration, the nonlinear change of the stiffness of the variable-section beam is directly processed, avoiding errors caused by simplified assumptions; using the displacement influence line data, the structural response can be obtained through quasi-static loading (such as moving vehicles) without interrupting traffic, thereby reducing testing costs.

[0143] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

Claims

1. A method for inverse analysis of the bending stiffness of a beam bridge based on displacement influence lines, characterized in that: The following steps are involved: S1: Introducing stiffness distribution function The bending stiffness distribution function is introduced into the virtual work principle, and the force method equation is reconstructed to obtain the displacement influence line expression under the virtual work principle. S2: Create a conversion function According to the quadratic polynomial of the height change of the variable-section beam, the displacement influence line expression in step S1 is rewritten into an integral form, and then the Gauss-Legendre quadrature method is introduced to solve the integral value, thereby establishing a conversion function between the bending stiffness and the displacement influence line; S3: Bending Stiffness Inversion Based on the conversion function obtained in step S2, the bending stiffness of the bridge is solved through the displacement influence line, thereby realizing the bending stiffness inversion.

2. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 1, characterized in that: In step S1, the specific processing process is as follows: S11: For a beam with variable cross-section, the bending stiffness EI(x) varies with the position x. Assuming the moving unit force F = 1 and the action position is x0, the expression of the influence line of the measuring point displacement needs to be modified to: in, M is the bending moment function expression of the bridge structure when a unit moving force acts on the structure. p (x, x0) is the functional expression of the bending moment generated in the structure by the fictitious unit force at the measuring point; S12: Based on the basic equation of the force method, the redundant constraints of the system are released, and redundant forces X1, X2, and X3 are introduced. The redundant forces X1, X2, and X3 are solved by numerical integration, and then the functional expression of the bending moment of each span is obtained to achieve the reconstruction of the force method equation; S13: Based on the functional expression of the bending moment of each span, the displacement influence line expression under the virtual work principle is obtained: Among them, M k (x) is the bending moment function expression when the moving unit force acts on the kth span, M p,k (x, x0) is the bending moment function expression generated by the fictitious unit force at the measuring point in the structure, EI k (x) is the bending stiffness distribution function of the kth span of the bridge.

3. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 2, characterized in that: In step S12, the basic equation of the force method is as follows: Among them, the displacement of the j-th unit force point along its action direction caused by the unit force applied at the i-th point is δ ij , also known as the flexibility coefficient, the vertical displacement of the i-support position generated by the structure under load is Δ ip , that is, the free term, the values of i and j are 1, 2 or 3.

4. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 3 is characterized in that: In step S12, when reconstructing the force method equation, the flexibility coefficient δ in the force method basic equation is ij and the free term Δ ip It needs to be calculated by segmented integration, and the calculation formula is as follows:

5. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 4 is characterized in that: In step S2, the integral form of the displacement influence line expression is as follows: Where m and u are constants related to the bridge structure; D(x) is a sixth-order polynomial about x, which originates from the nonlinear variation of the section moment of inertia with the beam height.

6. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 5, characterized in that: Assume that the change of each span height with position x h(x) conforms to a quadratic parabola, and the moment of inertia I(x) is proportional to the cube of h(x): Where B is the cross-sectional width of the bridge; After substituting h(x) into the expansion, I(x) becomes a sixth-degree polynomial, which can be factored and extracted to obtain D(x).

7. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 5, characterized in that: In step S2, the conversion function between the bending stiffness and the displacement influence line is as follows: EI=Δ H ·δ(x) Among them, δ(x) is the bending stiffness EI and the displacement influence line Δ H The conversion function of .

8. The method for inverse analysis of bending stiffness of beam bridges based on displacement influence lines according to claim 7, characterized in that: In step S2, the Gauss-Legendre quadrature method can dynamically adjust the distribution density of the integration nodes according to the severity of the change of the integrand in the integration interval. The formula is as follows: When the integration interval is not [-1,1], the mapping relationship between the standard interval and the physical interval [a,b] is established through affine transformation: Use the Gauss-Redjean quadrature formula to integrate the right side of the integral form of the displacement influence line expression to obtain the integral value. The Gauss-Redjean quadrature formula is as follows: Among them, A k is the weighting coefficient.

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