Analytical method of thrust influence line for uniform cross-section parabolic arch based on approximate curve fitting

Abstract

The present invention discloses a method for analyzing the thrust influence line of a parabolic arch of uniform cross-section based on approximate curve fitting, which belongs to the technical field of arch thrust influence line derivation and analysis, and includes step S1: establishing a basic system of a parabolic arch of uniform cross-section, and then obtaining the parabolic arch axis equation, and obtaining the corresponding self-displacement and load-displacement expressions by fitting the arc length curve through the suspension line; step S2: deriving the analytical expression of the thrust influence line of the parabolic arch of uniform cross-section based on the basic equation of the force method. Based on the suspension line fitting, the present invention uses approximate differentials instead of exact differentials to derive the analytical solutions of the thrust influence lines of a two-hinged and hingeless parabolic arch of uniform cross-section, and proves that the approximate integral method can be used to deduce the analytical solutions of the influence lines of the parabolic arch; and proposes a high-precision formula analytical solution for the thrust influence line of the parabolic arch of uniform cross-section based on the approximate integral fitting, which greatly avoids a large amount of trial calculation work before finite element modeling, and specifically improves the analytical accuracy and universality of the formula.

Description

Technical Field

[0001] The present invention relates to the technical field of arch thrust influence line derivation and analysis, and in particular to a method for analyzing the arch thrust influence line of a parabola with a uniform cross-section based on approximate curve fitting. Background Art

[0002] Arches are widely used in civil engineering. Common arch types include single-hinged, two-hinged, three-hinged, and hingeless arches. Two-hinged and hingeless arches are widely used in practical engineering. Parabolas are often used as reasonable arch axes in mid-through arch bridges. Thrust, a support reaction inherent in arch structures, is of great value for controlling displacement and settlement and is a key indicator in arch structure design and evaluation. The influence line index, a quasi-static response indicator, is a key parameter reflecting the characteristic bending stiffness of a structural section and the structural boundary characteristics.

[0003] Therefore, the analytical derivation of the formula for the thrust influence line of a parabolic arch with uniform cross-section has practical engineering significance. In current engineering practice, arch structures require extensive trial calculations during both the modeling and construction phases. However, due to the cumbersome and complex arch axis equations and curve fitting integrals, explicit solutions are difficult to obtain, and traditional analytical mechanics methods suffer from large errors. Therefore, an analytical method for the thrust influence line of a parabolic arch with uniform cross-section is proposed based on approximate curve fitting. Summary of the Invention

[0004] The technical problem to be solved by the present invention is: how to solve the problem that the arch axis equation and the curve fitting integral formula are cumbersome and complicated, it is difficult to obtain an explicit solution, and the traditional mechanical analytical method has large errors. A method for analyzing the thrust influence line of a parabolic arch with equal cross-section based on approximate curve fitting is provided.

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

[0006] S1: Establish the basic system of parabolic arch with uniform cross-section, and then obtain the equation of the parabolic arch axis. Fit the arc length curve by the catenary line to obtain the corresponding expressions of self-deflection and load-deflection.

[0007] S2: Based on the basic equation of the force method and the principle of approximate integral fitting, the analytical expression of the thrust influence line of a parabolic arch with uniform cross-section is derived.

[0008] Furthermore, in step S1, the parabola axis equation is:

[0009] y=(4f / l 2 )·x 2

[0010] Where l is the arch axis span, f is the sag, l is the arch span, and x is the position of the moving load.

[0011] Furthermore, in step S1, the catenary line fitting formula is as follows:

[0012] ds=ch(x / a)dx

[0013] Where a is the arch parameter.

[0014] Furthermore, in step S1, when the uniform cross-section parabolic arch is a two-hinged arch, the two-hinged arch is converted into a statically determinate structure with redundant forces under the action of a unit force according to the force law equation. When the unit force moving load moves along the arc length, the force law equation of the uniform cross-section parabolic two-hinged arch is:

[0015] δ 11 H P +Δ 1P =0

[0016] According to the force method equation, the autovariable expression is obtained:

[0017]

[0018] The load displacement expression is:

[0019]

[0020] Among them, M1 is the bending moment generated by the support of any section, is the bending moment generated by unit load of any section, EI is the section stiffness, H P It is the arch foot thrust;

[0021] Using the cable line fitting curve integral to integrate the parabola along the arc, according to the static equilibrium condition, The segmentation is represented as:

[0022] when hour,

[0023] when hour,

[0024] Where b is the position of any cross section;

[0025] Then the load displacement expression is obtained as follows:

[0026]

[0027] Furthermore, in step S2, the unknown structural force is the thrust of the two hinged arches, and the basic equation of the force method is:

[0028] F H =H P =-Δ 1P / δ 11

[0029] Among them, F H That is, arch-foot thrust;

[0030] Then, the analytical expression of the thrust influence line of a parabolic two-hinged arch with uniform cross-section is obtained as follows:

[0031]

[0032] Furthermore, in step S1, when the uniform cross-section parabolic arch is a hingeless arch, the elastic center method is used to convert the hingeless arch under the action of a unit force into a statically determinate structure with redundant force. Under the action of a unit force moving load moving along the arc length, the force law equation of the uniform cross-section parabolic hingeless arch is:

[0033]

[0034] The internal forces of the parabolic hingeless arch with uniform cross-section under all loads are statistically analyzed, and the self-displacement expression is obtained by solving the force method equation:

[0035]

[0036] The load displacement expression is:

[0037] Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds

[0038] Where M2 is the section bending moment caused by the lateral restraint, M p is the bending moment generated by the moving load; N2 is the section axial force generated by the lateral constraint, N p is the axial force generated by the moving load; EI is the section stiffness, EA is the elastic modulus of the material;

[0039] Substitute into the internal force expression Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds, the parabola is integrated along the arc by fitting the cable line, and the load displacement expression is obtained as follows:

[0040]

[0041] Furthermore, in step S2, the horizontal redundant force at the elastic center is a force equal to and opposite to the thrust of the hingeless arch, and the basic equation of the force method is:

[0042]

[0043] Then, the analytical expression of the thrust influence line of the uniform cross-section parabolic hingeless arch is obtained as follows:

[0044]

[0045] Compared with the existing technology, the present invention has the following advantages: the analytical method for the thrust influence line of a parabolic arch of equal cross-section based on approximate curve fitting is based on the fitting of the suspension cable, and uses approximate differentials instead of exact differentials to derive the analytical solutions of the thrust influence lines of a two-hinged parabolic arch and a hingeless arch of equal cross-section, proving that the approximate integral method can be used to deduce the analytical solutions of the influence lines of a parabolic arch; a high-precision formula analytical solution for the thrust influence line of a parabolic arch of equal cross-section based on approximate integral fitting is proposed, which greatly avoids a large amount of trial calculation work before finite element modeling, reduces the investment of manpower and material resources, and specifically improves the analytical accuracy and universality of the formula. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 1 is a flow chart of a method for analyzing the thrust influence line of a parabolic arch of uniform cross-section based on approximate curve fitting in the first embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram of the basic structure of a medium-section parabolic two-hinged arch according to a first embodiment of the present invention;

[0048] Figure 3 This is a schematic diagram of the basic structure of a parabolic hingeless arch with a medium cross-section according to a first embodiment of the present invention;

[0049] Figure 4 Schematic diagram of arch axes with five different rise-to-span ratios for the five-arch structure in Example 2 of the present invention;

[0050] Figure 5a This is a thrust surface diagram of a two-hinge arch in the second embodiment of the present invention;

[0051] Figure 5b This is a thrust surface diagram of a hingeless arch in the second embodiment of the present invention;

[0052] Figure 6a 1 is a comparison chart of the formula solution and the finite element solution results of Example 1 in Example 2 of the present invention;

[0053] Figure 6b 1 is a comparison chart of the formula solution and the finite element solution results of Example 2 in Example 2 of the present invention;

[0054] Figure 6c 1 is a comparison chart of the formula solution and the finite element solution results of Example 3 in Example 2 of the present invention;

[0055] Figure 6d 1 is a comparison chart of the formula solution and the finite element solution results of Example 4 in Example 2 of the present invention;

[0056] Figure 6e This is a comparison chart of the formula solution and the finite element solution results of Example 5 in Example 2 of the present invention. DETAILED DESCRIPTION

[0057] 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.

[0058] Example 1

[0059] like Figure 1 As shown, this embodiment provides a technical solution: a method for analyzing the thrust influence line of a parabolic arch with a uniform cross-section based on approximate curve fitting, comprising the following steps:

[0060] S1: Establish the basic system of parabolic two-hinged arches and hingeless arches with uniform cross-section, and then obtain the equation of the parabolic arch axis. Fit the arc length curve by the catenary line to obtain the corresponding expressions of self-deflection and load-deflection.

[0061] S2: Based on the derivation of the basic equations of the force method and the principle of approximate integration, the analytical expressions of the thrust influence lines of parabolic two-hinged arches and hingeless arches with uniform cross-sections are obtained.

[0062] In this embodiment, for the basic structure of a parabolic two-hinged arch with a uniform cross-section, the specific process of deriving the analytical expression of the thrust influence line is as follows:

[0063] In step S1, the arch bridge curve fitting is usually performed using a suspension line. The suspension line fitting formula is as follows:

[0064] ds=ch(x / a)dx

[0065] Among them, a is the arch parameter;

[0066] The thrust influence line is solved by the static method. The unit force P is applied to any section b to solve the thrust influence line at the arch foot A. According to the force method equation, the two-hinged arch under the unit force is converted into a statically determinate structure with redundant force. The basic structure of the two-hinged arch with equal cross-section parabola is as follows: Figure 2 As shown;

[0067] The parabola arch axis equation can be established as:

[0068] y=(4f / l 2 )·x 2

[0069] Where l is the arch axis span, f is the sag, l is the arch span, and x is the position of the moving load;

[0070] Calculating the redundant force of the basic structure under the action of unit force is the premise of solving the analytical solution of thrust. Under the action of unit force moving load moving along the arc length, the force method equation of the uniform cross-section parabola two-hinged arch is:

[0071] δ 11 H P +Δ 1P =0

[0072] Among them, H P It is the arch foot thrust;

[0073] The inflected expression is:

[0074]

[0075] The load displacement expression is:

[0076]

[0077] Among them, M1 is the bending moment generated by the support of any section, is the bending moment generated by unit load on any section, and EI is the section stiffness;

[0078] Using the cable line fitting curve integral to integrate the parabola along the arc, according to the static equilibrium condition, It can be expressed in segments as:

[0079] when hour,

[0080] when hour,

[0081] Where b is the position of any cross section;

[0082] Then the load displacement expression is obtained as:

[0083]

[0084] In step S2, the unknown structural force is the thrust of the two-hinged arch, and the basic equation of the force method is:

[0085] F H =H P =-Δ 1P / δ 11

[0086] Among them, F H That is, arch-foot thrust;

[0087] Then, the analytical expression of the thrust influence line of a parabolic two-hinged arch with uniform cross-section is obtained as follows:

[0088]

[0089] In this embodiment, for the basic structure of a parabolic two-hinged arch with a uniform cross-section, the specific process of deriving the analytical expression of the thrust influence line is as follows:

[0090] In step S1, the arch bridge curve fitting is usually performed using a suspension line. The suspension line fitting formula is as follows:

[0091] ds=ch(x / a)dx

[0092] The hingeless arch is a cubically indeterminate structure. The elastic center method is now used to simplify the force equation. By removing the constraints at the arch crown and making all the redundant forces act at the elastic center, the redundant unknown forces can be solved. The statically indeterminate structure method is used to solve the hingeless arch with a parabola of uniform cross-section. The unit force P is applied to any section b, and the thrust influence line at the arch foot A is then solved.

[0093] Using the elastic center method, the hingeless arch under unit force is converted into a statically determinate structure with redundant force. The basic structure of the hingeless arch with uniform cross-section is as follows: Figure 3 As shown;

[0094] The parabola arch axis equation can be established as:

[0095] y=(4f / l 2 )·x 2

[0096] Specifically, the tensile and compressive stiffness and bending stiffness of the uniform cross-section parabola hingeless arch are set as EA and EI, the redundant forces include bending moment X1, axial force X2 and shear force X3, and the distance between the elastic center and the arch top is y s ;

[0097] The thrust of the hingeless arch is equal to the horizontal redundant force at the elastic center in value, and opposite in direction to the horizontal redundant force (for the redundant force on one side). Therefore, solving the thrust of the arch foot of the hingeless arch can be converted into solving the horizontal redundant force at the elastic center.

[0098] The distance from the elastic center to the arch top is defined by the elastic center of the hingeless arch, and we can get:

[0099]

[0100] When the unit force moving load moves along the arc length, the force equation of the uniform cross-section parabola hingeless arch is:

[0101]

[0102] Among them, the displacement symbol has two subscripts. The first subscript indicates the direction of the displacement, and the second subscript indicates the force that produces the displacement.

[0103] The internal forces of the hingeless arch basic structure under all loads are listed in Table 1.

[0104] Table 1 Internal forces of basic structure of hingeless arch

[0105]

[0106] Its self-displacement can be solved in accordance with the force method equation as follows:

[0107]

[0108] The load displacement is also solved according to the force method equation, that is:

[0109] Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds

[0110] Where M2 is the section bending moment caused by the lateral restraint, M p is the bending moment generated by the moving load; N2 is the section axial force generated by the lateral constraint, N p is the axial force generated by the moving load; EI is the section stiffness, EA is the elastic modulus of the material;

[0111] Substitute into the internal force expression Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds, the parabola is integrated along the arc by fitting the cable line, and the load displacement expression is obtained as follows:

[0112]

[0113] In step S2, the horizontal redundant force at the elastic center is the force equivalent to and opposite to the thrust of the hingeless arch. The basic equation of the force method is:

[0114]

[0115] Where M2 is the section bending moment caused by the lateral restraint, M p is the bending moment generated by the moving load; N2 is the section axial force generated by the lateral constraint, N p is the axial force generated by the moving load; EI is the section stiffness, EA is the elastic modulus of the material;

[0116] Then, the analytical expression of the thrust influence line of the uniform cross-section parabolic hingeless arch is obtained as follows:

[0117]

[0118] Example 2

[0119] In this example, to investigate the accuracy and practicality of the analytical solution of the thrust influence line derived in Example 1, two parabolic arches with equal cross-sections were used as examples. A finite element model was established using MIDAS / Civil, and the derived thrust influence line calculation results were compared. The relative errors between the analytical solution and the finite element solution were also compared. The spans of the five arch structures were all 117.5m, and the rise-to-span ratios were 1 / 7, 1 / 5, 1 / 4, 1 / 3, and 1 / 2, respectively. Figure 4 The specific parameters of each example are listed in Table 2.

[0120] Table 2 Structural parameters of the example

[0121]

[0122] Taking the finite element solution as the true value, the change law of the thrust influence line in the two dimensions of span and span ratio is shown in Figure 5a 、 Figure 5b .

[0123] The five examples of the two-hinge arch and the hingeless arch are compared with the finite element solution (i.e., the analytical solution, calculated by the analytical expression of the thrust influence line in the embodiment) Figure 6a-Figure 6e .

[0124] The relative errors between the analytical solution and the finite element solution of the thrust influence line for the five calculation examples of two-hinged arches are all less than 3%, and the maximum error of the thrust peak is 1.36%. The relative errors between the analytical solution and the finite element solution of the thrust influence line for the five calculation examples of hingeless arches are all less than 7%, and the maximum error of the thrust peak is 3.51%, indicating that the analytical solution of the thrust influence line proposed in this invention has high accuracy.

[0125] Based on the fitting of the suspension cable, the present invention uses approximate differentials instead of exact differentials to derive the analytical solutions of the thrust influence lines of parabolic two-hinged arches and hingeless arches with uniform cross-sections, proving that the approximate integral method can be used to deduce the analytical solutions of the influence lines of parabolic arches. A high-precision formula analytical solution for the thrust influence lines of parabolic arches with uniform cross-sections based on approximate integral fitting is proposed.

[0126] In summary, the present invention fills the gap in high-precision approximate analysis that has not yet been reported, and puts forward higher requirements for arch thrust influence line testing. By comparing five examples, the effectiveness and accuracy of the thrust influence line analytical solution of the present invention are verified. Among them, the error between the two-hinged arch thrust influence line formula solution and the finite element solution is less than 3%, and the error between the hingeless arch thrust influence line formula solution and the finite element solution is less than 7%. This verifies the accuracy and universality of the analytical solution proposed by the present invention for different span ratios and different boundary conditions.

[0127] 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. The analytical method of the thrust influence line of the uniform cross-section parabolic arch based on approximate curve fitting is characterized by: The following steps are involved: S1: Establish the basic system of parabolic arch with uniform cross-section, and then obtain the equation of the parabolic arch axis. Fit the arc length curve by the catenary line to obtain the corresponding expressions of self-deflection and load-deflection. S2: Based on the basic equation of the force method and the principle of approximate integral fitting, the analytical expression of the thrust influence line of the uniform cross-section parabolic arch is derived; In step S1, the parabola axis equation is: y=(4f / l 2 )·x 2 Where l is the arch axis span, f is the sag, l is the arch span, and x is the position of the moving load; In step S1, the catenary line fitting formula is as follows: ds=ch(x / a)dx Among them, a is the arch parameter; In step S1, when the uniform cross-section parabolic arch is a two-hinged arch, the two-hinged arch is converted into a statically determinate structure with redundant forces under the action of a unit force according to the force law equation. When the unit force moving load moves along the arc length, the force law equation of the uniform cross-section parabolic two-hinged arch is: d 11 H P +D 1P =0 According to the force method equation, the autovariable expression is obtained: The load displacement expression is: Among them, M1 is the bending moment generated by the support of any section, is the bending moment generated by unit load of any section, EI is the section stiffness, H P It is the arch foot thrust; Using the cable line fitting curve integral to integrate the parabola along the arc, according to the static equilibrium condition, The segmentation is represented as: when hour, when hour, Where b is the position of any cross section; Then the load displacement expression is obtained as follows: In step S2, the arch foot thrust is an unknown structural force, and the basic equation of the force method is: F H =H P =-D 1P / d 11 Among them, F H That is, arch-foot thrust; Then, the analytical expression of the thrust influence line of a parabolic two-hinged arch with uniform cross-section is obtained as follows:

2. The method for analyzing the thrust influence line of a parabolic arch with a uniform cross-section based on approximate curve fitting according to claim 1 is characterized in that: In step S1, when the uniform cross-section parabolic arch is a hingeless arch, the elastic center method is used to convert the hingeless arch into a statically determinate structure with redundant force under the action of a unit force. Under the action of a unit force moving load moving along the arc length, the force law equation of the uniform cross-section parabolic hingeless arch is: The internal forces of the parabolic hingeless arch with uniform cross-section under all loads are statistically analyzed, and the self-displacement expression is obtained by solving the force method equation: The load displacement expression is: Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds Where M2 is the section bending moment caused by the lateral restraint, M p is the bending moment generated by the moving load; N2 is the section axial force generated by the lateral constraint, N p is the axial force generated by the moving load; EI is the section stiffness, EA is the elastic modulus of the material; Substitute into the internal force expression Δ 2P =∫ s (M2M p / EI)ds+∫ s (N2N p / EA)ds, the parabola is integrated along the arc by fitting the cable line, and the load displacement expression is obtained as follows:

3. The method for analyzing the thrust influence line of a parabolic arch with a uniform cross-section based on approximate curve fitting according to claim 2 is characterized in that: In step S2, the horizontal redundant force at the elastic center is a force equal to and opposite to the thrust of the hingeless arch. The basic equation of the force method is: Then, the analytical expression of the thrust influence line of the uniform cross-section parabolic hingeless arch is obtained as follows:

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