Vehicle arrangement method for load test of cable-suspended arch bridge based on internal force influence line

Abstract

The application discloses a cable-catenary arch bridge load test vehicle arrangement method based on internal force influence line, comprising the following steps: step one, deriving a catenary arch structure expression; step two, expanding the arch axis curve differential by using Taylor formula to obtain an approximate expression of the catenary equation of the arch bridge and the accurate differential of the arch axis; step three, obtaining an internal force analytical expression of the arch section; step four, calculating the section bending moment of the bridge by using the internal force analytical expression, and drawing an arch foot bending moment influence line, and the position of the load test vehicle arrangement corresponding to the most unfavorable load can be obtained according to the influence line image. The application derives the basic internal force analytical solution of the cable-arch combined structure, has the advantages of high precision, convenient calculation, wide applicability and the like, and is applied to the bridge load experiment, can quickly and accurately determine the load test loading vehicle position, does not need complicated finite element simulation calculation, and reduces the time consumption of the field test.

Description

Technical Field

[0001] This invention relates to the field of bridge safety monitoring, specifically to a method for arranging test vehicles for cable-stayed arch bridges based on the influence line of internal forces. Background Technology

[0002] Arch bridges, with their strong spanning capacity, material savings, and ease of construction, are widely used by designers in the valleys, large rivers, and deep ravines of central and western my country. However, with increasing traffic volume, service life, and material aging, these damaged arch bridges can no longer meet normal operational requirements. Because the main arch ring of an arch bridge is a compression-bending member, excessive bending moments often lead to an increase in arch ring cracks, especially at the arch foot, causing a decrease in cross-sectional resistance and thus reducing the structural bearing capacity, ultimately endangering the safety of the arch bridge structure. Furthermore, real-world bridge traffic scenarios are characterized by high traffic volume, diverse vehicle types, and highly random vehicle movement. Efficiently and accurately obtaining vehicle load flow and the mechanical effects of traffic flow vehicle loads is a challenge in bridge structural performance evaluation. Therefore, methods for determining vehicle measurement point locations are crucial for conducting bridge load experiments.

[0003] Bridge load testing includes static load testing and dynamic load testing. Generally, only static load testing is performed, but dynamic load testing is added when necessary, such as for extra-large bridges or new types of bridges. Static load testing involves applying a static load to a designated location on the bridge to measure the structure's static strain, static displacement, and cracks, thereby inferring the bridge structure's working condition and serviceability under load. Dynamic load testing involves applying dynamic loads, such as the load of a moving vehicle or other dynamic loads, to the bridge structure to measure its dynamic characteristics, such as vibration deformation, thereby determining the impact and vibration effects on the bridge structure under dynamic loads.

[0004] The cable-catenary arch combined structure, with cables symmetrically arranged on both sides of the arch rib, forms a new force-bearing system with the cables, thereby altering the force transmission path of the catenary arch structure and ultimately reducing the internal forces within the arch. Based on existing research, many scholars have focused on the stability, bearing capacity, dynamic response, and practical bridge applications of cable-arch structures. However, research on the analytical solution of the influence line of internal forces in cable-arch structures is relatively lacking, which greatly hinders the conduct of load experiments on arch bridges. Currently, in practical engineering, when evaluating the structural performance of bridges, load experiments require extensive simulations and tests of vehicle positions, making it difficult to identify the most unfavorable load location, and the actual simulations and tests are extremely time-consuming. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing bridge load testing techniques, which require tedious simulation and testing to determine vehicle positions, consuming significant resources and time. This invention provides a method for arranging load testing vehicles for cable-stayed arch bridges based on internal force influence lines, simplifying the tedious simulation and testing operations and facilitating the determination of the load testing vehicle placement positions.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] The method for arranging test vehicles for cable-stayed arch bridge load tests based on internal force influence lines includes the following steps:

[0008] Step 1: Establish the force model of the cable-catenary arch structure. Assuming the arch is a catenary arch with a uniform cross section, derive the structural expression of the catenary arch based on the force method principle of structural mechanics.

[0009] Step 2: Establish the basic equations for the basic structure of the cable-catenary arch combined structure. Based on the elastic center theory and numerical integration method, expand the differential of the arch axis curve using Taylor formula to obtain approximate expressions for the catenary equation of the arch bridge and the precise differential of the arch axis.

[0010] Step 3, change the constant displacement δ ij , load displacement Δ iP Substituting into the basic equation, we obtain the redundant forces X1 to X4 at the top of the arch. Due to the symmetry of the structure, taking the internal forces of the right half of the arch section as the representative, i.e., x≥0, we derive the analytical expression of the internal forces of the arch section.

[0011] Step four: Calculate the bending moment of the arch bridge section using the internal force analytical expression from step three, and draw the bending moment influence line of the arch. Based on the influence line image, the location of the load test vehicle corresponding to the most unfavorable load can be obtained, which is the location of the maximum bending moment of the arch.

[0012] Furthermore, the expression for the catenary arch structure in step one is as follows:

[0013]

[0014] In the formula, y is the ordinate of the arch axis section, f is the rise, m is the arch axis coefficient, cosh(·) is the hyperbolic cosine function, and the catenary equation variables are... ξ is the relative coordinate of the arch axis, and ξ = 2x / L, where x is the abscissa of the arch axis and L is the span of the arch.

[0015] Furthermore, the basic equation in step two is:

[0016]

[0017] In the formula, δ ij For constant displacement, Δ iPThe displacement is denoted by X, representing the elastic center of the basic structure along the X-axis under the action of a unit force or a unit external load. j Displacement in direction; X j (j=1,2,3) represents the redundant forces at the crown, where X1 is the bending moment, X2 is the axial force, and X3 is the shear force; X j (j=4) represents the cable force.

[0018] Furthermore, the displacement of the elastic center in the basic structure is constituted by the deformation of the arch axis. Here, the axial deformation of the arch ribs is considered, but shear deformation is not considered. δ ij and Δ iP The basic expression is:

[0019]

[0020]

[0021] In the formula, and M P N P T P These represent the section bending moment, axial force, and cable force of the basic structure under the action of redundant force and unit load alone, respectively; ds represents the differential of the arch axis arc length; EI and EA are the arch bending stiffness and axial stiffness, respectively, and the arch bending-compression stiffness ratio b = EI / EA; E c A c L c Let represent the axial stiffness and initial length of the cable, respectively, and let the ratio of the axial stiffness of the cable and the arch be t = EcAc / EA.

[0022] Furthermore, the approximate expressions for the catenary equation and the exact differential of the arch axis in step two are as follows:

[0023]

[0024]

[0025] In the formula, It is called the approximate differential function, where λ i and n i (i = 1 to 3) are all constant coefficients, and the calculation formula is as follows:

[0026]

[0027] In the above formula, the coefficients in; f is the rise, m is the arch axis coefficient, L is the arch span, and o(ξ) 6 ) represents the remainder term of Taylor's formula.

[0028] constant displacement δ ij , load displacement Δ iPSubstituting into the basic equations, the internal forces of the arch bridge's basic structure are calculated. Depending on the location of the moving load, two cases can be considered: positive and negative half-axis. We will analyze the case where the moving load is located on the positive half-axis (right half-arch), i.e., 0 ≤ x. P When ≤L / 2 and when the moving load is located on the negative half-axis (left half-arch), i.e., -L / 2 < x P From the analytical expression of the internal forces when <0, the analytical expression of the internal forces of the entire structure can be derived.

[0029] Furthermore, taking the internal forces of the right semi-arch (positive semi-axis) section as an example, the analytical expression for the internal forces in step three is:

[0030]

[0031] In the formula, M is the bending moment of the arch structure; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p x is the abscissa of the vertical moving load P; x1 is the abscissa of the cable constraint position; α is the inclination angle of the cable constraint; X i (i = 1, 2, 3) represents the excess force at the top of the arch, where X1 is the bending moment, X2 is the axial force, X3 is the shear force, and X4 is the cable force.

[0032] Furthermore, taking the internal forces of the left semi-arch (negative semi-axis) section as an example, the analytical expression for the internal forces in step three is:

[0033] M = X1 + X2(yy) s )-X3x;

[0034] In the formula, M is the bending moment of the arch structure; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p X1 is the abscissa of the vertical moving load P; X2 is the bending moment, X3 is the axial force, and X4 is the shear force.

[0035] The present invention has the following advantages:

[0036] 1. This invention proposes an analytical solution for the influence line of internal forces in a cable-arch combined structure, derived using an approximate line integral method. This fills a gap in research on analytical solutions for the influence line of internal forces in cable-arch structures under vertical moving loads, providing reference data for theoretical analysis of future inspection, monitoring, and load testing of cable-catenary arch bridges. Furthermore, the analytical solution in this paper is in good agreement with the finite element results, with an overall relative error within 1%, fully meeting engineering accuracy requirements. This also verifies the reliability and accuracy of the calculation formula presented in this paper.

[0037] 2. This invention derives the analytical solution of the basic internal forces of a cable-cable arch, which has the advantages of high precision, convenient calculation and wide applicability. When applied to bridge load tests, it can quickly and accurately determine the position of the loaded vehicle without the need for cumbersome finite element simulation calculations, thus reducing the time spent on field testing. Attached Figure Description

[0038] Figure 1 This is a calculation diagram of a cable-arch composite structure under vertical moving load.

[0039] Figure 2 This is the influence line of the arch bending moment between the analytical solution and the finite element solution in Embodiment 2 of the present invention.

[0040] Figure 3 This is the influence line of the arch foot bending moment between the analytical solution and the finite element solution in Embodiment 2 of the present invention.

[0041] Figure 4 This is the influence line of the arch foot bending moment when ξ1=0.4 in Embodiment 3 of the present invention.

[0042] Figure 5 This is a schematic diagram of the vehicle arrangement for the symmetrical loading test with negative bending moment at the arch foot section in Embodiment 3 of the present invention (Test Scheme 1).

[0043] Figure 6 This is a schematic diagram of the arrangement of the test vehicle for the symmetrical loading test with negative bending moment at the arch foot section in Embodiment 3 of the present invention (Test Scheme 2).

[0044] Figure 7 This is a diagram illustrating the technical specifications of the three-axle heavy-duty truck used in Embodiment 3 of the present invention. Detailed Implementation

[0045] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0046] Example 1:

[0047] The method for arranging test vehicles for cable-stayed arch bridge load tests based on internal force influence lines includes the following steps:

[0048] Step 1: Establish the force model of the cable-catenary arch structure. Assuming the arch is a catenary arch with a uniform cross section, derive the structural expression of the catenary arch based on the force method principle of structural mechanics.

[0049] Step 2: Establish the basic equations for the basic structure of the cable-catenary arch combined structure. Based on the theory of elastic center and the method of numerical integration, expand the differential of the arch axis curve using Taylor formula to obtain approximate expressions for the catenary equation of the arch bridge and the precise differential of the arch axis.

[0050] Step 3, change the constant displacement δ ij, load displacement Δ iP Substituting into the basic equation, we obtain the redundant forces X1 to X4 at the top of the arch. Due to the symmetry of the structure, taking the internal forces of the right half of the arch section as the representative, i.e., x≥0, we derive the analytical expression of the internal forces of the arch section.

[0051] Step four: Calculate the cross-sectional bending moment of the arch bridge using the internal force analytical expression from step three, and draw the bending moment influence line of the arch. Based on the influence line image, the location of the load test vehicle corresponding to the most unfavorable load can be obtained, which is the location of the maximum bending moment of the arch.

[0052] The stress conditions of the catenary arch structure are analyzed, and the analytical expression of the influence line of the arch moment of the cable-catenary arch under load test is further explained:

[0053] like Figure 1 As shown, Figure 1 The diagram shows the calculation of a cable-arch combined structure under vertical moving load. In the diagram, a is a simplified calculation diagram of the cable-arch combined structure, b is the basic system of the cable-arch combined structure, and c is a detailed enlarged diagram of the cable force.

[0054] A and C represent the arch foot, B represents the arch crown, and E (E′) represents the intersection of the cable and the arch. One end of the cable is anchored at point D (D′) at the same height as the arch crown B. Using the force method, the arch is divided into two symmetrical cantilever arches, with a redundancy force at the arch crown of X. i (i = 1, 2, 3), cable force X4, using a simplified system with elastic centers, the force method equations are simplified into easily solvable canonical equations. The arch span is L, the rise is f, and the rigid arm length is y. s Let the coordinates of the vertical moving load P be x. P The position coordinates of the cable constraint are x1, and the inclination angle of the cable constraint is α. Let K be the horizontal inclination angle of the tangent at any point K on the arch axis. Without considering prestress, based on the deformation characteristics of the arch structure, under the action of a moving load, the deformation increment of the cable in the half-arch where the load is located is positive, and the cable force is X4; the deformation increment of the cable in the other half-arch is negative, the cable is no longer working, i.e., the cable force is zero. For ease of calculation, a relative coordinate ξ = 2x / L is adopted, where the relative coordinate of the moving load ξ is... P =2x P / L, the relative coordinates of the cable constraint position ξ1=2x1 / L.

[0055] The expression for the catenary arch axis is:

[0056]

[0057] In the formula, y is the ordinate of the arch axis section, f is the rise, m is the arch axis coefficient, cosh(·) is the hyperbolic cosine function, and the catenary equation variables are... ξ is the relative coordinate of the arch axis, and ξ = 2x / L, where x is the abscissa of the arch axis and L is the span of the arch.

[0058] right Figure 1 The basic structure (basic system) shown establishes the basic equations:

[0059]

[0060] In the formula, δ ij For constant displacement, Δ iP The displacement is denoted by X, representing the elastic center of the basic structure along the X-axis under the action of a unit force or a unit external load. j Displacement in direction; X j (j=1,2,3) represents the redundant forces at the crown, where X1 is the bending moment, X2 is the axial force, and X3 is the shear force; X j (j=4) represents the cable force.

[0061] The displacement of the elastic center in the basic structure is constituted by the deformation of the arch axis. Here, we consider the axial deformation of the arch ribs (ignoring shear deformation), δ ij and Δ iP The basic expression is:

[0062]

[0063]

[0064] In the formula, and M P N P T P These represent the section bending moment, axial force, and cable force of the basic structure under the action of redundant force and unit load alone, respectively; ds represents the differential of the arch axis arc length; EI and EA are the arch bending stiffness and axial stiffness, respectively, and the arch bending-compression stiffness ratio b = EI / EA; E c A c L c Let represent the axial stiffness and initial length of the cable, respectively, and let the ratio of the axial stiffness of the cable and the arch be t = EcAc / EA.

[0065] Based on the approximate line integral method, the catenary equation and the exact differential of the arch axis are expanded using Taylor's formula, and higher-order remainders above the 7th order are ignored, then we have:

[0066]

[0067]

[0068] In the formula, This is called the approximate differential function, where λi and ni (i = 1 to 3) are constant coefficients, and the calculation formula is as follows:

[0069]

[0070] In the above formula, the coefficients in; f is the rise, m is the arch axis coefficient, L is the arch span, and o(ξ) 6 ) represents the remainder term of Taylor's formula.

[0071] Based on the simplified system using the elastic center method, its basic structure is a symmetrical structure. Since the redundant forces X1 and X2 are positively symmetrical and X3 is anti-symmetrical, the secondary coefficient δ... 13 =δ 31 =0, δ 23 =δ 32 =0. And the secondary coefficient δ 12 The expression is:

[0072]

[0073] The calculation of the internal forces of the basic structure is the basis for solving the influence line of the internal forces of the statically indeterminate structure. Referring to the contents of Table 1, Table 1 gives the expression of the internal forces of the basic structure under the action of redundant force and external load alone. After substituting them into the analytical expressions of constant displacement and load displacement and performing approximate curve integration, the explicit solutions of the analytical expressions of constant displacement and load displacement considering the axial deformation of the arch are obtained. The results are shown in Table 2.

[0074] Table 1. Redundant forces and internal forces of the basic structure under external loads.

[0075]

[0076] Note: When the moving load is located on the right half of the arch (left half of the arch), the upper (lower) symbol is used; the bending moment of the arch rib is positive when the inner side is under tension, positive when the axial force is under compression, and positive when the cable force is under tension.

[0077] Table 2 Calculation of Redundant Internal Forces

[0078]

[0079] Note: Constant displacement δ ij and load displacement Δ iP This was deduced based on a unit force acting on the right half of the arch. When the moving load acts on the left half of the arch, i.e., the negative half of the x-axis, δ 34 and Δ 3P Take -1 times the corresponding right half-arch value; when ξ1>ξ P At that time, Δ 4P ξ in P Replace with ξ1.

[0080] The constant displacement δ in Table 2 ij , load displacement Δ iPSubstituting the calculation formulas into the basic equations yields the redundant forces X1 to X4. However, according to the basic principle of the force method, the cable force is replaced by the redundant force X4. But since the redundant force X4 becomes negative under the action of moving load (compression), and without considering the application of prestress, the cable force only exists when the deformation increment of the cable is positive. Therefore, the case of a negative cable force does not exist, and the cable force has a zero point.

[0081] According to the superposition principle, the cable-cable arch can be regarded as two cantilever arches under the combined action of external load and redundant force. Due to the symmetry of the structure, taking the internal force of the right half-arch section as the representative (i.e., x≥0), its internal force analytical solution can be divided into two cases, positive and negative half-axis, depending on the position of the moving load.

[0082] When the moving load is located on the positive half-axis (right half-arch), i.e., 0 ≤ x P When L / 2 ≤ L, the analytical expression for the internal forces of the arch section is:

[0083]

[0084] In the formula, M is the bending moment of the arch structure; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p x is the abscissa of the vertical moving load P; x1 is the abscissa of the cable constraint position; α is the inclination angle of the cable constraint; X i (i = 1, 2, 3) represents the excess force at the top of the arch, where X1 is the bending moment, X2 is the axial force, X3 is the shear force, and X4 is the cable force.

[0085] When the moving load is located on the negative half-axis (left half-arch), i.e. -L / 2 < x P When < 0, the analytical expression for the internal forces of the arch section is:

[0086] M = X1 + X2(yy) s )-X3x

[0087] In the formula, M is the bending moment of the arch structure; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p X1 is the abscissa of the vertical moving load P; X2 is the bending moment, X3 is the axial force, and X4 is the shear force.

[0088] Example 2:

[0089] Taking a cable-stayed arch bridge with uniform cross-section as an example, a finite element model was established using ANSYS. The derived internal force calculation results were compared, and the relative errors between the analytical solution and the finite element results were also compared. The arch bridge has a span L = 30m, a rise f = 5m, a rise-to-span ratio of 1 / 6, an arch axis coefficient m = 2.514, and an EI = 7.26 × 10⁻⁶. 8 (N·m 2), EA = 1.07 × 10 10 (N), the parameters of the cable are: E c =1.95×10 11 (N / m 2 ), A c =2.22×10 -3 (m 2 Under the concentrated load case of L / 4, plot the influence line curves of bending moment at the arch crown and arch foot sections under a unit moving load when the rise-to-span ratio is 1 / 6, as shown below. Figure 2 , Figure 3 As shown.

[0090] Table 3 Comparison of key section values ​​for cable-stayed arches under different rise-to-span ratios.

[0091]

[0092] Note: ξ = 0 and ξ = 1.0 represent the arch crown section and arch foot section, respectively.

[0093] Comparing the relative errors of the analytical solution and finite element results under different rise-to-span ratios under the L / 4 concentrated load condition, the relative errors in the arch foot region all increase with the increase of the rise-to-span ratio, while the relative errors in the arch crown region are almost zero. The relative errors of the bending moments are all within 1%, indicating that the calculation method has high accuracy.

[0094] Example 3:

[0095] Taking a cable-stayed arch bridge with uniform cross-section as an example, the span of the arch bridge is L = 30m, the rise is f = 5m, the rise-to-span ratio is 1 / 6, the arch axis coefficient is m = 2.514, and EI = 7.26 × 10⁻⁶. 8 (N·m 2 ), EA = 1.07 × 10 10 (N), the parameters of the cable are: E c =1.95×10 11 (N / m 2 ), A c =2.22×10 -3 (m 2 The coordinates of the cable constraint position are x1, and the relative coordinates of the cable constraint position are ξ1 = 2x1 / L. The method of this invention is used to calculate the bending moment of the bridge section, and the influence line of the bending moment at the arch foot is plotted, as shown below. Figure 4 The figure shows the influence line of the arch foot bending moment when ξ1=0.4. Figure 5 and Figure 6 This is a schematic diagram of the vehicle arrangement for a symmetrical load test with negative bending moment at the arch foot section.

[0096] The static load test of this bridge used three three-axle trucks. The main technical specifications are as follows: Figure 7 As shown, the wheelbase and axle load are shown in Table 4.

[0097] Table 4. Wheelbase and Axle Load of Three-Axle Trucks

[0098] vehicle number Wheelbase 1 (m) Wheelbase 2 (m) Front axle load (kN) Rear axle total weight (kN) Vehicle gross weight (kN) Car #1 3.60 1.35 70 300 370 Car #2 3.60 1.35 70 300 370 Car #3 3.60 1.35 70 300 370

[0099] On highway bridges, when a vehicle load is applied, the formula used to calculate the maximum bending moment is:

[0100] M max (x)=n1*[(p1*(y(x)+p2*(y(x+Δx)]

[0101] Where: p1 is the rear axle force of the standard vehicle, p2 is the front axle pressure of the standard vehicle, Δx is the distance between the front and rear axles of the standard vehicle, and n1 is the number of vehicles;

[0102] Based on the above parameters, n1 = 3, p1 = 300kN, p2 = 70kN, Δx = 3.6m.

[0103] Test scheme 1: y(x)=-1.58, y(x+Δx)=-1.02, substituting into the formula, the maximum negative bending moment at the arch foot is calculated to be -1636.2kN·m.

[0104] Test scheme 2: y(x)=-1.58, y(x+Δx)=-0.81, substituting into the formula, the maximum negative bending moment at the arch foot is calculated to be -1592.1kN·m.

[0105] Comparing the results of test scheme 1 and test scheme 2, the maximum absolute value of the bending moment is taken, that is, the maximum negative bending moment at the arch foot is -1636.2kN·m.

[0106] Obviously, the above embodiments are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art will recognize that other variations or modifications can be made based on the above description; it is neither necessary nor possible to exhaustively list all possible implementations; however, obvious variations or modifications derived therefrom are still within the scope of protection of the present invention.

Claims

1. A method for arranging test vehicles for cable-stayed arch bridges based on internal force influence lines, characterized in that... Includes the following steps: Step 1: Establish the force model of the cable-catenary arch structure. Assuming the arch is a catenary arch with a uniform cross section, derive the structural expression of the catenary arch based on the force method principle of structural mechanics. Step 2: Establish the basic equations for the basic structure of the cable-catenary arch combined structure. Based on the theory of elastic center and the method of numerical integration, expand the differential of the arch axis curve using Taylor formula to obtain approximate expressions for the catenary equation of the arch bridge and the precise differential of the arch axis. Step 3, constant displacement , load displacement Substituting into the basic equations, we obtain the excess force at the crown. ~ Due to the symmetry of the structure, the internal forces of the right semi-arch section are taken as representative, that is... Thus, the analytical expression for the internal forces of the arch section is derived; Step 4: Calculate the cross-sectional bending moment of the arch bridge using the internal force analytical expression from Step 3, and draw the bending moment influence line of the arch. Based on the influence line image, the location of the load test vehicle corresponding to the most unfavorable load can be obtained, which is the location of the maximum bending moment of the arch. The expression for the catenary arch structure in step one is: ； In the formula, The ordinate of the arch axis section is... For the arrow height, This is the arch axis coefficient. For hyperbolic cosine functions, the catenary equation variables are... , Let be the relative coordinates of the arch axis, and , The x-coordinate of the arch axis The span of the arch; The basic equation in step two is: ； In the formula, For constant displacement, The displacement is denoted by load, representing the elastic center of the basic structure under the action of a unit force or a unit external load alone, along the direction of displacement. Displacement in direction; To provide excess force for the vault, ,in For bending moment, For axial force, For shear force; X j=4 For cable force; The displacement of the elastic center in the basic structure is determined by the deformation of the arch axis. Here, the axial deformation of the arch ribs is considered, but shear deformation is not. and The basic expression is: ； ； In the formula, , These are respectively represented as the section bending moments of the basic structure under the action of the redundant force alone; , These are respectively represented as the axial forces of the basic structure under the action of the redundant force alone; , These are respectively represented as the cable forces of the basic structure under the action of the redundant force alone; , , These are respectively represented as the section bending moment, axial force, and cable force of the basic structure under a single unit load; Represents the differential of the arc length of the arch axis; , These are the arch's bending stiffness and axial stiffness, respectively. Let the arch's bending-compression stiffness ratio be... ; , Let the axial stiffness of the cable and its initial length be respectively, and let the ratio of the axial stiffness of the cable to that of the arch be... ; The approximate expressions for the catenary equation and the exact differential of the arch axis in step two are as follows: ； ； In the formula, It is called an approximate differential function, where and All are constant coefficients. The calculation formula is as follows: ； In the above formula, the coefficients ,in; , For the arrow height, This is the arch axis coefficient. For the arch span, This is the remainder term of Taylor's formula; In step three, when the moving load is located on the positive half-axis, i.e., 0 ≤ x P When L / 2 ≤ L, the analytical expression for internal forces is: ； In the formula, Let x be the bending moment of the arch structure section; x and y be the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p x is the abscissa of the vertical moving load P; x1 is the abscissa of the cable constraint position; α is the inclination angle of the cable constraint; X i For the redundant forces at the top of the arch, i=1,2,3, where X1 is the bending moment, X2 is the axial force, X3 is the shear force, and X4 is the cable force; In step three, when the moving load is located on the left half of the arch, i.e., -L / 2 < x P When < 0, the analytical expression for the internal forces of the arch section is: ； In the formula, Let x be the bending moment of the arch structure section; x and y be the abscissa and ordinate of the arch axis, respectively; y s X is the length of the rigid arm; i For the redundant forces at the top of the arch, i=1,2,3, where X1 is the bending moment, X2 is the axial force, and X3 is the shear force.

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