Vehicle arrangement method for load test of catenary arch bridge based on internal force influence line

Abstract

The application discloses a catenary arch bridge load test vehicle arrangement method based on internal force influence line, comprising the following steps: step one, deducing 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 arch equation 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 internal force of the bridge by using the internal force analytical expression, drawing the bending moment influence line of the arch, and determining the position of the vehicle arrangement according to the influence line image. The application deduces the basic internal force analytical solution of the catenary arch, has the advantages of high precision, convenient calculation, wide applicability and the like, is applied to the bridge load experiment, can quickly and accurately determine the 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 catenary arch bridge load tests 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] Catenary arches are statically indeterminate structures with multiple inconsistencies. Traditional calculation methods for determining influence lines require consulting tables and performing numerical calculations, resulting in a large workload, tedious calculations, and low efficiency. Consequently, it is difficult to guarantee a fast and accurate determination of the internal force influence lines for catenary arch bridges, which greatly hinders the conduct of load experiments on arch bridges. Currently, only analytical solutions for the internal forces of catenary arches with uniform cross-sections are provided under the self-weight conditions of the main arch ring and the bridge deck system. Analytical solutions for the internal forces of the arch under vertical moving loads are relatively lacking. Therefore, in current engineering practice, when evaluating the structural performance of bridges, load experiments require extensive simulations and tests of vehicle positions, making it difficult to find the most unfavorable load location. The actual simulations and tests are 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 vehicles in catenary arch bridge load tests based on internal force influence lines, simplifying the tedious simulation and testing operations and facilitating the determination of 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 catenary arch bridge load tests based on internal force influence lines includes the following steps:

[0008] Step 1: Establish the force model of the 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 catenary arch. Based on the theory of elasticity center and the method of numerical integration, expand the differential of the arch axis curve using Taylor's 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 X3 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: Use the internal force analytical expression from step three to calculate the bending moment internal force of the arch bridge section, and draw the influence line of the bending moment at the arch crown and the bending moment influence line at the arch foot. Based on the influence line image, the location of the vehicle arrangement can be obtained, namely the location of the maximum positive bending moment at the arch crown and the location of the maximum negative bending moment at the arch foot.

[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 any section of the arch axis, 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 top of the arch, where X1 is the bending moment, X2 is the axial force, and X3 is the shear 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 Let ds represent the bending moment and axial force of the basic structure under the action of redundant force and unit load alone; ds represents the differential of the arc length of the arch axis; EI and EA are the bending stiffness and axial stiffness of the arch, respectively, and let the bending-compression stiffness ratio of the arch be b = EI / 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, This is called the approximate differential function, where ξ is the relative coordinate of the arch axis, and ξ = 2x / L, x is the abscissa of the arch axis, L is the arch span, and λ i and n i (i = 1 to 3) are all constant coefficients, and the calculation formula is as follows:

[0026]

[0027] In the 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 iP Substituting 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. PWhen -L / 2 ≤ x 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. The following will take the internal forces of the right half-arch section as an example.

[0029] Furthermore, the analytical expression for the internal forces in step three is:

[0030]

[0031]

[0032] In the formula, M and N are the bending moment and axial force of the arch section, respectively; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p Let x be the x-coordinate of the vertical moving load P; X1 is the horizontal inclination angle of the tangent at any point on the arch axis; X2 and X3 are the bending moment, axial force, and shear force at the arch crown section, respectively.

[0033] The present invention has the following advantages:

[0034] 1. This invention proposes an analytical solution for the influence line of internal forces in a catenary arch structure under moving loads, derived using an approximate line integral method. This fills a gap in research on analytical solutions for the influence line of internal forces in catenary arch structures under vertical moving loads, providing reference data for future theoretical analysis of catenary arch bridge inspection, monitoring, and load testing. 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 formulas presented in this paper.

[0035] 2. This invention derives the analytical solution of the basic internal forces of a catenary arch under moving load, which has the advantages of high precision, convenient calculation and wide applicability. When applied to bridge load experiments, 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

[0036] Figure 1 This is a schematic diagram of the force model analysis of the catenary arch structure in this invention.

[0037] 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.

[0038] Figure 3 This is the influence line of the axial force at the crown of the arch in the analytical solution and the finite element solution of Embodiment 2 of the present invention.

[0039] Figure 4 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.

[0040] Figure 5 This is the influence line of the axial force at the arch foot in the analytical solution and the finite element solution of Embodiment 2 of the present invention.

[0041] Figure 6 This is a curve comparing the axial forces of the two-hinged arch bridge deck system under its own weight in Embodiment 3 of the present invention.

[0042] Figure 7 This is a curve comparing the bending moments under the self-weight condition of the two-hinged arch bridge deck system in Embodiment 3 of the present invention.

[0043] Figure 8 This is the influence line of the arch bending moment in Embodiment 4 of the present invention.

[0044] Figure 9 This is a schematic diagram of the wheelbase of the test vehicle in Embodiment 4 of the present invention.

[0045] Figure 10 This is a schematic diagram of the vehicle arrangement under symmetrical loading of positive bending moment at the arch section A in Embodiment 4 of the present invention. Detailed Implementation

[0046] 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.

[0047] Example 1:

[0048] The method for arranging test vehicles for catenary arch bridge load tests based on internal force influence lines includes the following steps:

[0049] Step 1: Establish the force model of the 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.

[0050] Step 2: Establish the basic equations for the basic structure of the catenary arch. Based on the theory of elasticity center and the method of numerical integration, expand the differential of the arch axis curve using Taylor's formula to obtain approximate expressions for the catenary equation of the arch bridge and the precise differential of the arch axis.

[0051] Step 3, change the constant displacement δ ij , load displacement Δ iP Substituting into the basic equation, we obtain the redundant forces X1 to X3 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.

[0052] Step four: Use the internal force analytical expression from step three to calculate the bending moment internal force of the arch bridge section, and draw the influence line of the bending moment at the arch crown and the bending moment influence line at the arch foot. Based on the influence line image, the location of the vehicle arrangement can be obtained, namely the location of the maximum positive bending moment at the arch crown and the location of the maximum negative bending moment at the arch foot.

[0053] The stress conditions of the catenary arch structure are analyzed, and the analytical expression of the influence line of the arch bending moment of the catenary arch under load test is further explained:

[0054] like Figure 1 As shown, Figure 1 The diagram shows the calculation of a catenary arch structure under vertical moving load, where a is a simplified calculation diagram of the arch structure and b is the basic system of the arch structure.

[0055] A and C represent the arch feet, and B represents the arch crown. Using the force method, the arch is divided into a cantilever arch with left and right symmetry. The excess force at the arch crown is X. j (j=1,2,3), the force method equations are simplified into easily solvable canonical equations using a simplified system with elastic centers. 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 .

[0056] The expression for the catenary arch axis is:

[0057]

[0058] In the formula, y is the ordinate of any section of the arch axis, 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.

[0059] right Figure 1 The basic system shown establishes the basic equations:

[0060]

[0061] Where: 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 top of the arch, where X1 is the bending moment, X2 is the axial force, and X3 is the shear force.

[0062] 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:

[0063]

[0064]

[0065] In the formula, and MP N P Let ds represent the bending moment and axial force of the basic structure under the action of redundant force and unit load alone; ds represents the differential of the arc length of the arch axis; EI and EA are the bending stiffness and axial stiffness of the arch, respectively, and let the bending-compression stiffness ratio of the arch be b = EI / EA.

[0066] 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:

[0067]

[0068]

[0069] In the formula, This is called the approximate differential function, where ξ is the relative coordinate of the arch axis, and ξ = 2x / L, x is the abscissa of the arch axis, L is the arch span, and λ i and n i (i = 1 to 3) are all constant coefficients, and the calculation formula is as follows:

[0070]

[0071] In the 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.

[0072] 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:

[0073]

[0074] 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 Equation (3) and Equation (4) 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.

[0075] Table 1. Redundant forces and internal forces of the basic structure under external loads.

[0076]

[0077] 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 defined as positive when the inner side is under tension and positive when the axial force is under compression.

[0078] Table 2 Calculation of Redundant Internal Forces

[0079]

[0080] constant displacement δ ij , load displacement Δ iP Substituting 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 ≤ x 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 internal forces when <0, a near-analytical expression of the internal forces of the entire structure can be derived. Taking the internal forces of the right semi-arch section as an example, the analytical expression of its internal forces under vertical moving load is as follows:

[0081]

[0082]

[0083] In the formula, M and N are the bending moment and axial force of the arch section, respectively; x and y are the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p Let x be the x-coordinate of the vertical moving load P; X1 is the horizontal inclination angle of the tangent at any point on the arch axis; X2 and X3 are the bending moment, axial force, and shear force at the arch crown section, respectively.

[0084] Example 2:

[0085] Taking a catenary arch bridge with a 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, and an arch axis coefficient m = 2.514. EA = 1.07 × 10 10 (N). Plot the influence line curves of bending moment and axial force 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. Figures 2-5 As shown.

[0086] Comparative analysis of the relative errors between the analytical solution and the finite element method results shows that the relative error in the arch foot region is relatively small, at only 0.30%, while the relative error in the arch crown region is almost zero. The maximum absolute error is only 0.004, and this section is located near the zero bending moment. The relative errors of other sections are all within 1%; the relative error of axial force is also within 1%. The results indicate that the internal force curves of the analytical solution and the finite element method solution have a high degree of fit, verifying the reliability and accuracy of the calculation formula presented in this paper.

[0087] Example 3:

[0088] Regarding the analytical solution of internal forces in catenary arches, Hu Changfu et al. [Hu Changfu, Lu Xiaoyu, Gan Huihui et al. Practical analytical solution of catenary arches based on approximate integration [J]. Journal of Central South University (Natural Science Edition) 2015, 46(3):1058-1065] proposed an approximate integration method to obtain the practical analytical solution of catenary arches. Under the structural diagrams of two-hinged arches and hingeless arches, based on this approximate method, the practical analytical solution of the integral constant of the catenary arch, the practical analytical solution of the internal forces of the main arch ring self-weight and the uniformly distributed load on the bridge deck are solved. However, in the study of analytical solutions of internal forces in catenary arches with uniform cross-sections, the analytical solution of internal forces of the arch under vertical moving loads is relatively blank. This invention proposes to derive an analytical solution for the influence line of internal forces in an arch structure using an approximate curve integral method. This fills a gap in research on the analytical solution for the influence line of internal forces in a catenary arch structure under vertical moving loads. It can be used to calculate the influence line of internal forces in a top-bearing catenary arch bridge, providing reference data for the theoretical analysis of future monitoring of catenary arch bridges.

[0089] In terms of accuracy, the analytical solution for internal forces obtained by Hu et al. showed a maximum relative error of 4.81% for axial force and a maximum relative error of 11.67% for extreme bending moment under the self-weight condition of the two-hinged arch bridge deck system. Figure 6 and Figure 7 As shown. The analytical solution derived in this invention is in good agreement with the finite element results, and the overall relative error is within 1%, fully meeting the engineering accuracy requirements, such as... Figures 2-5 As shown.

[0090] Example 4:

[0091] Taking a catenary 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 method of this invention was used to calculate the bending moment and internal forces at the cross-sections of this bridge, and the influence line of the bending moment at the arch crown was plotted, as shown below. Figure 8 As shown.

[0092] The catenary arch experiences a maximum positive bending moment of 491.3 kN·m and a maximum negative bending moment of -101.9 kN·m at the crown.

[0093] The static load test of this bridge used three three-axle trucks. The main technical specifications are as follows: Figure 9 As shown, the wheelbase and axle load are shown in Table 3.

[0094] Table 3. Wheelbase and Axle Load of Three-Axle Trucks

[0095] 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

[0096] On highway bridges, when a vehicle load is applied, the formula used to calculate the maximum bending moment is:

[0097]

[0098] 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;

[0099] Based on the above parameters, n1 = 3, p1 = 300 kN, p2 = 70 kN, Δx = 3.6 m, y(x) = 1.65, y(x+Δx) = 0.19. Substituting these parameters into the formula, we obtain the maximum positive bending moment at the arch crown as 1524.9 kN·m. Thus, we obtain the vehicle arrangement scheme for the maximum positive bending moment at the arch crown section.

[0100] 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 catenary arch bridge loads based on internal force influence lines, characterized in that... Includes the following steps: Step 1: Establish the force model of the 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 catenary arch. Based on the theory of elasticity center and the method of numerical integration, expand the differential of the arch axis curve using Taylor's 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: Use the internal force analytical expression from Step 3 to calculate the bending moment internal force of the arch bridge section, and draw the bending moment influence line of the arch. Based on the influence line image, the location of the load test vehicle can be obtained, namely the location of the maximum positive bending moment at the top of the arch and the location of the maximum negative bending moment at the bottom of the arch. The expression for the catenary arch structure in step one is: ； In the formula, Let be the ordinate of any cross section of the arch axis. 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; 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 expressed as the bending moment and axial force of the basic structure under a single unit external load; Represents the differential of the arc length of the arch axis; , Let b be the arch's bending stiffness and axial stiffness, respectively, and let b = EI / EA be the ratio of the arch's bending and compressive stiffness. The approximate expressions for the catenary equation and the exact differential of the arch axis in step two are as follows: ； ； In the formula, This is called the approximate differential function, where, Let be the relative coordinates of the arch axis, and , The x-coordinate of the arch axis For the arch span, and All are constant coefficients. The calculation formula is as follows: ； In the 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; The analytical expression for the internal forces in step three is: ； ； In the formula, N and N represent the bending moment and axial force of the arch section, respectively; x and y represent the abscissa and ordinate of the arch axis, respectively; y s x is the length of the rigid arm; p Let x be the x-coordinate of the vertical moving load P; X1 is the horizontal inclination angle of the tangent at any point on the arch axis; X2 and X3 are the bending moment, axial force, and shear force at the arch crown section, respectively.

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