Cross section self-adaptive design method for automatically matching strength and stability of main arch of arch bridge

Abstract

The invention discloses a section self-adaptive design method for automatically matching the strength and stability of an arch bridge main arch, and relates to the technical field of bridge engineering structure design and optimization. Comprising the following steps: taking vault section damage as a failure mode of a structure, and establishing a main arch strength and stability coupling correlation model; setting constraint conditions, and solving the main arch strength and stability coupling correlation model under the constraint conditions; performing optimization solution in the solution set meeting the constraint condition by taking the minimization of the material consumption of the main arch as a target; according to the area of the box-shaped section and an inertia moment calculation formula, solving size parameters of the width of the box-shaped section and the thickness of the concrete slab; and determining a final section design scheme of the main arch of the arch bridge based on the size parameters of the width of the box-shaped section and the thickness of the concrete slab. According to the method, the material utilization rate, the structural safety and the design efficiency of the large-span arch bridge can be improved, and the dual purposes of economy and safety are achieved.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering structural design and optimization technology, and in particular to an adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge. Background Technology

[0002] The cross-sectional design of the main arch of a long-span arch bridge is crucial to the structural safety and economy. An unreasonable distribution of materials will not only waste materials but also become a burden on the structure. Therefore, the cross-sectional design of the main arch must not only meet the strength requirements of the structure but also ensure a reasonable distribution of the arch rib stiffness in order to effectively resist structural stress, deformation, and instability under various loads.

[0003] Existing methods for designing the cross-sections of arch bridges rely on elastic theory and empirical formulas. These methods fail to address the balance between material strength and structural stability under strong nonlinear influences, and cannot obtain precise cross-sectional dimensions under complex load conditions and geometric nonlinear effects. Instead, they simply add safety factors to empirical formulas, resulting in bulky arch structures that hinder further development of arch bridge spans. Furthermore, existing methods employ iterative optimization, repeatedly adjusting the main arch cross-section to achieve strength and stability goals. This cumbersome process is inefficient and cannot guarantee globally optimal results, ultimately relying on the designer's experience.

[0004] Therefore, proposing an adaptive cross-section design method that automatically matches the strength and stability of the main arch of an arch bridge to solve the problems existing in the prior art is an urgent problem to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, the present invention provides an adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge, which can improve the material utilization rate, structural safety and design efficiency of long-span arch bridges, thereby achieving the dual goals of economy and safety.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: An adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge includes: S1. Taking the failure of the arch crown section as the failure mode of the structure, a coupling correlation model between the strength and stability of the main arch is established, which includes the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. S2. Set constraints and, based on the relationship between the design axial compressive stress and the design arch crown cross-sectional area, solve the coupling relationship model of the main arch strength and stability under constraints to obtain the design combination solution set of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height that satisfies all constraints. S3. With the goal of minimizing the amount of material used in the main arch, optimize the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness and arch crown cross-sectional height. S4. Based on the optimal design combination of arch crown section area, arch crown section stiffness and arch crown section height, the dimensional parameters of the box section width and concrete slab thickness are obtained according to the calculation formula of box section area and moment of inertia. S5. Based on the dimensional parameters of the box section width and the concrete slab thickness, determine the final cross-sectional design scheme of the main arch of the arch bridge.

[0007] Optionally, in the above method, in S1, the failure of the arch crown section is taken as the failure mode of the structure, and a coupled correlation model between the strength and stability of the main arch is established, including the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. Specifically: The failure of the arch section is defined as the local concrete compressive stress reaching the design value of the compressive strength. f c It satisfies the following formula (1): (1) In the formula, To design axial compressive stress, This refers to the nonlinear bending stress at the point of failure of the arch bridge. This is the nonlinear stability coefficient; According to the nonlinear bending moment calculation formula Represented as equation (2): (2) In the formula, The nonlinear bending moment at the arch crown section. h 0 is the height of the arch crown section. For the stiffness of the arch crown section; Combining equations (1) and (2), we obtain the coupled correlation model between the strength and stability of the main arch: (3) According to the design axial compressive stress With the design vault area The correlation formula between them (4): (4) Substituting equation (4) into equation (3), we obtain equation (5): (5).

[0008] Optionally, in S2, constraints can be set as follows: A type of stability coefficient and , , The relationship between them is given by equation (6): (6) In the formula, E The elastic modulus of the main arch material. I 0 is the moment of inertia of the arch crown section. The main arch span, μ is the length coefficient for calculating the main arch span:

[0009] in, For arches with varying widths at the same height: In the formula, s= al The horizontal inclination angle of the arch foot along the arch axis. a For the curvature of the vault, l Half the span of the main arch; For arches with equal width and varying height It can be calculated using the following formula; A class of stability coefficients is greater than the nonlinear stability coefficient. As the constraint condition for the stiffness of the arch rib, equation (7) is as follows: (7) Based on engineering experience, we select the following constraint formula for the height of the arch crown section: (8) (8).

[0010] Optionally, in S3 of the above method, the goal is to minimize the amount of material used in the main arch. The optimization solution is then performed within the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height. Specifically: S301, Input design parameters are lost. Main arch span The magnitude of the vertical concentrated force and location The density of the main arch material Elastic modulus of main arch material and nonlinear stability coefficient ; S302, Input design material strength f c With design axial compressive stress s m The larger value is calculated using the following formula.

[0011] make , The stress iteration coefficient is selected based on the convergence result. k 1 The initial value is 0.9; S303. Based on the shortest force transmission path, determine the reasonable bridge completion state of the arch bridge through static equilibrium conditions. Hy''=q (x) Calculate the reasonable arch axis with the cross-sectional area of ​​the arch , H The horizontal reaction force of the arch, q (x) represents the dead load distribution of the arch, which is obtained after the arch structure design parameters are initially determined through step S301. S304. The allowable bending stress is obtained according to equation (1). ; S305, Input the arch cross section I For smaller values ​​of the moment of inertia, the following formula is used for calculation:

[0012] In the formula or 2 represents the nonlinear stability coefficient, with a reference value of 1.75. S306. Obtain the bending moment of the arch crown section using the nonlinear bending moment analytical calculation method. The height of the arch section is calculated according to equation (3):

[0013] In the formula, w s The nonlinear redundant force coefficient is solved using the following formula:

[0014] in, e m= s m / E represents the compressive strain value of the arch crown section. l Half the span of the main arch; In the formula, π1 to π4 are the calculation coefficients, which are solved according to the following formula:

[0015] in β s、 s represents the loading process coefficient and the horizontal inclination angle of the arch foot along the arch axis, respectively. , s= al , a The curvature of the vault.

[0016] The above method, optionally, includes the following formulas for calculating the area and moment of inertia of the box-shaped section in S4: In obtaining , , Then, the area and moment of inertia of the box-shaped section are calculated using the following formulas: (9) (10) In the formula, the unknown quantity is the cross-sectional width. With the thickness of concrete slab ; Combine equations (9) and (10) and solve numerically.

[0017] Optionally, in step S4 of the above method, the dimensional parameters of the box section width and concrete slab thickness are obtained by calculating the area and moment of inertia of the box section, specifically: S401, At this time, , , Given that, calculate according to equations (9) and (10) , ,make ; S402, if or Then repeat steps S302 to S306; S403, if Then let , The stress correction factor is set to 1.01. Repeat steps S302 to S306, S401 and S404. S404, if or , , The stiffness correction factor is set to 1.01. Repeat steps S302 to S306, S401 and S404. S405 Repeat steps S403 and S404. ; End the calculation and output the cross-sectional dimensions. , , .

[0018] As can be seen from the above technical solutions, compared with the prior art, the present invention provides an adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge, which has the following beneficial effects: The present invention solves the problem of difficulty in matching the two under nonlinear effects by establishing a coupled mathematical model of strength and stability. Considering the alignment of large-span arch bridges with strength and stability matching, it can obtain the main arch alignment and cross-section parameters under the conditions of strength and stiffness by means of an optimization algorithm for the main arch cross-section under any preset stress and elastic stability coefficient. The traditional design process that relies on manual experience is transformed into an executable iterative algorithm, realizing the automation of cross-section design, greatly shortening the design cycle and reducing the dependence on the experience of designers. With the goal of minimizing material usage, the material utilization rate is improved. The lightweight design not only saves materials but also significantly reduces the lifting weight and improves the convenience of construction. The present invention is applicable to arch bridges with different spans and different design stress levels, and can be flexibly applied to various box-section forms such as equal width and variable height, and equal height and variable width, and has broad engineering application prospects. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0020] Figure 1 A flowchart of an adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge, provided by the present invention; Figure 2 The comparison results between the new and original design lines in a specific embodiment of the cross-section adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge provided by the present invention. Detailed Implementation

[0021] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] In this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. The terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0023] Reference Figure 1 As shown, this invention discloses an adaptive cross-section design method for automatically matching the strength and stability of the main arch of an arch bridge, comprising: S1. Taking the failure of the arch crown section as the failure mode of the structure, a coupling correlation model between the strength and stability of the main arch is established, which includes the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. S2. Set constraints and, based on the relationship between the design axial compressive stress and the design arch crown cross-sectional area, solve the coupling relationship model of the main arch strength and stability under constraints to obtain the design combination solution set of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height that satisfies all constraints. S3. With the goal of minimizing the amount of material used in the main arch, optimize the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness and arch crown cross-sectional height. S4. Based on the optimal design combination of arch crown section area, arch crown section stiffness and arch crown section height, the dimensional parameters of the box section width and concrete slab thickness are obtained according to the calculation formula of box section area and moment of inertia. S5. Based on the dimensional parameters of the box section width and the concrete slab thickness, determine the final cross-sectional design scheme of the main arch of the arch bridge.

[0024] Furthermore, in S1, taking the failure of the arch crown section as the failure mode of the structure, a coupled correlation model between the strength and stability of the main arch is established, which includes the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. Specifically: The failure of the arch section is defined as the local concrete compressive stress reaching the design value of the compressive strength. f c It satisfies the following formula (1): (1) In the formula, To design axial compressive stress, This refers to the nonlinear bending stress at the point of failure of the arch bridge. This is the nonlinear stability coefficient; According to the nonlinear bending moment calculation formula Represented as equation (2): (2) In the formula, The nonlinear bending moment at the arch crown section. h 0 is the height of the arch crown section. For the stiffness of the arch crown section; Combining equations (1) and (2), we obtain the coupled correlation model between the strength and stability of the main arch: (3) According to the design axial compressive stress With the design vault area The correlation formula between them (4): (4) Substituting equation (4) into equation (3), we obtain equation (5): (5).

[0025] Furthermore, in S2, constraints are set as follows: A type of stability coefficient and , , The relationship between them is given by equation (6): (6) In the formula, E The elastic modulus of the main arch material. I 0 is the moment of inertia of the arch crown section. The main arch span, μ is the length coefficient for calculating the main arch span:

[0026] in, For arches with varying widths at the same height: In the formula, s= al The horizontal inclination angle of the arch foot along the arch axis. a For the curvature of the vault, l Half the span of the main arch; For arches with equal width and varying height It can be calculated using the following formula; A class of stability coefficients is greater than the nonlinear stability coefficient. As the constraint condition for the stiffness of the arch rib, equation (7) is as follows: (7) Based on engineering experience, we select the following constraint formula for the height of the arch crown section: (8) (8).

[0027] Furthermore, in S3, with the objective of minimizing the material usage of the main arch, optimization is performed within the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height, specifically: S301, Input design parameters are lost. Main arch span The magnitude of the vertical concentrated force and location The density of the main arch material Elastic modulus of main arch material and nonlinear stability coefficient ; S302, Input design material strength f c With design axial compressive stress s m The larger value is calculated using the following formula.

[0028] make , The stress iteration coefficient is selected based on the convergence result. k 1 The initial value is 0.9; S303. Based on the shortest force transmission path, determine the reasonable bridge completion state of the arch bridge through static equilibrium conditions. Hy''=q (x) Calculate the reasonable arch axis with the cross-sectional area of ​​the arch , H The horizontal reaction force of the arch, q (x) represents the dead load distribution of the arch, which is obtained after the arch structure design parameters are initially determined through step S301. S304. The allowable bending stress is obtained according to equation (1). ; S305, Input the arch cross section I For smaller values ​​of the moment of inertia, the following formula is used for calculation:

[0029] In the formula or 2 represents the nonlinear stability coefficient, with a reference value of 1.75. S306. Obtain the bending moment of the arch crown section using the nonlinear bending moment analytical calculation method. The height of the arch section is calculated according to equation (3):

[0030] In the formula, w s The nonlinear redundant force coefficient is solved using the following formula:

[0031] in, e m= s m / E represents the compressive strain value of the arch crown section. l Half the span of the main arch; In the formula, π1 to π4 are the calculation coefficients, which are solved according to the following formula:

[0032] in β s、 s represents the loading process coefficient and the horizontal inclination angle of the arch foot along the arch axis, respectively. , s= al , a The curvature of the vault.

[0033] Furthermore, in S4, the formulas for calculating the area and moment of inertia of the box-shaped section are as follows: In obtaining , , Then, the area and moment of inertia of the box-shaped section are calculated using the following formulas: (9) (10) In the formula, the unknown quantity is the cross-sectional width. With the thickness of concrete slab ; Combine equations (9) and (10) and solve numerically.

[0034] Furthermore, in S4, based on the formulas for calculating the area and moment of inertia of the box section, the dimensional parameters of the width of the box section and the thickness of the concrete slab are obtained, specifically: S401, At this time, , , Given that, calculate according to equations (9) and (10) , ,make ; S402, if or Then repeat steps S302 to S306; S403, if Then let , The stress correction factor is set to 1.01. Repeat steps S302 to S306, S401 and S404. S404, if or , , The stiffness correction factor is set to 1.01. Repeat steps S302 to S306, S401 and S404. S405 Repeat steps S403 and S404. ; End the calculation and output the cross-sectional dimensions. , , .

[0035] In one specific embodiment, the design method of the present invention is used to optimize the cross-section of the main arch ring of a reinforced concrete arch bridge with a main span of 386 meters and a rise of 70.18 meters. span It is 386m, with a rise of 386m. The column spacing is 70.18m. The arch is 31m long, with no columns at the crown. The arch ribs are made of C80 concrete. The original design used a catenary structure with an arch axis coefficient of 2.4. The total volume of concrete for the arch ribs was 8914 cubic meters, with a total weight of 23000 tons. Construction employed segmental prefabrication, cantilever assembly, and a combined cable-stayed system. The average segment length was 6m, and the maximum lifting weight was 242 tons.

[0036] The original design scheme was calculated using the finite element method (FEM). The maximum stress level of the structure under dead load was 18.06 MPa. For ease of comparison, the new scheme pre-sets the axial compressive stress of the arch ribs under dead load to 18 MPa, maintaining the same stress level as the original scheme. The arch ribs are designed as parallel double ribs, with each rib adopting a constant width and variable height cross-section. The pre-set in-plane first-order stability coefficient of the arch ribs is 4. A comparative analysis of the original and optimized schemes was conducted using the FEM method, and the results are as follows: Reference Figure 2 As shown, the new alignment is higher at L / 4 than the original alignment. The maximum alignment deviation occurs at 140m, with a maximum deviation value of 1.17m.

[0037] Table 1 shows the calculation results of the arch rib section in the new scheme. In the new scheme, the height of the medium-width variable-height arch section changes from 4.81m at the arch crown to 6.35m at the arch foot, with a section width of 4m. The thickness of the top and bottom slabs changes from 261.5mm at the arch crown to 344.9mm at the arch foot, while the top slab thickness remains unchanged at 261.5mm. In the original design, the concrete slab thickness was 1m. Compared to the original design, the new scheme has a smaller slab thickness. The new scheme adopts the design concept of equal stress, reducing the cross-sectional area. However, to meet stiffness requirements, the material needs to be distributed as much as possible at the outer edge of the section, thus reducing the section thickness.

[0038] Table 1 Calculation results of cross-sectional parameters

[0039] By employing the design method of this invention, without altering the stress level, the entire arch is kept at the same stress value under constant load through equal-stress arch design and constant-load bending moment adjustment. Theoretically, this increases material utilization to 100%, reducing the material consumption of the main arch ring from 22,000 tons to 10,209.2 tons, a reduction of 54.6%. The average concrete material consumption per square meter of bridge deck main arch ring is 1.95 tons. The reduction in the self-weight of the arch ribs allows for a 91.7% increase in hoisting length or a 52.6% reduction in hoisting weight.

[0040] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, for system or system embodiments, since they are basically similar to method embodiments, the description is relatively simple, and relevant parts can be referred to the descriptions in the method embodiments. The systems and system embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0041] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge, characterized in that, include: S1. Taking the failure of the arch crown section as the failure mode of the structure, a coupling correlation model between the strength and stability of the main arch is established, which includes the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. S2. Set constraints and, based on the relationship between the design axial compressive stress and the design arch crown cross-sectional area, solve the coupling relationship model of the main arch strength and stability under constraints to obtain the design combination solution set of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height that satisfies all constraints. S3. With the goal of minimizing the amount of material used in the main arch, optimize the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness and arch crown cross-sectional height. S4. Based on the optimal design combination of arch crown section area, arch crown section stiffness and arch crown section height, the dimensional parameters of the box section width and concrete slab thickness are obtained according to the calculation formula of box section area and moment of inertia. S5. Based on the dimensional parameters of the box section width and the concrete slab thickness, determine the final cross-sectional design scheme of the main arch of the arch bridge.

2. The cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge according to claim 1, characterized in that, In S1, the failure of the arch crown section is taken as the failure mode of the structure. A coupled correlation model between the strength and stability of the main arch is established, which includes the nonlinear stability coefficient, the height of the arch crown section, the design axial compressive stress, and the stiffness of the arch crown section. Specifically: The failure of the arch section is defined as the local concrete compressive stress reaching the design value of the compressive strength. f c It satisfies the following formula (1): （1） In the formula, To design axial compressive stress, This refers to the nonlinear bending stress at the point of failure of the arch bridge. This is the nonlinear stability coefficient; According to the nonlinear bending moment calculation formula Represented as equation (2): （2） In the formula, The nonlinear bending moment at the arch crown section. h 0 is the height of the arch crown section. For the stiffness of the arch crown section; Combining equations (1) and (2), we obtain the coupled correlation model between the strength and stability of the main arch: （3） According to the design axial compressive stress With the design vault area The correlation formula between them (4): （4） Substituting equation (4) into equation (3), we obtain equation (5): （5）。 3. The cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge according to claim 2, characterized in that, In S2, the constraints are set as follows: A type of stability coefficient and , , The relationship between them is given by equation (6): （6） In the formula, E The elastic modulus of the main arch material. I 0 is the moment of inertia of the arch crown section. The main arch span, μ is the length coefficient for calculating the main arch span: in, For arches with varying widths at the same height: In the formula, s= al The horizontal inclination angle of the arch foot along the arch axis. a For the curvature of the vault, l Half the span of the main arch; For arches with equal width and varying height It can be calculated using the following formula; A class of stability coefficients is greater than the nonlinear stability coefficient. As the constraint condition for the stiffness of the arch rib, equation (7) is as follows: （7） Based on engineering experience, we select the following constraint formula for the height of the arch crown section: (8) （8）。 4. The cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge according to claim 3, characterized in that, In S3, with the objective of minimizing the material usage of the main arch, optimization is performed within the solution set that satisfies the constraints to obtain the optimal design combination of the arch crown cross-sectional area, arch crown cross-sectional stiffness, and arch crown cross-sectional height, specifically: S301, Input design parameters are lost. Main arch span The magnitude of the vertical concentrated force and location The density of the main arch material Elastic modulus of main arch material and nonlinear stability coefficient ; S302, Input design material strength f c With design axial compressive stress σ m The larger value is calculated using the following formula. make , The stress iteration coefficient is selected based on the convergence result. k 1 The initial value is 0.9; S303. Based on the shortest force transmission path, determine the reasonable bridge completion state of the arch bridge through static equilibrium conditions. Hy''=q (x) Calculate the reasonable arch axis with the cross-sectional area of ​​the arch , H The horizontal reaction force of the arch, q (x) represents the dead load distribution of the arch, which is obtained after the arch structure design parameters are initially determined through step S301. S304. The allowable bending stress is obtained according to equation (1). ; S305, Input the arch cross section I For smaller values ​​of the moment of inertia, the following formula is used for calculation: In the formula η 2 represents the nonlinear stability coefficient, with a reference value of 1.

75. S306. Obtain the bending moment of the arch crown section using the nonlinear bending moment analytical calculation method. The height of the arch section is calculated according to equation (3): In the formula, w s The nonlinear redundant force coefficient is solved using the following formula: in, ε m= σ m / E represents the compressive strain value of the arch crown section. l Half the span of the main arch; In the formula, π1 to π4 are the calculation coefficients, which are solved according to the following formula: in β s、 s represents the loading process coefficient and the horizontal inclination angle of the arch foot along the arch axis, respectively. , s= al , a The curvature of the vault.

5. The cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge according to claim 4, characterized in that, In S4, the formulas for calculating the area and moment of inertia of the box-shaped section are as follows: In obtaining , , Then, the area and moment of inertia of the box-shaped section are calculated using the following formulas: （9） （10） In the formula, the unknown quantity is the cross-sectional width. With the thickness of concrete slab ; Combine equations (9) and (10) and solve numerically.

6. The cross-sectional adaptive design method for automatically matching the strength and stability of the main arch of an arch bridge according to claim 5, characterized in that, In S4, based on the formulas for calculating the area and moment of inertia of the box section, the dimensional parameters of the box section width and concrete slab thickness are obtained, specifically: S401, At this time, , , Given that, calculate according to equations (9) and (10) , ,make ; S402, if or Then repeat steps S302 to S306; S403, if Then let , The stress correction factor is set to 1.

01. Repeat steps S302 to S306, S401 and S404. S404, if or , , The stiffness correction factor is set to 1.

01. Repeat steps S302 to S306, S401 and S404. S405 Repeat steps S403 and S404. ; End the calculation and output the cross-sectional dimensions. , , .

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