A method for load bearing design of main arch after arch forming combined with multi-scale finite element model

Abstract

The application discloses a kind of main arch ring arching after load design method combined with multi-scale finite element model, it is related to building design technical field, first, main arch ring multi-scale finite element model is constructed, then coupling construction technology is simulated construction whole process with life and death element method, output construction stress accumulation data and permanent stress nephogram;Then, joint above-mentioned data is based on force flow theory Quantitative analysis arching after force transmission mechanism, output force transmission path, component collaborative load bearing proportion and so on result;Finally, in combination with construction stress data, with stress uniformity and the like as target optimization main arch ring section, material and interface structure, by multi-scale model iteration checking and calculating output compliance design scheme.The application quantifies component collaborative load bearing proportion, fully considers the influence of construction course, solves the contradiction between traditional design precision and efficiency, the problem of insufficient force transmission mechanism quantization, provides technical support for super-large span stiff skeleton arch bridge construction.

Description

Technical Field

[0001] This invention relates to the field of architectural design technology, and more specifically to a method for designing the load-bearing capacity of the main arch ring after its formation by combining a multi-scale finite element model. Background Technology

[0002] Reinforced steel frame arch bridges, with their significant advantages such as high stiffness, aesthetically pleasing design, strong adaptability to mountainous areas, and low maintenance costs, have become one of the core preferred bridge types for the construction of long-span mountainous bridges. As transportation infrastructure construction expands towards larger spans and more complex geological conditions, ultra-large-span reinforced steel frame arch bridges with main spans of 600 meters or more are emerging. The post-arch load-bearing capacity of these bridges directly determines the structural safety, durability, and long-term operational reliability. The main arch ring, as the core load-bearing component, is formed collaboratively by a reinforced steel frame, internal concrete, and external concrete through segmented construction. Its post-arch load-bearing state depends not only on the material properties and cross-sectional design of each component but also on the coupling influence of multiple factors during construction, such as asynchronous stiffness formation, stress accumulation, and the force transmission characteristics at the interfaces between rings and segments. This makes accurate post-arch load-bearing design a challenging engineering problem.

[0003] The current load-bearing design of the main arch ring of stiffened arch bridges mainly relies on traditional finite element simulation and engineering experience analogy, and has not yet formed a systematic approach. Either beam or plate elements are used to simplify the simulation of the entire bridge, which reduces computational complexity but fails to accurately capture the local stress distribution in key areas such as the arch crown and arch foot, and is particularly difficult to reflect the force transmission details at the interface between the main steel pipe and concrete, and between the outer concrete rings / segments, leading to significant deviations between the design results and the actual load-bearing state; or refined solid element modeling is used for the entire bridge, which improves the accuracy of local stress calculation but consumes massive computational resources, has a lengthy modeling and solution cycle, and is susceptible to distortions in boundary condition assumptions, making it difficult to apply to parameter optimization and scheme iteration in the engineering design stage. Existing load-bearing designs mostly focus on overall load-bearing capacity verification, relying on traditional theories such as the equivalent beam-column method and limit analysis, failing to fully reveal the stress transmission path inside the main arch ring after arch formation and the collaborative load-bearing ratio of each component. For the composite stress system formed by the rigid steel frame, the inner concrete and the outer concrete, the longitudinal, lateral and vertical force transmission mechanism has not been accurately quantified. This makes it impossible to optimize the component size, material matching and interface structure in a targeted manner during the design process. Either the overly conservative design will lead to a waste of resources, or the neglect of local weak stress links will leave safety hazards.

[0004] Therefore, it is urgent to propose a design method for the load-bearing capacity of the main arch ring after its completion, which combines a multi-scale finite element model, quantifies the collaborative load-bearing ratio of each component, and fully considers the impact of the construction process. This is a problem that needs to be solved by those skilled in the art to provide technical support for the safe and reliable construction of ultra-large span stiffness skeleton arch bridges. Summary of the Invention

[0005] In view of this, the present invention provides a design method for the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model, which quantifies the collaborative load-bearing ratio of each component and fully considers the impact of the construction process, providing technical support for the safe and reliable construction of ultra-large span stiffness skeleton arch bridges.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for designing the load-bearing capacity of the main arch ring after arch formation, combining a multi-scale finite element model, includes: S1, construct the multi-scale finite element model of the main arch ring and output the multi-scale finite element model file; S2. Based on the multi-scale finite element model file, coupled with the segmented construction process of the main arch ring, the birth and death element method is used to simulate the entire construction process, and output the stress accumulation data of the construction process and the permanent stress distribution cloud map after the arch is completed. S3, using the stress accumulation data of the construction process and the permanent stress distribution cloud map after the arch is formed as joint inputs, quantitatively analyzes the force transmission mechanism after the arch is formed based on the force flow theory, and outputs stress transmission path data, component collaborative bearing ratio value and interface force transmission reliability evaluation results. S4 combines the output of S3 with the stress accumulation data during the construction process, and optimizes the cross-sectional dimensions, material matching parameters and interface structure of the main arch ring with the goals of stress uniformity, load-bearing ratio matching and structural stability. Through multi-scale model iterative verification, it outputs a load-bearing design scheme that meets the design specifications after the arch is completed.

[0007] Preferably, the multi-scale finite element model of the main arch ring is divided into a local fine region and a global simplified region. The local fine region selects the key stress section of the arch crown, and uses solid elements to simulate the concrete inside the pipe and the outer concrete, and shell elements to simulate the main steel pipe. The contact areas of each component are connected by common nodes. The global simplified region uses beam elements to simulate the main steel pipe, the concrete inside the pipe and the web members, and plate elements to simulate the outer concrete. The mechanical parameters of the two regions are coupled through the rigid region.

[0008] Preferably, the length of the local fine-grained region is determined based on the principle of full coverage of the key stress range of the arch. Where L is the total length of the local fine region. The length of the arch is precisely simulated on one side; the thickness of the plate unit is designed based on the gradual change characteristics of the actual cross-section of the main arch ring. ,in, Longitudinal coordinates along the arch axis The thickness of the plate unit at that location, The thickness of the reference section. k The thickness gradient coefficient is... x The coordinates are the longitudinal coordinates along the arch axis.

[0009] Preferably, in the quantitative analysis of the force transmission mechanism after arch formation based on force flow theory, the stress accumulation data during the construction process is used to quantify the stress superposition effect of each component during the construction stage, and the total stress tensor after arch formation is corrected. The correction formula is as follows: ; Based on the Gurtin tensor decomposition theory, the corrected total stress is decomposed into a fully self-balancing stress component and an irrotational stress component. ; The above-mentioned fully self-equilibrium stress components are expressed using the Airy stress function, and the formula is as follows: ; ; in, This is the corrected total stress tensor after arch formation. To preserve the stress tensor after the arch is formed, Let be the stress increment at the k-th construction stage, n be the total number of construction stages, and i and j be the coordinate direction indices. For completely self-balancing stress components, For irrotational stress components, , These are Levi-Civita sequence symbols used to describe the arrangement of coordinate directions. For Airy stress function; and for The partial derivatives, This is the Kronecker delta function, which takes the value 1 when i=j and 0 when i≠j. The subscript "," indicates the partial derivative with respect to the corresponding coordinate. It is a potential vector; The load function is defined by the partial derivatives of the Airy stress function with respect to the x and y coordinates. The load function is used to characterize the distribution of the corrected stress in the plane, and the formula is as follows: ; in, , Let x and y be the load functions, respectively, in the x and y directions. Let x and y be the Airy stress function, and x and y be the coordinate axes of the plane coordinate system. The formula for the stress transfer path gradient is: ; in, For gradient operators, , respectively the load function , The partial derivatives along the x and y directions; by calculating the gradient of the load function, the direction vector of the stress transfer path is obtained, thus clarifying the direction and distribution law of stress transfer within the structure after correction, i.e., stress transfer path data.

[0010] Preferably, the component cooperative load-bearing ratio specifically includes: Axial force bearing ratio formula: ; in, The axial force bearing ratio of the stiffened steel frame. This represents the axial force bearing capacity ratio of the concrete inside the pipe. The axial force bearing ratio of the outer concrete. , , These represent the axial forces borne by each component after the arch is formed. These represent the cumulative axial force increments of each component during the construction phase, in N. 总 The corrected total axial force at the cross section; Bending moment bearing ratio formula: ; in, The bending moment bearing ratio of the stiffened steel frame. This represents the bending moment bearing capacity ratio of the concrete inside the pipe. The bending moment bearing ratio of the outer concrete. , , These represent the bending moments borne by each component after the arch is formed. These represent the cumulative bending moment increments of each component during the construction phase, M. 总 The corrected total bending moment of the section; Formula for the proportion of principal stress channels: ; in, η The proportion of stress in the main stress transmission channel. The corrected stress value is the main stress transmission channel composed of the main steel pipe, the concrete inside the pipe, and the adjacent outer concrete. This is the corrected total stress value of the cross section.

[0011] Preferably, in step S3, when volume forces are present, the stress balance equation is corrected by combining the potential function with the stress increment corresponding to the volume forces in the accumulated stress data of the construction process. The relationship between volume forces and the potential function is as follows: Where b is the volume force vector and V is the volume force potential function. The gradient operator is used; the modified stress balance equation is: ; ; in, , , , These are the corrected anisotropic stress components. , ^V、 The cumulative stress increment generated by volume forces during the construction phase is represented by the potential function and the cumulative stress increment of volume forces during the construction phase. This eliminates the influence of volume forces and the construction process on the equilibrium equations.

[0012] Preferably, in step S4, the post-arch load-bearing design scheme is evaluated based on stress uniformity, load-bearing safety, and structural stiffness, specifically including: ; ; ; in, This refers to the stress non-uniformity coefficient of the outer concrete. This represents the maximum compressive stress in the outer concrete. The minimum compressive stress of the outer concrete. The average compressive stress of the outer concrete. This represents the maximum reduction in tensile stress in the outer concrete. To determine the maximum tensile stress before optimization, To optimize the maximum tensile stress, For the vertical deformation of the arch, The bending moment is distributed along the arch axis. The equivalent elastic modulus of the material. The moment of inertia for the cross section is the integral interval of the entire span of the main arch ring.

[0013] As can be seen from the above technical solution, compared with the prior art, this invention discloses a method for designing the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model. By constructing a multi-scale finite element model of the main arch ring that combines local refinement and global simplification, it takes into account both the accuracy of stress calculation and the efficiency of modeling and analysis. At the same time, it adopts the birth and death element method to couple the construction process to simulate the entire process, accurately capturing the stress accumulation effect during construction. Based on the force flow theory and combined with construction stress data to correct the total stress tensor, it realizes the accurate quantification of the force transmission path and the proportion of collaborative load-bearing of components after arch formation. Furthermore, it corrects the stress balance equation under the action of volume force through the potential function, further improving the accuracy of mechanical analysis. Finally, it optimizes the cross-section, material and interface structure of the main arch ring with the goals of stress uniformity, load-bearing ratio matching and structural stability, and determines the design scheme through multi-scale model iterative verification. It effectively solves the problems of deviation between traditional design and actual load-bearing state and insufficient quantification of force transmission mechanism, avoids overly conservative design or safety hazards, and greatly improves the scientificity, accuracy and engineering practicality of the load-bearing capacity design of the main arch ring of ultra-large span stiffness frame concrete arch bridge after arch formation, providing important technical support for its safe and reliable construction. Attached Figure Description

[0014] 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.

[0015] Figure 1 A flowchart of the method steps provided by the present invention; Detailed Implementation

[0016] 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.

[0017] This invention discloses a method for designing the load-bearing capacity of the main arch ring after arch formation, combining a multi-scale finite element model, such as... Figure 1 As shown, it includes: S1, construct the multi-scale finite element model of the main arch ring and output the multi-scale finite element model file; S2, based on a multi-scale finite element model file, couples the main arch ring segmented construction process, and uses the birth and death element method to simulate the entire construction process, outputting the stress accumulation data of the construction process and the permanent stress distribution cloud map after the arch is completed; the entire construction process includes the closure of the stiffened steel frame, the pouring of concrete inside the pipe, and the segmented pouring of the outer concrete (bottom plate, web plate, top plate). S3 uses the cumulative stress data during construction and the permanent stress distribution cloud map after arch formation as joint inputs. Based on the force flow theory, it quantitatively analyzes the force transmission mechanism after arch formation and outputs stress transmission path data, component collaborative bearing ratio value and interface force transmission reliability evaluation results. The quantitative analysis includes stress tensor decomposition, load function solution and gradient calculation. S4 combines the output of S3 with the stress accumulation data during the construction process, and optimizes the cross-sectional dimensions, material matching parameters and interface structure of the main arch ring with the goals of stress uniformity, load-bearing ratio matching and structural stability. Through multi-scale model iterative verification, it outputs a load-bearing design scheme that meets the design specifications after the arch is completed.

[0018] In one specific embodiment, the multi-scale finite element model of the main arch ring is divided into a locally refined region and a globally simplified region. The locally refined region selects the key stress-bearing section of the arch crown, using solid elements to simulate the inner and outer concrete of the pipe, and shell elements to simulate the main steel pipe. Contact areas of all components are connected using common nodes. The globally simplified region uses beam elements to simulate the main steel pipe, the inner concrete, and the web members, and plate elements to simulate the outer concrete. The mechanical parameters of the two regions are coupled through a rigid region. The transfer of mechanical parameters in the rigid region is based on the principle of force balance, ensuring stress continuity at the junction of the two regions. ,in, For the stress at the end nodes of a local fine region, This refers to the effective force transmission cross-sectional area at the end of a localized, fine-grained region. To simplify the stress at the corresponding connection nodes in the global region, This is the effective force transmission cross-sectional area at the connection point corresponding to the global simplified region. This formula ensures that the mechanical parameters are transmitted without interruption between the local refined region and the global simplified region by making the product of stress and effective force transmission area equal, thus avoiding sudden stress changes at the connection point.

[0019] In one specific embodiment, the length of the local fine-grained region is determined based on the principle of full coverage of the key stress range of the arch. Where L is the total length of the local fine region. To accurately simulate the length of one side of the arch, the length of one side is doubled to ensure coverage of the core load-bearing section of the arch; the thickness of the plate elements is designed based on the gradual variation characteristics of the actual cross-section of the main arch ring. ,in, Longitudinal coordinates along the arch axis The thickness of the plate unit at that location, The thickness of the reference section. k The thickness gradient coefficient is... x Using the longitudinal coordinate along the arch axis, this formula realizes that the thickness of the plate element changes linearly with the position of the arch axis, matching the variation law of the cross-sectional dimensions of the main arch ring.

[0020] In one specific embodiment, based on the force flow theory, the force transmission mechanism after arch formation is quantitatively analyzed. Stress accumulation data from the construction process is used to quantify the stress superposition effect of each component during the construction stage, and the total stress tensor after arch formation is corrected. The correction formula is as follows: ; Based on the Gurtin tensor decomposition theory, the corrected total stress is decomposed into a fully self-balancing stress component and an irrotational stress component. ; The above-mentioned fully self-equilibrium stress components are expressed using the Airy stress function, and the formula is as follows: ; ; in, This is the corrected total stress tensor after arch formation. To preserve the stress tensor after the arch is formed, Let represent the stress increment at the k-th construction stage (taken from cumulative stress data throughout the construction process), n be the total number of construction stages, and i and j be the coordinate direction indices. For completely self-balancing stress components, For irrotational stress components, , These are Levi-Civita sequence symbols used to describe the arrangement of coordinate directions. For Airy stress function; and for The partial derivatives, This is the Kronecker delta function, which takes the value 1 when i=j and 0 when i≠j. The subscript "," indicates the partial derivative with respect to the corresponding coordinate. It is a potential vector; The load function is solved using the weighted residual method from the reference materials. The core idea is to transform the partial differential equation of the load function into an integral equation, avoiding direct handling of the discontinuity of the stress derivative. The formula is as follows: ; ; Where D is the computational domain of the load function, and Φ is the weighting function used to balance the numerical errors at various points within the computational domain. , , These are the fully self-equilibrium stress components after stress tensor correction. This is a planar computational infinitesimal element; using this set of integral formulas, the load function can be stably solved based on the modified total stress tensor.

[0021] The load function is defined by the partial derivatives of the Airy stress function with respect to the x and y coordinates. The load function is used to characterize the distribution of the corrected stress in the plane, and the formula is as follows: ; in, , Let x and y be the load functions, respectively, in the x and y directions. Let x and y be the Airy stress function, and x and y be the coordinate axes of the plane coordinate system. The formula for the stress transfer path gradient is: ; in, For gradient operators, , respectively the load function , The partial derivatives along the x and y directions; by calculating the gradient of the load function, the direction vector of the stress transfer path is obtained, thus clarifying the direction and distribution law of stress transfer within the structure after correction, i.e., stress transfer path data.

[0022] In one specific embodiment, the component cooperative load-bearing ratio specifically includes: Axial force bearing ratio formula: ; in, The axial force bearing ratio of the stiffened steel frame. This represents the axial force bearing capacity ratio of the concrete inside the pipe. The axial force bearing ratio of the outer concrete. , , These represent the axial forces borne by each component after the arch is formed. These represent the cumulative axial force increments of each component during the construction phase (taken from cumulative stress data throughout the construction process), in N. 总 The corrected total axial force at the cross section; Bending moment bearing ratio formula: ; in, The bending moment bearing ratio of the stiffened steel frame. This represents the bending moment bearing capacity ratio of the concrete inside the pipe. The bending moment bearing ratio of the outer concrete. , , These represent the bending moments borne by each component after the arch is formed. These represent the cumulative bending moment increments of each component during the construction phase (taken from cumulative stress data throughout the construction process), M 总 The corrected total bending moment of the section; Formula for the proportion of principal stress channels: ; in, η The proportion of stress in the main stress transmission channel. The corrected stress value is the main stress transmission channel composed of the main steel pipe, the concrete inside the pipe, and the adjacent outer concrete. This is the corrected total stress value of the cross section.

[0023] All stresses, construction-accumulated stresses, and interface stresses calculated from the multi-scale model must satisfy the stress balance equation; otherwise, the calculation results are invalid and lack engineering design basis. Gurtin stress tensor decomposition, load functions, and stress transfer paths must all satisfy the stress balance equation to correctly identify the actual force transmission direction and channels within the main arch ring. Construction-process stress accumulation data considers volume forces, and the balance equation is used to integrate these superimposed construction stresses and volume forces into mechanical equilibrium, ensuring the total stress state after arch completion. The calculation of axial force, bending moment, and the bearing ratio of principal stress channels all rely on stress balance. The sum of the forces borne by each component must equal the total external force, thus ensuring that the force distribution of the steel frame, the concrete inside the pipe, and the outer concrete is realistic and usable for design optimization. In S3, when volume forces (structural self-weight, internal stress generated by concrete hydration heat, etc.) are present, the stress balance equation is corrected using a potential function combined with the stress increment corresponding to the volume forces in the construction-process stress accumulation data. The relationship between volume forces and the potential function is: Where b is the volume force vector and V is the volume force potential function. The gradient operator is used; the modified stress balance equation is: ; ; in, , , , These are the corrected anisotropic stress components. , ^V、 The cumulative stress increment generated by volume forces during the construction phase (taken from the stress accumulation data of the construction process) is used to eliminate the influence of volume forces and the construction process on the equilibrium equation by introducing the potential function and the cumulative stress increment of volume forces during the construction phase into the stress term.

[0024] In one specific embodiment, S4 evaluates the load-bearing design scheme after arch formation based on stress uniformity, load-bearing safety, and structural stiffness, specifically including: ; ; ; in, This refers to the stress non-uniformity coefficient of the outer concrete. This represents the maximum compressive stress in the outer concrete. The minimum compressive stress of the outer concrete. The average compressive stress of the outer concrete. This represents the maximum reduction in tensile stress in the outer concrete. To determine the maximum tensile stress before optimization, To optimize the maximum tensile stress, For the vertical deformation of the arch, The bending moment is distributed along the arch axis. The equivalent elastic modulus of the material. The moment of inertia for the cross section is the integral interval of the entire span of the main arch ring.

[0025] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. The methods disclosed in the embodiments are described simply because they correspond to the methods disclosed in the embodiments; relevant parts can be found in the method section.

[0026] 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 method for designing the load-bearing capacity of the main arch ring after arch formation, combining a multi-scale finite element model, characterized in that, include: S1, construct the multi-scale finite element model of the main arch ring and output the multi-scale finite element model file; S2. Based on the multi-scale finite element model file, coupled with the segmented construction process of the main arch ring, the birth and death element method is used to simulate the entire construction process, and output the stress accumulation data of the construction process and the permanent stress distribution cloud map after the arch is completed. S3, using the stress accumulation data of the construction process and the permanent stress distribution cloud map after the arch is formed as joint inputs, quantitatively analyzes the force transmission mechanism after the arch is formed based on the force flow theory, and outputs stress transmission path data, component collaborative bearing ratio value and interface force transmission reliability evaluation results. S4 combines the output of S3 with the stress accumulation data during the construction process, and optimizes the cross-sectional dimensions, material matching parameters and interface structure of the main arch ring with the goals of stress uniformity, load-bearing ratio matching and structural stability. Through multi-scale model iterative verification, it outputs a load-bearing design scheme that meets the design specifications after the arch is completed.

2. The method for designing the load-bearing capacity of the main arch ring after arch formation, combining a multi-scale finite element model as described in claim 1, is characterized in that... The multi-scale finite element model of the main arch ring is divided into a local fine region and a global simplified region. The local fine region selects the key stress section of the arch crown, and uses solid elements to simulate the concrete inside the pipe and the outer concrete, and shell elements to simulate the main steel pipe. The contact areas of each component are connected by common nodes. In the global simplified region, beam elements are used to simulate the main steel pipe, the concrete inside the pipe and the web members, and plate elements are used to simulate the outer concrete. The mechanical parameters of the two regions are coupled through the rigid region.

3. The method for designing the load-bearing capacity of the main arch ring after arch formation, combining a multi-scale finite element model as described in claim 2, is characterized in that... The length of the local fine-grained region is determined based on the principle of full coverage of the key stress range of the arch. Where L is the total length of the local fine region. The length of the arch is precisely simulated on one side; the thickness of the plate unit is designed based on the gradual change characteristics of the actual cross-section of the main arch ring. ,in, Longitudinal coordinates along the arch axis The thickness of the plate unit at that location, The thickness of the reference section. k The thickness gradient coefficient is... x The coordinates are the longitudinal coordinates along the arch axis.

4. The method for designing the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model according to claim 1, characterized in that, In the quantitative analysis of the force transmission mechanism after arch formation based on force flow theory, the stress accumulation data during the construction process is used to quantify the stress superposition effect of each component during the construction stage, and correct the total stress tensor after arch formation. The correction formula is as follows: ； Based on the Gurtin tensor decomposition theory, the corrected total stress is decomposed into a fully self-balancing stress component and an irrotational stress component. ； The above-mentioned fully self-equilibrium stress components are expressed using the Airy stress function, and the formula is as follows: ； ； in, This is the corrected total stress tensor after arch formation. To preserve the stress tensor permanently after the arch is formed, Let be the stress increment at the k-th construction stage, n be the total number of construction stages, and i and j be the coordinate direction indices. For completely self-balancing stress components, For irrotational stress components, , These are Levi-Civita sequence symbols used to describe the arrangement of coordinate directions. For Airy stress function; and for The partial derivatives, This is the Kronecker delta function, taking the value 1 when i=j and 0 when i≠j. The subscript "," indicates the partial derivative with respect to the corresponding coordinate. It is a potential vector; The load function is defined by the partial derivatives of the Airy stress function with respect to the x and y coordinates. The load function is used to characterize the distribution of the corrected stress in the plane, and the formula is as follows: ； in, , Let x and y be the load functions, respectively, in the x and y directions. Let x and y be the Airy stress function, and x and y be the coordinate axes of the plane coordinate system. The formula for the stress transfer path gradient is: ; in, For gradient operators, , respectively the load function , The partial derivatives along the x and y directions; by calculating the gradient of the load function, the direction vector of the stress transfer path is obtained, thus clarifying the direction and distribution law of stress transfer within the structure after correction, i.e., stress transfer path data.

5. The method for designing the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model according to claim 1, characterized in that, The specific components' collaborative load-bearing ratio includes: Axial force bearing ratio formula: ; in, The axial force bearing ratio of the stiffened steel frame. This represents the axial force bearing capacity ratio of the concrete inside the pipe. The axial force bearing ratio of the outer concrete. , , These represent the axial forces borne by each component after the arch is formed. These represent the cumulative axial force increments of each component during the construction phase, in N. 总 The total axial force of the corrected cross section; Bending moment bearing ratio formula: ; in, The bending moment bearing ratio of the stiffened steel frame. This represents the bending moment bearing capacity ratio of the concrete inside the pipe. The bending moment bearing ratio of the outer concrete. , , These represent the bending moments borne by each component after the arch is formed. These represent the cumulative bending moment increments of each component during the construction phase, M. 总 The corrected total bending moment of the section; Formula for the proportion of principal stress channels: ; in, η The proportion of stress in the main stress transmission channel. The corrected stress value is the main stress transmission channel composed of the main steel pipe, the concrete inside the pipe, and the adjacent outer concrete. This is the corrected total stress value of the cross section.

6. The method for designing the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model according to claim 1, characterized in that, In S3, when volume forces are present, the stress balance equation is corrected by combining the potential function with the stress increment corresponding to the volume forces in the cumulative stress data of the construction process. The relationship between volume forces and the potential function is as follows: Where b is the volume force vector and V is the volume force potential function. The gradient operator is used; the modified stress balance equation is: ； ； in, , , , These are the corrected anisotropic stress components. , ^V、 The cumulative stress increment generated by volume forces during the construction phase is represented by the potential function and the cumulative stress increment of volume forces during the construction phase. This eliminates the influence of volume forces and the construction process on the equilibrium equations.

7. The method for designing the load-bearing capacity of the main arch ring after arch formation by combining a multi-scale finite element model according to claim 1, characterized in that, In step S4, the post-arch load-bearing design scheme is evaluated based on stress uniformity, load-bearing safety, and structural stiffness, specifically including: ； ； ； in, This refers to the stress non-uniformity coefficient of the outer concrete. This represents the maximum compressive stress in the outer concrete. The minimum compressive stress of the outer concrete. The average compressive stress of the outer concrete. This represents the maximum tensile stress reduction of the outer concrete. The maximum tensile stress before optimization, To optimize the maximum tensile stress, This is due to the vertical deformation of the arch. The bending moment is distributed along the arch axis. The equivalent elastic modulus of the material. The moment of inertia for the cross section is the integral interval of the entire span of the main arch ring.

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