A Multi-Objective Optimization Method for Finite Element Model of a Small-Radius Curved Cable-Stayed Bridge

Abstract

This invention discloses a multi-objective optimization method for a finite element model of a small-radius curved cable-stayed bridge, belonging to the field of bridge engineering technology. The method includes the following steps: Step 1: Determine the structural optimization variables for a small-radius multi-tower wide-beam curved cable-stayed bridge. These variables include the bridge's curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, main girder torsional stiffness, crossbeam stiffness, and initial cable tension. Step 2: Combining the inherent bending-torsional coupling force characteristics and spatial force characteristics of small-radius curved cable-stayed bridges, establish multi-objective optimization constraints and a structural performance evaluation system based on bridge structural design specifications. Considering the significant bending-torsional coupling and complex spatial force characteristics of small-radius multi-tower wide-beam curved cable-stayed bridges, seven core parameters—curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, main girder torsional stiffness, crossbeam stiffness, and initial cable tension—are used as optimization variables. Insensitive redundant parameters are discarded, achieving precise matching of optimization dimensions.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, and more specifically, to a multi-objective optimization method for a finite element model of a small-radius curved cable-stayed bridge. Background Technology

[0002] Due to the curvature of the planar alignment, small-radius curved cable-stayed bridges inevitably generate additional torque on the main girder while bearing vertical loads, resulting in a bending-torsional coupled stress state. When such bridges employ multiple towers and wide main girders, the spatial stress behavior becomes even more complex, with problems such as torsional deformation of the main girder, lateral displacement of the towers, and uneven distribution of cable forces becoming particularly prominent.

[0003] The prior art, disclosed in CN117364632A, presents a construction method for a cable-stayed bridge pylon. This method includes: erecting the lower formwork of the pylon on the pier abutment and pouring concrete to the level of the beam's bottom slab; using the lower formwork as support, installing the middle formwork of the pylon and pouring concrete; dismantling the lower formwork to form multiple layers of column formwork; installing multiple layers of column formwork on the top surface of the already poured middle section of the pylon and pouring concrete; and using a formwork-changing construction method based on the multiple layers of column formwork, performing the construction cyclically. After completing the lower part of the pylon using the lower formwork, dismantling it and dismantling it into multiple layers of column formwork allows for subsequent reuse of the formwork, avoiding the need for custom-made formwork, reducing construction costs, and improving construction efficiency.

[0004] The existing technology, with publication number CN120493354A, discloses a rapid modeling method for bridge truss elements using finite element analysis. Combining truss structure modeling with BIM modeling logic, it constructs a clearer and more reasonable modeling method. Users can effectively integrate with other software at a lower cost by modifying simple underlying data or utilizing the results already formed in the method. This method uses Python and includes: 1. Using Python to edit class attributes; 2. Creating material and section instances; 3. Dividing the bridge structure into different component instances according to the bridge type, binding boundary information between different components through boundary classes, and determining the loads acting on different components through load classes; 4. Encapsulating axis classes and binding component instances to axis instances; 5. Adding a calculation module for mesh generation; 6. Obtaining each meshing point on the axis and connecting these points to generate the corresponding element set; 7. Establishing boundary and load instances.

[0005] While the aforementioned methods achieve rapid parametric modeling of truss elements, they primarily focus on the general modeling process for conventional bridges, failing to specifically optimize for the unique bending-torsional coupling stress characteristics and spatial stress features of small-radius curved cable-stayed bridges. This method lacks a systematic identification and screening mechanism for the core sensitive parameters of curved cable-stayed bridges, and it also lacks an intelligent iterative optimization framework linked to the evaluation system of bending-torsional coupling effects. Therefore, directly applying this modeling method to the structural optimization of small-radius multi-tower wide-beam curved cable-stayed bridges still faces the following problems: firstly, redundant modeling variables lead to an explosion in optimization dimensions, making it difficult to converge to the global optimum; secondly, the lack of an evaluation orientation oriented towards bending-torsional coupling control means the optimization process cannot specifically address the weakest points of torsional deformation and uneven spatial stress in curved cable-stayed bridges. Summary of the Invention

[0006] (a) Technical problems to be solved To address the shortcomings of existing technologies, this invention provides a multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge, thus solving the aforementioned problems.

[0007] (II) Technical Solution To achieve the above-mentioned objectives, the present invention provides the following technical solution: a multi-objective optimization method for a finite element model of a small-radius curved cable-stayed bridge, comprising the following steps: Step 1: Determine the structural optimization variables for a small-radius, multi-tower, wide-beam curved cable-stayed bridge. The optimization variables include the bridge curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, torsional stiffness of the main beam, stiffness of the crossbeam, and initial tension of the cables. Step 2: Combining the inherent bending-torsional coupling stress characteristics and spatial stress characteristics of small-radius curved cable-stayed bridges, establish multi-objective optimization constraints and structural performance evaluation system based on bridge structural design specifications; Step 3: Use an intelligent iterative optimization algorithm to generate multiple combinations of structural optimization variables, and use the finite element rapid modeling interface to realize the automatic updating of the overall finite element model of the bridge and mechanical numerical calculation. Step 4: Collect the core mechanical indicators of structural displacement, internal force, and torsional deformation after finite element calculation, and complete the quantitative evaluation and convergence determination of each variable combination based on the evaluation system; Step 5: If the convergence condition is not met, automatically iterate and update the combination of optimized variables, repeat the modeling calculation and performance evaluation process until the optimization convergence criterion is met, and output the optimal combination of structural parameters.

[0008] Preferably, the selection of optimization variables is adapted to the structural stress characteristics of small-radius multi-tower wide-beam curved cable-stayed bridges. Considering the significant bending-torsional coupling effect, complex spatial stress, and large differences in the sensitivity of structural parameters, the core structural parameters that control the overall stress state and deformation characteristics of the bridge are selected, while non-sensitive redundant structural parameters are discarded, so as to achieve accurate matching and simplified layout of optimization variables.

[0009] Preferably, the curve radius and the width-to-span ratio are the core variables for controlling the bending-torsional coupling stress of a small-radius curved cable-stayed bridge. The curve radius determines the overall curve shape and curvature characteristics of the bridge, while the width-to-span ratio reflects the spatial scale matching characteristics of the wide beam main girder. Together, they affect the torsional stress distribution of the main girder and the overall spatial deformation law, and are key parameters for regulating the bending-torsional coupling response of the bridge.

[0010] Preferably, the height-to-span ratio and tower-to-span ratio are sensitive variables that adapt to the spatial force system of a multi-tower curved cable-stayed bridge. The height-to-span ratio regulates the vertical stiffness and vertical force distribution of the bridge, while the tower-to-span ratio determines the load transfer path and overall stiffness distribution of the multi-tower structure, directly affecting the collaborative force-bearing performance of the multi-tower structure and the overall stability of the bridge.

[0011] Preferably, the torsional stiffness of the main beam and the stiffness of the crossbeam are the core variables for controlling the local stress of the wide beam main beam. The torsional stiffness of the main beam constrains the torsional deformation development trend of the wide beam structure, while the stiffness of the crossbeam ensures the lateral integrity of the main beam and the lateral load transfer capacity, effectively improving the problem of local stress concentration and uneven deformation of the main beam of a small-radius curved bridge.

[0012] Preferably, the initial tension of the cable is a key control variable for the spatial stress of the cable-stayed bridge with a balanced curve. By adjusting the initial tension of the cable, the load-bearing ratio of the cable can be optimized, the internal force distribution of the main beam and the bridge tower can be improved, and the eccentric force and additional torsional effect caused by the small radius curve can be offset.

[0013] Preferably, the multi-objective optimization constraints are constructed based on the strength limits, stiffness limits, and stability limits of the bridge structure design code, while coupling the bending-torsional coupling force constraints and spatial deformation constraints specific to small-radius curved cable-stayed bridges, taking into account both the overall structural stress safety and the rationality of the stress on local components.

[0014] Preferably, the structural performance evaluation system takes the overall stress uniformity, torsional deformation controllability, displacement stability, and multi-tower coordinated stress balance of the bridge as the core evaluation dimensions. In view of the spatial stress characteristics of small-radius multi-tower wide-beam curved cable-stayed bridges, a multi-dimensional quantitative evaluation standard is established to achieve accurate comparison and screening of the performance of different parameter combinations.

[0015] Preferably, the intelligent iterative optimization algorithm sets up an iterative update mechanism based on the multi-parameter coupling and correlation characteristics. According to the performance evaluation results of each group of optimization variables, it directionally corrects the iterative adjustment direction of the optimization variables, avoids the randomness defects of manual parameter tuning, and gradually approaches the global optimal parameter combination.

[0016] Preferably, the finite element rapid modeling interface is adapted to the spatial modeling logic of small-radius multi-tower wide-beam curved cable-stayed bridges, which can realize integrated automatic modeling and synchronous parameter updates of curved lines, multi-tower layout, wide-beam structure and cable-stayed system, and construct a closed-loop automated optimization process for parameter adjustment, modeling calculation and evaluation.

[0017] (III) Beneficial Effects Compared with existing technologies, this invention provides a multi-objective optimization method for finite element models of cable-stayed bridges with small radius curves, which has the following beneficial effects: 1. This paper presents a multi-objective optimization method for finite element models of small-radius curved cable-stayed bridges. Addressing the significant bending-torsional coupling and complex spatial stress characteristics of small-radius, multi-tower, wide-beam curved cable-stayed bridges, this method uses seven core parameters—curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, main girder torsional stiffness, crossbeam stiffness, and initial cable tension—as optimization variables. It eliminates insensitive redundant parameters, achieving precise matching of optimization dimensions. Through the linkage of intelligent iterative algorithms and a rapid modeling interface, a closed-loop process is formed, encompassing parameter adjustment, automatic modeling, mechanical calculations, and result evaluation, completely replacing the traditional, inefficient manual parameter tuning method. In cases of strong multi-parameter coupling, this method can automatically search for the globally optimal parameter combination, effectively avoiding the local optimum dilemma caused by human experience, and significantly improving optimization efficiency and structural design quality.

[0018] 2. This paper presents a multi-objective optimization method for finite element models of small-radius curved cable-stayed bridges. This method assigns higher weights to the torque and torsional deformation angle evaluation terms in the evaluation system and adds a bending-torsional coupling penalty term, directly targeting the weakest bending-torsional stress links of the small-radius curved cable-stayed bridge. Simultaneously, by independently extracting the torsional deformation angle of the main girder segment corresponding to each bridge tower and expanding the crossbeam stiffness into multiple independent parameters distributed along the bridge length, the spatial stress balance of the multi-tower structure can be finely controlled. In the post-processing step, the optimal parameter combination is checked under additional working conditions. If necessary, additional indicators are incorporated into a new round of optimization to ensure that the final parameter combination still meets the acceptance criteria for bending-torsional stiffness efficiency and tower efficiency under complex working conditions such as moving loads, temperature, wind, and earthquakes. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of the method flow of the present invention. Detailed Implementation

[0020] 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 of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0021] Please see Figure 1 The present invention provides a technical solution: A multi-objective optimization method for a small-radius curved cable-stayed bridge is proposed, which optimizes the structural parameters of a small-radius, multi-tower, wide-beam curved cable-stayed bridge. The basic structural parameters of the bridge are as follows: the main bridge consists of multiple spans, multiple bridge towers are provided, the bridge plane is located on a small-radius circular curve, the main girder is a thin-walled steel box girder, the bridge towers are reinforced concrete towers, and the cables are arranged in a fan-shaped cable surface.

[0022] Step 1: Determine the structural optimization variables.

[0023] Based on the structural stress characteristics of a small-radius, multi-tower, wide-beam curved cable-stayed bridge, this bridge exhibits significant bending-torsional coupling effects, complex spatial stress, and large differences in the sensitivity of structural parameters. Therefore, core structural parameters controlling the overall stress state and deformation characteristics of the bridge were selected as optimization variables, while non-sensitive redundant parameters were discarded to achieve precise matching and simplified layout of the optimization variables. Specifically, seven optimization variables were determined: bridge curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, main girder torsional stiffness, crossbeam stiffness, and initial cable tension.

[0024] The height-to-span ratio is defined as the ratio of the main girder height to the main span diameter; the tower-to-span ratio is defined as the ratio of the tower height to the main span diameter; and the width-to-span ratio is defined as the ratio of the main girder width to the main span diameter. The curve radius determines the overall curvature characteristics of the bridge's curve, while the width-to-span ratio reflects the spatial scale matching characteristics of the wide-beam main girder. Both factors jointly influence the torsional stress distribution and overall spatial deformation of the main girder, making them key parameters for controlling the bridge's bending-torsional coupling response. The height-to-span ratio and tower-to-span ratio are sensitive variables for adapting the spatial force system of multi-tower curved cable-stayed bridges. The height-to-span ratio regulates the bridge's vertical stiffness and vertical force distribution, while the tower-to-span ratio determines the load transfer path and overall stiffness distribution of the multi-tower structure, directly affecting the collaborative force-bearing performance of the multi-tower structure and the overall stability of the bridge. The torsional stiffness of the main girder and the stiffness of the crossbeams are core variables for controlling the local stress of the wide-beam main girder. The torsional stiffness of the main girder constrains the torsional deformation development trend of the wide-beam structure, while the crossbeam stiffness ensures the lateral integrity and lateral load transfer capacity of the main girder, effectively improving the problems of local stress concentration and uneven deformation in the main girder of small-radius curved bridges. The initial tension of the cables is a key control variable for the spatial stress of a cable-stayed bridge with a equilibrium curve. By adjusting the initial tension of the cables, the load-bearing ratio of the cables can be optimized, the internal force distribution of the main girder and the bridge tower can be improved, and the eccentric force and additional torsional effect caused by the small radius curve can be offset.

[0025] Step 2: Establish multi-objective optimization constraints and structural performance evaluation system by combining stress characteristics and specifications.

[0026] Based on the limits for strength, stiffness, and stability specified in the bridge structural design code, and coupled with the bending-torsional coupling force constraints and spatial deformation constraints specific to small-radius curved cable-stayed bridges, multi-objective optimization constraints are constructed. These constraints include, but are not limited to: For strength, the maximum normal stress, maximum shear stress, maximum compressive stress of the main girder, and maximum cable force of the towers are all no greater than their respective code limits; for stiffness, the vertical deflection at mid-span of the main girder, the lateral deflection of the main girder, and the horizontal displacement at the top of the towers are all no greater than the corresponding limits specified in the code; for torsion, the maximum torsional deformation angle of the main girder is no greater than the code limit for torsion angle, and the maximum torque of the main girder is no greater than the code's torsional bearing capacity; and for bending-torsional coupling, the bending-torsional coupling coefficient is no greater than a preset threshold.

[0027] The structural performance evaluation system focuses on the overall uniformity of stress on the bridge, the controllability of torsional deformation, displacement stability, and the balanced stress distribution among multiple towers. Specifically, it establishes an evaluation system with seven evaluation items: displacement, bending moment, torque, torsional deformation angle, cable force uniformity, beam strength, and tower strength. Each evaluation item is assigned a preset weight, with the sum of the weights of the torque and torsional deformation angle items exceeding the weights of the other individual evaluation items to emphasize the control of bending-torsional coupling.

[0028] Each scoring item uses a percentage system or an equivalent standardized scoring method, scoring based on the degree of deviation of each mechanical indicator from the specification limit. The specific scoring rules are as follows: a full score is awarded when the mechanical indicator is less than a first preset percentage of the specification limit; a higher score is awarded when it is greater than the first preset percentage but less than or equal to a second preset percentage; a passing score is awarded when it is greater than the second preset percentage but less than or equal to the specification limit; and a score of zero is awarded when it exceeds the specification limit. The overall score is a weighted sum of all scoring items, with a maximum score of one hundred points or a normalized value.

[0029] Step 3: Use an intelligent iterative optimization algorithm to generate multiple combinations of structural optimization variables, and use the finite element rapid modeling interface to achieve automated model updates and mechanical calculations.

[0030] An intelligent iterative optimization algorithm is employed, specifically, one of the following can be selected: particle swarm optimization, genetic algorithm, or differential evolution algorithm. The particle swarm optimization algorithm will be used as an example. The algorithm parameters are set as follows: The particle swarm size is defined, and the position vector of each particle is a seven-dimensional vector, corresponding sequentially to the curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, main beam torsional stiffness, crossbeam stiffness, and initial cable tension. Inertia weight, individual learning factor, social learning factor, and velocity upper limit are set. The velocity upper limit is limited to a preset percentage of the value range of each parameter variable. The position boundary constraints are the physical value ranges of each parameter set in step one.

[0031] The initial value ranges of each optimization variable are set with lower and upper limits based on engineering experience. For cable-stayed bridges with multiple towers, the range of the tower-to-span ratio is further narrowed, and differences in the tower heights of adjacent towers are allowed, with the difference controlled within a preset percentage of the smaller tower height. For small-radius curved cable-stayed bridges with a width-to-span ratio exceeding a predetermined threshold, the lower limits of the torsional stiffness of the main girder and the crossbeam stiffness are correspondingly increased. At the same time, the stiffness of each crossbeam on the curve is independently set as a variable parameter, that is, the crossbeam stiffness parameter is expanded from a single value to multiple independent values ​​distributed along the bridge length.

[0032] A set of values ​​is randomly selected from the above range to form an initial parameter variable combination, denoted as the initial particle swarm. This initial parameter variable combination is then input into the rapid modeling interface. This rapid modeling interface embeds a parameter-driven finite element template, adapted to the spatial modeling logic of small-radius multi-tower wide-beam curved cable-stayed bridges, enabling integrated automatic modeling and synchronous parameter updates of curved lines, multi-tower layouts, wide-beam structures, and cable-stayed systems. The specific modeling process is as follows: First, a planar curve is generated based on the curve radius value. The curve shape uses a circular curve or a transition curve plus a circular curve, and the curve length must at least cover the range of a multi-span continuous beam. Then, the cross-sectional dimensions of the main beam are generated based on the height-to-span ratio and width-to-span ratio values. The main beam cross-section uses a single-box single-cell or single-box multi-cell thin-walled closed section to provide torsional stiffness. Next, the tower height and tower column cross-section are generated based on the tower-to-span ratio value. The tower type can be single-column, A-shaped, or diamond-shaped. Then, the initial strain or initial stress of each cable is set based on the initial cable tension value. The coordinates of the cable anchorage points on the beam and tower are automatically calculated for lateral offset based on the curve shape to ensure spatial force transfer. Finally, the torsional constant of the main beam cross-section is adjusted based on the main beam torsional stiffness value, and the elastic modulus or moment of inertia of the crossbeam elements is set based on the crossbeam stiffness value. When the curve radius is less than a preset threshold, the crossbeam arrangement is automatically densified. For cases involving multiple bridge towers, the geometric model of each tower is generated separately, and corresponding tower height differences are set.

[0033] After modeling is completed, the finite element method (FEM) solver is invoked to perform static and dynamic characteristic calculations. The calculations consider the basic combination of dead load, live load, temperature load, and wind load. The following mechanical parameters are extracted from the calculation results: vertical displacement of the main girder, lateral displacement of the main girder, longitudinal displacement of the bridge tower top, transverse displacement of the bridge tower top, maximum positive bending moment of the main girder, maximum negative bending moment of the main girder, maximum torque of the main girder, maximum shear force of the main girder, maximum torsional deformation angle of the main girder, ratio of maximum to minimum cable force, maximum bending moment at the end of the crossbeam, and maximum axial force at the bottom of the bridge tower. In extracting the torsional deformation angle of the main girder, multiple equally spaced cross-sectional sampling points are set along the entire length of the main girder. The torsional radii around the beam axis are calculated at each sampling point, and the maximum absolute value of the torsional radii among all sampling points is taken as the maximum value of the torsional deformation angle of the main girder. Simultaneously, the torque value at each sampling point is extracted to generate a torque distribution curve along the bridge length. Locations where the first derivative of the torque distribution curve exceeds a preset threshold are marked as torque abrupt change sections, and the torque value at these sections is used as one of the candidate values ​​for the maximum torque of the main girder. For multi-tower cable-stayed bridges, the independent torsional deformation angle of the main girder segment corresponding to each bridge tower is extracted, and the maximum value of each independent torsional deformation angle is used as the overall evaluation criterion.

[0034] Furthermore, the bending-torsional coupling coefficient is calculated. The calculation method is as follows: a unit vertical load and a unit torque are applied simultaneously at the mid-span section of the main beam. The resulting vertical displacement and torsional angle are extracted from this section, and the product of the vertical displacement and the torsional angle is defined as the bending-torsional coupling coefficient. This coupling coefficient is then compared with a preset bending-torsional coupling threshold.

[0035] Step 4: Based on the evaluation system, complete the quantitative evaluation and convergence determination of each group of variable combinations.

[0036] Substitute the mechanical indices extracted in step three into the evaluation system established in step two to calculate the comprehensive score for each particle. For example, after comparing each mechanical index of a particle with the specification limit, obtain the score for each scoring item according to the scoring rules, and then obtain the comprehensive score through weighted summation.

[0037] A convergence score threshold and a fluctuation threshold are preset. The fluctuation threshold is the limit of the rate of change or standard deviation of the comprehensive score over multiple consecutive iterations. It is determined whether the comprehensive score of the current globally optimal particle is greater than or equal to the convergence score threshold, and simultaneously whether the standard deviation of the comprehensive score over multiple consecutive iterations is less than the fluctuation threshold. If both conditions are met, convergence is determined, and the process proceeds to step six; otherwise, the process proceeds to step five.

[0038] Step 5: If the convergence condition is not met, automatically iterate and update the combination of optimized variables, and repeat the modeling calculation and performance evaluation process.

[0039] For unconverged particle swarms, the intelligent iterative algorithm updates the position or gene of each particle based on its overall score and the deviation of each mechanical index from its limits. Taking particle swarm optimization as an example, the position of the next generation of particles is calculated according to the velocity update formula and the position update formula. During the update process, when a particle's position exceeds the boundary, the position in that dimension is reset to the boundary value, and the velocity is multiplied in reverse by the decay coefficient. Simultaneously, if the bending-torsional coupling coefficient is greater than a preset threshold, a bending-torsional coupling penalty term is added to the evaluation system. This penalty term occupies a preset weight in the overall score, and the weights of the corresponding other scoring items are proportionally reduced. The specific value of the bending-torsional coupling penalty term is the ratio of the bending-torsional coupling coefficient to the bending-torsional coupling threshold minus one, multiplied by the penalty coefficient.

[0040] For the genetic algorithm, the population size, mutation factor, and crossover probability are set, and differential mutation or uniform mutation strategies are adopted. Single-point crossover or binomial crossover strategies are used. The selection operation is based on the fitness function value, i.e., the comprehensive score, using greedy retention or roulette wheel selection. A certain proportion of the best individuals in each generation are retained to enter the next generation. For cases where the width-to-span ratio is greater than a preset high value, the correlation mutation probability between the width-to-span ratio and the torsional stiffness of the main beam is increased during crossover and mutation operations.

[0041] The new combination of parameter variables is automatically written into the parameter table of the finite element template through the data interface, triggering the rapid modeling interface to regenerate the finite element model and replace the model from the previous iteration. Then, static calculations, mechanical index extraction, and comprehensive scoring calculations are re-executed on the new model.

[0042] Repeat the above iterative process. After each iteration, the system automatically records the current combination of parameter variables, the corresponding comprehensive score, and various mechanical indicators, and stores them in a table in a local or remote database. If the convergence condition is not met even after the number of iterations exceeds the preset maximum number of iterations, the system retrieves the combination of parameter variables with the highest comprehensive score from the historical iterations from the database as the suboptimal solution and outputs a convergence failure message along with the difference between the current optimal score and the convergence score threshold.

[0043] Step 6: Output the optimal combination of structural parameters.

[0044] When the convergence condition is met, the optimal combination of parameters at convergence is output. This optimal combination of parameters includes at least the optimal curve radius, optimal height-to-span ratio, optimal tower-to-span ratio, optimal width-to-span ratio, optimal main beam torsional stiffness, optimal crossbeam stiffness, and optimal initial cable tension. The comprehensive score under this optimal combination, along with detailed comparison data of each mechanical index with the code limits, is also output.

[0045] Post-processing steps: After outputting the optimal parameter variable combination, the finite element model is automatically reconstructed using this optimal parameter variable combination, and at least one additional load case calculation is performed from moving load analysis, temperature load analysis, wind load analysis, and seismic load analysis. The ratio of the maximum torque of the main beam to the torsional deformation angle of the main beam under the additional load case is extracted as the bending-torsional stiffness efficiency index, and the ratio of the bending moment at the base of each bridge tower to the tower height is extracted as the tower body efficiency index. These bending-torsional stiffness efficiency indices and tower body efficiency indices are compared with the corresponding acceptance criteria. If both indices meet the acceptance criteria, the current optimal parameter variable combination is locked as the final design parameters; if either index does not meet the acceptance criteria, the mechanical index under the additional load case is included as a new evaluation item in the multi-objective evaluation system of step two, and the iterative optimization process of steps one to six is ​​restarted.

[0046] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely preferred examples and are not intended to limit the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A multi-objective optimization method for a finite element model of a small-radius curved cable-stayed bridge, characterized in that, Includes the following steps: Step 1: Determine the structural optimization variables for a small-radius, multi-tower, wide-beam curved cable-stayed bridge. The optimization variables include the bridge curve radius, height-to-span ratio, tower-to-span ratio, width-to-span ratio, torsional stiffness of the main beam, stiffness of the crossbeam, and initial tension of the cables. Step 2: Combining the inherent bending-torsional coupling stress characteristics and spatial stress characteristics of small-radius curved cable-stayed bridges, establish multi-objective optimization constraints and structural performance evaluation system based on bridge structural design specifications; Step 3: Use an intelligent iterative optimization algorithm to generate multiple combinations of structural optimization variables, and use the finite element rapid modeling interface to realize the automatic updating of the overall finite element model of the bridge and mechanical numerical calculation. Step 4: Collect the core mechanical indicators of structural displacement, internal force, and torsional deformation after finite element calculation, and complete the quantitative evaluation and convergence determination of each variable combination based on the evaluation system; Step 5: If the convergence condition is not met, automatically iterate and update the combination of optimized variables, repeat the modeling calculation and performance evaluation process until the optimization convergence criterion is met, and output the optimal combination of structural parameters.

2. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The selection of optimization variables is adapted to the structural stress characteristics of small-radius multi-tower wide-beam curved cable-stayed bridges. In view of the significant bending-torsional coupling effect, complex spatial stress, and large differences in the sensitivity of structural parameters, the core structural parameters that control the overall stress state and deformation characteristics of the bridge are selected, and non-sensitive redundant structural parameters are discarded to achieve accurate matching and simplified layout of optimization variables.

3. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The curve radius and the width-to-span ratio are the core variables for controlling the bending-torsional coupling stress of a small-radius curved cable-stayed bridge. The curve radius determines the overall curve shape and curvature characteristics of the bridge, while the width-to-span ratio reflects the spatial scale matching characteristics of the wide beam main girder. Together, they affect the torsional stress distribution and overall spatial deformation law of the main girder, and are key parameters for regulating the bending-torsional coupling response of the bridge.

4. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The height-to-span ratio and tower-to-span ratio are sensitive variables for adapting the spatial force system of multi-tower curved cable-stayed bridges. The height-to-span ratio regulates the vertical stiffness and vertical force distribution of the bridge, while the tower-to-span ratio determines the load transfer path and overall stiffness distribution of the multi-tower structure, directly affecting the collaborative force-bearing performance of the multi-tower structure and the overall stability of the bridge.

5. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The torsional stiffness of the main beam and the stiffness of the crossbeam are the core variables for controlling the local stress of the wide beam main beam. The torsional stiffness of the main beam constrains the torsional deformation development trend of the wide beam structure, while the stiffness of the crossbeam ensures the lateral integrity of the main beam and the lateral load transfer capacity, effectively improving the problem of local stress concentration and uneven deformation of the main beam of small-radius curved bridges.

6. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The initial tension of the cable is a key control variable for the spatial stress of a cable-stayed bridge with a balanced curve. By adjusting the initial tension of the cable, the load-bearing ratio of the cable can be optimized, the internal force distribution of the main beam and the bridge tower can be improved, and the eccentric force and additional torsional effect caused by the small radius curve can be offset.

7. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The multi-objective optimization constraints are constructed based on the strength limits, stiffness limits, and stability limits of the bridge structure design code. At the same time, they are coupled with the bending-torsional coupling force constraints and spatial deformation constraints specific to small-radius curved cable-stayed bridges, taking into account both the overall structural stress safety and the rationality of the stress on local components.

8. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The structural performance evaluation system takes the overall uniformity of bridge stress, controllability of torsional deformation, displacement stability, and balanced stress distribution among multiple towers as core evaluation dimensions. It establishes multi-dimensional quantitative evaluation standards for the spatial stress characteristics of small-radius multi-tower wide-beam curved cable-stayed bridges, enabling precise comparison and screening of performance with different parameter combinations.

9. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The intelligent iterative optimization algorithm sets up an iterative update mechanism based on the multi-parameter coupling and correlation characteristics. According to the performance evaluation results of each group of optimization variables, it directionally corrects the iterative adjustment direction of the optimization variables, avoids the randomness defects of manual parameter tuning, and gradually approaches the global optimal parameter combination.

10. The multi-objective optimization method for the finite element model of a small-radius curved cable-stayed bridge according to claim 1, characterized in that, The finite element rapid modeling interface is adapted to the spatial modeling logic of small-radius multi-tower wide-beam curved cable-stayed bridges. It can realize integrated automatic modeling and synchronous parameter updates of curved lines, multi-tower layout, wide-beam structure, and cable-stayed system, and construct a closed-loop automated optimization process for parameter adjustment, modeling, calculation, and evaluation.

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