Bridge deck and pier interactive calculation method based on ML and PSO

Abstract

The invention discloses a bridge deck and pier interactive calculation method based on ML and PSO, and the method comprises the steps: inputting the initial parameters of a bridge, carrying out the modeling of a bridge deck and a pier through a finite element analysis method, and obtaining a finite element model to calculate a mechanical response; secondly, simulating time-varying response of the bridge deck and the bridge pier under the dynamic load action through a time-history analysis method to obtain acceleration, speed and displacement parameters, and analyzing a natural vibration mode through a mode analysis method to obtain natural frequency and vibration mode; and then, optimizing the load transfer model. Thirdly, optimizing design parameters of a bridge deck and a bridge pier by using PSO, predicting a dynamic load by using an ML model, and automatically adjusting the design parameters in combination with an optimization result; and finally, calculating safety coefficients of the bridge floor and the bridge pier based on the automatically adjusted design parameters, and judging whether the design meets a safety standard or not. According to the method, the interaction effect can be accurately simulated, design parameters are optimized, the design precision and safety are improved, meanwhile, the design efficiency is improved, and the cost is reduced.

Description

Technical Field

[0001] The present invention relates to the field of bridge structure calculation and optimization, and in particular to a method for calculating the interaction effect between bridge decks and piers based on ML (machine learning) and PSO (particle swarm optimization algorithm). Background Art

[0002] As an important transportation infrastructure, bridge structures are subject to dynamic and static loads from various sources, including traffic loads, wind loads, and seismic loads. With the increase in traffic demand and the increasing complexity of bridge structures, the difficulty of bridge design and analysis has gradually increased. In traditional bridge design, designers rely on static analysis and finite element analysis (FEM) to predict the mechanical response of bridges under static and dynamic loads. However, traditional calculation methods often fail to effectively consider the interaction between the bridge deck and piers. The limitations of existing methods become particularly apparent, especially when faced with complex loads and long-span bridges.

[0003] At present, in the field of bridge design, the following technical means are usually used to analyze and optimize the mechanical properties of bridges:

[0004] (1) Static analysis

[0005] Static analysis is a basic method for bridge design. By calculating the stress, deformation and other parameters of the bridge under static loads, it helps engineers determine the basic load-bearing capacity of the bridge. Common static analysis methods include:

[0006] 1) Beam Theory: Based on elastic beam theory, this method simplifies the model into a one-dimensional form to calculate the bridge's response to dead and live loads. This method is suitable for simpler bridges but cannot effectively handle complex bridge designs, especially when the interaction between the deck and piers becomes critical. Beam Theory cannot provide accurate results.

[0007] 2) Finite Element Analysis (FEM): The finite element method (FEM) predicts the mechanical response of a bridge by dividing the structure into multiple finite elements and performing numerical calculations. FEM is widely used in bridge design, but it typically treats different bridge components (such as the deck and piers) as independent elements, ignoring the interaction between the deck and piers. While FEM can simulate the mechanical response of a bridge, its limitations remain significant when considering dynamic loads and interaction effects.

[0008] (2) Kinetic analysis

[0009] As bridge design becomes increasingly complex, the analysis of bridge response under dynamic loads such as traffic flow, wind, and earthquakes has gradually attracted attention. Dynamic response analysis can predict the vibration characteristics and structural response of bridges under dynamic loads. The main methods are:

[0010] 1) Modal Analysis: This study examines the bridge's natural frequencies and vibration modes by analyzing its vibration modes. Modal analysis helps engineers understand the inherent characteristics of a bridge, but it is insufficient for deeply exploring the complex interactions between the bridge deck and piers.

[0011] 2) Time-History Analysis: Time-history analysis calculates the time-varying response of a bridge by simulating its response to various dynamic loads (such as traffic, wind, and earthquakes). While time-history analysis effectively captures dynamic effects, it is computationally intensive and can produce inaccurate results when dealing with complex bridge structures and load transfer.

[0012] (3) Coupling effect analysis

[0013] In recent years, researchers have proposed methods for analyzing the coupling effects between bridge decks and piers, attempting to simulate the interaction between the two through a coupling model. This method primarily relies on finite element analysis, establishing a coupling interface between the bridge deck and piers to reflect the mechanical relationship between them. Common coupling effect analysis methods include:

[0014] 1) Deck-Pier Coupled Model: This model couples the stresses, displacements, and other variables of the deck and piers to reflect their interaction. This approach can describe the joint response of the deck and piers under dynamic loads. However, in practice, the model is complex and computationally challenging, making accurate results difficult to obtain, especially for long-span bridges or under complex loading conditions.

[0015] 2) Simplified coupling method: Some studies have proposed to approximate the interaction effect between the bridge deck and the pier through simplified coupling equations. Although it is computationally simple, it is not accurate enough under complex load conditions and cannot fully consider all possible dynamic effects.

[0016] Although existing technologies have made some progress in bridge design and analysis, there are still many shortcomings and problems in solving the complex interaction between bridge decks and piers. The main problems include:

[0017] (1) Problems with independent modeling of bridge deck and piers

[0018] Existing finite element analysis and traditional design methods typically model the bridge deck and piers separately, assuming that the interaction between them is negligible. While this approach is simple, it neglects the mechanical coupling between the deck and piers, potentially leading to significant errors in the calculation results. In actual engineering, the interaction between the deck and piers has a significant impact on the stability and durability of bridges, especially under high loads or extreme conditions. Independent modeling cannot provide an accurate design basis.

[0019] (2) Limitations of dynamic response analysis

[0020] Existing dynamic response analysis methods, such as modal analysis and time-history analysis, often fail to fully consider the dynamic interaction between the bridge deck and piers. Especially under complex traffic flows, wind forces, or seismic loads, existing dynamic analysis methods fail to fully simulate the complex response of bridges under actual operating conditions, resulting in an inability to accurately assess bridge safety and stability.

[0021] (3) Computational complexity in coupling effect analysis

[0022] While some advanced bridge deck-pier coupling models can account for the interaction between the two, this approach is computationally complex and requires significant computing resources and time, especially when dealing with long-span or complex bridge structures. Furthermore, existing coupling methods often simplify the interaction between the deck and piers, leading to significant uncertainty in the analysis results under certain conditions.

[0023] (4) Insufficient load transfer mechanism

[0024] While existing technologies consider stress transfer in bridges under static loads, the load transfer mechanisms under dynamic loads are often overlooked. Load transfer between the bridge deck and piers is influenced not only by factors like deck type and span, but also by a combination of dynamic loads, environmental factors, and more. Current methods often fail to comprehensively account for these factors, resulting in oversimplified load transfer models that fail to accurately reflect the actual response of bridges under complex loads.

[0025] (5) Insufficient optimization

[0026] While existing computational methods can provide basic mechanical analysis of bridges, they remain insufficient for optimal design. Traditional methods often lack global optimization of bridge design and fail to consider the complex interactions between the bridge deck and piers. Consequently, existing design methods can lead to over- or under-design, impacting the economics and long-term durability of bridges. Summary of the Invention

[0027] To address the aforementioned challenges in the existing technologies, this paper proposes a method for calculating the interaction between bridge decks and piers based on machine learning and particle swarm optimization (PSO). By combining machine learning with the particle swarm optimization (PSO) algorithm, this method aims to automatically optimize design parameters and adjust load transfer mechanisms in real time, accurately predicting the dynamic response of bridges. This method not only addresses the shortcomings of existing technologies in modeling the interaction effects between bridge decks and piers, but also improves design accuracy and computational efficiency through an intelligent optimization process. By incorporating machine learning techniques into dynamic load prediction and load transfer mechanism optimization, this method avoids the simplification issues inherent in traditional methods and provides a novel solution for bridge design.

[0028] In order to achieve the above object, the specific scheme of the present invention is as follows:

[0029] The bridge deck and pier interaction calculation method based on ML and PSO includes the following steps:

[0030] Step 1: Input the initial parameters of the bridge and use the finite element analysis method to model the bridge deck and piers separately. Connect the bridge deck and piers through nodes and elements to obtain the finite element model to calculate the mechanical response of the bridge deck and piers.

[0031] Step 2: Perform dynamic response analysis on the finite element model from Step 1: Use time history analysis to simulate the time-varying response of the bridge deck and piers under dynamic loads such as traffic flow, wind, and earthquakes, and obtain the acceleration, velocity, and displacement parameters of the bridge deck and piers. Simultaneously, use modal analysis to analyze the inherent vibration modes of the bridge deck and piers and obtain their natural frequencies and vibration modes.

[0032] Step 3: Based on the dynamic response analysis results of Step 2, and according to the structural characteristics, type, load type and environmental factors of the bridge deck and piers, optimize the load transfer model between the bridge deck and piers;

[0033] Step 4: Based on the load transfer model optimized in step 3, the design parameters of the bridge deck and piers are optimized using a particle swarm optimization algorithm. The dynamic loads of the bridge deck and piers are predicted using a machine learning model. The design parameters are automatically adjusted based on the results of the load transfer model optimization.

[0034] Step 5: Calculate the safety factors of the bridge deck and piers based on the design parameters automatically adjusted in step 4 to determine whether the designs of the bridge deck and piers meet safety standards.

[0035] Furthermore, the initial parameters of the bridge in step 1 include the material, span, deck type, pier height, and pier cross-sectional dimensions of the bridge deck and piers. The basic equations for obtaining the finite element model to calculate the mechanical responses of the bridge deck and piers are as follows:

[0036]

[0037] Where, is the mass matrix; is the damping matrix; is the stiffness matrix; is the displacement vector of the node; represents the acceleration vector of the node; represents the velocity vector of the node; is the load vector under the action of external load.

[0038] Furthermore, the formula of the time course analysis method in step 2 is as follows:

[0039]

[0040] Where, is the displacement, For in time the speed of the moment; Indicates time; is the integral variable; is the time increment.

[0041] The modal analysis obtains the natural frequencies and vibration modes of the structure by solving the following eigenvalue problem:

[0042]

[0043] Where, is the stiffness matrix; is the mass matrix; is the modal circular frequency; is the corresponding vibration mode vector.

[0044] Furthermore, the formula for optimizing the load transfer model between the bridge deck and the piers in step 3 is as follows:

[0045]

[0046] Where: Indicates time Total load transferred from the bridge deck to the piers; Indicates the Type of original acting loads (such as dead load, live load, wind load, earthquake load); Indicates the Class load at time The transfer coefficient of Indicates the number of load types considered.

[0047] Furthermore, the formula for optimizing the design parameters of the bridge deck and piers using the particle swarm optimization algorithm in step 4 is as follows:

[0048]

[0049]

[0050] Where: Indicates the At the first iteration, The design variable position vector of each particle; Represents the updated particle position; Indicates the updated particle velocity; Indicates the At the first iteration, The velocity vector of each particle; Indicates the The best historical position of each particle; Indicates the current global optimal position of the entire particle swarm; represents the inertia weight; 、 They represent individual learning factors and social learning factors respectively; 、 Represents a uniform random number between [0,1].

[0051] Furthermore, the formula for predicting the dynamic loads of the bridge deck and piers using the machine learning model is as follows:

[0052]

[0053] Where:

[0054] Time to get predictions for the machine learning model Dynamic load values ​​at the site, including traffic load, wind load, earthquake load, etc.;

[0055] A trained machine learning model function, such as a neural network (NN) or support vector machine (SVM);

[0056] is the input feature vector of the model, at time It includes the following: historical dynamic load data, specifically F at past moments; bridge structure response data, including displacement, velocity, acceleration; bridge design parameters including span, material, and support stiffness; environmental factors including wind speed, temperature, and traffic density;

[0057] is the set of model parameters obtained through the training process.

[0058] Furthermore, the machine learning model described in step 4 utilizes historical bridge design data and adopts a neural network or a support vector machine to predict dynamic loads. The historical bridge design data includes dynamic load data during and after construction, bridge response data, bridge design parameters, structural characteristic data, and environmental factor data. The dynamic load data includes traffic loads, wind loads, and seismic loads, and the corresponding bridge response data includes displacement, velocity, and acceleration.

[0059] Furthermore, the formula for prediction using the neural network model is as follows:

[0060]

[0061] Where: Time to get predictions for the machine learning model Dynamic load value at ; For the The weight matrix of the layer; For the The weight matrix of the layer; is the weight matrix of the first layer; is the bias vector of the first layer; For the The bias vector of the layer; For the The bias vector of the layer; is the total number of layers in the neural network; For time The input feature vector at ; The activation function, specifically ReLU, Sigmoid, and Tanh, introduces nonlinearity.

[0062] Furthermore, the formula for calculating the safety factor of the bridge deck and piers in step 5 is as follows:

[0063]

[0064] Where, is the safety factor; is the design strength; is the bridge response under actual load.

[0065] The bridge deck and pier interaction calculation method based on ML and PSO is applied to steel structure bridges, concrete bridges, composite material bridges or long-span bridges.

[0066] Advantages of the present invention

[0067] 1. Improve design accuracy and safety: While conventional finite element analysis (FEM) methods can provide the basic mechanical response of bridge structures, they often overlook the complex interactions between the bridge deck and piers. This is particularly true under dynamic loads (such as traffic flow, earthquakes, and wind loads), which limits the accuracy of conventional methods. Conventional finite element methods typically treat the bridge deck and piers as independent units, ignoring the mechanical coupling between them. This leads to significant errors in the calculation results under high loads or extreme conditions. This invention, however, introduces an intelligent optimization method that combines machine learning (ML) with particle swarm optimization (PSO). This method accurately simulates the interactions between the bridge deck and piers, while simultaneously considering multiple influences such as dynamic loads, load transfer mechanisms, and environmental factors. It leverages historical data to predict changes in dynamic loads and adjust design parameters in real time, thereby ensuring the accuracy of the design solution, avoiding the errors associated with conventional methods, and improving design accuracy and safety.

[0068] 2. Improved design efficiency: Existing design processes typically rely on manual adjustments and repetitive calculations for optimization, resulting in long design cycles. Furthermore, when faced with complex load conditions, traditional methods require significant computational resources and time, resulting in low efficiency. This invention automates the design process by combining the particle swarm optimization (PSO) algorithm with machine learning (ML). PSO automatically searches the design space to find the optimal solution, while the machine learning model predicts dynamic load changes and provides real-time data support, reducing manual intervention and repetitive calculations, significantly improving design efficiency.

[0069] 3. Improved Economy: Traditional design methods can easily lead to over-design or material waste, especially when the interaction between the bridge deck and piers is not fully considered. This often results in overly conservative designs, increasing construction costs. This invention leverages machine learning to predict dynamic loads and a particle swarm optimization algorithm to automatically optimize design parameters. This avoids over-design or under-design, reduces unnecessary redundant structures and material waste, and reduces material waste and construction costs through precise optimization of design and load transfer mechanisms, while ensuring safety and stability.

[0070] 4. Enhance the long-term durability and maintainability of the design: Traditional design methods often focus on the bridge construction phase, ignoring changes during long-term use. They fail to fully consider the long-term impact of factors such as dynamic changes, load effects, and environmental changes on the bridge, which can easily lead to premature structural problems. This invention combines machine learning technology and particle swarm optimization algorithms to fully consider the long-term impact of dynamic loads during the design phase. Through an intelligent optimization process, the health status of the bridge is monitored and adjusted in real time. This effectively reduces fatigue damage during long-term use of the bridge, reduces manual intervention and maintenance costs, significantly shortens the design cycle, and improves design efficiency.

[0071] 5. Addressing deficiencies in load transfer mechanisms: Traditional load transfer models are overly simplified and fail to account for the actual impact of dynamic loads, environmental changes, and other factors on load transfer. They are only applicable to static or simple dynamic load conditions and cannot accurately reflect the load transfer mechanisms of bridges in complex environments. This invention introduces a load transfer coefficient and machine learning technology to establish a more accurate load transfer model. This model comprehensively considers the interaction between the bridge deck and piers, as well as the dynamic load transfer mechanism. It can adjust the load transfer coefficient based on real-time load changes, making the design more accurate and reducing load calculation errors.

[0072] 6. Broad Applicability and Flexibility: Traditional bridge design methods are often only applicable to specific bridge types or sizes. Finite element analysis methods require individual modeling for different bridges, and traditional optimization methods are also limited to specific design conditions. However, the calculation method of this invention is applicable to the design of various bridge types, including highway bridges, railway bridges, long-span bridges, cable-stayed bridges, and suspension bridges. By combining machine learning and particle swarm optimization algorithms, it can automatically adjust parameters and provide flexible and efficient optimization solutions to meet different design requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 This is a flow chart of the bridge deck and pier interaction calculation method based on ML and PSO of the present invention. DETAILED DESCRIPTION

[0074] The present invention will be further explained and illustrated below in conjunction with the accompanying drawings and specific embodiments. It should be noted that this specific embodiment is not intended to limit the scope of rights of the present invention.

[0075] like Figure 1 As shown, this specific embodiment provides a bridge deck and pier interaction calculation method based on ML and PSO. The method is applicable to steel structure bridges, concrete bridges, composite bridges or long-span bridges, and specifically includes the following steps:

[0076] Step 1: Input the initial parameters of the bridge, including the material, span, deck type, pier height, and pier cross-sectional dimensions of the bridge deck and piers. Based on these initial parameters, the bridge deck and piers are modeled separately using finite element analysis. The deck and piers are connected by nodes and elements. Nodes are important locations in the structure, typically connecting key components. Elements are line segments or patches connecting nodes, simulating the material and stiffness of the structure. The finite element model is then used to calculate the mechanical response of the bridge deck and piers.

[0077] The basic equations for obtaining the finite element model to calculate the mechanical response of the bridge deck and piers are as follows:

[0078]

[0079] Where, is the mass matrix; is the damping matrix; is the stiffness matrix; is the displacement vector of the node; represents the acceleration vector of the node; represents the velocity vector of the node; is the load vector under the action of external load.

[0080] Step 2: Perform dynamic response analysis on the finite element model in Step 1: Use the time-history analysis method to simulate the time-varying response of the bridge deck and piers under the dynamic loads of traffic flow, wind, and earthquake, and obtain the acceleration, velocity, and displacement parameters of the bridge deck and piers. At the same time, use the modal analysis method to analyze the natural vibration modes of the bridge deck and piers to obtain their natural frequencies and vibration modes to help predict the response mode of the bridge under load.

[0081] The formula of the time course analysis method is as follows:

[0082]

[0083] Where, is the displacement, For in time the speed of the moment; Indicates time; is the integral variable; is the time increment.

[0084] The modal analysis obtains the natural frequencies and vibration modes of the structure by solving the following eigenvalue problem:

[0085]

[0086] Where, is the stiffness matrix; is the mass matrix; is the modal circular frequency; is the corresponding vibration mode vector.

[0087] Step 3: Based on the dynamic response analysis results of Step 2, the concept of load transfer coefficient is introduced. The load transfer model between the bridge deck and piers is optimized according to the structural characteristics, type, load type and environmental factors of the bridge deck and piers, including wind, traffic flow, earthquake, etc.

[0088] Specifically, in order to achieve accurate modeling of the interactive response between the bridge deck and the piers, this embodiment constructs a dynamic load transfer model with structural and environmental adaptive capabilities. The dynamic load transfer model sets the load transfer coefficient , express the Class load at time The specific modeling and optimization methods are as follows:

[0089] 1. Basic structure of load transfer model

[0090] The formula for the load transfer model between the optimized bridge deck and the pier is as follows:

[0091]

[0092] Where: Indicates time Total load transferred from the bridge deck to the piers; Indicates the Class original action loads, including dead load, live load, wind load, and earthquake load; Indicates the Class load at time The transfer coefficient of Indicates the number of load types considered.

[0093] 2. Optimization method of load transfer coefficient

[0094] Load transfer coefficient The calculation includes three parts: structural influence function, load type influence function, and environmental adjustment function. The specific definitions are as follows:

[0095] (1) Structural influence function

[0096] The structural influence function is used to reflect the mechanical relationship between the bridge deck and the pier in terms of stiffness and inertia:

[0097]

[0098] Where: represents the longitudinal equivalent stiffness of the bridge deck, kN / m; It represents the longitudinal equivalent stiffness of the pier, kN / m; represents the moment of inertia of the bridge deck section, m 4 ; represents the moment of inertia of the pier section, m 4 ; It represents the structural influence coefficient, and its value range is approximately 0.2~0.9.

[0099] (2) Load type influence function

[0100] Based on a large amount of finite element simulation and bridge monitoring data statistics, the typical transfer capacity of various loads is determined as follows:

[0101] Load Type#timg# #timg#Typical value (recommended value) Dead Load 0.90 Live Load 0.70 Wind Load 0.45 Seismic Load 0.60

[0102] (3) Environmental regulation function

[0103] This function is used to modify the transfer path based on real-time environmental factors, including wind speed, temperature, and earthquake acceleration:

[0104]

[0105] Where: Indicates time Wind speed, m / s; Indicates time Ambient temperature, °C; Indicates time Peak earthquake acceleration, m / s²; Indicates the wind speed adjustment coefficient (the recommended value is 0.01), s / m; Indicates the temperature adjustment coefficient (the recommended value is 0.005), 1 / ℃; Indicates the earthquake acceleration adjustment factor (the recommended value is 0.015), s² / m.

[0106] (4) Final load transfer coefficient formula

[0107] In summary, the load transfer coefficient The calculation formula is as follows:

[0108]

[0109] Where, represents the load transfer coefficient; represents the structural influence function; represents the load type influence function; represents the environmental adjustment function.

[0110] All factors are defined deterministic functions, and the formula does not contain any empirical interpolation or subjective fitting components.

[0111] 3. Optimization Basis and Adjustment Methods

[0112] The coefficients of the load transfer model are determined and verified by the following process:

[0113] (1) Initial value calculation: Calculate based on structural design parameters, load type and environmental initial values .

[0114] (2) Finite element simulation verification: A bridge deck-pier coupling model was established on the ANSYS platform to simulate the reaction forces on the piers under different load combinations. .

[0115] (3) Optimization objective function setting:

[0116]

[0117] Where, Indicates time Total load transferred from the bridge deck to the piers; Indicates time The pier reaction force values ​​calculated by the finite element simulation model; Represents the time step index variable; represents the total number of time steps; Indicates the Class load at time The dynamic load transfer coefficient transmitted from the bridge deck to the bridge pier at any time is the main parameter variable in the optimization process of the present invention; represents the minimization objective function; It represents the sum of the errors of all time steps as the overall error indicator.

[0118] (4) Numerical adjustment method: Newton descent method is used The gradual adjustment of the iteration accuracy is set to .

[0119] The load transfer model proposed in this embodiment can accurately calculate the load transfer path between the bridge deck and the piers under known structural parameters and environmental boundary conditions. Compared with the traditional fixed coefficient method, it has higher physical realism, adjustability and safety assurance capabilities, and is applicable to different bridge types such as concrete bridges, steel structure bridges, and composite bridges.

[0120] Step 4: Based on the load transfer model optimized in Step 3, the design parameters of the bridge deck and piers are optimized using the particle swarm optimization (PSO) algorithm. PSO is used to optimize design variables. By simulating the foraging behavior of bird flocks, PSO efficiently searches for the optimal solution in the design space. Each particle represents a design solution, and the particle adjusts its position in the search space based on the value of the fitness function.

[0121] Specifically, to automatically optimize the design parameters of the bridge deck and piers, this embodiment uses the particle swarm optimization algorithm (PSO) to globally optimize the structural parameters. The following is the basis for PSO parameter setting and the optimization performance evaluation results:

[0122] 1. Overview and Applicability of PSO Algorithm

[0123] The particle swarm optimization algorithm simulates the collaborative search behavior of a group to find the best solution in a continuous space. It has the following advantages in structural design problems:

[0124] Able to handle high-dimensional continuous optimization problems;

[0125] Does not rely on problem gradient information;

[0126] It is easy to link with the neural network prediction module and is suitable for embedding in the multi-parameter design of bridges.

[0127] In this example, PSO is used to optimize the following bridge design variables:

[0128] Parameter name symbol Value range (constraint) illustrate Bridge deck thickness #timg# [0.25 m, 0.60 m] Influence on bridge deck deadweight and deflection Pier section width #timg# [1.5 m, 4.0 m] Affects the stiffness and stability of bridge piers Pier section height #timg# [2.0 m, 6.0 m] Related to bending resistance Support stiffness #timg# [1e5, 1e7] kN / m Influence load transfer path

[0129] The optimization goal is to minimize the self-weight and stress peak of the bridge structure while meeting safety constraints to achieve lightweight and uniform stress distribution.

[0130] 2. Particle Swarm Optimization Algorithm Parameter Setting

[0131] In PSO, each particle represents a design solution. The formula for optimizing the design parameters of the bridge deck and piers using the particle swarm optimization algorithm is as follows:

[0132]

[0133]

[0134] Where: Indicates the At the first iteration, The design variable position vector of each particle; Represents the updated particle position; Indicates the updated particle velocity; Indicates the At the first iteration, The velocity vector of each particle; Indicates the The best historical position of each particle; Indicates the current global optimal position of the entire particle swarm; represents the inertia weight; 、 They represent individual learning factors and social learning factors respectively; 、 Represents a uniform random number between [0,1].

[0135] The parameter values ​​are as follows:

[0136] parameter Numerical Setting basis Population size#timg# 30 Commonly recommended values ​​for engineering optimization Maximum number of iterations 100 Set an upper limit on error convergence Inertia weight#timg# 0.9 → 0.4 Linear decreasing strategy to promote convergence Individual learning factor#timg# 2.0 Recommended value, balancing local search capabilities Group learning factor#timg# 2.0 Recommended value, enhanced global search Speed ​​limit#timg# 20% of the upper and lower limits of each variable Prevent the search from jumping out of the feasible domain

[0137] The inertia weight adopts a linear decreasing strategy:

[0138]

[0139] in =0.9, =0.4, is the current iteration number.

[0140] 3. Optimization objective function definition

[0141] Taking into account the structural deadweight, maximum stress and material consumption, the objective function is set as follows:

[0142]

[0143] Where: represents the optimization objective function; Represents the set of design variables The objective function defined Perform the minimize operation; represents the set of bridge structure design variables; represents the deadweight of the bridge structure; Indicates the maximum stress value of the structure under load; 、 is the normalized weighting coefficient, which is used to reflect the emphasis of the design goal on lightweight and strength control. , .

[0144] During implementation, to improve the accuracy of predicting the response of bridge structures to multiple dynamic loads (such as traffic, wind, and earthquakes), a machine learning model was used to predict the dynamic loads on the bridge deck and piers, providing accurate input load data for the particle swarm optimization (PSO) algorithm. The machine learning model was used to predict dynamic loads and adjust the optimization model in real time. The model leveraged changes in historical bridge design data and employed a neural network or support vector machine to predict the dynamic loads on the bridge deck and piers, which served as input data for the PSO optimization algorithm. Changes in load types such as traffic flow, wind speed, and earthquake intensity were predicted based on the historical bridge design data, further adjusting the design parameters in the optimization algorithm. The historical bridge design data included dynamic load data during and after construction, bridge response data, bridge design parameters, structural characteristics, and environmental factor data. Dynamic load data included traffic loads, wind loads, and earthquake loads, and the corresponding bridge response data included displacement, velocity, and acceleration.

[0145] The formula for predicting the dynamic loads on the bridge deck and piers using the machine learning model is as follows:

[0146]

[0147] Where:

[0148] Time to get predictions for the machine learning model Dynamic load values ​​at the location (including traffic load, wind load, earthquake load, etc.);

[0149] A trained machine learning model function, such as a neural network (NN) or support vector machine (SVM);

[0150] is the input feature vector of the model, at time It includes the following contents: historical dynamic load data, bridge structure response data, bridge design parameters, and environmental factors; historical dynamic load data is F at past moments, bridge structure response data includes displacement, velocity, and acceleration, bridge design parameters include span, material, and support stiffness, and environmental factors include wind speed, temperature, and traffic density, etc.

[0151] is the set of model parameters obtained through the training process.

[0152] The formula using the neural network model is as follows:

[0153]

[0154] Where: Time to get predictions for the machine learning model Dynamic load value at ; For the The weight matrix of the layer; For the The weight matrix of the layer; is the weight matrix of the first layer; is the bias vector of the first layer; For the The bias vector of the layer; For the The bias vector of the layer; is the total number of layers in the neural network; For time The input feature vector at ; The activation function, specifically ReLU, Sigmoid, and Tanh, introduces nonlinearity.

[0155] The machine learning algorithm can provide dynamic load prediction data for the PSO optimization process and automatically adjust the design parameters based on the results of the load transfer model optimization, thereby improving the accuracy of the fitness function and achieving the global optimal bridge design.

[0156] The following are the specific implementation steps of the machine learning model:

[0157] 1. Model type and selection basis

[0158] This example uses a multi-layer feedforward neural network (MLP) model to predict bridge dynamic loads. The model selection is based on the following technical features:

[0159] (1) The dynamic response of bridges has strong nonlinear characteristics and is affected by both structural parameters and environmental parameters, making it suitable for neural network modeling.

[0160] (2) The MLP model can input historical time series data in a fixed dimension and adapt to the static fitness function interface in the particle swarm optimization algorithm.

[0161] (3) Compared with recurrent neural network (RNN) models, MLP is more suitable for embedding into engineering simulation processes in terms of training efficiency and deployment complexity.

[0162] 2. Training Data Source and Input Feature Definition

[0163] The training data is collected from two sources: one is the actual bridge health monitoring system (including acceleration, displacement, wind speed and other data recorded by sensors); the other is the time-history response data generated based on the simulation of the bridge finite element model.

[0164] Model input features , in time The moments are defined as follows:

[0165]

[0166] The symbols are defined as follows:

[0167] symbol meaning unit #timg# The total load value at time #timg# (known), where #timg# kN #timg# Displacement (m), velocity (m / s), acceleration (m / s²) at the previous moment m, m / s, m / s² #timg# span m #timg# elastic modulus GPa #timg# Sectional moment of inertia <![CDATA[m 4 ]]> #timg# Equivalent stiffness of bridge pier kN / m #timg# Current wind speed m / s #timg# Current temperature ℃ #timg# Current traffic density vehicles / km

[0168] The sample input window length is set to , that is, each input includes the load sequence of the past 5 moments and the current response, structure and environment information. The total input dimension is .

[0169] 3. Model Structure and Mathematical Expression

[0170] The machine learning model uses a three-layer feedforward neural network with the following network structure:

[0171] Input layer dimension: 13

[0172] First hidden layer: 64 neurons, activation function ReLU

[0173] Second hidden layer: 32 neurons, activation function ReLU

[0174] Output layer: 1 node, outputs predicted load value .

[0175] The prediction formula is as follows:

[0176]

[0177]

[0178]

[0179] Where:

[0180] , : first layer weights and biases; , : Second layer weights and biases;

[0181] , : Output layer parameters; represents the output vector of the first hidden layer; represents the output vector of the second hidden layer; Indicates the predicted load value; represents the input feature vector.

[0182] 4. Training Process and Loss Function

[0183] The model training adopts supervised learning method, and the training data is ,in is the target true load value.

[0184] The loss function uses the mean square error (MSE):

[0185]

[0186] Where, Represents the total loss function value, which is used to measure the error between the model prediction value and the true value; Indicates the total number of training samples; Represents the model at time Predicted values ​​for dynamic loads; Indicates time The actual target dynamic load value at .

[0187] The optimization method is Adam optimizer, and the initial learning rate is , decaying by half every 10 rounds.

[0188] To prevent overfitting, add L2 regularization term and regularization coefficient .

[0189] 5. Training Hyperparameter Settings

[0190] Parameter name Numerical illustrate batch size 128 The number of samples per gradient update Maximum number of training rounds (epochs) 100 If the early stopping condition is met, terminate early learning rate 0.001 Adam optimizer initial value Dropout probability 0.2 Applied to hidden layers to enhance robustness Regularization coefficient #timg# #timg# Controlling model complexity Sliding window length#timg# 5 To construct the input sequence

[0191] 6. Model Performance Evaluation

[0192] A five-fold cross-validation was performed on a mixed dataset (with a ratio of 3:2 between simulated data and measured data). The model prediction performance is as follows:

[0193] index value Mean Absolute Error (MAE) < 4.2% Mean Squared Error (MSE) < 0.007 Coefficient of determination (#timg#) > 0.93 Peak recall (Recall@Peak) > 0.89

[0194] The model demonstrates high accuracy and robustness under both common load ranges and extreme loads, meeting the safety and reliability requirements of bridge design.

[0195] (1) Mean absolute error (MAE)

[0196] Mean Absolute Error (MAE) is a commonly used indicator to measure the degree of deviation between the predicted value of the regression model and the true value. Its definition formula is as follows:

[0197]

[0198] Where: is the total number of samples; For the model The predicted value on samples; For the The true target value of samples; is the absolute error between the predicted value and the true value.

[0199] (2) Mean Square Error (MSE)

[0200] Mean Squared Error (MSE) is one of the commonly used indicators to measure the degree of deviation between the predicted value and the true value. Its definition formula is as follows:

[0201]

[0202] Where, Represents the total loss function value, which is used to measure the error between the model prediction value and the true value; Indicates the total number of training samples; Represents the model at time Predicted values ​​for dynamic loads; Indicates time The actual target dynamic load value at .

[0203] The unit of MSE is the square of the unit of the target variable (e.g. if the target is load, in kN, then the unit of MSE is kN²). The smaller the value, the more accurate the model prediction. Compared with MAE, MSE is more sensitive to outliers because the error is amplified by the square. In the prediction of bridge structure response, MSE can be used to measure the overall fitting error of the model within the normal load range, and can be used together with Use it together with MAE for a more comprehensive effect.

[0204] (3) Coefficient of determination ( )

[0205] Coefficient of Determination, usually expressed as , which is used to measure the ability of the regression model to explain the variation of the observed values. Its calculation formula is as follows:

[0206]

[0207] Where: is the total number of samples; For the The predicted value of samples; For the The true value of the samples; is the average of the true values; is the residual sum of squares (RSS, residual error); is the total sum of squares (TSS, representing the overall fluctuation of the data).

[0208] Features:

[0209] : indicates perfect prediction by the model;

[0210] , the model has the same prediction effect as using the mean;

[0211] , the model is worse than using the average directly (indicating that the model is not appropriate).

[0212] This indicator is often used to evaluate the goodness of fit of regression models and is suitable for regression tasks such as bridge load prediction.

[0213] (4) Peak recall rate (Recall@Peak)

[0214]

[0215] Where:

[0216] : Under real load In the sample of The number of peaks successfully identified.

[0217] : Under real load In the sample of The number of peaks that could not be identified.

[0218] Threshold selection: Typically, peak threshold Select the value of the upper 5%~10% quantile in the real load series:

[0219]

[0220] In bridge safety assessments, high-load moments often determine the ultimate structural response capacity. Recall@Peak is used to determine whether the model can timely predict these critical points. In this paper, Recall@Peak can be used as an evaluation model to assess the reliability assurance capability under extreme load conditions when input into the PSO optimization algorithm.

[0221] Step 5: Calculate the safety factors of the bridge deck and piers based on the design parameters automatically adjusted in step 4 to determine whether the designs of the bridge deck and piers meet safety standards.

[0222] The formula for calculating the safety factor of the bridge deck and pier is as follows:

[0223]

[0224] Where, is the safety factor; is the design strength; is the bridge response under actual load.

Claims

1. The bridge deck and pier interaction calculation method based on ML and PSO is characterized by: The steps include: Step 1: Input the initial parameters of the bridge and use the finite element analysis method to model the bridge deck and piers separately. Connect the bridge deck and piers through nodes and elements to obtain the finite element model to calculate the mechanical response of the bridge deck and piers. Step 2: Perform dynamic response analysis on the finite element model from Step 1: Use time history analysis to simulate the time-varying response of the bridge deck and piers under dynamic loads such as traffic flow, wind, and earthquakes, and obtain the acceleration, velocity, and displacement parameters of the bridge deck and piers. Simultaneously, use modal analysis to analyze the inherent vibration modes of the bridge deck and piers and obtain their natural frequencies and vibration modes. Step 3: Based on the dynamic response analysis results of Step 2, and according to the structural characteristics, type, load type and environmental factors of the bridge deck and piers, optimize the load transfer model between the bridge deck and piers; Step 4: Based on the load transfer model optimized in Step 3, the design parameters of the bridge deck and piers are optimized using a particle swarm optimization algorithm. The dynamic loads of the bridge deck and piers are predicted using a machine learning model. The design parameters are automatically adjusted based on the results of the load transfer model optimization. Step 5: Calculate the safety factors of the bridge deck and piers based on the design parameters automatically adjusted in step 4 to determine whether the designs of the bridge deck and piers meet safety standards.

2. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The initial parameters of the bridge in step 1 include the material, span, deck type, pier height, and pier cross-sectional dimensions of the bridge deck and piers. The basic equations for calculating the mechanical responses of the bridge deck and piers are as follows: ， Where, is the mass matrix; is the damping matrix; is the stiffness matrix; is the displacement vector of the node; represents the acceleration vector of the node; represents the velocity vector of the node; is the load vector under the action of external load.

3. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The formula for the time course analysis method described in step 2 is as follows: ， Where, is the displacement, For in time the speed of the moment; Indicates time; is the integral variable; is the time increment; The formulas for the natural frequencies and vibration modes are as follows: ， Where, is the stiffness matrix; is the mass matrix; is the modal circular frequency; is the corresponding vibration mode vector.

4. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The formula for optimizing the load transfer model between the bridge deck and the piers described in step 3 is as follows: ， Where: Indicates time Total load transferred from the bridge deck to the piers; Indicates the Class original action loads, including dead load, live load, wind load and earthquake load; Indicates the Class load at time The transfer coefficient of Indicates the number of load types considered.

5. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The formula for optimizing the design parameters of the bridge deck and piers using the particle swarm optimization algorithm in step 4 is as follows: ， ， Where: Indicates the At the first iteration, The design variable position vector of each particle; Represents the updated particle position; Indicates the updated particle velocity; Indicates the At the first iteration, The velocity vector of each particle; Indicates the The best historical position of each particle; Indicates the current global optimal position of the entire particle swarm; represents the inertia weight; 、 They represent individual learning factors and social learning factors respectively; 、 Represents a uniform random number between [0,1].

6. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The formula for predicting the dynamic loads on the bridge deck and piers using the machine learning model described in step 4 is as follows: ， Where: Time to get predictions for the machine learning model Dynamic load values ​​including traffic load, wind load and earthquake load; A trained machine learning model function, such as a neural network model or a support vector machine model; is the input feature vector of the model, at time It includes the following: historical dynamic load data, specifically F at past moments; bridge structure response data, including displacement, velocity, and acceleration; bridge design parameters including span, material, and support stiffness; and environmental factors such as wind speed, temperature, and traffic density. is the set of model parameters obtained through the training process.

7. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The machine learning model described in step 4 uses the historical design data of the bridge and adopts a neural network model or a support vector machine model to predict the dynamic load. The historical design data of the bridge includes dynamic load data during the construction process and after completion, bridge response data, bridge design parameters, structural characteristic data and environmental factor data. The dynamic load data includes traffic load, wind load and seismic load, and the corresponding bridge response data includes displacement, velocity and acceleration.

8. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 7 is characterized in that: The formula for predicting dynamic load using the neural network model is as follows: ， Where: Time to get predictions for the machine learning model Dynamic load value at ; For the The weight matrix of the layer; For the The weight matrix of the layer; is the weight matrix of the first layer; is the bias vector of the first layer; For the The bias vector of the layer; For the The bias vector of the layer; is the total number of layers in the neural network; For time The input feature vector at ; The activation function, specifically ReLU, Sigmoid, and Tanh, introduces nonlinearity.

9. The bridge deck and pier interaction calculation method based on ML and PSO according to claim 1 is characterized in that: The formula for calculating the safety factor of the bridge deck and piers described in step 5 is as follows: ， Where, is the safety factor; is the design strength; is the bridge response under actual load.

10. Application of the bridge deck and pier interaction calculation method based on ML and PSO as described in any one of claims 1 to 9 to steel structure bridges, concrete bridges, composite material bridges or long-span bridges.

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