# Research for Pedestrian Steel Bridge Design of Neural Network in Structural Model Updating

Authors: Rui Zhao; Yuhang Wu; Zehua Feng
Journal: Shock and Vibration
 (2022)

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## Abstract

The application of the neural network method in health monitoring and structural system identification has received extensive attention. A reasonable neural network structure is very important for its performance. This paper takes the pedestrian bridge of the Xingfu intersection in Urumqi, China, as the research object and uses MIDAS/Civil to establish a finite element analysis model. Taking the natural vibration frequency obtained from the dynamic test of the actual bridge as the target, two kinds of neural networks are used to predict the structural material parameters. An appropriate bridge model correction method is selected by comparing the prediction results of the BP neural network and the GRNN. The test results show that the pedestrian bridge model based on MIDAS/Civil has a high accuracy, but it still does not meet the actual needs. The modified model based on the BP neural network is close to the actual measured results, and a more accurate finite element analysis model can be established by this method, which makes the modified model closer to the real stress state of the structure.


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## Body

## 1. Introduction

The long-term health inspection of steel structure bridge is of great significance to its safety, and the current measures to ensure the long-term safety performance of the bridge mainly focus on the inspection and reinforcement. The health monitoring and prediction of bridges are based on finite element analysis. Firstly, the measuring points are arranged at each node of the bridge, and the measured data (concrete strength, deflection value, strain value, and low-order natural frequency of the bridge) are collected. Then, the finite element model is established according to the existing design drawings and combined with external conditions. The state of the bridge is judged by the correlation analysis between the measured value and the calculated value. However, the measured results are often different from the calculated results of the bridge finite element model, and the degree of difference often determines the evaluation results of the bridge. Therefore, the establishment of a more accurate finite element model is the basis of structural analysis. Due to the discreteness degree of the finite element model, the setting of modeling parameters, some simplified assumptions are used in modeling, and the calculated values of the finite element model bring errors. In addition, the initial finite element model established strictly in accordance with the design drawings cannot actually reflect the mechanical behavior of the structure due to the influence of material performance deviation, structural degradation or damage, and other factors [1]. Therefore, it is necessary to modify the bridge finite element model according to the measured results of bridge detection to improve the accuracy and reliability of the bridge finite element model. The essence of finite element model modification is an approximation problem, and one or several indexes of the finite element model are infinitely close to the real value by modifying various parameters of the finite element model.Using a neural network to modify the finite element model can not only reduce the calculation times but also make the predicted results tend to close the actual value [2]. This paper mainly adopts BP neural network and GRNN. BP neural network (backpropagation neural network) is a kind of multilayer feedforward neural network. The error between the actual output and the expected output of the system is reduced by changing the connection weights among neurons, and the weights and thresholds between the hidden layer and the output layer are adjusted forward and backward according to the gradient descent method. GRNN (general regression neural network) is a feedforward neural network model based on the nonlinear regression principle. GRNN is improved based on RBF and has a similar structure. The difference lies in the addition of a summation layer and the removal of the weight connection between the hidden layer and the output layer (the least-square superposition of the Gaussian weight).The core factor of the stability of a building structure is determined by its own material and structure. It is predicted by the neural network, which makes the model reference. Alkayem et al. [3–7] proposed that improving the accuracy of the intelligent algorithm for computing model correction is a direction worthy of study and summarized the model correction methods based on the neural network comprehensively. In 2004, Fei et al. [8–10] used RBF neural network to conduct a revision study on the aircraft model, and the results showed that the design parameter error was within 2%, and the modal frequency error was within 1%. In 2012, Wang et al. [11, 12] applied a neural network to the finite element model modification of a reinforced concrete bridge and verified that the modified finite element model could reflect the physical state of the structure more truly and better reflect the real dynamic characteristics of the bridge structure. In 2017, Jia et al. [13, 14] modeled the PC hollow beam that has been in service for more than 20 years and modified it by using a neural network intelligent algorithm. The physical state between the modified finite element model and the actual structure is very close. The elastic modulus of the steel strand is consistent with the experimental value, and all errors are controlled within 5%. In 2018, Bao et al. [15–17] used MIDAS to establish a bridge model and BP neural network to modify the model based on the deflection. The deflection error of the modified bridge finite element model was less than 10%, and a good correction effect was obtained.Previous studies prove that neural network has a good effect on model modification, but few scholars apply a neural network to model modification of pedestrian steel bridge. In this paper, BP neural network and GRNN in radial basis neural network are used as the correction method. Urumqi Xingfu Road pedestrian bridge is taken as the model to be modified. Taking the characteristic quantity as the independent variable and the steel design parameters as the dependent variable, the neural network intelligent algorithm is used to approximate the nonlinear mapping relationship between the two [18]. Then, the generalization property of the neural network is used to solve the target value of design parameters directly. Through the reliability comparison and error analysis of BP neural network and GRNN training test results, a model modification method suitable for the pedestrian bridge was found.


## 2. Project Overview

Xinjiang Urumqi Xingfu Road pedestrian steel bridge was built in 2002 and is located at the intersection of the Urumqi beltway and Xingfu Road in Urumqi, China, as shown in Figure1. The main pedestrian bridge has a net height of 5 m and a span of 23.763 m and 25.762 m, respectively. The main girder adopts a uniform cross-section single-cell box girder structure, which is 1 m high and 4 m wide, and the thickness of the roof is 10 mm, the thickness of the bottom plate is 16 mm, the thickness of the web is 12 mm, and the box girder internal welding 10 mm thick stiffening ribs. The bridge is equipped with 4 escalators, which are welded with the main beam. The width of the staircase is 2.5 m, and the height-width ratio of the steps is 1 : 3. The ladder beams are all steel box girder structure, with a height of 280 mm and the plate thickness of 10 mm. The piers are made of hot-rolled seamless steel tubes, with a wall thickness of 16 mm and a diameter of 426 mm. The foundation is an expanded concrete foundation, the balustrade is a carbon stainless steel composite steel pipe, and the bridge deck pavement adopts a 15 mm thick rubber plate; Q235B killed steel is used for the pier column and main beam steel plate, and Q235C killed steel is used for other members. The main components are welded in the factory and then transported to the site for welding and assembly.Figure 1
Urumqi Xingfu Road pedestrian bridge.


## 3. Model Establishment and Dynamic Test

### 3.1. Initial Model Establishment

According to the design drawing size of the pedestrian bridge, the finite element model of the pedestrian steel bridge is established by using MIDAS/Civil. Semirigid rubber bearings were set for the main beam, escalator beam, and column of the pedestrian bridge, and consolidation boundary conditions were selected for the supports and top pillar. Mechanical properties of materials refer to Q235 steel in the General Specifications for Design of Highway Bridges and Culverts [19] (D60-2015). The finite element model is shown in Figure 2.Figure 2
Finite element model of the pedestrian bridge.The dynamic characteristics of the pedestrian bridge model were analyzed by using the subspace iteration method. The first 5-order natural vibration frequencies of the pedestrian bridge are calculated as shown in Table1.Table 1
Calculation table of the first five natural vibration frequencies of the pedestrian bridge.
OrderFirst orderSecond orderThird orderFourth orderFifth orderFrequency (Hz)4.46685.54816.24278.185710.4471According to the study of Fan et al. [20, 21], the walking frequency of human is 2∼4 Hz, and the first 2 vibration modes of the bridge need only be considered in model modification and dynamic test of the real bridge for the pedestrian bridge.


### 3.2. Self-Vibration Characteristics Test

The test was conducted in the early morning in order to minimize the influence of external factors such as the stream of people and vehicles on the dynamic test results. JMC2-2091 acceleration-type vibration pickup and JMYD-1016 dynamic information acquisition instrument from Jinma High-tech Institute Changsha China were used. The acceleration-type vibration pickup is fixed at each measuring point to record the acceleration waveform when the pedestrian bridge vibrates. It is connected to the dynamic information acquisition instrument through the cable, and the acceleration signal is amplified, recorded, and analyzed. Finally, the acceleration signal is connected to a laptop computer and stored on a disk for spectrum analysis (Figure3).Figure 3
Dynamic test instrument. (a) Acceleration vibration pickup. (b) Dynamic information acquisition instrument.
(a)(b)11 measuring points are selected in this dynamic test, and the layout is shown in Figure4. Cylindrical semirigid rubber bearing is set on the top of the column, it is considered that no displacement occurs at this point, so no measuring point is set on the top of the column. The test was divided into two groups, and the first group was the measuring points 1-1 to 1-7 (Figure 4). In order to connect the test data of the two groups, position 1-7 of the first group was kept unchanged as the position 2-7 of the second group. The test system is shown in Figure 5.Figure 4
Layout of pedestrian bridge dynamic test points.Figure 5
Self-vibration test system.DASP software was used to analyze the measured data after the test [22, 23]. DASP software shows that the first-order vertical natural vibration frequency obtained from the dynamic test of the pedestrian bridge is 4.9804 Hz, and the second-order vertical natural vibration frequency is 5.9765 Hz. There is some error between the test results and the finite element calculation results.It can be seen from Table2 that the maximum error of the initial model of the pedestrian bridge established by MIDAS/Civil is only 10% in strict accordance with the design drawings, which has high accuracy. The initial model error is small, the degradation of material properties causes the error, and the parameters of materials in the structure can be predicted using neural networks [24, 25].Table 2
Comparison table of test results and calculation results.
OrderTest results (Hz)Calculation result (Hz)Error (Hz)Relative error (%)First order4.98044.46680.513610.3Second order5.97655.54810.42847.16


## 3.1. Initial Model Establishment

According to the design drawing size of the pedestrian bridge, the finite element model of the pedestrian steel bridge is established by using MIDAS/Civil. Semirigid rubber bearings were set for the main beam, escalator beam, and column of the pedestrian bridge, and consolidation boundary conditions were selected for the supports and top pillar. Mechanical properties of materials refer to Q235 steel in the General Specifications for Design of Highway Bridges and Culverts [19] (D60-2015). The finite element model is shown in Figure 2.Figure 2
Finite element model of the pedestrian bridge.The dynamic characteristics of the pedestrian bridge model were analyzed by using the subspace iteration method. The first 5-order natural vibration frequencies of the pedestrian bridge are calculated as shown in Table1.Table 1
Calculation table of the first five natural vibration frequencies of the pedestrian bridge.
OrderFirst orderSecond orderThird orderFourth orderFifth orderFrequency (Hz)4.46685.54816.24278.185710.4471According to the study of Fan et al. [20, 21], the walking frequency of human is 2∼4 Hz, and the first 2 vibration modes of the bridge need only be considered in model modification and dynamic test of the real bridge for the pedestrian bridge.


## 3.2. Self-Vibration Characteristics Test

The test was conducted in the early morning in order to minimize the influence of external factors such as the stream of people and vehicles on the dynamic test results. JMC2-2091 acceleration-type vibration pickup and JMYD-1016 dynamic information acquisition instrument from Jinma High-tech Institute Changsha China were used. The acceleration-type vibration pickup is fixed at each measuring point to record the acceleration waveform when the pedestrian bridge vibrates. It is connected to the dynamic information acquisition instrument through the cable, and the acceleration signal is amplified, recorded, and analyzed. Finally, the acceleration signal is connected to a laptop computer and stored on a disk for spectrum analysis (Figure3).Figure 3
Dynamic test instrument. (a) Acceleration vibration pickup. (b) Dynamic information acquisition instrument.
(a)(b)11 measuring points are selected in this dynamic test, and the layout is shown in Figure4. Cylindrical semirigid rubber bearing is set on the top of the column, it is considered that no displacement occurs at this point, so no measuring point is set on the top of the column. The test was divided into two groups, and the first group was the measuring points 1-1 to 1-7 (Figure 4). In order to connect the test data of the two groups, position 1-7 of the first group was kept unchanged as the position 2-7 of the second group. The test system is shown in Figure 5.Figure 4
Layout of pedestrian bridge dynamic test points.Figure 5
Self-vibration test system.DASP software was used to analyze the measured data after the test [22, 23]. DASP software shows that the first-order vertical natural vibration frequency obtained from the dynamic test of the pedestrian bridge is 4.9804 Hz, and the second-order vertical natural vibration frequency is 5.9765 Hz. There is some error between the test results and the finite element calculation results.It can be seen from Table2 that the maximum error of the initial model of the pedestrian bridge established by MIDAS/Civil is only 10% in strict accordance with the design drawings, which has high accuracy. The initial model error is small, the degradation of material properties causes the error, and the parameters of materials in the structure can be predicted using neural networks [24, 25].Table 2
Comparison table of test results and calculation results.
OrderTest results (Hz)Calculation result (Hz)Error (Hz)Relative error (%)First order4.98044.46680.513610.3Second order5.97655.54810.42847.16


## 4. Determine the Correction Parameters

### 4.1. Factors Affecting Natural Vibration Frequency

According to equations (1) and (2) for the approximate solution of the natural vibration frequency of the structure by the energy method given in the study of Ye et al. [26–28], the influencing factors of the natural vibration frequency of the structure can be determined.(1)ω2=∫01EIY″x2∫01m¯xYx2dx+∑miYi2,(2)f=ω2π,where f is the frequency; ω is the circular frequency; E is the elastic modulus of material; I is the section moment of inertia; Y(x) is the sectional shape function; Yi is the amplitude of the particle mi; m is the weight; l is the structural span.The natural vibration frequency of the structure is related to the elastic modulus of the material, sectional structure, material density, and the structural form of the bridge according to (1) and (2). According to the size, shape, weight, and allowable deviation of hot-rolled steel plate and strip [29, 30], a certain size deviation is allowed in the actual production process of components. The elastic modulus and density of components are the same, and the specific range is shown in Table 3.Table 3
Allowable range of material ex-factory performance.
Specification rangeThickness of steel plateH (mm)Elastic modulusE (Pa)Material densityρ (kg/m3)Upper limitH + 11.9 × 10117800Lower limitH − 12.1 × 10118000


### 4.2. Sensitivity Analysis of Influencing Factors

The sensitivity of candidate parameters should be identified first, and insensitive factors should be eliminated when selecting correction parameters. Because the nonsensitive coefficient has no obvious correction result to the model during the correction, the correction has little significance. Therefore, the model should be modified by modifying the sensitivity coefficient.Figure6 shows that the elastic modulus and density of materials are the main factors affecting the natural vibration frequency of structures. The sensitivity of the same parameter is not the same under different vibration modes. If only one mode is used as the correction target, the model modification will cause a certain deviation from the rest modes. Therefore, different vibration modes of the model to be modified should be taken as the correction target at the same time. In the first- and second-order cases, the elastic modulus and density of the material are still the main factors affecting the natural vibration frequency of the structure. Hence, the elastic modulus and density of the material can be used as the variables of model modification to modify the first- and second-order modes simultaneously.Figure 6
Sensitivity analysis of candidate parameters.


## 4.1. Factors Affecting Natural Vibration Frequency

According to equations (1) and (2) for the approximate solution of the natural vibration frequency of the structure by the energy method given in the study of Ye et al. [26–28], the influencing factors of the natural vibration frequency of the structure can be determined.(1)ω2=∫01EIY″x2∫01m¯xYx2dx+∑miYi2,(2)f=ω2π,where f is the frequency; ω is the circular frequency; E is the elastic modulus of material; I is the section moment of inertia; Y(x) is the sectional shape function; Yi is the amplitude of the particle mi; m is the weight; l is the structural span.The natural vibration frequency of the structure is related to the elastic modulus of the material, sectional structure, material density, and the structural form of the bridge according to (1) and (2). According to the size, shape, weight, and allowable deviation of hot-rolled steel plate and strip [29, 30], a certain size deviation is allowed in the actual production process of components. The elastic modulus and density of components are the same, and the specific range is shown in Table 3.Table 3
Allowable range of material ex-factory performance.
Specification rangeThickness of steel plateH (mm)Elastic modulusE (Pa)Material densityρ (kg/m3)Upper limitH + 11.9 × 10117800Lower limitH − 12.1 × 10118000


## 4.2. Sensitivity Analysis of Influencing Factors

The sensitivity of candidate parameters should be identified first, and insensitive factors should be eliminated when selecting correction parameters. Because the nonsensitive coefficient has no obvious correction result to the model during the correction, the correction has little significance. Therefore, the model should be modified by modifying the sensitivity coefficient.Figure6 shows that the elastic modulus and density of materials are the main factors affecting the natural vibration frequency of structures. The sensitivity of the same parameter is not the same under different vibration modes. If only one mode is used as the correction target, the model modification will cause a certain deviation from the rest modes. Therefore, different vibration modes of the model to be modified should be taken as the correction target at the same time. In the first- and second-order cases, the elastic modulus and density of the material are still the main factors affecting the natural vibration frequency of the structure. Hence, the elastic modulus and density of the material can be used as the variables of model modification to modify the first- and second-order modes simultaneously.Figure 6
Sensitivity analysis of candidate parameters.


## 5. Construct Neural Network

### 5.1. Sample Selection

A neural network algorithm mainly simulates the neural structure and working principle of biology to carry out mathematical modeling of the complex mapping relationship between input and output values [31]. Using a neural network to modify the computational model can not only avoid the iterative solution based on inverse problem model modification but also avoid the uncertainty and complexity of nonlinear optimization. The input layer, hidden layer, and output layer together constitute the basic structure of the neural network, and its working principle is shown in Figure 7.Figure 7
Working principle of the neural network.It is the set of right samples that are needed to provide for network optimization when you are building a neural network. The optimization process is to compare the calculated output value with the input sample value and then adjust the weight and threshold value under the training of learning rules to make the model output close to the sample value [32–34]. Sample selection was carried out in combination with the modified parameters selected in Section 3.2 and the material performance interval in Table 3. 10 equal parts are divided according to the difference between upper and lower limits of parameters; that is, each sample value increases the difference by 10% compared with the last sample. This gives 11 values for each correction parameter; the 2 parameters of the model and 11 values of each parameter can be regarded as a common combination of all parameters of a test (112 times). Sample selection is shown in Table 4.Table 4
Sample selection table.
OrderElasticity modulusE (Pa)Material densityρ (kg/m3)11.9 × 1011780021.92 × 1011782031.94 × 1011784041.96 × 1011786051.98 × 1011788062.0 × 1011790072.02 × 1011792082.04 × 1011794092.06 × 10117960102.08 × 10117980112.1 × 10118000The 121 training samples were divided into two parts, 111 training sets and 10 test sets, respectively. All samples were selected according to the principle of randomness in order to ensure the reliability of network training.


### 5.2. BP Neural Network

As a basic multilayer forward neural network, BP neural network has one input layer and one output layer. There are several hidden layers between the input and output layers, each adopts full connection mode, and the neurons of the same layer are independent [35, 36]. The neurons of the hidden layer adopt the S-type function and linear transfer function in the output layer. A three-layer neural network with a hidden layer can approximate any nonlinear function theoretically [37, 38]. BP neural network correction adopts an error backpropagation algorithm, which takes the weight and threshold as the cause of the error and apportions the error to the weight and threshold of each neuron through backpropagation algorithm, as shown in Figure 8. There are m neurons in the input layer, l neurons in the hidden layer, and n neurons in the output layer. The hidden layer adopts the S-type transfer function tansig, and the output layer adopts the linear function purelin. According to statistics, about 75% of neural networks are derived from BP neural network [39, 40], and BP neural network is considered to be a relatively reliable neural network.Figure 8
Structure of BP neural network.BP neural network can be created by calling thenewff function in MATLAB neural network toolbox. The test results after training are shown in Table 5.Table 5
Test results of BP neural network.
OrderTheoretical valuePredicted valueDensityρ (N/m3)Elastic modulusE (Pa)Densityρ (N/m3)Elastic modulusE (Pa)17.800 × 1041.940 × 10117.807 × 1041.943 × 101127.860 × 1042.000 × 10117.867 × 1042.002 × 101137.940 × 1042.060 × 10117.938 × 1042.059 × 101147.880 × 1041.960 × 10117.879 × 1041.962 × 101157.900 × 1042.100 × 10117.895 × 1042.099 × 101167.840 × 1041.980 × 10117.848 × 1041.982 × 101177.960 × 1041.960 × 10117.974 × 1041.963 × 101187.860 × 1042.020 × 10117.867 × 1042.023 × 101197.940 × 1042.100 × 10117.939 × 1042.100 × 1011107.920 × 1041.980 × 10117.919 × 1041.983 × 1011


### 5.3. GRNN

GRNN, also known as a generalized neural network, is a feedforward neural network model based on the nonlinear regression principle. The structure of the GRNN is shown in Figure9 [41, 42]. The input layer of the GRNN only sends samples into the hidden layer, and the number of neurons in the hidden layer is the same as the number of samples in the training set [43, 44]. In the hidden layer, the weight can be expressed by the Euclidean distance function IIdistII, and the radial basis function can be expressed by the Gaussian function. The output layer adopts linear output, and its weight function adopts normalized dot product weight function nprod. At last, the calculation network vector n2 is provided to the line transfer function purlin to obtain the calculation result [45, 46]. GRNN has the advantages of simple structure, simple training, and fast learning convergence.Figure 9
Structure of GRNN.GRNNs can be created by calling the MATLAB neural network toolboxnewgrnn function. The test results after training are shown in Table 6.Table 6
Test results of GRNN.
OrderTheoretical valuePredicted valueDensityρ (N/m3)Elastic modulusE (Pa)Densityρ (N/m3)Elastic modulusE (Pa)17.800 × 1041.940 × 10117.801 × 1041.937 × 101127.860 × 1042.000 × 10117.861 × 1041.999 × 101137.940 × 1042.060 × 10117.941 × 1042.071 × 101147.880 × 1041.960 × 10117.890 × 1041.959 × 101157.900 × 1042.100 × 10117.901 × 1042.101 × 101167.840 × 1041.980 × 10117.839 × 1041.979 × 101177.960 × 1041.960 × 10117.951 × 1041.958 × 101187.860 × 1042.020 × 10117.871 × 1042.020 × 101197.940 × 1042.100 × 10117.921 × 1042.080 × 1011107.920 × 1041.980 × 10117.891 × 1041.999 × 1011


### 5.4. Reliability Evaluation

The test results of the two neural networks are all good through the preliminary judgment [47, 48]. The relative error Ei, relative error variance S2, and determination coefficient R2 of test samples were selected to evaluate the reliability and error analysis of the two neural networks [49, 50]. The calculation methods of each coefficient are shown in equations (3)–(5), respectively.(3)Ei=Yi−yiyi,(4)S2=∑i=1nEi−E´i2n−1,(5)R2=n∑i=1nYiyi−∑i=1nYi∑i=1nyin∑i=1nYi2−∑i=1nYI2n∑i=1nyi2−∑i=1nyi2,where Yi (i = 1, 2, ···, n) is the predicted value of the sample i; Yi (i = 1, 2, ···, n) is the theoretical value of the sample i; n is the total number of samples.The prediction of density and elastic modulus by the neural network is shown in Figure10, where the abscissa is the sample ordinal number, and the ordinate is density and elastic modulus, respectively [51, 52].Figure 10
Reliability of neural network. (a) Density prediction reliability. (b) Reliability of elastic modulus prediction.
(a)(b)Figure10 shows that the prediction curves of the two neural networks have a good coincidence with the real value curves. This indicates that the predicted value is very close to the real value, which proves that BP neural network and GRNN achieve good prediction results in density and elastic modulus. In terms of reliability, the R2 values of the BP neural network in density and elastic modulus prediction are 0.99, and the GRNN is 0.92 and 0.97, respectively, which indicates the reliability of the BP neural network is better than GRNN in model modification [53, 54].Relative error and relative error value variance are calculated from (3) and (4), as shown in Figure 11.Figure 11
Relative error analysis. (a) Density relative error. (b) Relative error of elastic modulus.
(a)(b)As can be seen from Figure11, in terms of model modification and prediction, BP neural network has better stability than GRNN, and the amplitude and incidence of mutation are also smaller.The reliability and error fluctuation of the BP neural network in the model modification are better than that of the GRNN after comparison, which BP neural network model is better used to modify the pedestrian bridge. Using BP neural network, the predicted material density and elastic modulus areρ = 7.84 × 104N/m3 and E = 2.15 × 1011Pa, respectively. The results are shown in Table 7.Table 7
Comparison of corrected errors.
OrderTest result (Hz)Calculation result (Hz)Relative error (%)Correction results (Hz)Relative error (%)Correction value (%)First order4.98044.466810.34.63177.52.8Second order5.97655.54817.165.77163.63.56


## 5.1. Sample Selection

A neural network algorithm mainly simulates the neural structure and working principle of biology to carry out mathematical modeling of the complex mapping relationship between input and output values [31]. Using a neural network to modify the computational model can not only avoid the iterative solution based on inverse problem model modification but also avoid the uncertainty and complexity of nonlinear optimization. The input layer, hidden layer, and output layer together constitute the basic structure of the neural network, and its working principle is shown in Figure 7.Figure 7
Working principle of the neural network.It is the set of right samples that are needed to provide for network optimization when you are building a neural network. The optimization process is to compare the calculated output value with the input sample value and then adjust the weight and threshold value under the training of learning rules to make the model output close to the sample value [32–34]. Sample selection was carried out in combination with the modified parameters selected in Section 3.2 and the material performance interval in Table 3. 10 equal parts are divided according to the difference between upper and lower limits of parameters; that is, each sample value increases the difference by 10% compared with the last sample. This gives 11 values for each correction parameter; the 2 parameters of the model and 11 values of each parameter can be regarded as a common combination of all parameters of a test (112 times). Sample selection is shown in Table 4.Table 4
Sample selection table.
OrderElasticity modulusE (Pa)Material densityρ (kg/m3)11.9 × 1011780021.92 × 1011782031.94 × 1011784041.96 × 1011786051.98 × 1011788062.0 × 1011790072.02 × 1011792082.04 × 1011794092.06 × 10117960102.08 × 10117980112.1 × 10118000The 121 training samples were divided into two parts, 111 training sets and 10 test sets, respectively. All samples were selected according to the principle of randomness in order to ensure the reliability of network training.


## 5.2. BP Neural Network

As a basic multilayer forward neural network, BP neural network has one input layer and one output layer. There are several hidden layers between the input and output layers, each adopts full connection mode, and the neurons of the same layer are independent [35, 36]. The neurons of the hidden layer adopt the S-type function and linear transfer function in the output layer. A three-layer neural network with a hidden layer can approximate any nonlinear function theoretically [37, 38]. BP neural network correction adopts an error backpropagation algorithm, which takes the weight and threshold as the cause of the error and apportions the error to the weight and threshold of each neuron through backpropagation algorithm, as shown in Figure 8. There are m neurons in the input layer, l neurons in the hidden layer, and n neurons in the output layer. The hidden layer adopts the S-type transfer function tansig, and the output layer adopts the linear function purelin. According to statistics, about 75% of neural networks are derived from BP neural network [39, 40], and BP neural network is considered to be a relatively reliable neural network.Figure 8
Structure of BP neural network.BP neural network can be created by calling thenewff function in MATLAB neural network toolbox. The test results after training are shown in Table 5.Table 5
Test results of BP neural network.
OrderTheoretical valuePredicted valueDensityρ (N/m3)Elastic modulusE (Pa)Densityρ (N/m3)Elastic modulusE (Pa)17.800 × 1041.940 × 10117.807 × 1041.943 × 101127.860 × 1042.000 × 10117.867 × 1042.002 × 101137.940 × 1042.060 × 10117.938 × 1042.059 × 101147.880 × 1041.960 × 10117.879 × 1041.962 × 101157.900 × 1042.100 × 10117.895 × 1042.099 × 101167.840 × 1041.980 × 10117.848 × 1041.982 × 101177.960 × 1041.960 × 10117.974 × 1041.963 × 101187.860 × 1042.020 × 10117.867 × 1042.023 × 101197.940 × 1042.100 × 10117.939 × 1042.100 × 1011107.920 × 1041.980 × 10117.919 × 1041.983 × 1011


## 5.3. GRNN

GRNN, also known as a generalized neural network, is a feedforward neural network model based on the nonlinear regression principle. The structure of the GRNN is shown in Figure9 [41, 42]. The input layer of the GRNN only sends samples into the hidden layer, and the number of neurons in the hidden layer is the same as the number of samples in the training set [43, 44]. In the hidden layer, the weight can be expressed by the Euclidean distance function IIdistII, and the radial basis function can be expressed by the Gaussian function. The output layer adopts linear output, and its weight function adopts normalized dot product weight function nprod. At last, the calculation network vector n2 is provided to the line transfer function purlin to obtain the calculation result [45, 46]. GRNN has the advantages of simple structure, simple training, and fast learning convergence.Figure 9
Structure of GRNN.GRNNs can be created by calling the MATLAB neural network toolboxnewgrnn function. The test results after training are shown in Table 6.Table 6
Test results of GRNN.
OrderTheoretical valuePredicted valueDensityρ (N/m3)Elastic modulusE (Pa)Densityρ (N/m3)Elastic modulusE (Pa)17.800 × 1041.940 × 10117.801 × 1041.937 × 101127.860 × 1042.000 × 10117.861 × 1041.999 × 101137.940 × 1042.060 × 10117.941 × 1042.071 × 101147.880 × 1041.960 × 10117.890 × 1041.959 × 101157.900 × 1042.100 × 10117.901 × 1042.101 × 101167.840 × 1041.980 × 10117.839 × 1041.979 × 101177.960 × 1041.960 × 10117.951 × 1041.958 × 101187.860 × 1042.020 × 10117.871 × 1042.020 × 101197.940 × 1042.100 × 10117.921 × 1042.080 × 1011107.920 × 1041.980 × 10117.891 × 1041.999 × 1011


## 5.4. Reliability Evaluation

The test results of the two neural networks are all good through the preliminary judgment [47, 48]. The relative error Ei, relative error variance S2, and determination coefficient R2 of test samples were selected to evaluate the reliability and error analysis of the two neural networks [49, 50]. The calculation methods of each coefficient are shown in equations (3)–(5), respectively.(3)Ei=Yi−yiyi,(4)S2=∑i=1nEi−E´i2n−1,(5)R2=n∑i=1nYiyi−∑i=1nYi∑i=1nyin∑i=1nYi2−∑i=1nYI2n∑i=1nyi2−∑i=1nyi2,where Yi (i = 1, 2, ···, n) is the predicted value of the sample i; Yi (i = 1, 2, ···, n) is the theoretical value of the sample i; n is the total number of samples.The prediction of density and elastic modulus by the neural network is shown in Figure10, where the abscissa is the sample ordinal number, and the ordinate is density and elastic modulus, respectively [51, 52].Figure 10
Reliability of neural network. (a) Density prediction reliability. (b) Reliability of elastic modulus prediction.
(a)(b)Figure10 shows that the prediction curves of the two neural networks have a good coincidence with the real value curves. This indicates that the predicted value is very close to the real value, which proves that BP neural network and GRNN achieve good prediction results in density and elastic modulus. In terms of reliability, the R2 values of the BP neural network in density and elastic modulus prediction are 0.99, and the GRNN is 0.92 and 0.97, respectively, which indicates the reliability of the BP neural network is better than GRNN in model modification [53, 54].Relative error and relative error value variance are calculated from (3) and (4), as shown in Figure 11.Figure 11
Relative error analysis. (a) Density relative error. (b) Relative error of elastic modulus.
(a)(b)As can be seen from Figure11, in terms of model modification and prediction, BP neural network has better stability than GRNN, and the amplitude and incidence of mutation are also smaller.The reliability and error fluctuation of the BP neural network in the model modification are better than that of the GRNN after comparison, which BP neural network model is better used to modify the pedestrian bridge. Using BP neural network, the predicted material density and elastic modulus areρ = 7.84 × 104N/m3 and E = 2.15 × 1011Pa, respectively. The results are shown in Table 7.Table 7
Comparison of corrected errors.
OrderTest result (Hz)Calculation result (Hz)Relative error (%)Correction results (Hz)Relative error (%)Correction value (%)First order4.98044.466810.34.63177.52.8Second order5.97655.54817.165.77163.63.56


## 6. Conclusion

In this paper, the dynamic characteristics of the pedestrian bridge in Urumqi China were tested to obtain the dynamic parameters of the bridge, and the neural network was used to modify the pedestrian bridge model established by MIDAS/Civil. The following conclusions are drawn.(1)
The pedestrian bridge model established by MIDAS/Civil has high accuracy, which is found by comparing the dynamic test results with the calculation results of the initial finite element model. The degraded material parameters can be simulated by the neural network, so that the simulation results can more truly reflect the actual stress state of the structure.(2)
Through the calculation formula of natural vibration frequency, the factors affecting the natural vibration frequency of the structure are as follows: the dimensions, elastic modulus, and density of the material. The sensitivity analysis shows that the elastic modulus and material density have the greatest influence on the natural vibration frequency of the structure. The sensitivity of the same material is different in each order natural frequency.(3)
Based on reliability analysis and error analysis of calculation results of the BP neural network and GRNN, it can be seen that the two neural networks have a good effect on model modification, but the BP neural network has better reliability and smaller error fluctuation in model modification. BP neural network can be used to modify the structure model, and the modified model is closer to the real stress state of the structure.



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*Source: 1057422-2022-05-02.xml*