Application of BP Neural Network Improved by Fireworks Algorithm on Suspender Damage Prediction of Long-Span Half-Through Arch Bridge

. In recent decades, with the large-scale construction and rapid development of half-through arch bridges, as well as the increase of bridge service time, the suspender damage of arch bridge has become increasingly prominent. Terefore, real-time monitoring and regular detection of the health of arch bridge suspenders and timely detection and accurate judgment of the damage location and extent of suspenders are of great engineering signifcance for evaluating the reliability and residual life of arch bridge structures. By analyzing the main difculties and existing problems of suspender damage identifcation, this paper takes the change rate of modal curvature as the damage index, introduces freworks algorithm into the neural network model, optimizes the optimization process of neural network weight and threshold, and proposes a prediction model based on improved BP neural network by freworks algorithm. According to the measured data of the damage degree of a long-span arch bridge in daily monitoring and on-site inspection, the proposed prediction method is applied to verify the efectiveness and accuracy in engineering health detection. On this basis, the improved BP neural network by freworks algorithm is used to predict the suspender damage of a certain long-span half-through arch bridge, which provides an important basis for the actual bridge safety assessment.


Introduction
With the development of smart cities and the progress of science and technology, countries all over the world have increased their investment in infrastructure construction. A large number of complex structural forms have emerged, such as super-high-rise buildings, long-span spatial structures, super-large bridges, large dams, nuclear power plants, and large marine structures [1,2]. Tese structural forms have brought great convenience to economic development and people's lives, but once damaged, they will do great harm to our cities. Large scale and complexity are the development direction of the structure, and its service life is often decades or even hundreds of years. In this service process, the structure will be damaged to varying degrees due to congenital defects such as design and construction, catastrophic factors such as external load, environmental factors, material aging, corrosion efect, fatigue efect, and other uncertain factors [3]. After the damage, the bearing capacity and durability of the structure will be afected, the ability to resist external forces will be signifcantly reduced, and then accidents will occur. Tis will lead to heavy casualties and economic losses and cause adverse social impact [4]. Taking the bridge structure as an example, the data statistics show that various deterioration phenomena occur on the bridge decks of about 253000 concrete bridges in the United States. Some of the bridges have been damaged in diferent forms and degrees in less than 20 years of service. Moreover, 35000 bridges will be added every year, and the average number of partially or completely collapsed bridges is about 200 every year. Te construction cost of 11 concrete viaducts located on the ring expressway in the middle of England Island is 28 million pounds. However, the maintenance cost reached over 120 million pounds, which will be close to six times the original cost. According to the Yangcheng Evening News in China, the Department of Communications of Guangdong Province organizes a large number of personnel to conduct a general survey on the technical status of existing and under-construction roads and bridges in the province. Te results showed that of the 18700 bridges in Guangdong Province, 4244 were in category III and IV poor conditions and had insufcient bearing capacity, accounting for 22.7% of the total census, with a cumulative length of 109616 linear meters. For example, arch bridges have been widely used due to their beautiful appearance, simple construction methods, and strong spanning capacity. Te suspenders of arch bridges in service commonly sufer from diseases, which directly afect the safety and durability of bridge structures. Te suspender can be damaged or even broken due to corrosion, fatigue, and other reasons, which can greatly shorten the service life of the bridge and increase the risk of bridge collapse, as shown in Figure 1. Terefore, it is particularly important to identify the damage of bridge suspenders [5].
It can detect and predict the performance of the structure in real time, fnd and judge the damage location and degree of the structure in time, and then predict the performance change and remaining life of the structure to obtain the maintenance decision and the evacuation of local residents, which is of great signifcance to improve the service efciency of engineering structures and ensure the safety of people's lives and property [6]. As the core of structural health monitoring, the successful research of damage identifcation has essential guiding signifcance for how to establish the health monitoring system of engineering structures. Terefore, the research on structural damage identifcation has become a hot issue in the feld of structural health detection [7]. Structural damage detection technology and its identifcation methods have made great progress in academic or practical application research in recent years, but there are still many problems to be further studied and solved in the damage detection of complex civil engineering structures such as high-rise buildings and bridges [8]. A considerable part of the existing damage detection technologies and methods for civil engineering structures are copied from aerospace, aerospace, and mechanical structures. When the same technology and method are introduced into another discipline, we should pay attention to its applicability and the characteristics of this discipline.
At present, there are also some technical difculties in the feld of engineering structure damage detection [9]. Firstly, civil engineering structures are diferent from aviation, aerospace, and mechanical structures. Relatively large model error is allowed in design, analysis, and calculation. However, if a large error exists in the model used for detection, it will lead to a great diference between the calculated and actual dynamic characteristics of the damaged structure, so the error of the detection results based on the dynamic characteristics will be very large. Secondly, noise is unavoidable due to the infuence of many factors of the engineering structure. In the process of longterm health detection, noise may be introduced at every step and link from data acquisition to transmission. Terefore, a good ability to flter noise is what the identifcation method used should have. At present, damage identifcation in engineering structures is a nonunique problem, and if it cannot be well distinguished, it will result in unpredictable results of damage location and degree.
Because of the existence of these factors, many damage identifcation methods become invalid, which makes the research of damage identifcation face bottleneck. Terefore, a new method to overcome the above difculties has become an urgent need [10][11][12]. Te dynamic characteristics and response of the structure will change with the damage of the structure. In other words, there is a complex nonlinear relationship between the dynamic characteristics and response changes before and after the damage and the damage location and degree of the structure. Te traditional acoustic emission method, ultrasonic method, infrared method, and other nondestructive testing techniques are not only timeconsuming and expensive but also cannot detect some parts of large structures [13][14][15]. However, the dynamic characteristics and response of structures can be obtained through various detection methods and modal analysis. Te artifcial neural network (ANN) can take the damage index related to the dynamic characteristics and a certain response of the structure in various states as the input vector and take the damage diagnosis results (whether the damage exists, the damage location, damage degree, etc.) in various states as the output vector [16]. By learning to form a mapping, the neural network weights containing this mapping relationship can be saved, and there is no need to call the analysis model in the back analysis process. Trough the efective and fast forward operation of the weight obtained by learning and the damage index obtained by detection, the damage diagnosis results can be obtained during the online diagnosis [17]. In a word, the achievements and research of artifcial neural network in this feld are still in the basic exploration stage, and it still needs the continuous eforts and exploration of relevant personnel to make the structural health monitoring technology and damage identifcation method better serve the feld of engineering structures [18]. Since the performance of the neural network-based prediction model largely depends on the network structure and the weights and thresholds of each node of the network, the neural network structure, initial weights, and thresholds of the network will greatly restrict the prediction accuracy and convergence of the neural network-based prediction model.
To sum up, aiming at the problem that the single neural network model has slow convergence speed and is easy to fall into local optimization, the existing research mainly focuses on the optimization and improvement of the neural network using intelligent optimization algorithms such as particle swarm optimization (PSO) and genetic algorithm (GA), which solves the problem of low prediction accuracy of the single neural network prediction model to a certain extent [19][20][21][22][23]. However, with the deepening of research, scholars found that the above algorithm itself also has aspects to be improved [24]. Improper setting of genetic operator will afect the search performance of the algorithm and make the algorithm easy to fall into local optimal solution [25], and too large initial population in PSO algorithm will lead to the problem of slow search speed of the algorithm, which will restrict the performance of optimizing BP neural network prediction model based on swarm intelligence algorithm [26]. For example, the BP neural network prediction model optimized based on GA algorithm is limited to the large sample data model, while the prediction ability of the sample model with small samples and uneven distribution is not signifcantly improved [27]. At the same time, the diversity of particles will be lost due to the too fast particle optimization speed, which restricts the accuracy of the prediction results of BP neural network model optimized based on PSO algorithm.
Fireworks algorithm (FWA) is a new swarm intelligence optimization algorithm proposed by Tan et al., which works by simulating the mechanism of simultaneous explosion and difusion of frework at multiple points in the air [28]. It shows high optimization performance in solving optimization problems and has attracted the attention of scholars at home and abroad. Compared with GA and PSO algorithms, FWA simulates the mechanism of simultaneous explosion and difusion of frework explosion operators and ensures the diversity of frework population; at the same time, the freworks algorithm has stronger global search ability by introducing the idea of immune concentration and the distributed information sharing mechanism. Terefore, this paper introduces the freworks algorithm (FWA) into the BP neural network model to optimize the weight and threshold of BP neural network, proposes a prediction model based on the freworks algorithm to improve BP neural network (FWA-BPNN), to solve the problem that the traditional BP neural network prediction model has slow convergence speed and is easy to fall into the local optimal solution in the training process, and applies it to the damage prediction of long-span arch bridges.

Classical Model of BP Neural Network
BP neural network shows good self-learning and selfadjusting ability in solving nonlinear problems and is widely used to solve complex system prediction problems with many factors interlaced. Te essence of the BP neural network prediction model is to train the model through a large number of data in the fnite solution space and then fnd the weight w ij and threshold between network neurons θ i and other parameters to establish the mapping relationship between input and output and minimize the network error, as follows: ① Initialize the network weights and thresholds: the initial weights and thresholds of the network are initialized randomly in the interval [1,1]. ② Feed forward calculation: assuming that the weight value w (l) ij (k) of the network in the kth iteration process, the threshold value θ (l) i (k) of the ith neuron, and the expected output t i (k) of the ith neuron node in the lth layer are known, then is the input of the ith neuron in the lth layer of the neural network; y (l) i (k) is the output of the lth layer; l is the layer number of the network and l � 1, 2, . . . L; 1 ≤ i ≤ S L ; and y (0) (k) � x(k). ③ Error backpropagation: calculate the error δ (l) i (k) of the lth layer in the kth iteration of the neural network through equations (2) and (3).
where t i (k) is the expected output value of the ith neuron node, and 1 ≤ i ≤ m � S l . Based on this, δ (l) i (k) can be calculated by recursion formula (2) and ④ Update network weights and thresholds: use equation (4) to update the weights and thresholds of the neural network.
where w (l) ij (k + 1) and θ (l) i (k + 1) are the weights and thresholds of the network in the k + 1 iteration process, respectively; α is the momentum factor; 1 ≤ i ≤ S l ; 1 ≤ j ≤ S l− 1 ; and 1 ≤ l ≤ L.

Fireworks Algorithm.
Fireworks algorithm is a new swarm intelligence optimization algorithm. For the optimization problem min f(x) ∈ R, x ∈ Ω, to be solved, freworks algorithm is used to solve the optimization problem. Te specifc steps are as follows: ① Initialize the population: Some freworks are randomly generated in a specifc solution space. Each frework individual x i represents a solution in the solution space, that is, x i ∈ Ω. ② Calculate the ftness value: for each frework individual x i in the initial population, calculate the ftness value f(x i ) according to the ftness function f(x) and calculate the number of freworks produced by each frework explosion S i and the explosion radius A i according to the following equations: where y max � max(f(x i )) (i � 1, 2, . . ., n) is the ftness value of the individual with the worst ftness value of all freworks in the current population; y min � min (f(x i )) (i � 1, 2, . . ., n) is the ftness value of the best individual in the current population; c and d are constants, which are, respectively, used to limit the total number of sparks and represent the maximum explosion radius; and ε is a constant used to avoid the denominator being zero. ③ Generate sparks: Randomly select z dimensions to form a set Z, where z � rand (1, d × rand (0, R i )), and rand (0, R i ) is a random number generated within the explosion radius A i . In set Z, for each dimension k, use equations (7) and (8) to perform explosion mutation on freworks, map sparks beyond the boundary through the Gaussian mutation mapping rules in equation (9), and save them in the spark population.
where A i is the explosion radius of the ith frework; h is the position ofset; x ik is the kth dimension of the ith frework in the population; ex ik is the spark generated by the explosion of the ith frework; cx ik is the Gaussian variation spark of x ik after Gaussian variation; and r follows the Gaussian distribution.
④ Select the next generation group: Te next generation frework population is selected by using the selection strategy, that is, N frework individuals are selected from the frework explosion sparks and Gaussian variation spark populations to form the next generation candidate frework population. For the candidate frework population K, the selection strategy is as follows: select the individual x k with the minimum ftness value min (f(x i )) as the next generation of frework population individuals directly, and the remaining N − 1 frework individuals adopt the roulette gambling method. For the candidate individual x i , the probability formula (10) is adopted for its selection.
where R(x i ) is the sum of the distances between frework individual x i and other individuals, which is calculated by the following formula: ⑤ Determine termination conditions: if the termination conditions are met, stop the iteration; otherwise, continue with step ②.

Improved BP Neural Network Prediction Model Using
Fireworks Algorithm for Suspender Damage. Te weight and threshold of BP neural network are the key factors, which afect the prediction performance of the BP neural network model. Terefore, the freworks algorithm is introduced into the neural network model, and the position x ik of frework individuals in the frework population is used to represent the weight coefcient of network nodes and the threshold of network neurons. Based on the above rules, the specifc improvement strategies are as follows: ① Key parameter code: because the weights, thresholds, and frework individuals in the neural network are composed of a series of vectors, the real vector coding strategy is selected to code the key parameters in the model.
represents a set of parameters to be optimized, in which each dimension is composed of network weights and thresholds. In the neural network, note n IW (1,1) as the number of weight values between the input layer and the hidden layer, n b (1,1) as the number of neuron thresholds in the hidden layer, n IW (2,1) as the number of weights between the hidden layer and the output layer, and n b (2,1) as the number of neuron thresholds in the output layer; then, D � n IW(1,1) + n b(1,1) + n IW(2,1) + n b (2,1) .
② Calculate the ftness value: Initialize weight coefcient and threshold. Initialize the weight coefcient and threshold between nodes in the neural network in the interval [−1, 1], i.e., x i ∼U[−1, 1], and use the position of frework individual x i in the freworks algorithm to represent the weight coefcient of network nodes and the threshold of neurons, and then each frework individual represents a neuron in the neural network model. ③ Select the ftness function: Te goal of algorithm model training is to make the network output layer result as close as possible to the expected result through continuous iterative calculation, to obtain the weight parameter w (l) ij (k) and threshold value θ i between nodes when the network output result is optimal. In the FWA-BP neural network, the square error function is introduced to calculate the ftness value of individual freworks.
where t is the expected output of the network; P is the number of layers of the network; S is the number of network output units; and y is the actual output value of the network. Te actual output value of the network is specifcally expressed in the following equation: where x j is the input of the network; w ij is the weight of network nodes; θ i is the threshold value of the ith neuron in the network; and θ i � −w i (n + 1).
Te ftness function f i (x) is given in the following equation: ④ Optimize frework population: For each frework individual x, calculate its ftness value f(x i ) with equation (14), and calculate the number of explosive freworks S i and explosion radius A i with equations (5) and (6). At the same time, based on equations (7)-(9), each frework individual is operated with explosion, displacement, and mutation, and the selection strategy of equations (10) and (11) is used to select the best frework individual to form the next generation of frework population. ⑤ Determine termination conditions: According to equations (11) and (14), calculate the ftness value f(x i ) of the frework individuals in the frework population and the Euclidean distance R(x i ) between the frework individuals and judge whether the termination condition of the maximum number of iterations is satisfed. If it is satisfed, the new frework population is composed of the frework individuals with the minimum ftness value min (f(x i )) and the frework individuals with the maximum distance max (R(x i )) in the current frework population, and take the current frework population as the optimal frework population X best ; otherwise, continue with step ⑤. ⑥ Update network weights and thresholds: Use the optimal frework population X best obtained in step ⑤ to initialize and update the weight and threshold vector X in the network model. Based on the above steps, the fowchart of the whole FWA-BP algorithm can be obtained, as shown in Figure 2. provided for the entire bridge, including 5 above the bridge deck and 2 below the bridge deck. Te cross bracing of the arch crown adopts a meter shaped lattice structure, and the rest adopts a straight shaped lattice structure. Te bridge deck system consists of 41 cast-in-situ sections and 38 prefabricated beams, and continuous bridge decks are formed between adjacent beams through castin-situ fange joints. Prefabricated beams are divided into upper column beams and suspender beams, both of which are prestressed concrete open box girders. Te bridge deck of the column beam and suspender beam on the arch is a simply supported structure, which releases the temperature change and shrinkage and creep displacement of the entire continuous bridge deck system structure through the expansion joints at the simply supported ends. A box shaped steel beam is provided at the intersection of the bridge deck system and the arch ribs, with a total of two sets for the entire bridge. Te bridge deck is paved with 9 cm thick steel fber concrete and 4 cm thick asphalt concrete. Te bridge deck adopts a 3% two-way longitudinal slope and a 1.5% two-way transverse slope. Te longitudinal slope of the bridge deck is adjusted by the inclination of the beam fange plate, and the transverse slope is adjusted by the height change of the beam.

Health Inspection Analysis of Long-Span
In order to improve stability, column top tie beams have been added to No. 1 and No. 2 columns with column heights exceeding 8 m above the arch. Each suspender beam is equipped with double suspenders with a spacing of 1.5 m, and the suspension rod is composed of 55 galvanized highstrength steel wires with a diameter of 5mm, and the outer layer is protected by hot extruded polyethylene. Both ends are cold casting heading anchorage. To avoid direct exposure to the atmosphere, the upper and lower anchor heads are protected by protective covers. To protect the exposed steel wire (approximately 50 cm long) near the anchor head under the suspender, cement mortar is poured into the conduit under the suspender. At the same time, the suspender is wrapped with stainless steel within 3 m above the bridge deck to avoid man-made damage.
Te vertical web, diagonal web, and lateral connection systems of this bridge adopt hollow steel tube structures that are not flled with concrete. Te inner surface of the empty pipe structure is required to be sealed and coated with two layers of rust-resistant paint. Te anti-corrosion treatment of the outer surface of the steel structure adopts the GCM polymer material protection system. Tere are two forms of anti-corrosion for the outer surface of the arch foot outer covering section structure: the surface in contact with the concrete (including the batten plate) is not subject to anticorrosion treatment, the shear key is welded, and rust removal treatment is conducted to ensure a good combination with the concrete. Te surface of the exposed part also adopts a GCM polymer material protection system.

Suspender Damage Experimental Analysis of the Long-Span Arch Bridge.
Te bridge is a key transportation hub connecting the north and south sides, with an average of more than 12000 vehicles passing through it every day. Since its completion and opening to trafc, it has been operating  for a long time, so long-term and comprehensive health testing and related scientifc experiments have been conducted on the bridge. Te outer surface of the arch rib steel structure adopts a GCM polymer material protection system. Upon inspection, it was found that the anti-corrosion coating surface of the steel pipe arch rib has anti-rust phenomenon, with severe peeling and peeling in some parts and cracks in some parts. Tere are cracks on the concrete surface wrapped around the arch rib, mainly along the arch axis and perpendicular to the arch axis. In addition, there are a small number of oblique cracks, and a small number of cracks are accompanied by bleeding phenomenon.
Tere are 120 suspenders in the whole bridge, all of which are high-strength steel wire bundles. Except for the eight suspenders whose upper anchor heads are anchored in the upper chord batten concrete, the remaining suspenders' upper anchor heads are all anchored in the lower chord batten concrete, and the anchor heads are all cold casting heading anchorage. Upon inspection of the protective cover, it was found that the anti-corrosion coating of the anchor head protective cover had pitting and spot corrosion, and in severe cases, there were peeling and peeling phenomena, as well as the lack of fxing bolts. Te statistical results of the diseases of the upper anchor head protective cover are shown in Table 1.
For the inspection of anchor heads, 16 anchor heads are selected from both upstream and downstream sides at the upper end for inspection, while the frst anchor head from both upstream and downstream sides and the midspan anchor head are selected for inspection at the lower end. Upon inspection, it was found that there was condensed water on the inner wall of the protective cover and the top of the anchor cup cover, there was rust on the outer side of the anchor cup, the butter in the anchor cup had dried and evaporated, and the steel wire pier head was exposed and corroded. Te inspection results of anchor head corrosion are shown in Table 2. For ease of expression, the anchor heads are numbered sequentially from south to north. Te upstream, downstream, and upper and lower ends are distinguished by UT, UB, DT, and DB. For example, UB2 represents the second lower anchor head on the upstream side, and so on.
As can be seen from Table 2, most of the upper and lower anchor head protective covers have condensed water. When there is no or a small amount of accumulated water inside the anchor cup of the upper anchor head, the steel wire pier head may experience whitening. When there is a large amount of accumulated water, the steel wire pier head will produce slight rust. Te corrosion condition of the suspender was inspected, and some suspenders were selected to inspect the cable body under the protection of the intermediate PE pipe. It was found that the suspender cable body was not corroded. After the inspection, the cable body is sealed with cellophane and epoxy resin.
Te surface of the precast beam concrete for the bridge deck system is uniform in color, and there are no cracks, peeling, and exposed reinforcement. However, there are alkalization and whitening phenomena on the local concrete surface. Te cast-in-place concrete has cracks, mainly longitudinal cracks. Some cracks exceed the limit in width, with a maximum of 0.41 mm. In addition, there are a few oblique cracks. Te bridge deck pavement shall be free of looseness, oil spillage, cracking, waves, ruts, pits, and subsidence. Most expansion joints are blocked by foreign objects and lose their expansion function. Te measurement of bridge geometry includes the measurement of bridge deck geometry and arch axis geometry, which is arranged during the period when the structural temperature tends to stabilize.
Te bridge deck alignment measurement is conducted using a precision electronic level combined with an indium steel ruler. Under the condition of closing all trafc on the bridge deck, it is divided into two zones, upstream and downstream, for round-trip closed leveling. Te measuring points are arranged at the eighth point of the bridge deck. Te permanent measurement is arranged inside the collision barrier of the upstream and downstream side trafc lanes. Comparing the design value and the measured value of the bridge deck alignment, it is found that the overall bridge deck alignment has decreased, with the diference between the measured value and the design value being between −0.095 m and 0.069 m.
Te arch axis is measured using a tunnel section detector. Due to site conditions, only the arch axis elevation within 68.6 m from the midspan was tested and compared with the design value. It was found that the measured arch rib alignment slightly changed compared to the design alignment, with a diference between −0.015 m and 0.094 m. According to the theory of string vibration, the cable force of the suspension rod of the entire bridge is measured using a dynamic cable force tester.
When measuring the natural vibration frequency of the suspension rod of the bridge using the vibration frequency method, two sets of equipment, a dynamic signal collector and a cable force tester, are used together. When using a dynamic signal collector, fx the acceleration sensor with black electrical tape at half of the suspension rod, connect the dynamic signal collector through the sensor cable, and synchronize the collector with the computer acquisition system. Ten, collect the natural frequency of the suspension rod under both environmental and manual excitation. For the bridge site during the operation period, environmental random vibration is selected, and the suspension rod is directly excited by using vehicle loads and wind loads in the environment as the vibration sources of the suspension rod. During measurement, if the suspension rod is stationary and there is no natural vibration or the measured natural vibration frequency is unclear, a certain amount of artifcial excitation is required for the suspension rod. A small wooden mallet is used to strike the suspension rod as artifcial excitation, which can compensate for the unstable and weak environmental vibration source.
Te random vibration of the environment is tested using a cable force dynamic tester. Te cable force dynamic tester is a portable single or dual-channel vibration detection analyzer for micro-vibration signals. Te accelerometer is fxed on the suspension rod to measure its lateral vibration. Te cable force dynamic tester can collect the multi-Shock and Vibration 7 harmonic vibration curve of the suspension rod and obtain the lateral vibration frequency of the suspension rod through spectral analysis. Te characteristic of the corresponding relationship between the cable force and the vibration frequency of the cable is utilized. When the length of the cable, the constraint conditions at both ends, and the distribution mass are known, the cable force of the suspension rod of the bridge can be obtained by measuring the vibration frequency of the cable. In order to reduce measurement errors, the test points of the same suspension rod will be selected at diferent heights for multiple measurements under time constraints, and the measured natural frequencies will be compared to avoid signifcant errors in the data. In Table 3, the comparison between the tested cable force results and the cable force at completion acceptance. For ease of expression, the suspenders are numbered sequentially from south to north, and the upstream and downstream are distinguished by U and D. Te frst upstream suspender on the south side is U1, the frst downstream suspender is D1, the frst upstream suspender on the north side is U60, the frst downstream suspender is D60, and so on.
In Table 3, the diference value � the measured value of regular inspection − the measured value of handover acceptance. Te diference value is positive when the cable force increases and negative when the cable force decreases. From Table 3, it can be seen that the cable force of the suspender tested this time is generally deviated from the cable force at the completion acceptance, with most of the diference between 10 kN and 40 kN, with a maximum diference of 106.2 kN.

Suspender Damage Prediction Modeling of Arch Bridge.
Using the FWA-BP neural network prediction model and taking the damage prediction of long-span arch bridge suspenders as an example, the damage prediction model based on FWA-BP neural network is established. Te steps are as follows: ① Select the input and output indicators: ten indexes, such as modal curvature change rate, elastic modulus, frequency, vibration mode, boom damage location, noise level, instantaneous bearing weight, lateral bending vibration displacement, beam linear mass, and bending stifness, are selected as the input indexes of the network prediction model, and the boom damage degree is selected as the output index of the network prediction model. ② Standardize the data: To eliminate the impact of diferent dimensions on the accuracy of the prediction model, standardize the data for each indicator to the same order of magnitude, to improve the comparability between the data. Tis model adopts 0-1 standardization method to standardize the experimental data.
where max (X) is the maximum value in the dataset and min (X) is the minimum value in the dataset. After the data are standardized, the training data are mapped to the interval [0, 1] for comparative analysis. ③ Set key parameters: on the basis of the input and output indexes of the prediction model, the main parameters based on FWA-BP neural network are set as follows-the number of nodes in the input layer m = 10, the number of nodes in the output layer n = 1, the number of hidden layers e = 1, and the number of neurons in the hidden layer s of the neural network, and then empirical formula (16) For the selection of activation function, tansig and purelin activation functions are selected in the input layer and output layer, respectively, and trainlm function is selected as the training function of the network model. In the process of network training, set the learning rate as 0.01, the momentum factor as 0.9, the maximum number of iterations as 20000, and the minimum training error as 0.001. For the weight value w ij and threshold between network nodes θ, the optimal frework population obtained by iterative selection of freworks algorithm is used to initialize the network weight and threshold.

Shock and Vibration
At the same time, according to the network weight value w ij and threshold to be optimized θ, the key parameters in the freworks algorithm are set as follows: population size n � 70, frework explosion radius adjustment constant d � 5, frework explosion spark number adjustment constant c � 40, upper limit of frework explosion spark number ub � 0.8, lower limit of frework explosion spark number lb � 0.04, Gaussian variation spark number g � 5, and maximum iteration times T �1000.

Experimental Results and Performance Analysis of Prediction Model.
Te internal force and deformation of the arch rib control section under dead load calculated by the model in this paper are shown in Table 4.
In Table 4, Calculation I is the calculation result of the testing agency, and Calculation II is the calculation result of the model in this article; the axial force in the table is positive with pressure; the side tension below the bending moment is positive; the defection is positive downward.
Te comparison between the cable force calculated by the model in this article and the detection mechanism and the measured cable force is shown in Table 5, taking the cable force of the upstream suspender as an example.
In Table 5, Calculation I is the calculation result of the testing agency, and Calculation II is the calculation result of the model in this article. Diference ① = Calculate the mean value of II − Calculate I, with an increase in the diference being positive and a decrease being negative, and a percentage of diference ① = Diference ①/Calculate I. Diference ② = calculated II mean − measured mean, if the calculated II mean is greater than the measured mean, it is positive, and if it is less than the measured mean, it is negative, and a percentage of diference ② = diference ②/measured mean.
From Tables 4 and 5, it can be seen that the axial force calculation results of the model and the detection mechanism in this paper are very similar under the dead load, the bending moment calculation results of each section are basically consistent, and the defection calculation results are also basically similar. Te maximum deviation between the upstream suspender cable force calculated by this model and the cable force calculated by the detection mechanism is 5.5%, and the maximum deviation from the measured cable force is 5.6%. Terefore, the calculation results of this model are reliable and can be used as the basis for further analysis and research.

Suspender Damage Calculation Prediction.
Arch ribs, transverse connections, and suspension structures are the general components of the span structure of half-through arch bridge and through arch bridges. Te suspender is composed of load-bearing steel wire and steel pipe sheathed outside it, which plays a key role in hanging the bridge deck. Te waterproof system of its upper and lower anchor heads is easy to age. Among the components of the arch bridge, the suspender is the most easily damaged component, so the research on damage identifcation of this kind of arch bridge should focus on the suspender health detection. Taking No. 30 suspender in the middle as an example, some training samples are listed in Tables 6 and 7.
Firstly, the 0-1 standardization strategy according to equation (15) in Section 4.2 is used to standardize the experimental data, and then the experimental dataset is established with the standardized data to train and test the FWA-BP neural network prediction model. Te frst 80% of the data in the experimental dataset is selected as the training data set, and the last 20% of the data in the dataset is selected as the test dataset. Te prediction model based on FWA-BP neural network proposed in this paper is tested and verifed.
Secondly, to further verify the prediction performance based on FWA-BP neural network, the same datasets are used to train the traditional BP neural network, genetic algorithmimproved BP neural network (GA-BPNN), and particle swarm optimization algorithm-improved BP neural network (PSO-BPNN). Te parameter setting of the GA is as follows: population size popu � 30, genetic algebra gen � 100, crossover probability pcross � 0.8, and mutation probability pmutation � 0.05. For PSO algorithm, the parameters are as follows: speed update parameter c 1 � c 2 � 1.49445, evolution times maxgen � 150, population size sizepop � 30, individual maximum pop max � 7, individual minimum pop min � −7, individual maximum speed v max , and individual minimum speed v min . Te parameters of BP neural network in BP neural network prediction model optimized by diferent algorithms are the same as those in FWA-BP neural network model described in Section 4.2.
Tirdly, to reduce the accidental factors in the experimental process, the same algorithm model is trained and tested for 3 times with the same data, and the average value of the prediction error and iteration time of 3 times are taken as the prediction error and iteration times of the algorithm. Specifcally, under the same experimental conditions, the GA-BP neural network and PSO-BP neural network models are simulated, and the prediction results of the neural network prediction models optimized by diferent algorithms are obtained. Te results are shown in Tables 8 and 9.    It can be seen from Tables 8 and 9 that in the three rounds of tests, the average relative error AE, maximum relative error AE max , similarity R, and single round cumulative time T are predicted by each model, and the prediction results of BP neural network optimized by diferent algorithms fuctuate in diferent test samples. However, on the whole, the error rate of the prediction model based on FWA-BP neural network is lower than that of the existing prediction models based on PSO-BP neural network and GA-BP neural network, and its results are closer to target values than the other models.

Conclusions
In view of the weak generalization ability and low prediction accuracy of the prediction model based on the traditional BP neural network, the freworks algorithm is introduced into the BP neural network, and the weights and thresholds of the BP neural network are optimized and improved with the help of the freworks algorithm. A prediction model based on the freworks algorithm-improved BP neural network (FWA-BP) is proposed, and the algorithm of the prediction model based on FWA-BP neural network is implemented. Ten, taking the damage prediction of long-span arch bridge as an example, the damage prediction model based on FWA-BP neural network is established, and the performance of damage prediction is simulated and tested. Compared with the prediction methods based on BP neural network, GA-BP neural network, and PSO-BP neural network, the results show that under the given training target value, the prediction method based on FWA-BP neural network proposed in this paper has smaller prediction error rate and fewer iterations, which can efectively improve the prediction performance of BP neural network. Terefore, the damage degree prediction method of long-span arch bridge proposed in this paper is feasible and provides a theoretical basis for related engineering research.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that there are no conficts of interest regarding the publication of this paper.