Fatigue Performance Prediction of RC Beams Based on Optimized Machine Learning Technology

The development of fatigue damage in reinforced concrete (RC) beams is affected by various factors such as repetitive loads and material properties, and there exists a complex nonlinear mapping relationship between their fatigue performance and each factor. To this end, a fatigue performance prediction model for RC beams was proposed based on the deep belief network (DBN) optimized by particle swarm optimization (PSO). The original database of fatigue loading tests was established by conducting fatigue loading tests on RC beams. The mid-span deflection, reinforcement strain, and concrete strain during fatigue loading of RC beams were predicted and evaluated. The fatigue performance prediction results of the RC beam based on the PSO-DBN model were compared with those of the single DBN model and the BP model. The models were evaluated using the R2 coefficient, mean absolute percentage error, mean absolute error, and root mean square error. The results showed that the fatigue performance prediction model of RC beams based on PSO-DBN is more accurate and efficient.


Introduction
As important components of the global land transportation network, bridges, the most common of which are reinforced concrete (RC), play a pivotal role in improving people's livelihoods and promoting regional economic development [1]. In the context of increasing traffic volume, RC beams are subjected to high-frequency vehicle loads for a long time. The service performance and durability performance of many bridges have deteriorated at an accelerated rate, and the fatigue life has been further reduced, which seriously affects the operational safety of RC bridge structures [2,3]. From a macroscopic perspective, the fatigue damage of RC beams is caused by the gradual accumulation of structural material damage and the gradual deterioration of properties, and the fatigue performance of structural materials determines the fatigue performance of the structure itself [4]. The adoption of scientific and reliable methods to effectively evaluate the fatigue performance of existing RC beams has become an urgent engineering problem to be solved.
At present, the fatigue performance of RC beams has been studied mainly through indoor fatigue loading tests [5][6][7]. Wang and Li [8] investigated the effect of material randomness on structural fatigue performance by conducting fatigue loading tests on RC beams. Yang et al. [9] studied the fatigue performance of RC beams after water freeze-thaw and salt freeze-thaw cycles by conducting four-point bending fatigue loading tests. There are problems such as high intensity, long cycle time, and difficulty in obtaining fast and accurate test results in a short period of conducting a large number of tests. With the continuous development of numerical simulation technology, more and more researchers have started to use numerical simulation methods to study the fatigue performance of RC beams [10][11][12]. Jin et al. [13] simulated the bond-slip behavior of longitudinal reinforcement by establishing a three-dimensional mesoscale numerical model and investigated the mechanical properties of carbon fiber reinforced polymer (CFRP)-reinforced RC beams with impact damage. He et al. [14] studied the fatigue performance of ordinary RC beams and reinforced beams by simulating impact tests, and the dynamic properties of RC beams before and after CFRP strengthening under different impact conditions were investigated. However, the fatigue life of RC beams can be predicted well by using a single fatigue analysis software, but the mechanical properties of RC beams after a certain number of cyclic loading cannot be accurately evaluated, and it is difficult to realize the whole process of fatigue analysis. Dobromil P. et al. [15] used the finite element method to simulate the fatigue damage process of concrete structures, but usually, the accurate mechanical performance analysis needs to be realized by writing special programs or secondary development of software, and the versatility is not strong. The number of fatigue loading is usually in the millions, and it is a huge workload to measure the mechanical properties of RC beams under different loading times in a turn. Therefore, it is important to combine the appropriate amount of indoor fatigue loading tests with efficient scientific calculation algorithms to effectively judge the mechanical properties of RC beams under different damage conditions after cyclic loading for engineering structural evaluation and fatigue durability research.
In recent years, with the continuous development of artificial intelligence technology, many scholars have applied artificial neural networks in engineering structure prediction or structural material research [16][17][18][19][20][21][22] with rich results. Liu et al. [23] established a BP neural network prediction model for blast safe vibration velocity of newly cast concrete structures based on BP neural network theory and selected key influencing factors such as Poisson's ratio, and predicted the blast safe vibration velocity of concrete at different ages under two different conditions. Asteris et al. [24] used the artificial neural network method to predict the ultimate shear strength of RC beams and compared the predicted values with the experimental values as well as the values calculated by existing formulas in the code provisions, which proved the reliability and validity of the predictive performance of artificial neural networks. Cong et al. [25] used neural networks to estimate the effluent water quality at frequent changes in conditions to improve the accuracy of water quality evaluation in the wastewater treatment process. Sahoo and Mahapatra [26] used an artificial neural network (ANN) model to predict the compressive strength of concrete at different water curing days, sulfate exposure time, and fly ash substitution levels. Onyari and Ikotun [27] used an artificial neural network to predict the compressive and flexural strength of modified zeolite additive mortar, and the results showed that the compressive and flexural strength of modified zeolite mortar could be very short using a neural network model and the prediction results are more accurate.
The deep learning method, as the frontier in the field of machine learning, is one of the commonly used methods in the field of artificial intelligence. It improves prediction accuracy by building models with multiple layers and mining the features implicit in the learned data. Zhang et al. [28] proposed a general method for component life prediction under creep, fatigue, and creep-fatigue conditions based on deep learning. It has better prediction accuracy and generalization ability than traditional machine learning models. Yang et al. [29] proposed a multi-axis fatigue life prediction model based on deep learning and analyzed six sets of existing fatigue data from different materials, respectively. Hinton et al. [30] proposed a deep learning method called deep belief network (DBN), which has attracted a lot of attention from the academic community. DBN is a stack of multiple restricted Boltzmann machines (RBMs), which can map complex nonlinear relationships and have a better-unsupervised feature learning capability, and can maintain strong stability when dealing with complex data prediction [31]. Currently, DBN has been successfully applied to problems such as identification or evaluation in engineering. Wang et al. [32] based a regional landslide sensitivity zoning model on a deep belief network model with example analysis and finally compared the evaluation results with logistic Materials 2022, 15, 6349 3 of 24 regression (LR) and artificial neural network (BPNN) model evaluation results by RO curve features. Chen et al. [33] evaluated the landslide susceptibility of a region using the DBN model, generated a landslide susceptibility zoning map of the region, and compared it with shallow neural networks and traditional logistic regression methods. Xu et al. [34] identified RC beam damage based on acoustic emission and DBN.
Although the deep belief network is widely used, it has problems in that the number of hidden layer nodes is not easily determined, and the parameters, such as learning rate and the number of iterations, are mainly determined by manual experience during the pre-training process, and improperly set parameter values can have an impact on the model prediction performance. Therefore, it is necessary to optimize the DBN model further to further improve the accuracy of prediction. In this paper, a particle swarm optimization algorithm is used to search for the optimal number of hidden layer nodes. At present, regarding the intelligent search algorithms, the common ones are the simulated annealing algorithm [35], genetic algorithm [36,37], and particle swarm optimization algorithm. Among them, the effect of global optimization and computational efficiency of simulated annealing algorithm is greatly affected by parameters, genetic algorithm is difficult to converge effectively in a limited time, and particle swarm optimization algorithm is a population-based evolutionary algorithm with simple and easy-to-understand principle, fast convergence speed and global optimal solution.
In this paper, the fatigue performance of RC specimen beams was analyzed by conducting a four-point bending fatigue loading test under constant amplitude load, and the original database was established. A deep belief network (DBN) model for fatigue performance prediction of RC beams was established for the first time by using deep learning with massively parallel processing and self-learning characteristics. After training the RBM layer by layer and extracting feature information from complex data, the DBN model parameters were adaptively adjusted using a particle swarm optimization algorithm. The results of the fatigue performance prediction of RC beams based on the PSO-DBN model were compared with those of the single DBN model and the BP model. The comparative analysis of the simulation model predictions verified the feasibility and accuracy of the fatigue performance prediction of RC beams based on the PSO-DBN model and provided new ideas for engineering structure evaluation and fatigue durability research.

Specimens
Six RC specimen beams were designed for this test, and the design strength grade of concrete for the specimen beams was divided into three types, C35, C40, and C60 (150 mm cubic compressive strengths are 35 MPa, 40 MPa, and 60 MPa at the age of 28 days), two of each type, one of which was used for a static loading test and the remaining one for a fatigue loading test. The cross-sectional dimensions of the specimen beam were 150 mm × 200 mm, with a total length of 600 mm, and the tensile reinforcement was CRB600H (a new type of cold-rolled ribbed steel bar developed in China in recent years) with high strength and high ductility. The specific arrangement of the reinforcement is shown in Figure 1. Figure 2 shows the specimen beams before and after concreting.

Material Properties
The mechanical properties of materials were tested, as shown in Figure 3. According to the Standard for Test Methods of Physical and Mechanical Properties of Concrete (GB/T 50081-2019) [38], three standard cubic specimens were reserved for each of the three types of concrete with different design strengths during the casting of the specimen beams, and after the specimens were cured for 28 days, the average values of their cubic compressive strengths were tested to be 36.9 MPa, 55 MPa, and 63.7 MPa, respectively. The concrete mix ratios are shown in Table 1. According to the Metal Axial Tensile Test Method (GB/T 228.1-2010) [39], three pieces of each type of reinforcement were reserved for material mechanical property tests. The average yield strength and ultimate tensile strength of CRB600H reinforcements with a nominal diameter of 12 mm were 619 MPa and 671 MPa, respectively, and the average yield strength and ultimate tensile strength of HRB400 reinforcements with a nominal diameter of 8 mm were 467 MPa and 560 MPa, respectively.

Material Properties
The mechanical properties of materials were tested, as shown in Figure 3. According to the Standard for Test Methods of Physical and Mechanical Properties of Concrete (GB/T 50081-2019) [38], three standard cubic specimens were reserved for each of the three types of concrete with different design strengths during the casting of the specimen beams, and after the specimens were cured for 28 days, the average values of their cubic compressive strengths were tested to be 36.9 MPa, 55 MPa, and 63.7 MPa, respectively. The concrete mix ratios are shown in Table 1

Material Properties
The mechanical properties of materials were tested, as shown in Figure 3. According to the Standard for Test Methods of Physical and Mechanical Properties of Concrete (GB/T 50081-2019) [38], three standard cubic specimens were reserved for each of the three types of concrete with different design strengths during the casting of the specimen beams, and after the specimens were cured for 28 days, the average values of their cubic compressive strengths were tested to be 36.9 MPa, 55 MPa, and 63.7 MPa, respectively. The concrete mix ratios are shown in Table 1. According to the Metal Axial Tensile Test Method (GB/T 228.1-2010) [39], three pieces of each type of reinforcement were reserved for material mechanical property tests. The average yield strength and ultimate tensile strength of CRB600H reinforcements with a nominal diameter of 12 mm were 619 MPa and 671 MPa, respectively, and the average yield strength and ultimate tensile strength of HRB400 reinforcements with a nominal diameter of 8 mm were 467 MPa and 560 MPa, respectively.

Test Loading Device and Loading System
After finishing the maintenance of the specimen beams, the PMS-500 hydraulic pulsation testing machine produced by China Jinan Times Assay Testing Machine Co., Ltd. was used for test loading, and the loading device is shown in Figure 4a. The test included a static loading test and fatigue loading test, using vertical force control loading and four-point bending cycle constant amplitude loading, respectively. The purpose of the static loading test was to determine the static ultimate bearing capacity of the RC specimen beams and then determine the upper limit value of the fatigue load of the RC specimen beams, and the static test loading is shown in Figure 4b. Before the official start of the static loading test, it was preloaded to 10 kN to promote sufficient contact between the couplings. After checking that the channels of each measurement point were normal, the loading was carried out in increments of 10 kN per level of load, and the load increment per level was appropriately reduced when the specimen beams were close to damage. The indicators, such as mid-span deflection, the strain of tensile reinforcement, and concrete strain in the compression zone at the top of the beam, were measured during the test loading. With the increasing load, the tensile reinforcement at the bottom of the RC-1, RC-3, and RC-5 specimen beams yielded, and the concrete at the top of the beams was crushed. Finally, the static ultimate bearing capacities of the three specimen beams were 205 kN, 232 kN, and 243 kN, respectively.
Materials 2022, 15, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/materials beams and then determine the upper limit value of the fatigue load of the RC specimen beams, and the static test loading is shown in Figure 4b. Before the official start of the static loading test, it was preloaded to 10 kN to promote sufficient contact between the couplings. After checking that the channels of each measurement point were normal, the loading was carried out in increments of 10 kN per level of load, and the load increment per level was appropriately reduced when the specimen beams were close to damage. The indicators, such as mid-span deflection, the strain of tensile reinforcement, and concrete strain in the compression zone at the top of the beam, were measured during the test loading. With the increasing load, the tensile reinforcement at the bottom of the RC-1, RC-3, and RC-5 specimen beams yielded, and the concrete at the top of the beams was crushed. Finally, the static ultimate bearing capacities of the three specimen beams were 205 kN, 232 kN, and 243 kN, respectively. The fatigue test used four-point bending equal amplitude load, and the fatigue test loading is shown in Figure 4c. According to the results of the static loading test, the upper limit of fatigue load for RC-2, RC-4, and RC-6 specimen beams was taken as 40 kN, 70 kN, and 90 kN, and the lower limit of fatigue load was taken as 10 kN, and the fatigue lives of the three specimen beams were 3.59 million, 2.76 million, and 3.28 million times, respectively. The test loading scheme is shown in Table 2.  The fatigue test used four-point bending equal amplitude load, and the fatigue test loading is shown in Figure 4c. According to the results of the static loading test, the upper limit of fatigue load for RC-2, RC-4, and RC-6 specimen beams was taken as 40 kN, 70 kN, and 90 kN, and the lower limit of fatigue load was taken as 10 kN, and the fatigue lives of the three specimen beams were 3.59 million, 2.76 million, and 3.28 million times, respectively. The test loading scheme is shown in Table 2.

Measurement Point Arrangement and Data Acquisition
Before the RC specimen beams were cast, reinforcement strain gauges were arranged on the tensile reinforcement in the middle of the span of the beam to measure the strain of the tensile reinforcement in the specimen beams. Before the test, two concrete strain gauges were arranged at the top of the span of the specimen beams to measure the strain in the concrete of the specimen beams. Displacement sensors were arranged at the bottom of the span of the specimen beams and the top surface of the beams above the support to record the vertical displacement and the support settlement, respectively. During the loading process of the fatigue test, the DH3820 high-speed static strain testing and analysis system produced by China Jiangsu Donghua Testing Technology Co., Ltd. was used to automatically collect the deflection, strain on the tensile reinforcement, and strain on the concrete top of the beams during the test.  Figure 5 shows the fatigue performance of RC-2, RC-4, and RC-6 specimen beams under different load cycles. The relationship curves between the maximum crack width and fatigue life ratio n/N of the specimen beams are shown in Figure 5a. From Figure 5a, it can be seen that the crack development process of the specimen beams shows a "three-stage" pattern. The first stage is the crack derivation stage, in which the crack width is about 0-0.175 mm, the second stage is the stable crack development stage, in which the crack width is about 0.175-0.25 mm, and the third stage is the fatigue damage stage, in which the crack width expands sharply with the increase of the number of cycles, and finally leads to the fatigue damage of the specimen beams.

Principle of the DBN
The deep belief network is a neural network model consisting of multiple stacked restricted Boltzmann machines, as shown in Figure 6. The DBN training model mainly consists of two processes: pre-training and reverse fine-tuning. The pre-training process is a top-down independent and unsupervised learning process, where the output vector of the previous layer of the RBM network is used as the input vector of the next layer of the RBM network, and the deep feature information of the input vector data is extracted layer by layer to realize the training of the model layer by layer, to obtain the initialized network parameters. Then, the BP neural network is established in the last layer of DBN, and the output vector of the RBM network is used as the input vector of BP. The BP algorithm adjusts network parameters such as weights and biases, and to finish training the entire DBN, the error between the actual output and the desired output is propagated backward layer by layer. The curves of the bottom tensile reinforcement strain and top compressive zone concrete strain versus fatigue life ratio n/N for the specimen beams loaded statically to the corresponding upper fatigue limit are shown in Figure 5b,c. It can be seen from the figures that the strains in the bottom tensile reinforcement and the concrete strains in the top compressive zone increase rapidly when the specimen beams are loaded to the corresponding fatigue upper limit within the first 10% of the cycles of fatigue life. Within 10-70% of the fatigue life, the strains develop steadily. After that, additional cyclic loading leads to rapid strain development.
The mid-span deflection versus fatigue life ratio n/N of the specimen beams when statically loaded to the corresponding upper fatigue limit is shown in Figure 5d. From Figure 5d, it can be seen that the change in the mid-span deflection of the specimen beams also shows a three-stage pattern. In the first stage, the mid-span deflection increases significantly. In the second stage, the mid-span deflection of the specimen beams changes relatively smoothly. After that, the mid-span deflection increases rapidly as the number of cycles continues to increase.

Principle of the DBN
The deep belief network is a neural network model consisting of multiple stacked restricted Boltzmann machines, as shown in Figure 6. The DBN training model mainly consists of two processes: pre-training and reverse fine-tuning. The pre-training process is a top-down independent and unsupervised learning process, where the output vector of the previous layer of the RBM network is used as the input vector of the next layer of the RBM network, and the deep feature information of the input vector data is extracted layer by layer to realize the training of the model layer by layer, to obtain the initialized network parameters. Then, the BP neural network is established in the last layer of DBN, and the output vector of the RBM network is used as the input vector of BP. The BP algorithm adjusts network parameters such as weights and biases, and to finish training the entire DBN, the error between the actual output and the desired output is propagated backward layer by layer. A single RBM is made up of a visible layer and a hidden layer. Figure 6 depicts the layout of a network made up of a three-layer RBM, where h stands for the hidden layer, v for the visible layer, and W for the link weight between the two. Neurons in the same layer of the RBM are independent of one another, whereas connection weights connect neurons in adjacent layers. Binary values 0 and 1, respectively, reflect the inactive and active states of neurons.
The RBM is a thermodynamics-based energy model whose energy function can be expressed as: where vi is the state of neuron i in the visible layer, ai is the offset corresponding to vi, hj is the state of neuron j in the hidden layer, bj is the offset corresponding to hj, wij is the connection weight of neurons i and j, θ =(wij,ai,bj) is the RBM parameter, m is the number of neurons in the hidden layer, and n is the number of neurons in the visible layer. When the parameters in the RBM model are determined, the following joint probability distribution of (v,h) can be obtained： A single RBM is made up of a visible layer and a hidden layer. Figure 6 depicts the layout of a network made up of a three-layer RBM, where h stands for the hidden layer, v for the visible layer, and W for the link weight between the two. Neurons in the same layer of the RBM are independent of one another, whereas connection weights connect neurons in adjacent layers. Binary values 0 and 1, respectively, reflect the inactive and active states of neurons.
The RBM is a thermodynamics-based energy model whose energy function can be expressed as: where v i is the state of neuron i in the visible layer, a i is the offset corresponding to v i , h j is the state of neuron j in the hidden layer, b j is the offset corresponding to h j , w ij is the connection weight of neurons i and j, θ =(w ij ,a i ,b j ) is the RBM parameter, m is the number of neurons in the hidden layer, and n is the number of neurons in the visible layer.
When the parameters in the RBM model are determined, the following joint probability distribution of (v,h) can be obtained: If the number of training samples is N, the parameter θ can be obtained by learning the maximum log-likelihood function of the samples: where The difficult normalizing factor Z(θ) computation is typically approximated using sampling techniques, such as Gibbs [40]. Hinton developed the contrastive divergence (CD) algorithm, which allows the jth neuron in the hidden layer to be determined from the state of the neuron in the visible layer and the ith neuron in the visible layer to be reconstructed from the hidden layer using the activation probabilities listed in Equations (4) and (5): where σ = 1 1+exp(−x) is the sigmoid activation function. The maximum value of the log-likelihood function can be solved by the stochastic gradient ascent method. The amount of variation of parameters such as RBM weights and bias can be calculated as follows: where • data is the distribution defined by the original data model, • reconstructed is the distribution defined by the reconstructed model, and ε is the learning rate.

Parameter Optimization based on the PSO Algorithm
The particle swarm optimization (PSO) originated from the study of the foraging behavior of bird flocks. The basic idea is to find the optimal solution through collaboration and information sharing among different individuals in a swarm. PSO treats each individual in a swarm as a particle without volume and mass in a multidimensional search space. The PSO algorithm first initializes a set of particles in the multidimensional search space and then iteratively updates its speed and direction according to its optimal value and the global optimal value of the population to output the optimal solution [41].
Set a population X = (X 1 , X 2 ,···X n ) consisting of n particles, and in the D-dimensional search space, each particle n i in this population has its velocity vector V i = (V i1 , V i2 ,···V iD ) T and position vector X i = (x i1 , x i2 ···x iD ) T , and the particle's superiority concerning the target is evaluated by its corresponding individual fitness value, which leads to the individual optimal solution P i = (P i1 , P i2 ···P iD ) T when the particle flies in the D-dimensional search space. In turn, the individual optimal solution P i = (P i1 , P i2 ···P iD ) T is obtained. The particle, when flying in the D-dimensional search space, will combine the flight experience of other particles and its previous flight state to adjust the next position and velocity, thus outputting the current global optimal solution P g = (P g1 , P g ···P gD ) T and obtaining the optimal solution by k iterations. The velocity and position of the particle are updated as follows: where c 1 and c 2 are acceleration values, which are used to adjust the maximum step of individual and group optimal positions, r 1 and r 2 are inertia factors, distributed between [0,1], ω is the inertia weight, which can be used to balance the global search ability and local search ability, the velocity and position of particles are generally limited to the interval: , to avoid blindly searching for particles.

Fatigue Performance Prediction of RC Beams based on the PSO-DBN Model
The flow chart of the fatigue performance prediction of RC beams based on the PSO-DBN model is shown in Figure 7. The specific steps of the prediction model are as follows: Materials 2022, 15, x FOR PEER REVIEW 10 of 28

Fatigue Performance Prediction of RC Beams based on the PSO-DBN Model
The flow chart of the fatigue performance prediction of RC beams based on the PSO-DBN model is shown in Figure 7. The specific steps of the prediction model are as follows: In the first step, the data collected during the fatigue loading test are organized and pre-processed. The data are normalized by normalizing the original data to ensure that the data are relatively undistorted. After the normalization is completed, the data set is divided into two mutually exclusive sets, one of which is used as the training data set and the other as the test data set. After that, DBN initialization is completed.
In the second step, the initialization of particle position and velocity is completed, the particle fitness value is calculated, and then the particle position and velocity are updated. When the fitness value satisfies the set condition or the number of iterations is equal to M, the second step of PSO optimization is finished. Otherwise, the process of calculating the particle fitness value and updating the position and velocity of the particles will be repeated until the determination condition is satisfied.
In the third step, the DBN model structure parameters after PSO optimization are used to calculate the test data set by the optimal DBN structure and complete the prediction of the fatigue performance of the RC specimen beams.

Test Data Pre-Process
The data in this section were selected from the loading time range data of the RC specimen beams fatigue test. For the detailed data, see Table A1 in Appendix A. A total of 300 original data samples were selected as the input data for the PSO-DBN model based on 100-time course data for each of the different damage stages of the fatigue loading tests of RC-2, RC-4, and RC-6 specimen beams. In this study, based on the 300 experimental data samples, 70% of them were randomly selected as training sets, and the remaining 30% of data samples were used as test sets.
Based on the a priori knowledge of fatigue loading of RC beams, four relevant factors were identified as the input vectors of the model, namely: load amplitude (kN), concrete strength (MPa), static loading values at different damage stages (kN), and fatigue life ratio(n/N). Since the crack width time-range data were not available for the time being, the three mechanical property indices of concrete strain (με), tensile reinforcement strain (με), and mid-span deflection (mm) at the top of the RC specimen beams were used as the output vectors of the model in this paper. In the first step, the data collected during the fatigue loading test are organized and pre-processed. The data are normalized by normalizing the original data to ensure that the data are relatively undistorted. After the normalization is completed, the data set is divided into two mutually exclusive sets, one of which is used as the training data set and the other as the test data set. After that, DBN initialization is completed.
In the second step, the initialization of particle position and velocity is completed, the particle fitness value is calculated, and then the particle position and velocity are updated. When the fitness value satisfies the set condition or the number of iterations is equal to M, the second step of PSO optimization is finished. Otherwise, the process of calculating the particle fitness value and updating the position and velocity of the particles will be repeated until the determination condition is satisfied.
In the third step, the DBN model structure parameters after PSO optimization are used to calculate the test data set by the optimal DBN structure and complete the prediction of the fatigue performance of the RC specimen beams.

Test Data Pre-Process
The data in this section were selected from the loading time range data of the RC specimen beams fatigue test. For the detailed data, see Table A1 in Appendix A. A total of 300 original data samples were selected as the input data for the PSO-DBN model based on 100-time course data for each of the different damage stages of the fatigue loading tests of RC-2, RC-4, and RC-6 specimen beams. In this study, based on the 300 experimental data samples, 70% of them were randomly selected as training sets, and the remaining 30% of data samples were used as test sets.
Based on the a priori knowledge of fatigue loading of RC beams, four relevant factors were identified as the input vectors of the model, namely: load amplitude (kN), concrete strength (MPa), static loading values at different damage stages (kN), and fatigue life ratio(n/N). Since the crack width time-range data were not available for the time being, the three mechanical property indices of concrete strain (µε), tensile reinforcement strain (µε), and mid-span deflection (mm) at the top of the RC specimen beams were used as the output vectors of the model in this paper.
Since the magnitudes of different variables were different, the data were normalized and mapped to the interval [0,1] before the formal experiments to prevent their variability from affecting the modeling effect, reduce the computational complexity, and accelerate the convergence speed. In this paper, the normalization method was used to linearly transform the original data, as in Equation (9): where x * i is the normalized value of the sample data, x i is the original value of the input variable, and x max and x min are the maximum and minimum values of the original data, respectively.

Evaluation Indicators
In this study, the model's performance was assessed using the root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R 2 ). The performance of the prediction increases with decreasing RMSE, MAE, and MAPE. R 2 measures the percentage of the dependent variable's total variation that the independent variable explains through the regression connection. It is generally accepted that the closer R 2 is to 1, the better the regression relationship fits the data. The formula for calculating the above valuation metrics is shown below: where y i is the actual sample value, y i is the model predicted value, and N is the number of samples.

Model Parameter Setting
The selection of optimal parameters was achieved by using a longitudinal comparison method. Any three different test sample data sets were selected to compare the effect of the different number of hidden layers on model prediction. Each test sample set contains 90 data samples. The number of hidden layer nodes was set to 105, and the effect of different hidden layer choices on model prediction was analyzed, as shown in Table 3. As can be seen from Table 1, when the number of hidden layers is less than four, the model error gradually decreases as the number of hidden layers increases, which indicates that the model prediction accuracy keeps improving. When the number of hidden layers is greater than four, the model error keeps increasing as the number of hidden layers increases, which is because too many hidden layers make the training of the model complicated and lead to the overfitting phenomenon. Therefore, the optimal number of hidden layers for this model is set to four.
The proper number of hidden layer nodes was discussed and decided upon when the number of hidden layers was established. The research discusses the implications of the prediction when there are 65-125 nodes, as indicated in Table 4. When there are fewer than 105 nodes, the model error gradually decreases as there are more nodes. When there are more than 105 nodes, the model error gradually rises with the number of nodes. Consequently, 105 nodes are needed for the best effect to be realized. Table 4. The effect of the number of nodes in the hidden layer on the prediction effect of the model. The parameters of the deep belief network model were automatically adjusted using the PSO algorithm, and the parameters of the model were output. The number of DBN hidden layer nodes was automatically output as (115, 129, 109, 105) after PSO completes the parameter optimization and converges. The number of particle swarm is 20, acceleration factor c1 = c2 = 1.49, learning rate is 0.01. The specific parameters of the model are taken as shown in Table 5. Table 5. PSO-DBN model parameters.

Description Symbol Value
The number of neurons in the input layer -4 The number of neurons in the output layer -3 Table 5. Cont.

Description Symbol Value
The

Prediction Capability of the PSO-DBN Model
In this section, the predicted results of the PSO-DBN model are evaluated. The comparison between the test values and the predicted values of the three output vectors (mid-span deflection, tensile reinforcement strain, and top compressive zone concrete strain) in the model is shown in Figure 8, including both the training and test data sets. The comparison shown in Figure 8 shows that the predicted values of mid-span deflection, tensile reinforcement strain, and concrete strain in the top compression zone of the RC specimens output from the model are in good agreement with the test values. The errors between the predicted and tested values for both the training and test datasets are relatively small, with the mean absolute percentage errors (MAPE) of 0.075, 0.111, and 0.124 for the test dataset part and 0.067, 0.075, and 0.071 for the training dataset part for the three output vectors, respectively. Error-values indicate that the PSO-DBN model proposed in this paper predicts well the mechanical properties of RC specimen beams under cyclic loading at different damage stages.
The regression plots of the two parts of the training set and the test set are shown in Figure 9. From Figure 9, it can be seen that the PSO-DBN model has a better prediction ability. For the training set, the coefficients of determination (R 2 ) of the three output vectors are 0.983, 0.991, and 0.993, respectively. For the test set, the coefficients of determination (R 2 ) of the three output vectors are 0.979, 0.986, and 0.989, respectively. The coefficients of determination of the three output vectors remain above 0.97 for both the training and test parts. Therefore, it is feasible to use the PSO-DBN model to predict the mechanical properties of RC beams under cyclic loading at different damage stages with high accuracy and low error. It can be applied to develop a numerical tool to evaluate the deterioration performance of RC structures.
To illustrate the accuracy of the model in this paper in predicting the fatigue performance of RC beams, the model predicts the mid-span deflection, tensile reinforcement strain, and concrete strain in the compression zone of RC-2 specimen beam under different fatigue life ratios and compares them with the test values under the same load level (25 kN). As shown in Figure 10  The regression plots of the two parts of the training set and the test set are shown in Figure 9. From Figure 9, it can be seen that the PSO-DBN model has a better prediction ability. For the training set, the coefficients of determination (R 2 ) of the three output vectors are 0.983, 0.991, and 0.993, respectively. For the test set, the coefficients of determination (R 2 ) of the three output vectors are 0.979, 0.986, and 0.989, respectively. The coefficients of determination of the three output vectors remain above 0.97 for both the training  The regression plots of the two parts of the training set and the test set are shown in Figure 9. From Figure 9, it can be seen that the PSO-DBN model has a better prediction ability. For the training set, the coefficients of determination (R 2 ) of the three output vectors are 0.983, 0.991, and 0.993, respectively. For the test set, the coefficients of determination (R 2 ) of the three output vectors are 0.979, 0.986, and 0.989, respectively. The coefficients of determination of the three output vectors remain above 0.97 for both the training  mance of RC beams, the model predicts the mid-span deflection, tensile reinforcement strain, and concrete strain in the compression zone of RC-2 specimen beam under different fatigue life ratios and compares them with the test values under the same load level (25 kN). As shown in Figure 10,

Models Comparison and Analysis
To highlight the efficiency of the PSO-DBN model, the prediction results of the PSO-DBN model, single DBN model, and BP model are compared.
As shown in Figure 11, the correlation values of the two model algorithms in the training part (Figure 11a,c) and the testing part (Figure 11b,d) are determined considering RMSE, MAE, MAPE, and R 2 as evaluation metrics. As can be seen in Figure 11, the PSO-DBN model is more accurate than the single DBN model and BP model, as evidenced by the reduction in the error values of RMSE, MAE, and MAPE, and the improvement in prediction accuracy is more pronounced in the training part than in the testing part. Considering the coefficient of determination (R 2 ) as the fitted regression error criterion, the PSO-DBN model has R 2 values closer to 1 compared with the DBN model without optimization and BP model and shows some superiority in both the training and test sets.
For comparison purposes, Table 6 and 7 show the exact values of the four error criteria when using the PSO-DBN and DBN models, respectively, where the numbers 1, 2, and 3 correspond to mid-span deflection, reinforcement strain, and concrete strain, respectively. Focusing on the test section, the average increase in RMSE, MAE, MAPE, and R 2 reached 53.7%, 59.6%, 63.3%, and 6.0%, respectively. Thus, the use of PSO to adjust the weights and biases of DBN greatly improved the accuracy of the predictions.

Models Comparison and Analysis
To highlight the efficiency of the PSO-DBN model, the prediction results of the PSO-DBN model, single DBN model, and BP model are compared.
As shown in Figure 11, the correlation values of the two model algorithms in the training part (Figure 11a,c) and the testing part (Figure 11b,d) are determined considering RMSE, MAE, MAPE, and R 2 as evaluation metrics. As can be seen in Figure 11, the PSO-DBN model is more accurate than the single DBN model and BP model, as evidenced by the reduction in the error values of RMSE, MAE, and MAPE, and the improvement in prediction accuracy is more pronounced in the training part than in the testing part. Considering the coefficient of determination (R 2 ) as the fitted regression error criterion, the PSO-DBN model has R 2 values closer to 1 compared with the DBN model without optimization and BP model and shows some superiority in both the training and test sets.
For comparison purposes, Tables 6 and 7 show the exact values of the four error criteria when using the PSO-DBN and DBN models, respectively, where the numbers 1, 2, and 3 correspond to mid-span deflection, reinforcement strain, and concrete strain, respectively. Focusing on the test section, the average increase in RMSE, MAE, MAPE, and R 2 reached 53.7%, 59.6%, 63.3%, and 6.0%, respectively. Thus, the use of PSO to adjust the weights and biases of DBN greatly improved the accuracy of the predictions.

Conclusion
In this paper, the fatigue performance of RC specimen beams was analyzed by conducting a four-point bending fatigue loading test under constant amplitude load, and the original database was established. A deep belief network (DBN) model for fatigue performance prediction of RC beams was established for the first time by using deep learning with massively parallel processing and self-learning characteristics. After training RBM layer by layer and extracting feature information from complex data, the fatigue performance prediction model of RC beams based on PSO-DBN was established by adaptively adjusting DBN model parameters using a particle swarm optimization (PSO) algorithm. The conclusions of this study are as follows.

1.
Under the action of constantamplitude four-point bending cyclic loading, the midspan deflection, tensile reinforcement strain, and concrete strain in the compression zone at the top of the beam of RC specimen beams with CRB600H for tensile reinforcement showed a three-stage trend at different damage stages with different static loading, i.e., rapid development at the initial stage, stable and slow development at the middle stage, and rapid development at the later stage until the fatigue fracture of the reinforcement.

2.
The PSO-DBN model describes the complex nonlinear mapping relationship between the RC specimen beams and their material properties, load magnitude, and other factors and accurately predicts and reflects the real process of fatigue damage evolution of the RC specimen beams. By collecting the static loading time data of RC specimen beams at different damage stages during the fatigue loading test, a database Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A