Reliability intelligence analysis of concrete arch bridge based on Kriging model and PSOSA hybrid algorithm

The traditional probabilistic reliability analysis method has problems such as poor convergence, low calculation accuracy, and long time consumption in calculating the reliability of concrete arch bridges due to factors such as the uncertainty of the structural parameters and the performance function being highly nonlinear. This paper proposes a method for calculating the reliability of concrete arch bridges based on the Kriging model and particle swarm optimization algorithm (PSOSA) of the simulated annealing algorithm. This method takes advantage of the Kriging model in small samples and high-dimensional nonlinear data processing capabilities and establishes a response surface model to approximate the actual limit state function. The optimization of the PSO algorithm is realized through the self-adaptive and variable probability mutation operation of the SA algorithm, which enhances the ability of the PSO algorithm to get rid of the local minimum, effectively avoids falling into the local minimum, and finally makes the calculation result tend to the global optimum. It overcomes the problems of slow convergence speed and premature maturity of traditional PSO algorithms. The correctness and effectiveness of the method proposed in this paper are verified through the example analysis and the actual engineering application of a concrete arch bridge. The research results show that the method proposed in this paper has obvious advantages in sample size, calculation accuracy, and iteration times compared with the existing reliability calculation methods for concrete arch bridges. This paper provides a fast and effective method for the structural reliability calculation of concrete arch bridges.


Introduction
Concrete arch bridges have been widely used recently due to their strong bearing capacity, lightweight, and good seismic performance advantages.With the continuous innovation of design concepts and the continuous improvement of construction technology, the span of concrete arch bridges has been increasing, and the bridge structure has become more and more flexible (Liu et al. 2022).Therefore, it is of great significance to accurately analyze the reliability of concrete arch bridges.Since the performance function of the current concrete arch bridge structure is high-dimensional nonlinear in the reliability calculation process, there is no explicit function expression and other problems (Zhang et al. 2020).Therefore, how to quickly and accurately analyze the reliability of concrete arch bridge structures has become a hot issue in current research.Although the traditional Monte Carlo simulation method (MCS) (Baisthakur and Chakraborty 2021) is suitable for solving the reliability problem of the implicit performance function and has high calculation accuracy, the calculation efficiency is low and time-consuming due to the large number of sampling required.The conventional first-order second-moment method (Kriegesmann and Ludeker 2019) (FORM) and second-order second-moment method (Chun 2021) (SORM) have the problem that the derivative of the implicit performance function is challenging to solve.However, the response surface method is currently a powerful tool for solving complex structural reliability problems.
The basic idea of the response surface method to solve the structural reliability is to use an easy-to-handle regression model as the response surface function and make the implicit function explicit to overcome the time-consuming problem of calculating the implicit function.Then combine the conventional reliability analysis method to solve the structural failure probability (Fang 2020).The key to solving the response surface method is the fitting effect of the response surface function.When the performance function is strongly nonlinear, the widely used classical response surface method (Yang et al. 2022a, b;Zhang and Qiu 2021) (RSM) based on quadratic polynomial is difficult to accurately approximate the real performance function, resulting in a significant calculation error.In recent years, some scholars have proposed using a regression model with better regression performance to construct a response surface and achieved good results, such as artificial neural network (Marugan et al. 2019;Han et al. 2019) (ANN), radial basis function (Zhang et al. 2021;Wang and Fang 2018) (RBF), support vector regression (Pan et al. 2020) (SVR).However, in practical applications, the above-mentioned response surface method still has many shortcomings.For example, ANN has problems such as difficulty determining the optimal network topology, poor generalization ability under minor sample conditions, and overfitting (Xu et al. 2020).SVR has the problem that the model's optimal parameters and loss function are difficult to determine (Pepper et al. 2022).In addition, there is a common problem with the above methods: the computational accuracy is overly dependent on the construction of preset training samples.When the preset training sample size is small or the distribution is not ideal, the regression model will generate a large fitting error, leading to a significant error in the reliability calculation result.Conversely, when the preset training sample size is large, the computational accuracy is high, but the computational efficiency is low.
As a machine learning method developed recently, the Kriging model has a strict statistical theoretical basis and is adaptable to complex regression problems such as high dimensions, small samples, and nonlinearity.It has been widely used in many fields (Sundar and Shields 2019;Yan et al. 2020).However, the concrete arch bridge structure has problems such as small sample size, high-dimensional nonlinearity, and low calculation accuracy in the reliability calculation.Therefore, using the Kriging model to construct a response surface model for the reliability calculation of concrete arch bridges to approximate the actual limit state function will overcome the above problems.The Kriging model includes two models: regression model and correlation model.The regression model is a global approximation in space, and the attribute value of the unknown point is estimated by assigning weights to the known points around the unknown point.Correlation models reflect spatial distribution structures or spatial correlation types.At the same time, the range of the spatial correlation is given, and the observed value of the sampling point can be used to estimate the variable value of the unsampled point in the study area.In addition, selecting and determining relevant model parameters in the Kriging model is a multivariate optimization process.The performance of the numerical optimization algorithm directly affects the accuracy and stability of the relevant model parameters, which in turn affects the performance of the Kriging model (Chu et al. 2020;Qin et al. 2019).Most of the current Kriging models use the pattern search method (Wang et al. 2020;Zhan and Xing 2021) to solve the parameters of the related models.Like other traditional numerical optimization methods (such as the fastest descent method and quadratic programming method), these optimization algorithms have the advantage of high efficiency.However, it is very sensitive to the starting point selection, and it is easy to fall into the trap of local optimum, which cannot effectively guarantee convergence to the global optimum.Therefore, in order to improve the shortcomings of the Kriging model, strive to ensure high computational efficiency while obtaining the global optimum.Therefore, it is necessary to combine better optimization algorithms to improve the optimization efficiency of model parameters.
The PSOSA optimization algorithm integrates two different optimization mechanisms, the simulated annealing algorithm (Lee and Kim 2020) (SA) and particle swarm optimization algorithm (Gu and Hao 2020) (PSO), which is conducive to optimizing the search process and enhancing searchability and search efficiency in the global and local senses.SA algorithm adopts serial optimization structure (Zhai and Feng 2022), while PSO adopts group parallel search (Jiang et al. 2021).Combining the two algorithms can make SA a parallel SA algorithm and improve its optimization performance.At the same time, SA is an adaptive and variable probability mutation operation, which enhances the ability of PSO to get rid of the local minimum, effectively avoids falling into the local minimum, and makes the algorithm eventually tend to the global optimum.Therefore, this paper proposes a Kriging-PSOSA hybrid algorithm.This method not only takes advantage of the advantages of the Kriging model in dealing with problems such as complex structures, small samples, high dimensions, and nonlinearity but also uses the characteristics of the PSOSA hybrid algorithm to better update the particle swarm coordinates and search for the global optimal solution faster.In order to realize the rapid optimization and determination of the relevant model parameters in the Kriging model, the accuracy and efficiency of the reliability calculation of complex structures are effectively improved.
In order to solve the above problems and put forward the application of Kriging-PSOSA method in solving the reliability problems of concrete arch bridges.This paper proposes to use the Kriging model with analytical uncertainty, high dimensionality, and nonlinear problems to construct the response surface of the implicit performance function.Furthermore, the PSOSA algorithm is used to improve the optimization efficiency of model parameters so that the constructed response surface model can truly approach the limit state function of the concrete arch bridge structure.Through the example analysis and the actual engineering application of a concrete arch bridge, the correctness and feasibility of the method proposed in this paper are verified in the reliability calculation.The main contributions and novelties of this paper are: (1) This paper proposes a Kriging-PSOSA hybrid algorithm applied to the reliability calculation of concrete arch bridges, which overcomes the limitations of the classical response surface method in solving small-sample, high-dimensional nonlinear reliability problems of concrete arch bridges.(2) Solve the problems of the low computational efficiency of the MCS method in the traditional concrete bridge reliability solution method and the excessive dependence of the calculation accuracy of the existing response surface method on the scale and distribution of preset samples.
The remainder of this paper is organized as follows: Sect. 2 presents the basic theory of the Kriging model and PSOSA algorithm and their connection with this paper.Section 3 proposes the basic framework and calculation process of the Kriging-PSOSA hybrid algorithm, and uses examples to carry out the method verification.Section 4 uses the method proposed in this paper to calculate the structural reliability of an actual concrete arch bridge and compares and verifies it with other methods.Finally, the conclusions are drawn in Sect. 5.

Theory of Kriging model
The Kriging model is an interpolation model formed by superimposing a non-parametric stochastic process with a parametric linear regression model (Yang et al. 2022a, b).The expression for the model is: In the formula: Γ( , x) is the polynomial regression model.β is the regression coefficient is a random Gaussian process with zero mean and variance 2 .At different locations in the design space, the correlation between these random variables is expressed by covariance as: In the formula: ℝ(x i , x j ; ) is the correlation function between x i and x j .In the classic Kriging model, common correlation function models include exponential model, Gaussian model, linear model, spline function model, etc.At present, the commonly used function model is the Gaussian model (Wang 2021): In the formula: θ is the parameter vector, = [ 1 , 2 , ..., m ] T .m is the m-th dimension element of the input vector; M is the total dimension of the input vector.
Define the correlation matrix ℝ[(x i , x j ; )] N 0 ×N 0 , then the estimated values of and 2 are: In the formula: F is the identity matrix of N 0 × 1 .N 0 represents the number of training sample points.From Eqs. (1) to (3), it can be known that a Kriging model can be completely defined by the regression coefficient vector , the variance 2 of the random process, and the parameter vector .From Eqs. ( 4) and ( 5), it can be known that the regression coefficient vector and the variance 2 of the random process depend on the parameter vector .Therefore, when constructing the Kriging model, the parameter vector should be obtained first according to the sample points, and this process can be realized by maximum likelihood estimation.which is: The mean and variance of the predicted value Ĝ(x) for the predicted point x is expressed as: In the formula: And Ĝ(x) is taken as the predicted value of point x.

The basic theory of particle swarm optimization algorithm (PSOSA) based on simulated annealing algorithm
Particle swarm optimization (PSO) is a global random search algorithm based on swarm intelligence, which is proposed by simulating birds' migration and flocking behavior during foraging.Based on the concepts of "population" and "evolution", it realizes the search for the optimal solution in complex space through cooperation and competition among individuals (Pawan et al. 2022).Suppose that at time t , in an n-dimensional search space S ∈ R n and a population consisting of m particles, the position of the i-th particle is repre- sented by an n-dimensional vector, namely Each particle represents a candidate solution of the problem to be sought, and the fitness value is calculated by substituting X i into the fitness function.The quality of each solution is determined by its cor- responding fitness value.The better the fitness value, the better the corresponding solution.
The closer to the true solution, the moving speed of the particle is also an n-dimensional vector, namely The optimal position searched by the i-th particle so far in the S space is called the extreme individual value, denoted as The optimal position searched by the particle swarm so far is the global extremum, marked as g best = [P q1 , P q2 , ..., P qn ] , to represent the position of the best particle in the swarm.Each particle updates its velocity and position during the optimization process according to the following formula. (5) In the formula: particle number i = 1, 2, ..., m .t is the current iteration number.is the inertia weight, representing the influence coefficient of the last speed on the particle.c 1 and c 2 are learning factors, and c 1 represents the cognitive ability of the particle's experience, which is used to adjust the progress of the particle flying toward its best position.c 2 repre- sents the cognitive ability of the particle to learn social experience and adjusts the step size of the particle to the optimal global position.r 1 and r 2 are random numbers uniformly dis- tributed in the interval [0, 1].The purpose is to allow the particle to fly to the best position of the particle itself and the global best position of the particle with an equal probability of acceleration.
The simulated annealing algorithm is an extension of the local search method.However, it differs from local search by selecting the state with the largest cost value in the neighborhood with a certain probability of jumping out of the local extreme point.The acceptance criterion allows the objective function to deteriorate within a limited range, accepting new solutions with a certain probability (Gao et al. 2022).In reference (Aslett et al. 2017), the acceptance criterion allows the objective function to deteriorate without choosing according to probability, but directly according to ΔE < e , where ΔE is the change in fitness value caused by two positions, and e is the allowable target function deterioration range.Therefore, this paper combines the core steps of the two and proposes a particle swarm optimization algorithm (PSOSA) based on simulated annealing algorithm.Initialize each particle, set the number of particles n , randomly generate n initial solutions or give n initial solutions, and randomly generate n initial velocities.According to the current position and speed, the new position of each particle is generated, and the fitness value of each particle's new position is calculated.For each particle, if the fitness value of the particle is better than the original individual extreme value p i , set the current fitness value to the individual extreme value p i .According to each particle's individual extreme value p i , find the global extreme value p g .Update itself speed according to formula (9), update the current position according to formula (10), and calculate the amount of adaptation value change ΔE caused by the two positions.If ΔE < e , accept the new value; otherwise, reject.If the conditions are not met, or the maximum number of iterations is not reached, go to step 3-otherwise, end.

Design of Kriging-PSOSA calculation method
Based on the finite element model and the MATLAB calculation program, the Kriging-PSOSA hybrid response surface method is proposed to calculate and analyze the structural reliability.The process is as follows: (1) Determine the statistical characteristics and probability distribution of random variables in the operating state of the bridge structure, and use the uniform design method to generate input sample points.( 2 In order to reflect the accuracy and efficiency of the method, this paper uses the response surface method based on the Kriging surrogate model, the support vector machine response surface method, and other types of machine learning methods to analyze the reliability of the same structure.Moreover, through the accuracy analysis, the number of iteration steps, and other aspects, different methods are compared and analyzed to verify the feasibility and efficiency of the structural reliability calculation based on the Kriging-PSOSA hybrid response surface method.
Fig. 1 Kriging-PSOSA hybrid algorithm reliability calculation design and process

Example 1 verification
As shown in Fig. 2, for the ten-bar truss structure, let the length of the member be L , the cross-sectional area of the member be A s , the elastic modulus is E , and the external loads are P 1 , P 2 , and P 3 .All random variables are normally distributed, and their distribution characteristics are shown in Table 1.Taking the vertical displacement limit V(x) of the No. 2 node as the control variable, the allowable displacement is 0.004 m, and the performance function g = 0.004 − V(x) is established.
It can be seen from the above performance functions that it is a structural reliability problem with high-dimensional nonlinear implicit performance functions.The vertical displacement in the performance function is obtained using the ANSYS commercial finite element analysis program.The uniform design sampling method was used to generate 30 groups of samples within the range of [ − 3 , + 3 ] , and the train- ing samples were normalized and entered into the Kriging model for training, and the PSOSA algorithm was used to optimize the parameters.The optimization process to obtain the optimal weight parameter B of the model is shown in Fig. 3a.The optimal weight parameter of the model can be obtained by searching the PSOSA algorithm as [ 1 , 2 , 3 , 4 , 5 , 6 ] = [0.8865,0.0640, 0.0003, 0.0668, 0.5028, 0.0513].
Figure 3b shows the prediction result of the sample regression based on the Kriging model.It can be seen that the Kriging model constructed by the sample learning can accurately reflect the actual response surface of the performance function.Table 2 shows the calculation results of this example by the method in this paper and different methods in  reference (Su et al. 2013).Taking the calculation result of Monte Carlo sampling 200,000 times as the exact solution, its failure probability is 5.2253 × 10 -3 , and the corresponding reliability index is 2.6013.The failure probability calculated by the method in this paper is 5.1087 × 10 -3 , the corresponding reliability index is 2.6133, and the relative error is only 0.461%.Its iteration times and calculation accuracy are better than the Kriging model response surface method and support vector machine method (SVM).The calculation results of the example can fully demonstrate that the Kriging-PSOSA hybrid response surface method can well solve the structural reliability problem of high-dimensional nonlinear implicit performance functions.

Example 2 verification
As shown in Fig. 4, a plane truss has a calculated span and height are 9.0 m and 1.5 m, respectively.The cross-sectional area of the member is a random variable A with a mean of 1.6 × 10 −3 m 2 .Moreover, it is assumed that the elastic modulus E and the concentrated loads P 0 , P 1 , and P 2 are random variables, and the mean values are 2.0 × 10 7 kN∕m 2 , 30kN , 50kN , and 20kN , respectively.And the coefficient of variation of each random variable is 0.1.The node's vertical displacement limit at the lower chord's midspan (node number 4) is 0.12 m.The Kriging-PSOSA hybrid response surface method is used to calculate the reliability index of the plane truss.The statistical parameters of each variable of the structure are shown in Table 3.According to the limit state space function containing five random variables, a uniform design experiment was used to generate 30 groups of samples within the range of [ − 3 , + 3 ] .At this time, the interval contains 99.73% of the points, which meets the requirements of reliability calculation.Substitute the input sample into the ANSYS finite element model to calculate the corresponding response value (the vertical displacement value of the midspan node of the lower chord).The sample input and output points are combined to form a training sample, and the sample points are normalized and fed into the Kriging model for training.The PSOSA algorithm is used to optimize the parameters, and the optimization process of the optimal weight parameter 1 ∼ 5 of the model is obtained, as shown in Fig. 5a.It can be seen from Fig. 5b that the response surface function The failure probability calculated by the method in this paper is 9.1046 × 10 -4 , and the corresponding reliability index is 3.1182.In addition, according to reference (Yang et al. 2014), the failure probability of the plane truss structure obtained by 200,000 Monte Carlo important sampling simulations is 9.3322 × 10 -3 , and the corresponding reliability index is 3.1107.Considering the Monte Carlo calculation result as the exact value, the relative error of the Kriging-PSOSA algorithm is only 0.241%, which is similar to the reference results.In addition, the calculation methods and processes of different reliability of the plane truss are shown in Table 4.
It can be seen from Table 4 that compared with the Monte Carlo method, Kriging model response surface method, and support vector machine method (SVM).The relative error of the Kriging-PSOSA method proposed in this paper is only 0.241%, and the convergence condition is reached when the number of sample iterations is 22.It can be seen that the method in this paper has certain advantages in the calculation accuracy and calculation efficiency, and it has the characteristics of high precision and high efficiency.
In addition, the structural model of the above two examples is relatively simple.Compared with the traditional reliability calculation method, it is difficult to reflect the advantages of the Kriging-PSOSA hybrid response surface method in the reliability calculation of complex bridge structures.In order to further illustrate the application effect and advantages of this method in practical complex bridge structures and high-dimensional nonlinear implicit performance functions.In this paper, the reliability analysis and calculation of the deflection system of an actual concrete tied arch bridge is carried out, and the application of the Kriging-PSOSA hybrid response surface method in the actual complex bridge structure is further verified.

Project overview and establishment of bridge finite element model
A concrete arch bridge is a bottom-bearing reinforced concrete tied arch bridge with a span of 70 m.The sag-span ratio is 1/5, the sag height is 14 m, the arch axis is a quadratic parabola, and the bridge deck width is 10.0 m.The main arch and longitudinal beams are made of C50 concrete, the wind bracing is made of C40 concrete, and the suspenders are flexible.The site photo of the concrete tied arch bridge is shown in Fig. 6.According to the design data, ANSYS commercial finite element software is used to establish the bridge's initial finite element analysis model.Arch ribs and longitudinal beams are simulated by the BEAM44 element.The suspender is simulated by a three-dimensional unidirectional force LINK10 element, and the tensile force of the suspender is applied in the form of initial strain.The bridge deck is simulated with SHELL181 elements.The entire bridge has a total of 2049 nodes and 2360 units.The finite element model of the entire bridge is shown in Fig. 7.

Analysis and calculation of bridge deflection reliability
According to "General Design Specification for Highway Bridges and Culverts" (JTG D60-2015) and reference (Chen et al. 2007).The maximum allowable vertical deflection of the main arch under the action of vehicle load (excluding impact) is = L∕1000 = 0.07m .The maximum allowable vertical deflection of the main girder and bridge deck under the action of vehicle load (excluding impact) is = L∕800 = 0.0875m .A performance function for establishing the limit state of regular use, namely: where: v1 (x) is the maximum vertical deflection of the main arch.v2 (x) is the maximum vertical deflection of the main girder and deck.During the service process of bridges, there are uncertainties due to the changes in parameters such as material properties, structural geometric dimensions, and external loads with environmental changes.As a result, there is a big difference between the actual structural and theoretical design states.Many factors affect the safety of concrete arch bridges during operation.In this paper, the geometric deformation of the structure is used as the control index, and the random factors shown in Table 5 are selected for analysis mainly from the factors that significantly influence the bridge-forming state of the structure.The parameters of random variables are obtained through design data and actual measurement, as shown in Table 5.
According to the "Unified Standard for Reliability Design of Engineering Structures" (GB 50153-2008) and reference (Yang and Qin 2008), it can be known that the engineering reliability requirements can be met when the engineering structure reliability index is greater than 4.0.A 15-dimensional space is formed with random parameter variable E 1 , E 2 , E 3 , E 4 , E 5 , r 1 , r 2 , r 3 , r 4 , r 5 , A 1 , A 2 , I 1 , I 2 , q 1 .The uniform design sampling method was used to generate 50 groups of random samples within the range of [ − 3 , + 3 ] to form random input samples, as shown in Table 6 (due to space limitations, only the first three groups are listed).Based on the finite element model of the concrete arch bridge, the above samples are substituted into the calculated response values in turn, and 50 sets of corresponding vertical deflection response values of the main arch and vertical deflection of the main beam are obtained.Finally, two sets of training samples of random parameterdeflection of main arch and random parameter-deflection of main beam are generated.The After the two sets of training samples of the main arch and the main beam are optimized by the PSOSA algorithm, the optimal weight parameter 1 ∼ 15 of the model is obtained, respectively.The optimization process and results are shown in Figs. 8 and 9.It can be seen from Figs. 8 and 9 that the minimum mean square error of the weight parameters optimized by the PSOSA algorithm is as low as 0.0174% and 0.0224%, respectively.It is sufficient to ensure the construction of high-precision response surfaces.
Based on the Kriging model, the deflection response surface functions of the main arch and the main beam were constructed, respectively.This paper gives the response surface diagrams of parameters E 1 , I 1 , and main arch and main beam, respectively, as shown in Fig. 10a, b.The response surface plot can reflect the fluctuation of the deflection response value within the sample threshold.In addition, based on the Kriging model, the training samples are used for regression prediction, and the prediction results are shown in Fig. 10c,  d.It can be seen that the response surface function constructed by the Kriging model can truly simulate the structural limit state functions of the main arch and the bridge deck and has good accuracy.
Using the response surface model obtained by the Kriging-PSOSA hybrid algorithm for iterative calculation, the failure probability of the main arch of the concrete arch bridge based on the vertical deflection index is 2.4253 × 10 -5 , and the corresponding main arch reliability index β is 4.1316.The failure probability of the main beam based on the vertical deflection index is 5.2891 × 10 -6 , and the corresponding main beam reliability index β is 4.4667.All meet the requirements of engineering structure reliability index β > 4.0.
In addition, the Kriging model response surface method and the support vector machine method (SVM) were used to calculate the reliability of the main arch and main beam of the concrete arch bridge.The calculation results of different methods are shown in Tables 7  and 8.It can be seen from the calculation results that the Kriging-PSOSA hybrid algorithm proposed in this paper achieves convergence with 24 and 28 iterations of the calculation of the main arch and the main girder in the reliability calculation process of the concrete arch bridge.Compared with the other two methods, the number of iterations is relatively small, and the calculation accuracy can meet the requirements.It can be seen that the Kriging-PSOSA hybrid algorithm proposed in this paper can be well applied to the reliability calculation and analysis of actual complex bridge structures, with high calculation efficiency and reliable calculation accuracy.

Conclusions
In calculating the reliability of the traditional concrete arch bridge structure, due to its complex structure, the performance function presents high-dimensional nonlinearity, and there is no clear analytical expression, resulting in low calculation accuracy and difficulty in convergence.This paper proposes a Kriging-PSOSA hybrid algorithm based on the Kriging model and the particle swarm optimization algorithm (PSOSA) to solve the structural reliability of concrete arch bridges.Through the example verification and the engineering application of the actual concrete arch bridge.It shows that the method in this paper has obvious advantages in the number of samples, calculation accuracy, and calculation efficiency, and it is easy to combine with the existing general-purpose finite element analysis software to achieve rapid and accurate analysis of bridge structure reliability.The main conclusions are as follows: ( (3) Compared with the traditional bridge reliability analysis method, the Kriging-PSOSA hybrid response surface method has the advantages of high calculation accuracy, fast iteration speed, and ease of combining with general finite element analysis software.
It is convenient for practical engineering applications, especially for solving complex  structural reliability problems with high structural analysis costs and highly nonlinear and implicit performance functions.It provides a new idea for the research on the reliability calculation method of large and complex bridges.
) Establish a structural finite element model based on the bridge design data and operating conditions, calculate the target variables corresponding to each input sample, and obtain an output sample.And then form a training sample with the input sample.(3) Normalize the sample points, and use the DACE toolbox(Huang et al. 2022) to establish the basic Kriging model.Through the unsupervised training and parameter optimization process model by inputting sample points, the structural Kriging model is obtained.(4) Standard normalization of random variables, using penalty function to transform the constrained optimization problem into an unconstrained optimization problem, and construct a fitness equation suitable for the PSOSA algorithm to solve.Furthermore, the PSOSA algorithm updates the optimal position of search particles and particle swarms.Iteratively obtains the optimal weights of random variables to support the unsupervised learning process of the Kriging model.(5) A mathematical model for solving the structural reliability index is established through the prediction results of the Kriging model.In this process, it is necessary to update and optimize the samples of each Kriging prediction model so that the Kriging prediction model can well approximate the sample points; until the model builds a sufficiently accurate response surface, it can realistically simulate the structural limit state function.The specific flow of the Kriging-PSOSA hybrid algorithm for bridge structural reliability analysis is shown in Fig.1.

Fig. 3
Fig.3The calculation process of Kriging-PSOSA mixed response surface method

Fig. 4
Fig. 4 Reliability calculation diagram of plane truss

Fig. 5
Fig. 5 Process diagram of Kriging-PSOSA mixed response surface method 1) A Kriging-PSOSA hybrid algorithm for the reliability calculation of concrete arch bridges is proposed, which combines the Kriging model with the particle swarm optimization algorithm (PSOSA) based on the simulated annealing algorithm.Using the Kriging model establishes a small sample, nonlinear, high-dimensional implicit performance function response surface model.Combined with the SA algorithm's selfadaptive and variable probability mutation operation, the problems of slow convergence and premature maturity of the traditional PSO algorithm are improved.It effectively avoids falling into the local minimum problem and makes the calculation result tend to the global optimum.(2) Numerical examples and actual concrete arch bridge engineering case analysis and research results show that the Kriging-PSOSA hybrid algorithm for bridge reliability calculation proposed in this paper is correct, and it effectively improves the accuracy and efficiency of reliability calculation for complex structures.It overcomes the limitations of the traditional response surface method on the reliability of highly nonlinear structures.It solves the problem of the low computational efficiency of the traditional Monte Carlo method and the excessive dependence of the calculation accuracy of other response surface methods on the preset sample size and distribution.

Table 1
Statistical parameters of random variables of cross truss structure

Table 2
Comparison of results of different reliability calculation methods for ten-bar truss

Table 3
Statistical parameters of random variables of plane truss structure

Table 4
Comparison of results of different reliability calculation methods for plane trusses groups of sample points were normalized and substituted into the Kriging model for training, and the PSOSA algorithm was used for parameter optimization. two

Table 5
Statistical characteristics of random variables of concrete arch bridges

Table 6
Training samples for uniform design generation variables

Table 7
Comparison of reliability results of different calculation methods for main arch

Table 8
Comparison of reliability results of different calculation methods for main beams