Computational cost reduction and best model updating method seeking are the key issues during model updating for different kinds of bridges. This paper presents a combined method, Kriging model and Latin hypercube sampling method, for finite element (FE) model updating. For FE model updating, the Kriging model is serving as a surrogate model, and it is a linear unbiased minimum variance estimation to the known data in a region which have similar features. To predict the relationship between the structural parameters and responses, samples are preselected, and then Latin hypercube sampling (LHS) method is applied. To verify the proposed algorithm, a truss bridge and an arch bridge are analyzed. Compared to the predicted results obtained by using a genetic algorithm, the proposed method can reduce the computational time without losing the accuracy.
Bridge structures are playing important roles in our lives for supporting essential social and economic functionalities. However, they are potentially prone to damage due to significant loads during their service lives. For this reason, structural health monitoring (SHM) technologies have been deeply studied with the goal to monitor the conditions of the bridges and detect the damages as well as assess their conditions [
For a long time, algorithm development for FE model updating is a hot research area in SHM, and the goal of FE model updating is to minimize the discrepancies between measured data of a real structure and FE model [
For complex structures, such as cablestayed bridges, suspension bridges, and the other composite bridges, the number of nodes, elements, materials, and boundary conditions are complex.
It is still an issue to obtain accurate results with less computational time. To achieve high computational efficiency, some studies on substructure methods are accounted for truncation approximation method proposed by Weng et al. [
For exploring Kriging model updating in bridge structures, lots of researchers play important roles during the development of this method. In order to select the inputs to predict efficiently, Sacks et al. introduced the Kriging model, which originated from a statistical approach to mine valuation by a South African geologist Krige [
This paper uses the Kriging model updating for a truss bridge and an arch bridge. The training points for the Kriging model are obtained by Latin hypercube Sampling (LHS) instead of the design of an experiment. For these the two cases, the acceleration data were measured and used for optimization. LHS is applied to produce training samples. Compared to genetic algorithm (GA), the model updating results of two cases showed that the proposed method has significant improvement to reduce the calculation time without losing the accuracy.
The Kriging model was proposed by a geologist D. G. Krige. When applied it as a surrogate model, the response function
In the Kriging model, the values of
Then, the prediction value
Above equation is equivalent to the minimum optimization problem:
Finally,
For FE model updating, one key issue is to obtain several sample points. The sampling methods mainly can be classified as simple random sampling (SRS), stratified sampling method, cluster sampling method, and systematic sampling. For SRS, it is an unbiased surveying technique and a subset of individuals chosen from a large set [
Latin hypercube sampling (LHS) is a stratified sampling technique proposed by Mckay et al. [
In LHS, it must be decided how many sample points are to be used, and for each sample point, it is important to remember in which row and column the sample point was taken. LHS can ensure that the ensemble of random numbers is representative of the real variability whereas random sampling is just an ensemble of random numbers without any guarantees.
For LHS, it can take all the variation of input variables into consideration, just with a small number of runs. The total range of the input variables is ensured to be represented. For example, if we want to sample
Divide the interval of each dimension into
Sample randomly from a uniform distribution, a point in each interval in each dimension
Pair randomly (equal likely combinations) the points from each dimension
Figure
Examples of two methods to generate a sample of size
In this example, we apply the Kriging model to a plane truss to do model updating [
Elevation view of the truss bridge (unit: cm).
In the literature [
Establish the FE model through ANSYS [
Assume the damping ratio is 1% and excite the structure at node 3 with vertical excitation, and the excitation force is simulated with bandlimited white noise (BLWN). The vertical acceleration response of each DOF of the structure was calculated by Newmarkbeta timehistory analysis method. Eigensystem realization algorithm (ERA) was applied to obtain the first sixth modal parameters.
The objective function is established and then optimized by GA. The number of populations is 100. Then, the updated results are shown in Table
Flowchart of GA.
Comparison of truss updating results (unit: Hz).
Mode number  Test  Literature [ 
Error (%)  Kriging model  Error (%) 

1  8.79  8.83  −0.55  8.79  0.00 
2  29.60  30.18  −1.48  29.78  −0.60 
3  43.39  41.65  2.23  42.66  1.69 
4  59.10  59.62  −0.24  59.55  −0.75 
5  90.62  91.34  −0.40  91.09  −0.52 
6  119.81  120.84  −0.11  120.86  −0.88 
When applying the Kriging model to do model updating, the procedure is as follows, and the flowchart of the Kriging model is shown in Figure
Set elastic modulus and density of the materials as design parameters. Assume these design parameters and follow normal distributions
According to the distribution range of each random variable, the initial training samples are obtained through LHS method, and the training samples can be ranged from 10 to 100.
With the training samples, the structural responses are obtained with the FE model, and then the mapping relation of inputoutput of the initial training samples is obtained.
The parameter
Based on the Kriging model, the prediction is performed on the test samples to obtain the corresponding modal parameters.
Flowchart of the Kriging model.
The comparison optimization results between the GA method in the literature [
The frequencies of the first 3 modes from the Kriging model are better than that of the GA from the literature, and the errors for the first 6 modes are all below 2%. Based on the updated FE model, the updated design parameters are
In this example, the FE model of a tiedarch bridge is taken as an object. This bridge is a longspan concretefilled steel tubular continuous girder arch composite system bridge with the span length of 58.4 + 128.0 + 58.4 m (Figure
Elevation of the bridge (unit: cm).
The ambient arbitrary vibration measurement was conducted on the bridge after the bridge construction and before its opening to the operation to obtain the eigenmodes (including natural frequencies, mode shapes, and damping ratios), which were very important parameters for health monitoring and malfunctioning diagnosis of bridges.
In order to obtain the vertical and lateral modal characteristics of the bridge, measuring points are arranged. For the vertical test of the girder, there are 28 sensors, and each of them is arranged on each boom of the girder. For the lateral test of the girder, 7 sensors are arranged, and each of them is placed at the boom of the downstream. For the test of arch rib, in total, 32 sensors are arranged, half for vertical test and the half for the lateral test. The sensors are placed at the position of lateral braces. The arrangement of measuring points is shown in Figure
The arrangement of the sensors (unit: m).
The space bar model of the bridge was established by ANSYS (Figure
The FE model of the bridge.
By using the multiple reference DOF stabilization diagram algorithms based on NExT/ERA (MNExT/ERA), the modal parameters (natural frequencies and mode shapes) of the bridge are identified. Through the comparison, the results are shown in Table
Comparison of testing results and FE model results (unit: Hz).
Measured frequency  FEM frequency  Error (%)  Position  Modal description 

0.5911  0.4894  17.21  Arch rib  1st order symmetrical lateral bending 
1.1804  1.0322  12.56  2nd order antisymmetrical lateral bending  
1.8387  1.6145  12.19  3rd order symmetrical lateral bending  
2.5883  2.3091  10.79  4th order antisymmetrical lateral bending  
3.4136  2.9343  14.04  5th order symmetrical lateral bending  
4.3012  3.8008  11.63  6th order antisymmetrical lateral bending  
5.1021  4.5375  11.07  7th order symmetrical lateral bending  


2.0143  2.0799  −3.26  Girder and arch rib  1st order antisymmetrical vertical bending 
2.9367  3.0691  −4.51  2nd order symmetrical vertical bending  
3.2499  3.5905  −10.48  3rd order symmetrical vertical bending  
3.8289  4.2221  −10.27  4th order antisymmetrical vertical bending  
4.7944  4.4713  6.74  5th order symmetrical vertical bending  
4.9524  5.0990  −2.96  6th order antisymmetrical vertical bending  


3.9760  3.2979  17.05  Girder  1st order antisymmetrical lateral bending 
4.6262  5.2794  −14.12  2nd order symmetrical lateral bending  


4.4507  4.6486  −4.45  Arch rib  1st order torsional 
5.4851  5.5923  −1.95  2nd order torsional 
According to the structural design parameters given in the design drawings, the FE model was established to obtain the original natural frequencies and mode shapes of the bridge. By comparing the results between initial FE model and the experimental results (Table
As the initial FE model cannot show the responses of the real structure, in order to ensure the accuracy of the prediction and analysis of the FE model, model updating is necessary. For this arch bridge, the Kriging model and GA are applied.
In the FE model updating, there are many candidate parameters that could be used to produce the required change. One way is to allow all the parameters to take part in the updating procedure, which could lead to expensive computational expense. The other is to select a certain number of updating parameters based on the sensitivity. Then, how to select updating parameters and how many should be selected become important issues. Referred to the literature [
Arch ribconcrete: elastic modulus (
Arch ribsteel tube: elastic modulus (
Hanger rod: elastic modulus (
Girderconcrete: elastic modulus (
Wind support rigid arm: elastic modulus (
Hanger rod rigid arm: elastic modulus (
Secondary dead load (
Based on the selected parameters, the two updating methods can be applied.
As mentioned above, the Kriging model is a learning theory which is based on small sample statistical learning and forecasting and is regarded as the best optimizing linear unbiased estimate method. Without affecting the accuracy of the results, the time of calculation and analysis is reduced, and the computational efficiency is improved. To apply the Kriging model, firstly, take 100 sets of parameters as the design variables, and then substitute them to the FE model to obtain 100 sets of results. Secondly, take the first two or three corresponding natural frequencies of the arch rib lateral mode, arch rib vertical mode, girder lateral mode, and arch rib torsional mode as training samples. Then, based on the obtained four groups of modified design parameters, the corresponding weight values are assigned according to the accuracy of the modified results, and the final update results are obtained. The weight coefficient is a random number with the sum of 1 and is distributed according to the accuracy.
When applying genetic algorithm (GA), assume the design parameters obey uniform distributions
By using the Kriging model and GA, the updated parameters are listed in Table
Updated results of design parameters.
Design parameter  Initial model  Kriging model  GA  Variation coefficient (%) 


3.45 
3.51 
3.76 
10 

2600.00  2394.30  2537.30  10 

2.06 
2.47 
2.58 
5 

7800.00  7992.40  7794.80  5 

1.96 
2.10 
1.99 
5 

7800.00  7738.20  7661.60  5 

1.50 
1.81 
1.71 
10 

3.45 
3.41 
3.25 
10 

2600.00  2850.90  2603.00  10 

1.50 
1.98 
1.83 
10 

1.45 
1.35 
1.45 
8 
Based on the updated parameters, the updated natural frequencies and mode shapes are also obtained. The specific results are shown in Table
Comparison of testing results and FE model results (unit: Hz).
Measured frequency  FEM frequency  Error (%)  Kriging model  GA  Position  Modal description 

0.5911  0.4894  17.21  0.5369  0.5408  Arch rib  1st order symmetrical lateral bending 
1.1804  1.0322  12.56  1.1307  1.1380  2nd order antisymmetrical lateral bending  
1.8387  1.6145  12.19  1.7664  1.7805  3rd order symmetrical lateral bending  
2.5883  2.3091  10.79  2.5228  2.5445  4th order antisymmetrical lateral bending  
3.4136  2.9343  14.04  3.1978  3.2306  5th order symmetrical lateral bending  
4.3012  3.8008  11.63  4.1338  4.1753  6th order antisymmetrical lateral bending  
5.1021  4.5375  11.07  4.9215  4.9764  7th order symmetrical lateral bending  


2.0143  2.0799  −3.26  2.0350  2.0455  Girder and arch rib  1st order antisymmetrical vertical bending 
2.9367  3.0691  −4.51  2.9730  3.0057  2nd order symmetrical vertical bending  
3.2499  3.5905  −10.48  3.4377  3.4942  3rd order symmetrical vertical bending  
3.8289  4.2221  −10.27  4.1427  4.1815  4th order antisymmetrical vertical bending  
4.7944  4.4713  6.74  4.4020  4.4282  5th order symmetrical vertical bending  
4.9524  5.0990  −2.96  5.2573  5.1614  6th order antisymmetrical vertical bending  


3.9760  3.2979  17.05  3.7637  3.8466  Girder  1st order antisymmetrical lateral bending 
4.6262  5.2794  −14.12  5.0266  5.1318  2nd order symmetrical lateral bending  


4.4507  4.6486  −4.45  4.8588  4.8203  Arch rib  1st order torsional 
5.4851  5.5923  −1.95  5.9617  5.8306  2nd order torsional 
In order to show the high correlation of mode shapes between the measurement and FE model, for the simplicity of calculation, only some modes are listed in Table
The MAC values of each model updating method.
Mode shape  FE model  Kriging model  GA 

Arch rib 1st order symmetrical lateral bending  0.9673  0.9678  0.9579 
Girder 1st order antisymmetrical bending  0.9931  0.9932  0.9932 
Arch rib 1st order antisymmetrical vertical bending  0.8185  0.8173  0.8193 
Girder 1st order antisymmetrical vertical bending  0.9931  0.9932  0.9932 
Arch rib 1st order torsional  0.9796  0.9791  0.9791 
In order to obtain the corresponding frequency of the bridge, the MAC was applied to match the mode shapes (Table
Comparison of the error 2norm.
Initial model  Kriging model 1  Kriging model 2  Kriging model 3  Kriging model 4  Kriging model  GA 

1.5215  1.6890  1.6019  1.3656  1.3468  1.0505  0.9852 
In Table
The updated results by using GA show that only the error of firstorder natural frequency is over 10%, and the updated results have been improved a lot (its error 2norm is smallest). Moreover, as enough population and genetic algebra are defined, GA can obtain higher accurate results, while the process takes a long time.
In this paper, a dynamic model updating procedure for bridges based on the Kriging model is presented. The Kriging model was established based on LHS, and the effectiveness of the Kriging model method was investigated in two typical bridges. Comparing the proposed method with GA, the following conclusions can be drawn:
After the FE model updating, the accuracy of modal frequencies and mode shapes is greatly improved. Therefore, the updated FE model can be used for a longtime structural health monitoring.
The updated FE model with the Kriging model just needs a certain number of measured natural frequencies, and then a high accurate updated model can be obtained. In terms of computational cost reduction, compared to GA, the proposed method can reduce the computational cost without losing accuracy, which implies further potential application in the engineering field.
The proposed method only applies to a truss bridge and a tiedarch bridge in the paper, and although the Kriging model has its advantages in reducing the computational expense compared to GA, the advantages of the proposed method should be further validated for the application in complex bridge structures, such as cablestayed bridges, suspension bridges, and composite bridges.
The proposed method is verified on a truss bridge and a tiearch bridge; for the other types of bridges, such as simple supported beam bridge, suspension bridge, and cablestayed bridge, the application can be done in the same manner.
The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The first author wishes to thank the support from National Natural Science Foundation of China Grants nos. 11672108, 11202080, and 51478193 and the innovation fund of South China University of Technology leading to an excellent Ph.D. dissertation. Shiping Huang is also supported by the Research Fund of The State Key Laboratory of Coal Resources and Safe Mining, CUMT, and by the Fundamental Research Funds for the Central Universities. The authors wish to thank all colleagues for helping in the survey and measurements. The support of China Railway Guangzhou Bureau Group Co., Ltd., Research Project is also gratefully appreciated.