Based on the basic theory of wavelet neural networks and finite element model updating method, a basic framework of damage prognosis method is proposed in this paper. Firstly, a damaged Isteel beam model testing is used to verify the feasibility and effectiveness of the proposed damage prognosis method. The results show that the predicted results of the damage prognosis method and the measured results are very well consistent, and the maximum error is less than 5%. Furthermore, Xinyihe Bridge in the BeijingShanghai Highway is selected as the engineering background, and the damage prognosis is conducted based on the data from the structural health monitoring system. The results show that the traffic volume will increase and seasonal differences will decrease in the next year and a half. The displacement has a slight increase and seasonal characters in the critical section of mid span, but the strain will increase distinctly. The analysis results indicate that the proposed method can be applied to the damage prognosis of girder bridge structures and has the potential for the bridge health monitoring and safety prognosis.
Currently, Structural Health Monitoring (SHM) systems are widely used in bridge monitoring and maintenance management. However, they focus on the data accumulation, safety assessment, and so forth. Then the SHM systems can not satisfy the demands of proprietors, because they are generally limited to the damage detection (DD) level, in which the damage in a structural or mechanical system is herein defined as intentional or unintentional changes to its material and/or geometric properties, including variations of its boundary conditions, which adversely affect its performance [
The structural uncertainty and prognosis reliability are the basic topics of DP. Current prognosis methods for predicting structural failures are classified as conventional reliability models, modelbased prognosis models, and datadriven prognosis model [
Datadriven approaches attempt to derive models directly from routinely collecting conditionmonitored (CM) data instead of building models based on comprehensive system physics and human expertise. They are built based on historical records and produce prediction outputs directly in terms of CM data [
In this paper, the theoretical basis of wavelet neural network algorithm and its implementation process are described. Then, the wavelet neural network method and model updating method are combined to conduct the DP of structure. Subsequently, the framework of DP is proposed and applied to the DP of a damaged Isteel beam and sixspan continuous beam bridge, which means the initial realization of the second step of DPpredicting the future structural loads and structural properties. It also laid a solid foundation to the realization of the third steppredicting the remaining life of the structure.
Many problems are nonlinear in the field of civil engineering. The relationships between variables are so complicated that a large number of practical engineering problems are difficult to have exact solutions in math. However, the wavelet neural networks model is relying on an artificial intelligence algorithm, which uses wavelet unit to replace neuron. The model is composed of many interconnected wavelet units, which are used to simulate the formal structure of the human mind. Then the nonlinear dynamical system based on wavelet network can be obtained. Due to the learning ability, the weight parameters of the network can be updated continuously according to the setting rules, and then the wavelet neural network has the abilities of fitting function and dealing with information [
A wavelet neural networks model is shown in Figure
The wavelet neural network model.
The learning algorithm is the core of wavelet neural network, and its purpose is to establish mapping relationship between inputs and outputs by function fitting. A widely applied iterative algorithm called Stochastic Gradient Algorithm (e.g., [
A learning sample is given as (
As shown in (
The flow chart of wavelet neural network. Note:
Theoretically, there should be an optimal number of wavelet units in hidden layer. However, these empirical formulas are always gained from experiments in other fields, and they can only get a range or approximation instead of an exact figure. Therefore, the experience formulas can only be taken as a reference to determine the number of wavelet units in hidden layer in the field of civil engineering.
Damage prognosis combines modelbased method with datedriven method. Namely, the structural responses are calculated based on the updated finite element model and used as sample values which are needed by datadriven method [
A general damage prognosis solution procedure based on wavelet neural network and model updating.
At present, there are rarely experiments about DP in the field of civil engineering. In this paper, through a damaged Isteel beam model testing, the damage evolution of the dynamic property is explored. Finally, the prediction results based on the wavelet neural networks method are compared with the ambient vibration testing results in order to verify the validity of the proposed DP method.
This testing includes four Isteel beams, each one with a length of 3 meters. The density of the steel material is 7800 kg/m^{3}, and the elasticity modulus is 2.1 × 10^{5} MPa. In the Isection, area is 14.33 cm^{2}, inertia moment is 223 cm^{4}, and the damage section inertia moment is 10.5 cm^{4}. Among the four beams in this testing, one beam is undamaged, marked as beam0; one has a notch in the
Comparison of updated frequencies and measured frequencies, MAC values (unit, Hz).
Model number  Measured frequencies ①  Updated frequencies ②  Relative error 
MAC values (%) 

beam0  
First order  33.62  32.78  −2.5  93.1 
Second order  135.03  130.78  −3.1  91.4 
Third order  293.03  293.02  0  95.5 
beam1  
First order  20.71  22.02  6.3  95.1 
Second order  131.02  130.04  −0.7  93.4 
Third order  231.18  232.69  −0.7  90.9 
beam2  
First order  18.90  19.46  2.9  96.7 
Second order  85.94  90.67  5.5  94.3 
Third order  202.54  203.07  0.3  90.8 
beam3  
First order  17.02  17.29  1.6  94.7 
Second order  66.69  67.14  0.7  94.9 
Third order  148.33  149.68  0.9  90.2 
The notch of steel beam.
The environmental vibration testing.
Using ANSYS finite element (FE) analysis software, these FE beam models are established by beam3 elements, and it is composed of 60 elements and 61 nodes, as shown in Figure
Numerical model (unit, m).
Wavelet neural network training is conducted according to the method of this paper, and Morlet wavelet is selected as the excitation function; the network training is effective when the number of wavelet units in hidden layer is 3 (as (
According to the standard normal distribution, the initial parameters are selected randomly as follows:
The future fundamental frequency of a damaged Isteel beam was selected as the output value in this paper. Firstly, the experimental samples were selected by Doptimal design method, and the damage degree was simulated by cutting section stiffness. Then fundamental frequencies of beams under different damage degrees were calculated through the updated FE model. In order to validate the network, three beams were employed as the test samples, as shown in Tables
Training samples of the experimental design.

Inputs of section stiffness (×10^{−6} m^{4})  Output of the fundamental frequency (Hz)  

Damage section1  Damage section2  Damage section3  
1  0.93  0.92  2.2  30.67 
2  0.05  0.05  1.58  14.07 
3  1.21  1.4  0.45  29.91 





28  0.05  2.23  0.72  20.18 
29  0.05  0.73  2.23  19.91 
30  0.05  0.05  0.05  12.56 
The test samples.

Inputs of section stiffness (×10^{−6} m^{4})  Outputs of fundamental frequency (Hz)  

1  2.23  0.105  2.23  22.018 
2  0.105  0.105  2.23  19.455 
3  0.105  0.105  0.105  17.288 
Note: “
The training samples and test samples are put into the wavelet neural networks solution procedure based on MATLAB 7.0 software. Then adaptive training was realized, and suitable weight parameters can be obtained. Furthermore, the DP function model can be established.
As the DP function model has been established, the fundamental frequencies of these three steel beams can be predicted relying on the function model. The DP results based on wavelet neural network are shown in Table
The results of damage prognosis.

The measured values 
Theoretical values 
Output 
The error rate with theoretical values 
The error rate with measured values 

1  20.713  22.018  21.72  1.35%  4.86% 
2  18.903  19.455  19.59  0.70%  3.63% 
3  17.017  17.288  16.76  3.02%  −1.51% 
Xinyihe Bridge (Figure
Xinyihe Bridge.
Xinyihe Bridge panorama photo
The weighinmotion (WIM)
The traffic data used were monthly average number of cars during June 2012 to June 2013 (Figure
Historical traffic intensity (2012.06–2013.06).
The future traffic intensity (2013.07–2014.12).
According to the design papers, the FE model was built. Based on the twice ambient vibration testing in 2012 and 2013, the FE model was updated according to the thirdorder response surface method [
Parameters change before and after updating based on the response surface method.
Updated parameters 







(×10^{4} MPa)  (×10^{4} MPa)  (×10^{4} MPa)  (×10^{4} MPa)  (×10^{6} N/m)  (×10^{6} N/m)  (×10^{6} N/m)  


2012  3.6  3.5  3.2  2.5  0.6  3.1  3.1 
2013  3.3  2.8  3.2  2.2  0.9  4.7  1.7 
Rate (%)  −8.8  −19.8  −1.9  −12.4  50.0  48.4  −43.3 
Notice:
Comparison of updated frequencies and measured frequencies (unit, Hz).
Vibration mode  Frequency (Hz)  

Measured 
Updated 
Error1 (%)  Measured 
Updated 
Error2 (%)  
Vertical  
First order  2.89  2.88  0.43  2.87  2.77  3.48 
Second order  3.03  3.081  1.84  3.16  2.89  8.66 
Third order  3.79  3.61  0.51  3.67  3.34  8.99 
Fourth order  4.26  4.32  1.27  4.25  4.02  5.37 
Transverse  
First order  0.83  0.73  0.16  0.92  0.87  4.90 
Second order  1.43  1.47  2.84  1.45  1.48  1.45 
Longitudinal  
First order  1.79  1.74  2.45  1.30  1.28  1.61 
Notice: measured value1, updated value1, and error1 were the updated results in February 2012; measured value2, updated value2, and error2 were the updated results in June 2013.
As mentioned above, the FE model could be updated continuously based on the results of ambient vibration testing in the future. If there are enough frequencies samples, the wavelet neural network will be used for the model parameters prediction, namely, the prediction of the FE model. Due to that there is no realtime dynamics monitoring, we could not have enough samples, and then the model parameters change rate between the two updated FE models will be used as the actual structure performance degradation rate in this paper.
The maximum values of displacement and strain were separately selected as outputs of the prognosis model in this paper, and the parameters (
Training samples.





Vehicle load 
The maximum value of displacement  The maximum value of strain 

1  2.5  1.5  2  1.5  2.0  35.700  443.51 
2  2.5  3.3  2  3  2.0  28.316  347.75 






 
30  3.5  1.5  2.6  1.5  1.1  16.164  198.27 
31  2.5  3.3  3.5  1.5  2.0  28.969  352.76 
Notice: the value of vehicle load (
Verification of the results of damage prognosis model.
Measure value  Measured values  Outputs of network  Error (%) 

The maximum value of displacement (mm)  15.37  16.77  9.11 
The maximum value of strain ( 
534  475.2  11.01 
The training samples are substituted into the MATLAB model based on wavelet neural network; the DP model can be established relying on the selftraining of the prognosis model, and the goal is the error parameter
The accuracy of the prognosis model is verified by comparing the prediction results with the static test results in June 2013, as shown in Figure
The static and dynamic test of Xinyihe Bridge.
As shown in Table
The maximum value of displacement (2013.07–2014.12).
The maximum value of strain (2013.07–2014.12).
A damage prognosis framework for bridge structure is proposed combining the wavelet neural network method with the finite element model updating. The effectiveness of the proposed method is verified through a damaged Isection steel beam testing, and the maximum error is less than 5%.
The prognosis method can be used to predict future traffic load volume based on the traffic load monitoring. The traffic intensity predictions of Xinyihe Bridge show that the traffic volume will increase and seasonal differences will decrease in the next year and a half.
The prognosis prediction results agree very well with those from static load testing of Xinyihe Bridge. The errors between the predicted and measured results were within the range of 12%. The displacement has slight seasonal characters in the mid span, and the strains increase distinctly. The prognosis model can be used for the bridge safety prognosis and future maintenance.
This paper only solves the problem of the predicting of the future structural loads and structural properties. The remaining life prediction of real bridge structure is the next attention of our research. The proposed damage prognosis method can be incorporated into the structure health monitoring system, for the purpose of the online safety prognosis, and costefficient condition based maintenance of bridge structures.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support provided by the Natural Science Foundation Committee of China (Grant nos. 51178101 and 51378112) and the University Graduate Student Scientific Research Innovation Plan of Jiangsu Province (Grant no. CXZZ13_0109).