This paper investigates field applicability of a new system identification technique of estimating tensile force for a cable of long span bridges. The newly proposed hSI method using the combination of the sensitivity updating algorithm and the advanced hybrid microgenetic algorithm can allow not only avoiding the trap of local minimum at initial searching stage but also finding the optimal solution in terms of better numerical efficiency than existing methods. First, this paper overviews the procedure of tension estimation through a theoretical formulation. Secondly, the validity of the proposed technique is numerically examined using a set of dynamic data obtained from benchmark numerical samples considering the effect of sag extensibility and bending stiffness of a sagcable system. Finally, the feasibility of the proposed method is investigated through actual field data extracted from a cablestayed Seohae Bridge. The test results show that the existing methods require precise initial data in advance but the proposed method is not affected by such initial information. In particular, the proposed method can improve accuracy and convergence rate toward final values. Consequently, the proposed method can be more effective than existing methods in terms of characterizing the tensile force variation for cable structures.
Recently, the number of cablestayed structures built in countries around the world is increasing. The cablestayed structures deteriorate due to a variety of reasons such as overloading, aging, manufacturing imperfection, and climate conditions. Localized damage in materials may make deleterious contributions to the tensile force of the cable system. Therefore, it is necessary to estimate the tension of the cables to ensure structural safety during the construction phase and maintenance work after completion.
The tensile force of the cable system can be characterized by means of a condition assessment technology based on a variety of disciplines. Among the many nondestructive evaluation (NDE) methods, this study is focused on the use of dynamic response in detecting tensile forces in the cable structure system. Of the many condition assessment techniques for cablestayed structures available today, system identification methods are based on detecting the changes in static or dynamic behavior of a cable [
These works, based on the local optimization algorithm (LOA), have limited capabilities in dealing with complex problems, primarily due to their limitations in handling assumed initial conditions in the analysis. They have several limitations, such as divergence and instability problems, during numerical calculations. In particular, the trap problem of false minimum is frequently observed for large and complicated structures. In recent years, global optimization algorithms (GOA), such as neural networks, genetic algorithms (GAs), and simulated annealing methods have been developed and promisingly applied to the field of structural identification. Among them, GAs attracted our attention because not a great deal of data was needed in advance. This is an advantage over natural frequencybased neural network methods that require prioriknowledge of both the modal frequencies and the modal shapes to train the neural network and to detect the structural damage. Lee and Wooh [
Despite the broad spectrum of applications, the conventional GAs usually require a large number of iterations and thus high computational cost. To solve an inverse problem using a GA, it is necessary to carry out iterative forward computations for each individual. Noh and Lee [
In this paper, we will focus on the practical applications of the proposed method to cablestayed longspan bridge suggested in the study of Noh and Hu [
Noh and Hu [
A schematic representation of the hSI method.
In the process of tension estimation using the hSI method based on the finite element model, the tensile force to be identified is a component of an identification vector of the following form:
First, to explain a new reproduction process of the hGA in GOA different compared to the existing GA algorithm, the reproduction process for mating pools is formulated, assuming the
Next, for the reproduction process, the string vector of (
To identify the tension of cables accurately, hGA should be combined with the nonlinear finite element model that can reflect the properties of cables in the present state. That is, the finite element model parameters that can be sensitive to changes in tension should be determined as the identification parameters in the system identification procedure. In this study, natural frequencies are produced from the finite element model, which are considered the deflection curve and tensile force distribution of cables. The produced frequencies are used for determining the object function (
In the next step, new identification parameters including the tension of a cable are investigated using (
It may be noted that the next search point
Once the parent genes are selected by a new hGA reproduction process through (
Then, the purpose of the final loop of the hGA operation is to introduce diversity by mutation, in order to explore other areas in the search space. It is suggested that the mutation probability should be lowered adaptively as the process converges. Finally, through the mutation process, the following group matrix for the
The group matrix
In order to decide a proper bifurcation between GOA and LOA, the number of generations in hGA and iterations in SUA should be assigned in advance. In this study, the relationship between the number of generations (
After iterative calculations for
Then, the static displacement and tensile force distribution can be produced for the identification vector, which is similar to the hGA process. In the next step, the natural frequencies are determined from the finite element vibration model using the static displacement curve and tensile force distribution. Using the change in natural frequencies for different identification parameters, the sensitivity matrix (
Then, from the produced natural frequency data, the rate of change (
Equation (
Finally, the
From (
Finally, the tension of a cable is determined from the identification parameters holding at the termination stage and relevant natural frequencies can also be determined through the finite element vibration analysis using the identification parameters.
Before the field application of the tension estimation method using the hSI method, the benchmark numerical tests are carried out to verify the development theory. In addition, the results of the estimations of the tension force using the characteristic equation of the mathematical model, the system identification method based on the LOA and GOA, and the proposed hSI method are compared to investigate the accuracy and applicability of the proposed hSI method.
Figure
Material and geometrical properties for four numerical inclined cable models.
Items  Material and geometrical properties  

B1  B2  B3  B4  

0.079  5075.8  1.41  0.508 

1923.5  3.0295  50.459  505.113 

30  30  30  30 

100  100  100  100 

400  400  400  400 

2.9036  0.7259  26.1325  0.7259 

1.5988  17.186  20826.0  0.00478 

7.8507  7.6110  7.8633  273.45 

0.1  0.984  0.1001  0.5901 

4.9535  4.6097  4.9204  5950.6 
The sagged cable mode for benchmark numerical tests.
The finite element models of the cables used in the benchmark numerical tests were modeled with 60 linear beam elements of the same length. Also, in the repetitive calculation phase of the hSI method, the finite element model used for vibration analysis is renewed at the time when the system identification parameters are updated. The process of renewal can be summarized as follows. First, for the identification parameters that are updated at each repetition, a geometric nonlinear finite element analysis under the weight of the cable is performed to calculate the static sag and distribution of the tension force. Then, a finite element model that has a static sag curve and a calculated distribution of tension force is constructed. After that, the vibration analysis is performed to calculate the natural frequency.
To estimate the tension force of benchmark numerical examples for a slope cable including sags using the hSI method, the first thing that should be done is to select the system identification parameters. In this benchmark numerical study, the tension of cables is characterized by three independent identification parameters of horizontal force (
The search range of identification variable.
Parameters  Search range 







Since we set the value of
Estimated identification parameters (total number of iterative computation: 113).
Parameters  Estimated results  

B1  B2  B3  B4  

2.749 
0.726 
26.058 
0.730 

0.097 
0.984 
0.100 
0.592 

378.735 
399.800 
398.856 
402.530 


Elapsed time (sec)  0.65  0.56  0.57  0.58 
( ): identification error (%).
Figure
The convergence processes to find the horizontal force for B1~4.
The proposed method using the hSI is compared with those of various existing approaches as listed in Table
Estimation results of cable tension using various existing approaches.
Methods  Estimated cable tension (MN)  

B1  B2  B3  B4  
Taut string theory  3.325 
112.346 
44.789 
0.838 
Triantafyllou and Grinfogel [ 
3.352 
0.815 
31.330 
0.841 
Linear regression method  3.352 
0.690 
30.160 
0.835 
Zui et al. [ 
3.354 
2.940 
29.031 
0.685 
Sensitivity updating algorithm  3.106 
0.838 
30.089 
0.848 
Microgenetic algorithm ( 
3.362 
1.398 
27.736 
0.723 
This study (hGA + SUA)  3.174 
0.838 
30.089 
0.843 


True cable tension ( 
3.354  0.843  30.175  0.843 
( ): identification error (%).
In the case of applying SUA and hSI, it shows excellent detectability as errors were within 0.63%, except for B1, because it uses the FE model which reflects the geometrical characteristics of cables such as the sag. For the same iteration (113 times), the detectability for the case of B1 is lower than others because of their complexity due to small sagged geometry and large flexural stiffness. It means that, for B1, it is not possible to guarantee the convergence of the final value with 113 repetitive calculations. Therefore, it is necessary to increase the number of calculations of SUA of LOA in order to obtain an optimal solution.
Table
Influence of initial information of identification parameters on the estimated results.
Case  Initial value  Identification parameters  






B1 

5.547  0.139  764.340 

2.690  0.096  370.587  



 


B2 





0.726  0.984  399.800  



 


B3 





26.058  0.100  398.856  



 


B4 

5.126  0.997  2828.500 

0.734  0.592  402.530  




Since the
Comparison of elapsed time of each system identification method after 113 iterative generations or computing.
System identification methods  Elapsed time (sec)  

B1  B2  B3  B4  
SUA 
0.62 


1.48 
SUA 
1.12  1.29  1.18  1.31 
SUA 




0.72  0.69  0.61  0.72  
hSI (This study)  0.65  0.56  0.57  0.58 
To investigate field applicability of the proposed hSI method, we considered the time series data measured by the four accelerometers that were installed by 2 on each side of an actual cablestayed bridge (cables number 1 and number 44, Seohae Bridge). As shown in Figure
The locations, number, and section of measured cables.
Idealized cable for FE model.
Cable number 1
Cable number 44
Using a Piezoelectric type accelerometer, time series acceleration data perpendicular to the longitudinal direction of the cables on both sides were obtained. Here, the sampling frequency was 0.01 sec, while the average wind velocity during the measurement was 4.23 m/s. The ambient temperature was −1.76°C. The time series data of the acceleration measurement was examined using Welch’s method (NFFT = 2048, Hanning window) to identify the acceleration spectrum. Also, by using the ERADC [
Extracted natural frequencies from measured data and calculated natural frequencies from FE models for cable number 1 (Hz).
Mode number  Upwardbound  Downwardbound  

Test  FEM  Error (%)  Test  FEM  Error (%)  
1  0.47  0.46  −1.71  —  0.46  — 
2  0.94  0.92  −1.87  0.92  0.92  0.48 
3  —  1.14  —  —  0.94  — 
4  1.38  1.38  −0.28  1.38  1.38  0.25 
5  1.84  1.85  0.42  1.84  1.84  0.25 
6  —  2.28  —  —  1.87  — 
7  2.30  2.31  0.30  2.29  2.29  −0.20 
8  2.76  2.77  0.19  2.75  2.75  −0.11 
9  3.23  3.23  0.06  —  2.80  — 
10  —  3.42  —  3.21  3.21  0.06 
11  3.69  3.69  −0.01  3.67  3.67  −0.07 
12  4.15  4.16  0.24  —  3.74  — 
13  —  4.56  —  4.13  4.13  0.04 
14  4.61  4.62  0.25  4.59  4.59  0.07 
15  5.08  5.08  0.09  —  4.68  — 
16  5.54  5.54  0.03  5.05  5.05  0.07 
17  —  5.70  —  5.51  5.51  −0.02 
18  6.00  6.00  0.08  —  5.61  — 
19  6.46  6.46  −0.01  5.98  5.97  −0.15 
20  —  6.84  —  6.44  6.43  −0.10 
—: not extracted.
Extracted natural frequencies from measured data and calculated natural frequencies from FE models for cable number 44 (Hz).
Mode number  Upwardbound  Downwardbound  

Test  FEM  Error (%)  Test  FEM  Error (%)  
1  —  0.75  —  —  0.76  — 
2  —  1.33  —  —  1.33  — 
3  1.50  1.51  0.50  1.51  1.51  0.33 
4  2.26  2.26  0.06  2.27  2.27  0.05 
5  —  2.66  —  —  2.67  — 
6  3.01  3.02  0.24  3.02  3.02  0.00 
7  3.77  3.77  −0.03  3.78  3.78  −0.12 
8  —  3.99  —  —  4.00  — 
9  4.52  4.52  −0.02  4.55  4.54  −0.13 
10  5.29  5.28  −0.18  5.31  5.29  −0.38 
11  —  5.32  —  —  5.34  — 
12  6.03  6.03  −0.05  6.05  6.05  0.02 
13  —  6.65  —  —  6.67  — 
14  6.79  6.79  −0.03  6.82  6.81  −0.21 
15  7.55  7.54  −0.19  7.58  7.56  −0.27 
16  —  7.99  —  —  8.01  — 
17  8.31  8.3  −0.09  8.33  8.32  −0.17 
18  9.06  9.05  −0.09  9.09  9.07  −0.25 
19  —  9.32  —  —  9.35  — 
20  9.81  9.81  −0.03  9.83  9.83  −0.02 
—: not extracted.
The material and geometrical properties of each cable.
Condition  Tests  

F1  F2  F3  F4  

0.0147  0.0093  0.0021  0.0017 

1455.3  873.1  800.5  4313.9 
Horizontal load (MN)  4.8940  4.8341  4.3652  4.3938 
Effective diameter (m)  0.280  0.280  0.225  0.225 
Elasticity (GPa)  2.1  2.1  2.1  2.1 
Remark  Number 1 (up)  Number 1 (down)  Number 44 (up)  Number 44 (down) 
The result of cable tension force estimation using the existing methods and the proposed hSI method based on the natural frequency obtained through the field measurement is shown in Table
Cable tension force estimated from existing methods and the hSI method.
Methods  Estimated cable tension (MN)  

F1  F2  F3  F4  
Taut string theory  5.586 
5.532 
5.226 
5.253 
Triantafyllou and Grinfogel [ 
5.578 
5.534 
5.226 
5.253 
Linear regression method  5.571 
5.518 
5.214 
5.267 
Shimada [ 
5.549 
5.486 
5.196 
5.204 
Sensitivity updating algorithm  5.583 
5.525 
5.233 
5.275 
Microgenetic algorithm ( 
5.592 
5.514 
5.250 
5.287 
This study (hGA + SUA)  5.583 
5.525 
5.233 
5.272 


Static jacking force ( 
5.643  5.611  5.308  5.193 
( ): identification error (%).
Estimation results of identification parameters through various system identification methods.
Case 
^{final}

Methods  

SUA (LOA)  hSI (GOA + LOA)  
F1 

4.892  4.887  4.892 

0.184  0.210  0.188  

1.182 
2.005 
1.288  

55.840  73.736  58.294  

0.033  0.043  0.034  

1,473  1,130  1,411  


F2 

4.841  4.832  4.841 

0.208  0.224  0.208  

1.929 
2.595 
1.929  

71.357  82.757  71.357  

0.043  0.051  0.043  

1,147  988  1,147  


F3 

4.362  4.887  4.892 

0.212  0.210  0.188  

2.082 
2.005 
1.288  

55.840  73.736  58.294  

0.014  0.010  0.014  

757  1,087  750  


F4 

4.892  4.887  4.892 

0.184  0.210  0.188  

1.182 
2.005 
1.288  

55.840  73.736  58.294  

0.014  0.013  0.012  

760  813  871 
The estimation of the tension force using the taut string theory yields average tension forces of 5.586 MN and 5.532 MN for cable number 1 on each side of the bridge, while cable number 44 yields 5.226 MN and 5.253 MN, respectively. Here, the effective lengths of the cable applied were 228.86 m and 165.33 m, while the mass per unit length was as per the reported actual physical properties of the actual cables. The method of Triantafyllou and Gingfogel, which estimates the average tension force from nonlinear characteristics equations of the sloped cables and takes sags into consideration, shows errors within 1.4% in all cases, indicating a moderate level of precision in estimation. The mass per unit length of the cable and data on the axial stiffness of the cable were assumed to be known in advance. The estimated tension force using the linear regression method, which considers bending, ignores the sag, and uses the relationships between the natural frequency, bending stiffness, and average tension forces, shows errors within 1.8% for a relatively higher level of precision. However, when the bending stiffness was estimated, F4 case showed a negative value of bending stiffness, which was
The estimation results by the SUA,
When compared to the tension forces measured using a hydraulic jack before completion [
In this paper, the characteristics of the hSI methods that could be applied to the estimation of the tension forces effectively are examined and the theoretical formulation process of the hSI method is suggested. Also, to verify the efficiency of the hSI tension force estimation method, benchmark numerical tests for various cases are conducted and the applicability of the hSI method through field tests is verified. During the benchmark numerical and field studies, comparative studies with the existing characteristics equation methods and system identification methods are also carried out. The key findings from this study are summarized as follows.
The hSI method is a new algorithm combined to use the advantages of both the LOA and GOA and adopts the posthybridization method where the GOA provides the initial values for the LOA. The most significant characteristic of this method is that the hGA, which is a GOA, helps solve the local convergence problem due to incorrect initial value settings, by using the sensitivity based system identification method, which is a LOA, to improve the precision and convergence to the final value.
The benchmark numerical tests that consider sag and bending stiffness showed that, in the case of string theory, it was more appropriate for the estimation of tension force for a simple cable system where the influence of sag and bending stiffness is not significant. Also, the method suggested by Triantafyllou and Grinfogel, which considers sags, showed errors within 4% in all cases, proving to be a relatively precise method. However, as the method uses a trial and error method, it is time consuming to find the tension compared to the string theory and initial information on the mass per unit length and axial stiffness of the cable is required to estimate the tension force properly. For estimating the tension force using the linear regression method, in which bending is considered but sags are not, except for the cable with large sags, the tension force can be estimated precisely within 1% errors. However, when applying the characteristic equation of the beam under axial loading for cables with large sags, model errors were confirmed in the results of tension estimation. For the low mode method suggested by Zui et al. [
In the field test that used cables applied to an actual cablestayed bridge, all methods showed errors smaller than 2.3%. When considering the fact that the bridge undergoes the tension loss by continuous changes of tension forces, with an accumulation of damage due to the environmental cyclic loadings such as thermal, traffic, and wind loads, the reduction rate in tension force mentioned above is insignificant. Unlike cables used in the benchmark numerical tests, the actual cables of cablestayed bridges used in field study are systems that are less influenced by sags and bending. Therefore, the existing estimation methods and the proposed hSI method do not differ significantly in terms of the estimation result. However, while the existing mathematical models and the SUA, which is LOA, require precise data in advance, it should be noted that the
Conclusively, the proposed hSI method could be effectively applied to estimate the tension force variation without initial information in actual cable bridges with inclined cables including sag.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by a National Research Foundation of Korea (NRF) Grant funded by the Korean government (MEST) (no. 20120008762).