Mathematical Problems in Engineering

Volume 2015, Article ID 512858, 12 pages

http://dx.doi.org/10.1155/2015/512858

## Feasibility Study on Tension Estimation Technique for Hanger Cables Using the FE Model-Based System Identification Method

Steel Solution Center, POSCO, 100 Songdogwahak-ro, Yeonsu-gu, Incheon 406-840, Republic of Korea

Received 8 September 2014; Accepted 13 October 2014

Academic Editor: Sang-Youl Lee

Copyright © 2015 Kyu-Sik Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Hanger cables in suspension bridges are partly constrained by horizontal clamps. So, existing tension estimation methods based on a single cable model are prone to higher errors as the cable gets shorter, making it more sensitive to flexural rigidity. Therefore, inverse analysis and system identification methods based on finite element models are suggested recently. In this paper, the applicability of system identification methods is investigated using the hanger cables of Gwang-An bridge. The test results show that the inverse analysis and systemic identification methods based on finite element models are more reliable than the existing string theory and linear regression method for calculating the tension in terms of natural frequency errors. However, the estimation error of tension can be varied according to the accuracy of finite element model in model based methods. In particular, the boundary conditions affect the results more profoundly when the cable gets shorter. Therefore, it is important to identify the boundary conditions through experiment if it is possible. The FE model-based tension estimation method using system identification method can take various boundary conditions into account. Also, since it is not sensitive to the number of natural frequency inputs, the availability of this system is high.

#### 1. Introduction

Recently, the number of long-span bridges built in countries around the world is increasing. Of these long-span bridges, the suspension bridge uses stiffening girders that are attached to the main cables to support the load. In Korea, starting from the completion of the Nam-Hae Bridge in 1973, Yeongjong, Gwang-An, Yi Sun-Sin, and many other suspension bridges have been constructed, and Ulsan Bridge is also under construction as a suspension bridge. Most long-span bridges, such as suspension bridges, support their loads by means of cables. 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 most accurate means to estimate the tension on the cables is to use the load cells to collect the data directly. However, this is not an option in many cases due to the conditions in the field. Therefore, the most commonly used method is to extract the natural frequency from the measurements of acceleration signal and enter the extracted data into an equation that shows the relationship between the natural frequency and tension, and thus the cable tension can be estimated indirectly. The two most widely used methods are the taut string theory [1], which does not take the flexural rigidity into consideration, and the linear regression method [2], which regards the cable as a beam under the axial load and considers the flexural rigidity. Yun et al. [3] analyzed the influence of the effective length in the linear regression method, while Ahn et al. [4] used the static method, which requires the minimal amount of measurements compared to the dynamic methods, to calculate the tension on the hanger cables.

The estimation methods for tension using the existing dynamic method put the natural frequency values to the equation on the natural frequency and tension. Therefore, if the mathematical model used differs significantly from the actual structure, a significant error is unavoidable. In particular, in the case of hanger cables that transfer the load applied to the stiffening girder to the main cable, they are, in most cases, short cables with higher flexural rigidity. Also, due to the influence from the clamps and boundary conditions, errors are significant when the tension is estimated using the existing methods [5–7].

Therefore, to estimate the tension of shorter cables that are greatly influenced by flexural rigidity, like the hanger cables in a suspension bridge, some new methods are proposed that are based on the finite element model of the hanger cable. Kim et al. [6, 7] suggested formula-based inverse analysis method, which defines the errors between the calculated frequency using the analysis models and the frequency actually measured as the objective functions, and uses an optimized algorithm, the univariate search method and modal participation factor. Park et al. [5] suggested a vibration-based system identification technique. This method uses the measured frequency as the input variable and the sensitivity equation to estimate the tension force through repeated calculations. Here, for a precise system identification of the hanger cables, a 3D finite element model is created, in which the physical properties including the tension force and the rigidities of hanger cable and clamp are set as the identification variables. The tension estimation technique for the hanger cable using the formula-based inverse analysis method was applied to the Gwang-An Bridge [6, 7] while the tension estimation technique for the hanger cables based on the FE model-based system identification method was applied to the theoretical development [5], Yeongjong Bridge [6, 7], and the test sample [8].

In this paper, through the hanger cables of Gwang-An Bridge, the reliability of the system identification method based on the finite element model, which has higher applicability in tensile force estimation for short cables that are more sensitive to the flexural rigidity, is compared with that of the existing tension estimation methods, and its applicability is verified.

#### 2. Tension Estimation Methods

##### 2.1. Methods Using Mathematical Equation on the Natural Frequency and Tension

Many dynamic methods to estimate tensile force for cables have been developed considering dynamic characteristics and physical properties of cables. Among them, the flat taut string theory for cables that neglect both sag-extensibility and flexural rigidity is as follows:where denotes the th natural frequency in Hz. The terms , , and denote tension force, mass density, and length of cable, respectively. The computation of tension force is straightforward with given measured frequency and mode number. However, the application of this formula is strictly limited to a flat long slender cable because it can not consider both sag-extensibility and bending stiffness of cables.

The modern cable theory [9] that takes account of the sag-extensibility without flexural rigidity requires additional information of the unstrained length of cable and involves solving a nonlinear characteristic equation by trial-and-error [10]. However, such additional information is often not available in practice, therefore the linear regression method [11] that considers cables as an axial load beam had been developed. This method considers the flexural rigidity but neglects the sag-extensibility as follows:where denotes the flexural rigidity of a cable. The unknown tension force and flexural rigidity can be identified through linear regression procedures with given measured frequency and mode number. This method is widely used by the field engineers because of its simplicity and speediness. To consider both sag-extensibility and bending stiffness, the practical formula [12] had been developed. But a priori data of the axial and flexural rigidities of the target cable system is required for the proper use of this practical method. However, the flexural rigidity of cable is often neither available nor valid because the shear and bending mechanisms of a cross section of cable could be different from those of beam.

##### 2.2. Methods Using System Identification Approach Based on FE Model

In the process of estimating the tension using the finite element model based system identification method, the tensile force of cables is illustrated by the identification vector which is composed of several unknown parameters. In this study, the nine unknown parameters () are used for identifying the tensile force in the system identification procedure and the identification vector is defined aswhere denotes the tensile force of a cable; , (), and denote axial, flexural, and torsional rigidities for a cable, respectively; and , (), and denote axial, flexural, and torsional rigidities for a clamp, respectively.

The identification vector for the th iteration in the sensitivity-based updating algorithm can be assumed aswhere mean the identification vector for the th iteration.

Then, the static displacement and tensile force distribution can be produced for the identification vector. In the next step, the natural frequency () is determined from the finite element vibration analysis using the static displacement curve and tension distribution. Using the change in natural frequencies for different identification variables, the sensitivity matrix () with size can be determined approximately as follows:

Then, from the produced natural frequency data, the rate of change () for the eigenvalue can be obtained aswhere and denote the th mode’s natural frequency measured from experiment or field test and the th mode’s frequency calculated from nonlinear finite element vibration analysis using in the th iteration, respectively. Equation (6) can be rewritten in the vector form as

Equation (7) is referred to as a linear sensitivity equation, and the rate of change for the identification vector by using (7) can be expressed aswhere means the pseudoinverse matrix for and can be determined as

Finally, the th identification variable in the ()th iteration can recalculated as

From (4) to (10), the loop is repeated until the convergence condition is satisfied. The convergence condition for the repetition analysis uses the square roots of the sum of square (SRSS) in the following:

Finally, tensile force is determined from the identification variables holding at the termination stage and relevant natural frequencies can be determined through the finite element vibration analysis with the identification variables.

#### 3. Field Application: Hanger Cables of Gwang-An Bridge

##### 3.1. Measurement of the Vibration Signal

Located in Busan, Gwang-An Bridge is 900 m in total length (center span = 500 m; 3 spans and 2 hinges with each side span = 200 m) and a width of 24 m. The height of the main tower (from sea level) is 116.5 m. The hanger cables of Gwang-An Bridge become shorter as they approach the center of the span from the location of the main tower. In this paper, we examined the two hanger cables located on the beach side, as shown in Figure 1. At each sector, one hanger cable band holds two groups of hanger cables as shown in Figure 2. Of these, the acceleration signals of the hanger cable installed on the bridge side were measured.