Vibration-based cable tension estimation using two iterative algorithms: Methodology and experimental validation
Subject Areas :
Latif Doosti
1
,
Omid Bahar
2
,
Mohsen Ghafory-Ashtiany
3
,
Mohsen Elmi
4
1 - Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES)
2 - International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran.
3 - International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran.
4 - Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES)
Keywords: System Identification, Post-tensioning, Cable Structures, Tension estimation, Vibration-based methods,
Abstract :
In this paper, a reliable method based on two iterative algorithms is proposed for the estimation of cable tension forces. In this method, a Finite Element model, considering the cable's geometric and mechanical characteristics, is used in order to obtain mass and stiffness matrices. The initial geometric stiffness matrix of the structure is calculated according to the taut string theory, assuming an initial value for the tension force. Furthermore, the natural frequency of the MDOF system is calculated using Ritz Method, and then compared to measured vibration frequencies, which can be obtained by output-only system identification methods. In this method, the difference between the computational and measured frequencies should not exceed the pre-defined threshold, otherwise the iteration process would be repeated after modifying the initial assumption of the cable force. To evaluate the accuracy and the effectiveness of the proposed method, an experimental study was performed on an external cable in the IIEES structural laboratory. Also, existing cable forces of the Hwamyeong cable-stayed bridge in South Korea have been employed. In both cases, the iterative method is compared with common theoretical and empirical equations in the literature. The results have shown that the iterative method is of high accuracy and great applicability, reducing the difference to even less than 2%.
Vibration-based cable tension estimation using two iterative algorithms: Methodology and experimental validation
Abstract
In this paper, a reliable method based on two iterative algorithms is proposed for the estimation of cable tension forces. In this method, a Finite Element model, considering the cable's geometric and mechanical characteristics, is used in order to obtain mass and stiffness matrices. The initial geometric stiffness matrix of the structure is calculated according to the taut string theory, assuming an initial value for the tension force. Furthermore, the natural frequency of the MDOF system is calculated using Ritz Method, and then compared to measured vibration frequencies, which can be obtained by output-only system identification methods. In this method, the difference between the computational and measured frequencies should not exceed the pre-defined threshold, otherwise the iteration process would be repeated after modifying the initial assumption of the cable force. To evaluate the accuracy and the effectiveness of the proposed method, an experimental study was performed on an external cable in the IIEES structural laboratory. Also, existing cable forces of the Hwamyeong cable-stayed bridge in South Korea have been employed. In both cases, the iterative method is compared with common theoretical and empirical equations in the literature. The results have shown that the iterative method is of high accuracy and great applicability, reducing the difference to even less than 2%.
Keywords: Cable Structures, Tension estimation, Post-tensioning, Vibration-based methods, Iterative algorithm, System Identification.
1. Introduction
Recent technological advances in material and construction industry have led to an increase in the number of long-span prestressed bridges, such as cable-stayed and post-tension concrete box girder bridges. Prestressed cables of bridges play a vital role as the main component in load transferring mechanism. Despite all the advantages, there are growing concerns over potential damage modes and prestress losses after the tensioning operation. These damages are not easily identifiable unless they are considerable that require an extensive repair or replacement. In such cases, there is no guarantee for the bridge to operate a safe service and maintain its integrity. Regarding to that issue, there have been several reports on collapse of prestressed concrete bridges, such as the collapse of a post-tension bridge in England in 1967 [1], a post-tension segmental concrete bridge in Wales in 1985 [2], a post-tension concrete bridge in Belgium in 1992 [3], the Punt Marandi cable-stayed bridge in Italy in 2018 [4], and the Nanfang'ao single-arch bridge in Taiwan in 2019 [5], which all reveal the vulnerability of these types of bridges.
Structural health monitoring of the post-tension system and taking necessary actions including repair, maintenance, and rehabilitation process is required to prevent the occurrence of severe damages, and consequent substantial human and financial losses.
A comprehensive study has been conducted on inspection methods of prestressing strands. The advantages and disadvantages of the introduced methods are described in details [6]. An extensive experimental investigation was also carried out on a full-scale post-tensioned girder model and four large-scale stay cable systems to develop guidelines and select a Non-Destructive Evaluation (NDE) method for assessing in-service post-tensioning and stay-cable systems. Models were analyzed to ensure the effectiveness of NDE methods for detecting the location and severity of either strand or grout defects [7].
Cable tension loss is one of the most important damages in post-tensioning system and researchers have introduced several cable tension force estimation techniques [8–10]. Fig. 1 shows the traditional and innovative methods of cable force monitoring reviewed by Zjang et al. [9].
Pressure gauge measuring method is the simplest method among all cable tension estimation methods. Cable force is obtained using the hydraulic pressure inside the jack cylinder. The advantages of this method are ease of use and simple calculation. However, it relies on heavy jack that is only suitable for construction stage and the installation also has a high initial cost. Also manual measurements can cause data instability.
Cables force can be obtained using classical strain measurement method according to the hook’s law and measured strain value. This method is also simple but it’s efficiency is low.
Magnetic flux method is based on measuring the elastic permeability using relative sensors and as a non-destructive method, can be used for cable tension monitoring during service life of structure [11,12]. A new non-destructive cable tension monitoring system has been proposed based on the elasto-magnetic test and the self-induction phenomenon [13]. This method has advantages of NDT methods while obviateing strain gauge disadvantages. The main drawback of this method is that there should be an access to the cable for calibration and reference testing which is costly [14,15].
Fig. 1 Cable force monitoring methods [9].
Vibration based tension estimation methods have the advantages of simplicity, ease of installation and also high efficency and precision which made them very useful for cable force monitoring.
Acoustic emission techniques (AET), are based on the strain produced by elastic waves after component fracture and are useful for long term monitoring of changes of cable force [16]. However, sensitivity of AET method to the noise is still one of the main drawbacks that needs to be solved.
After developing optical fiber technology, fiber-optic sensors (FOS) and especially, fiber Bragg grating (FBG) sensors have a vital role in cable tension estimation [17,18]. These sensors have been applied on many cable stayed bridges all over the world. Corrosion resistance and anti-electromagnetic induction are among the most important advantages of FOS. Nonetheless, high cost, time-consuming and low servival rate of sensors are the main problems of this method.
In recent years Intelligent devices and technologies like mobile phones, cameras, and computers that are fitted with high-performance sensors have advanced quickly to overcome disadvantages of traditional cable force estimation methods including laborious installation and data transfering by wire, which result in low efficiency and take a lot of time. A new idea for structural health monitoring based on mobile phones was introduced in 2012 by Zhao et al. [19]. After that, a software called Orion CC (Orion Cloud Cell) was created to quickly and effectively monitor cable force [20,21]. This technique, which has made some progress, encourages the creation of intelligent devices for monitoring the health of cable-stayed bridges [22]. The method of monitoring intelligent devices has advanced significantly with the advent of image processing and wireless sensing technology [23,24]. Using single and multiple point images captured by a camera, dynamic measurement of cable force can be determined based on digital image processing (DIP) and digital image correlation (DIC) [25,26]. Wireless sensor networks are also an exciting new technology that has the potential to significantly advance structural health monitoring [27]. Machine learning and deep learning have had a substantial impact on the monitoring of the structure health of cable-stayed bridges [28–30]. The development of bridge structural health monitoring is efficiently facilitated by intelligent devices and wireless communication mode. It has many important benefits, including inexpensive cost, simple operation, excellent precision, and stability. However, internal algorithms on some devices may have limitations. Additionally, one of the primary factors for precision is the impact of the external environment. However, it cannot be denied that intelligent devices will have a crucial role for cable force monitoring in future.
Three-point bending test and a laser-based method, can be used to estimate cable tension force [31]. Although they both have more developed theoretical foundations, the outcomes are less than optimal when used in practice. They are so infrequently employed in practice.
As discussed there are various techniques for measuring cable forces, but applying these methods may disrupt the bridge serviceability, and they are also expensive, and time-consuming without being quite accurate. Vibration based method is a quick and non-destructive method of cable force evaluation, with lower cost, ease of use, as well as higher speed and accuracy comparing with the other mentioned methods [32–35].
In this paper, the vibration-based cable tension estimation background will be reviewed. Afterward, by applying two iterative algorithms, a new reliable vibration-based method for the estimation of cable tension forces is presented. The main novelty of the present work is to propose a fast and reliable iterative procedure,which can be used in practical engineering problems. The proposed method is then validated by performing an experimental study on a prestressed mono-strand cable. Finally, the suggested method is used to estimate the tension force of several available prestressed cables in previous studies to illustrate its applicability.
2. Vibration-based cable tension estimation method
2.1. Background
Changes in the tension force alter vibration indicators of the cable system which in turn can be used to detect the location, type, and severity of damages. Monitoring changes of cable forces may be considered as an indirect method for detecting the stiffness reduction due to loss of pre-stressing tension, strand corrosion, or cable failure.
The cable tension is calculated according to the correlation between the cable tension and the natural vibration frequencies. Since the 1950s, efforts have been made to determine the cable forces based on the fundamental vibrational mode and the development of a simple relationship between the fundamental frequency and the tension force employing the taut string theory [36–41]. In this regard, a cable with low bending stiffness and simple boundary conditions can be considered as a taut string and the cable force is calculated using the simple taut string Eq. (1), presented in Table 1. The equations in Table 1 are among the widely used vibration-based equations; however, their accuracy has proved insufficient in some cases, particularly for cables with high bending stiffness and fixed-end boundary conditions [8].
Table 1
Theoretical and empirical equations used for comparison
|
|
In Table 1, T is the tension force, m represents the mass density, l denotes the effective length, 
is the nth measured frequency of the cable ( Hz) and n is the order of frequency, and EI indicates the bending stiffness of the cable, 
, 
Many studies have been done in the field of vibration based cable tension estimation. A completely noncontact video-based technique is proposed utilizing a moving handheld camera for vibration frequency identification of the cable, as well as taut string theory for tension estimate [44]. According to taut string theory, Zhao et al proposed a cable force measuring method using smartphone [45]. For estimating the cable tension in cable-stayed bridges, a region-based convolution neural network and wireless smart sensors, as an automated vibration-based system has been proposed [46]. In another study, a non-contact vision-based system has been developed, in which a feature-based video image processing technique is used to record dynamic responses of cables [47].
In the vibration-based methods, the natural frequencies of cables can be measured by high accuracy accelerometers after exciting the cable and recording its response using different methods including fast fourier transform (FFT), enhanced frequency domain decomposition (EFDD) and stochastic subspace identification (SSI) methods. These methods are among the most common and widely accepted methods of output-only system identification [54].
Fourier transform is a common frequency-based transform for linear system analysis. It breaks down a signal into sine waves of various frequencies, adding them together to form the original waveform while differentiating between the frequencies' sine waves and their corresponding amplitudes. FFT is an effective approach for computing the discrete Fourier transform and its inverse, by minimising the number of computations required. This system identification method is frequently employed to obtain the structures' natural frequencies [48].
The EFDD method is an extension of the FDD method, which basically involves choosing the modes by locating the peaks in singular value decomposition plots generated from the spectral density spectra of the responses. EFDD provides a good estimate of the natural frequencies and mode shapes while also taking damping into account [49].
SSI technique is considered to be the most powerful time domain identification method for natural input modal analysis. This method directly fits a parametric model to the raw time series data. A parametric model is a mathematical model with some parameters that can be adjusted to change the way the model fits to the data [50].
Hegeir et al. investigated the cause of damages in Pathein Suspension Bridge using vibration-based tension force estimation of hangers and corresponding distribution along the bridge’s span [51]. Kangas et al. proposed a multiple signal classification (MUSIC) algorithm for the accurate identification of natural frequencies of cable structures [35]. Using output-only measurements, Lardies and Taa presented the time domain and time-frequency domain approaches for the modal system identification of cables [52]. Subspace Identification using a new modal coherence indicator and a wavelet transform with a new analyzing wavelet have been used for system identification in the time domain and time-frequency domain, respectively. Joaquim et al., conducted an optimization procedure for a given free length, and presented an estimation method for tensile force and the cable bending stiffness simultaneously. This procedure were applied to stay-cables of the Salgueiro Maia Bridge to cross the Tagus River in Santarem, Portugal [53].
In this paper, a new simple, reliable and precision vibration-based cable force estimation method is proposed using two iterative algorithms. For the purpose of natural frequency identification of cable system, three above mentioned algorithms (FFT, EFDD and SSI) have been used.
2.2. Proposed vibration based method for cable tension estimation
In this paper, for adopting a more accurate approach for the tension force predictions in the cables, a simple method is developed, which employs the Ritz method and an iterative algorithm. In the proposed method, the tension force in the cable, T, is considered as an independent variable, so that it’s most appropriate value can be achieved through a step of ΔT in size. Flowchart of the proposed method is shown in Fig. 2.
Fig. 2 Flow chart of the proposed method
The natural frequencies and mode shapes can be determined considering the undamped free vibration problem of a cable system with the length of L. In this paper, ritz iteration method has been exploited to achieve the modal parameters [55]. The procedure of the proposed method is as follows:
1. The mass and stiffness matrices, and also initial geometric stiffness matrix of the structure,
, are calculated using finite element method and according to the geometric and mechanical characteristics of the cable structure.
2. Following the measurement of the natural frequency of the field cable (
), the initial value of the cable tension is calculated using the taut string theory. Then, the iterative process begins.
3. By placing 
in the vibration equation and deriving the computational frequency (
) using the Ritz method, if the 
is smaller than a predefined threshold the iteration is supposed to be completed and the estimated cable tension is equal to
. In contrast, if is larger than the predefined threshold, the initial cable tension is updated with the equation of 
and the iteration process continues until will be smaller than the threshold.
The main goal of the proposed iterative algorithm is to identify the tension force through minimizing the difference between the measured frequency with the model one. To this aim and based on this fact that applying tension force increases the vibration natural frequency, in the flowchart, it is tried to update the tension force according to the relation, , in each step.
As can be observed in the flowchart, the difference between the measured natural frequency with the model one is selected as the objective function. Also the Ritz method has been employed to solve the inverse problem. In addition by varying the applied tension force, the objective function is minimized. Finally it is helpful to be noted that the iteration procedure is terminated when the difference between the measured natural frequency with the model one reach to predefined tolerance, which in this paper the value of 0.001 is proposed.
Tension calculations in the present study have been carried out employing a Matlab-based code.
2.3. Experimental validation of the proposed method
An experimental study was conducted for the verification of the proposed cable tension force estimation method. A laboratory model of a cable has been setup in the structural laboratory of the International Institute of Earthquake Engineering and Seismology (IIEES). The cable is anchored using steel supports connected to the stiff floor of the laboratory, as shown in Fig. 2.
In order to determine the natural frequencies of the cable, three different output-only system identification (SI) methods have been applied, including Fast Fourier Transform (FFT), Enhanced Frequency Domain Decomposition (EFDD), and Stochastic Subspace Identification (SSI). Furthermore, six accelerometer sensors with a sampling frequency of 500 Hz have been employed. The duration of each vibration test is set to be 300 seconds. Fig. 4 shows acceleration time history that is recorded by a sensor and the corresponding Fast Fourier Transform. According to the Table 3, six levels of tensioning force are considered from 45 to 108 kN, and output acceleration time histories of the ambient vibration and impact load tests are recorded.
a
|
| b
|
|
c
|
|
Fig. 3 Laboratory sample of prestressed cable; a Overall view, b Wired Acceleration Sensors, and c Data Logger
Fig. 4 a Output acceleration time-history in the impact test, b FFT of the output acceleration time-history
Table 2
Actual cable tension force and natural frequencies of the first 4 modes
Level of tensioning | Tension force (kN) | SI Method | Frequency (Hz) | |||
|
|
|
| |||
Level 1 | 45 | FFT | 9.62 | 19.34 | 28.91 | 38.62 |
EFDD | 9.61 | 19.27 | 28.89 | 38.63 | ||
SSI | 9.62 | 19.26 | 28.87 | 38.47 | ||
Level 2 | 63 | FFT | 11.48 | 22.71 | 34.18 | 45.65 |
EFDD | 11.38 | 22.78 | 34.21 | 45.74 | ||
SSI | 11.38 | 22.80 | 34.21 | 45.73 | ||
Level 3 | 72 | FFT | 12.21 | 24.41 | 36.62 | 48.83 |
EFDD | 12.16 | 24.38 | 36.60 | 48.88 | ||
SSI | 12.15 | 24.31 | 36.58 | 48.87 | ||
Level 4 | 81 | FFT | 12.94 | 25.88 | 38.82 | 51.76 |
EFDD | 12.89 | 25.78 | 38.72 | 51.78 | ||
SSI | 12.88 | 25.76 | 38.72 | 51.76 | ||
Level 5 | 90 | FFT | 13.67 | 27.34 | 41.02 | 54.93 |
EFDD | 13.68 | 27.36 | 41.11 | 54.92 | ||
SSI | 13.68 | 27.27 | 41.10 | 54.80 | ||
Level 6 | 108 | FFT EFDD SSI | 14.89 14.92 14.95 | 29.79 29.89 29.92 | 44.92 44.84 44.88 | 59.83 59.80 59.83 |
Tension Force | 45.00 | 63.00 | 72.00 | 81.00 | 90.00 | 108.00 |
First mode natural frequency (Hz) | 9.62 | 11.48 | 12.21 | 12.94 | 13.67 | 14.89 |
ξ | 14.88 | 17.60 | 18.82 | 19.96 | 21.04 | 23.05 |
Taut string theory ignoring bending stiffness | 47.13 | 67.08 | 75.91 | 85.28 | 95.22 | 112.99 |
Taut string theory considering bending stiffness | 47.07 | 67.02 | 75.85 | 85.22 | 95.16 | 112.93 |
Empirical formulas of Zoe et al. | 44.78 | 64.28 | 72.93 | 82.13 | 91.89 | 109.36 |
Empirical formulas of Ren et al. | 44.66 | 64.12 | 72.76 | 81.94 | 91.69 | 109.14 |
Proposed method | 44.31 | 63.47 | 71.92 | 80.89 | 90.38 | 107.41 |
The ratio of estimated cable tension force obtained from existing methods to the known experimental one, is defined as 
and presented in Fig. 5. As shown in Fig. 5, in the first level of tensioning (45 KN force and 9.62 Hz natural frequency), the empirical formulas of Zoe et al. estimate the tension force without error, Ren et al. empirical formulas with an error of 1%, the iterative method with a 2% error, and taut string methods with a 5% error. However, for other tension levels (63 to 108 KN force), the iterative method reveals the highest accuracy compared to the other methods. In the worst case, the iterative method determines the cable tension force with a one percent error.
Fig. 5 Ratio of estimated tension force of the cable test to the experimental force
3. Applicability of the proposed method
3.1. Hwamyeong cable-stayed bridge
In this section, tension forces is calculated using the proposed method for an existing bridge in the literature, and then the results have been compared to common existing tension estimation techniques. A series of tests performed to conduct a comparative field study on Hwamyeong Bridge, a cable-stayed bridge under construction, (see Fig. 6) [56].
Fig. 6 Hwamyeong cable-stayed bridge and test cables [57]
The Hwamyeong Bridge, which connects Busan and Gimhae, is the longest prestressed concrete cable-stayed bridge having a box girder in South Korea. It has a total length of 500 meters, including a 270-meter main span and two 115-meter side spans. Two cables among 72 cables (BLC02 and BLC04 marked in Fig. 6), at the side span of the bridge towards Gimhae, were chosen for the field test. They comprise 49 galvanized strands inside a high-density polyethylene (HDPE) duct of 200-mm diameter, with helical strips to reduce rain and wind-induced vibration. The general specifications of BLC02 and BLC04 and also their geometrical characteristics and references are shown in Table 4.
Table 4
Hwamyeong cable-stayed bridge cables used in this paper
Row | Cable | Cable Length (m) | Mass Density (kg/m) | Bending Stiffness, EI (KN.m2) | Reference |
1 | BLC 02 | 45.57 | 67.47 | 3583 | Cho et al.[56] |
2 | BLC 04 | 55.92 | 67.47 | 3583 | Cho et al.[56] |
Values of the cable tension in several conditions were measured by electromagnetic sensors and lift-off tests. The test results is reported in five different conditions, BT1, AT1, BT2, AT2, and FIN, which respectively mean Before and After Tensioning of BLC04, Before and After Tensioning of BLC02, and Final pre-stressing of cables. Tension forces in these cables are calculated using different methods as mentioned earlier. Then, results have been compared to lift-off tests and electromagnetic sensors (or design forces in case of inadequate information), which are considered as known values of tensile forces (See Tables 5 and 6).
Table 5
Results of tests and different vibration-based tension estimation methods on BLC04 cable (force in kN)
BLC 04 | BT1 | AT1 | BT2 | AT2 | FINAL |
First mode natural frequency (Hz) | 2.063 | 2.557 | 2.527 | 2.515 | 2.515 |
ξ | 53.2 | 66.6 | 66.5 | 66.0 | 65.6 |
Lift-off Test method | 3245 | 5077 | _ | _ | 4859 |
Electromagnetic Sensor method | 3246 | 5076 | _ | _ | 4930 |
Design Force | 3261 | 5175 | 5064 | 4984 | 4911 |
Taut string theory ignoring bending stiffness | 3591 | 5517 | 5389 | 5337 | 5337 |
Taut string theory considering bending stiffness | 3580 | 5506 | 5377 | 5326 | 5326 |
Empirical formulas of Zoe et al. | 3307 | 5165 | 5040 | 4991 | 4991 |
Empirical formulas of Ren et al. | 3295 | 5148 | 5024 | 4974 | 4974 |
Cho et al. Vibration Method | 3341 | 5217 | 5099 | 5022 | 5082 |
Proposed method | 3292 | 5129 | 5006 | 4957 | 4957 |
Table 6
Results of tests and different vibration-based tension estimation methods on BLC02 cable (force in kN)
BLC 02 | BT1 | AT1 | BT2 | AT2 | FINAL |
First mode natural frequency (Hz) | 2.6 | 2.576 | 2.557 | 3.058 | 3.058 |
ξ | 43.6 | 42.8 | 43.0 | 52.1 | 51.9 |
Lift-off Test method | _ | _ | 3186 | 4693 | 4620 |
Electromagnetic Sensor method | _ | _ | 3188 | 4691 | 4690 |
Design Force | 3273 | 3162 | 3057 | 5022 | 4704 |
Taut string theory ignoring bending stiffness | 3788 | 3719 | 3664 | 5240 | 5240 |
Taut string theory considering bending stiffness | 3771 | 3702 | 3647 | 5223 | 5223 |
Empirical formulas of Zoe et al. | 3429 | 3362 | 3310 | 4818 | 4818 |
Empirical formulas of Ren et al. | 3416 | 3350 | 3298 | 4801 | 4801 |
Cho et al. Vibration Method | 3529 | 3417 | 3345 | 4851 | 4862 |
Proposed Method | 3418 | 3352 | 3301 | 4797 | 4797 |
Afterward, considering the average values of lift-off tests and electromagnetic sensors in the cases of BT2, AT2 and FINAL, as well as the design forces for BT1 and BT2 (no test results are available) as the known cable tension value, the ratio of estimated cable tension force obtained from utilized methods to the known one, is defined as 
and presented in Fig. 7.
As can be seen in Fig. 7, regardless of the bending stiffness, the error of methods employing string theory is greater than of the others and the accuracy of tension estimation will not be significantly improved even when the bending stiffness is considered, result in the error remaining in the range of 7 to 17%. On the other hand, showing an error of 8% in the worst case and a maximum error of 4% in most cases proves the reasonable accuracy of the other methods. However, according to Fig. 7, by using the iterative algorithm, the error is limited to 2% which shows that this method accurately estimates cable tension forces. This results show that the iterative algorithm can effectively lowered the errors in order to estimate the tension forces.
Fig. 7 Ratio of tension estimation results of using different methods: a cable BLC04 and b cable BLC02
a
b |
|
c
d |
|
Fig. 8 Ratio of the estimated cable tension force using different methods to the iterative method: a and b cable BLC04, c and d cable BLC02.
After showing the accuracy of the proposed method in estimating the amount of cable tension force for BLC04 and BLC02 and different natural frequency levels from 0.4-3.5 Hz, the tension force estimation of these cables is done using the mentioned methods. It has to be noted that the ξ parameter is assumed to be within the range of 1-61 and 7-93 for BLC02 and BLC04, respectively.
The ratio of estimated cable tension force obtained from previously mentioned methods to the proposed method value, is defined as 
and presented in Fig. 8. Results of this comparison, show that taut string theory is not generally accurate, whether or not the bending stiffness is considered. For the natural frequencies less than 2.6 Hz (ξ less than 68 in BLC04 and 44.5 in BLC02), the estimated values have a minimum variation of 10 percent. However, for values of natural frequencies over 2.6 Hz (ξ bigger than 68 in BLC04 and 44.5 in BLC02), the produced error reduces to less than 10 percent and values are inversely varied by frequency. On the contrary, the accuracy of the applied empirical formulas and subsequently the proposed iterative method are acceptable within the range of frequencies.
As can be observed from Fig.8, there are considerable difference between the present results and those of taut string theory with and without considering bending stiffness. To justify this behaviour, it can be said that in the cases of low tension forces (i.e. low natural frequencies), the role of bending rigidity, EI, is considerable and can not be neglected. According to this fact, the results of taut string theory with and without considering bending stiffness in which the influence of EI is completely neglected or considered approximatelty, have observable errors in the range of low tension forces. While, the empirical formulas of Zoe et al. and Ren et al. provide acceptable accuracy even in the cases of low tension forces, since the influence of EI is applied in these methods.
3.2. Other available cables
After verifying the accuracy of the proposed method, in this part, the estimated tension forces of all cables presented in Table 7 within a range of possible natural frequencies, have been compared to different methods.
Table 7
Properties and references of the used cables.
Row | Cable | Cable Length (m) | Mass Density (kg/m) | Bending Stiffness, EI (KN.m2) | Reference |
1 | Z1 | 7.15 | 12.03 | 23.5 | Zoe et al.[37]
|
2 | Z2 | 9.95 | 12.03 | 23.5 | |
3 | Z3 | 3.4 | 14.68 | 34.5 | |
4 | Z4 | 31.5 | 14.68 | 34.5 | |
5 | Z5 | 301.9 | 104.79 | 380.3 | |
6 | R01 | 29.3 | 61.3 | 731 | Chen et al.[58] |
7 | R33 | 126.4 | 48 | 577 | |
8 | PM1 | 68.47 | 31.86 | 1630.8 | Nam and Nghia. [59] |
9 | PM2 | 101.59 | 41.3 | 2675.4 | |
10 | PM3 | 145.22 | 53.1 | 4247.1 | |
11 | PM4 | 179.36 | 30.18 | 4812.6 |
Fig. 9 depicts the ratio of estimated cable tension force of all cables in Table 8, obtained by applying the empirical formulas of Zoe et al. and Ren et al., to the proposed method at various tension levels. As can be seen from Fig. 9, the results of the three mentioned methods have been compared in a wide range of values of the ξ parameter. It is observed that in the worst case, the maximum difference reaches approximately 2%, which again demonstrates the validity of the proposed method and also its applicability to a wide range of cable structures. Up to the ξ equal to 80, the results obtained from the three methods vary by less than 1%, and for the value of ξ above 120, the difference in results gradually increases to 2%.
Fig. 9 Results of tension force estimation of the cables using different methods.
4- Conclusions
One of the common methods for estimating cable tension in prestressed components, from cable-stayed bridges to post-tensioned concrete box bridges, is the vibration-based method. These methods that have been used widely in the literature so far, have encountered many errors in most applications, especially in the case of cables with high bending stiffness and fixed end boundary conditions. Accordingly, to enhance and improve the vibration-based methods, a new cable tension force estimation method based on two iterative algorithms has been developed in this paper. The accuracy, reliability, and applicability of the proposed method have been investigated by comparing with several experimental tests and also some available previously reported results in the literature. The main conclusions are as follows:
- In the cases of low tension forces in which the role of EI can not be ignored, the results of the proposed method is more accurate than the result of method in which the EI is neglected or approximately considered.
- The present approach is independent of ξ parameter, while the relations of empirical formulas must be varied in different ranges of ξ and trial and error should be performed.
- In the Hwamyeong Bridge cables case, in which the ξ parameter was investigated across the range of 43 to 66, the average error of estimated cable tensile force is calculated as less than 1.25% for the BLC-04 cable and less than 3% for the BLC-02 cable.
- For all other cases, the proposed method accurately estimated the amount of tensile force of the cable in a wide range of ξ from 5 to 200, demonstrating that this method can be used for a wide range of cable structures.
If the bending stiffness is evaluated more accurately by trial and error or through an iterative algorithm, the accuracy of the estimated cable tensile force will increase as well. In all cases studied in this research, the values of bending stiffness have been considered as a known variable to estimate the cable tension. Additional future research opportunities are needed to investigate this issue.
Acknowledgment:
The authors would like to thank the Structural Laboratory of the International Institute of Earthquake Engineering and Seismology, and also the FRB Post-tensioning company for their helpful cooperation and assistance.
References
[1] C. Naito, Construction and Field Evaluation of Electrically Isolated Tendons in a Prestressed Concrete Spliced Girder Bridge, Journal of Bridge Engineering. 25 (2020) 5020001.
[2] B. Mathy, P. Demars, F. Roisin, M. Wouters, Investigation and strengthening study of twenty damaged bridges: A Belgium case history, in: Bridge Management 3. Inspection, Maintenance and Repair. Papers Presented at the Third International Conference on Bridge Management, University of Surrey, Guildford, UK, 14-17 April 1996, 1996.
[3] M. Morgese, F. Ansari, M. Domaneschi, G.P. Cimellaro, Post-collapse analysis of Morandi’s Polcevera viaduct in Genoa Italy, Journal of Civil Structural Health Monitoring. 10 (2020) 69–85.
[4] G. Shao, H. Jin, R. Jiang, Y. Xu, Dynamic Response and Robustness Evaluation of Cable-Supported Arch Bridges Subjected to Cable Breaking, Shock and Vibration. 2021 (2021).
[5] D. Proske, Bridge collapse frequencies versus failure probabilities, Springer, 2018.
[6] A. Azizinamini, J. Gull, others, Improved inspection techniques for steel prestressing/post-tensioning strand: Volume I, (2012).
[7] T. Terzioglu, M.M. Karthik, S. Hurlebaus, M.B.D. Hueste, Systematic Assessment of nondestructive evaluation techniques for post-tensioning and stay cable systems, Journal of Infrastructure Systems. 25 (2019) 4019015.
[8] C.-C. Chen, W.-H. Wu, S.-Y. Chen, G. Lai, A novel tension estimation approach for elastic cables by elimination of complex boundary condition effects employing mode shape functions, Engineering Structures. 166 (2018) 152–166.
[9] L. Zhang, G. Qiu, Z. Chen, Structural health monitoring methods of cables in cable-stayed bridge: A review, Measurement. 168 (2021) 108343.
[10] Y. Yao, M. Yan, Y. Bao, Measurement of cable forces for automated monitoring of engineering structures using fiber optic sensors: A review, Automation in Construction. 126 (2021) 103687.
[11] J.-W. Kim, S. Park, Magnetic flux leakage sensing and artificial neural network pattern recognition-based automated damage detection and quantification for wire rope non-destructive evaluation, Sensors. 18 (2018) 109.
[12] Y. Qu, H. Zhang, R. Zhao, L. Liao, Y. Zhou, Research on the method of predicting corrosion width of cables based on the spontaneous magnetic flux leakage, Materials. 12 (2019) 2154.
[13] S. Zhang, J. Zhou, Y. Zhou, H. Zhang, J. Chen, Cable tension monitoring based on the elasto-magnetic effect and the self-induction phenomenon, Materials. 12 (2019) 2230.
[14] M.L. Wang, Z.L. Chen, S.S. Koontz, G.M. Lloyd, Magnetoelastic permeability measurement for stress monitoring in steel tendons and cables, in: Nondestructive Evaluation of Highways, Utilities, and Pipelines IV, 2000: pp. 492–500.
[15] R. Geier, G. De Roeck, R. Flesch, Accurate cable force determination using ambient vibration measurements, Structure and Infrastructure Engineering. 2 (2006) 43–52. https://doi.org/10.1080/15732470500253123.
[16] C.U. Grosse, M. Ohtsu, Acoustic emission testing, Springer Science \& Business Media, 2008.
[17] D. Hu, Y. Guo, X. Chen, C. Zhang, Cable force health monitoring of Tongwamen bridge based on fiber Bragg grating, Applied Sciences. 7 (2017) 384.
[18] R. Zheng, L. Liu, X. Zhao, Z. Chen, C. Zhang, X. Hua, Investigation of measurability and reliability of adhesive-bonded built-in fiber Bragg grating sensors on steel wire for bridge cable force monitoring, Measurement. 129 (2018) 349–357.
[19] Y. Yu, X. Zhao, J. Ou, A new idea: Mobile structural health monitoring using Smart phones, in: 2012 Third International Conference on Intelligent Control and Information Processing, 2012: pp. 714–716.
[20] X. Zhao, Y. Yu, W. Hu, D. Jiao, R. Han, X. Mao, M. Li, J. Ou, Cable force monitoring system of cable stayed bridges using accelerometers inside mobile smart phone, in: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2015, 2015: pp. 452–458.
[21] X. Zhao, R. Han, Y. Yu, W. Hu, D. Jiao, X. Mao, M. Li, J. Ou, Smartphone-based mobile testing technique for quick bridge cable--force measurement, Journal of Bridge Engineering. 22 (2017) 6016012.
[22] G. Morgenthal, S. Rau, J. Taraben, T. Abbas, Determination of stay-cable forces using highly mobile vibration measurement devices, Journal of Bridge Engineering. 23 (2018) 4017136.
[23] X.W. Ye, C.Z. Dong, T. Liu, Force monitoring of steel cables using vision-based sensing technology: methodology and experimental verification, Smart Structures and Systems. 18 (2016) 585–599.
[24] D. Feng, T. Scarangello, M.Q. Feng, Q. Ye, Cable tension force estimate using novel noncontact vision-based sensor, Measurement. 99 (2017) 44–52.
[25] S.-W. Kim, B.-G. Jeon, N.-S. Kim, J.-C. Park, Vision-based monitoring system for evaluating cable tensile forces on a cable-stayed bridge, Structural Health Monitoring. 12 (2013) 440–456.
[26] W. Du, D. Lei, P. Bai, F. Zhu, Z. Huang, Dynamic measurement of stay-cable force using digital image techniques, Measurement. 151 (2020) 107211.
[27] G. Feltrin, J. Meyer, R. Bischoff, M. Motavalli, Long-term monitoring of cable stays with a wireless sensor network, Structure and Infrastructure Engineering. 6 (2010) 535–548.
[28] L. Sun, Z. Shang, Y. Xia, S. Bhowmick, S. Nagarajaiah, Review of bridge structural health monitoring aided by big data and artificial intelligence: From condition assessment to damage detection, Journal of Structural Engineering. (2020) 4020073.
[29] H. Pan, M. Azimi, F. Yan, Z. Lin, Time-frequency-based data-driven structural diagnosis and damage detection for cable-stayed bridges, Journal of Bridge Engineering. 23 (2018) 4018033.
[30] J.P. Santos, C. Cremona, A.D. Orcesi, P. Silveira, L. Calado, Static-based early-damage detection using symbolic data analysis and unsupervised learning methods, Frontiers of Structural and Civil Engineering. 9 (2015) 1–16.
[31] A. Cunha, E. Caetano, Dynamic measurements on stay cables of cable-stayed bridges using an interferometry laser system, Experimental Techniques. 23 (1999) 38–43.
[32] A. Cunha, E. Caetano, R. Delgado, Dynamic tests on large cable-stayed bridge, Journal of Bridge Engineering. 6 (2001) 54–62.
[33] A.B. Mehrabi, In-service evaluation of cable-stayed bridges, overview of available methods and findings, Journal of Bridge Engineering. 11 (2006) 716–724.
[34] L. DOOSTI, M. ELMI, M. CHERAGHZADE, O. BAHAR, M. GHAFORY-ASHTIANY, VIBRATION-BASED TENSION FORCE ESTIMATION OF EXTERNAL POST-TENSIONED TENDONS: EXPERIMENTAL STUDY, (n.d.).
[35] S. Kangas, A. Helmicki, V. Hunt, R. Sexton, J. Swanson, Cable-stayed bridges: case study for ambient vibration-based cable tension estimation, Journal of Bridge Engineering. 17 (2012) 839–846.
[36] Z. Fang, J. Wang, Practical formula for cable tension estimation by vibration method, Journal of Bridge Engineering. 17 (2012) 161–164.
[37] H. Zui, T. Shinke, Y. Namita, Practical formulas for estimation of cable tension by vibration method, Journal of Structural Engineering. 122 (1996) 651–656.
[38] W.-X. Ren, G. Chen, W.-H. Hu, others, Empirical formulas to estimate cable tension by cable fundamental frequency, Structural Engineering and Mechanics. 20 (2005) 363–380.
[39] H.M. Irvine, T.K. Caughey, The linear theory of free vibrations of a suspended cable, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 341 (1974) 299–315.
[40] L. Ma, A highly precise frequency-based method for estimating the tension of an inclined cable with unknown boundary conditions, Journal of Sound and Vibration. 409 (2017) 65–80. https://doi.org/10.1016/j.jsv.2017.07.043.
[41] L. Ma, H. Xu, T. Munkhbaatar, S. Li, An accurate frequency-based method for identifying cable tension while considering environmental temperature variation, Journal of Sound and Vibration. 490 (2021) 115693.
[42] B.H. Kim, T. Park, Estimation of cable tension force using the frequency-based system identification method, Journal of Sound and Vibration. 304 (2007) 660–676.
[43] J. Humar, Dynamics of structures, CRC press, 2012.
[44] D. Jana, S. Nagarajaiah, Computer vision-based real-time cable tension estimation in Dubrovnik cable-stayed bridge using moving handheld video camera, Structural Control and Health Monitoring. 28 (2021) e2713.
[45] X. Zhao, R. Han, Y. Ding, Y. Yu, Q. Guan, W. Hu, M. Li, J. Ou, Portable and convenient cable force measurement using smartphone, Journal of Civil Structural Health Monitoring. 5 (2015) 481–491.
[46] S. Jeong, H. Kim, J. Lee, S.-H. Sim, Automated wireless monitoring system for cable tension forces using deep learning, Structural Health Monitoring. 20 (2021) 1805–1821.
[47] C. Chu, F. Ghrib, S. Cheng, Cable tension monitoring through feature-based video image processing, Journal of Civil Structural Health Monitoring. 11 (2021) 69–84.
[48] L. Qiao, A. Esmaeily, H.G. Melhem, Signal pattern recognition for damage diagnosis in structures, Computer-Aided Civil and Infrastructure Engineering. 27 (2012) 699–710.
[49] N. Jacobsen, Using Enhanced Frequency Domain Decomposition as a Robust Technique to Harmonic Excitation in Operational Modal Analysis, (2006).
[50] B. Peeters, G. De Roeck, Reference based stochastic subspace identification in civil engineering, Inverse Problems in Engineering. 8 (2000) 47–74.
[51] O. Hegeir, T. Mizutani, K. Matsumoto, K. Nagai, The Cause Estimation of Damages in Pathein Suspension Bridge Based on Vibration, (2018) 1–7. https://doi.org/10.3390/ICEM18-05209.
[52] J. Lardies, M. Ta, J. Lardies, M.T. Modal, Modal parameter identification of stay cables from output-only measurements To cite this version : HAL Id : hal-02300174, (2020) 0–18.
[53] A.R. Joaquim, C. Cismasiu, F.A. Santos, E. CAETANO, Estimation of the tensile force in the stay-cables of Salgueiro Maia bridge using ambient vibration tests, in: Proceedings of the International Symposium on the Dynamics and Aerodynamics of Cables (ISDAC’2017), 2017.
[54] P. Van Overschee, B.L. De Moor, Subspace identification for linear systems: Theory—Implementation—Applications, Springer Science \& Business Media, 2012.
[55] A.K. Chopra, Dynamics of structures, Pearson Education India, 2007.
[56] S. Cho, J. Yim, S.W. Shin, H.-J. Jung, C.-B. Yun, M.L. Wang, Comparative field study of cable tension measurement for a cable-stayed bridge, Journal of Bridge Engineering. 18 (2013) 748–757.
[57] Dasan, 2012. http://www.dasan93.co.kr/en/portfolio-item/hwamyeong-bridge/?ckattempt=3.
[58] C. Chen, W. Wu, M. Leu, G. Lai, Tension determination of stay cable or external tendon with complicated constraints using multiple vibration measurements, MEASUREMENT. (2016). https://doi.org/10.1016/j.measurement.2016.02.053.
[59] H. Nam, N.T. Nghia, Estimation of cable tension using measured natural frequencies, Procedia Engineering. 14 (2011) 1510–1517.
The rights to this website are owned by the Raimag Press Management System.
Copyright © 2021-2024