Lift-off-validated assessment of cable force prediction methods for cable-stayed bridge using vibration-based approaches

This study investigates cable force estimation in cable-stayed bridges through a vibration-based approach, utilizing experimental data measured using an accelerometer sensor. In the initial phase of the research, the frequency data measured by accelerometers is validated through numerical modeling using the Midas Civil software. Additionally, besides employing the string formula, this study adopts formulas proposed by [1] to predict cable forces in two cable-stayed bridges in Indonesia. The estimated cable forces using both formulas are then compared with the actual cable forces measured during the lift-off test.The analysis results indicate that most of the cable frequency data is valid, with differences of less than 7% between the measured frequencies and numerical results. However, a significant difference is observed in one cable, BA-M11, with differences of up to 50.9%. This suggests that the mode order and frequency values measured for this cable are not valid. Through a numerical approach, accurate mode orders and frequencies are determined, enabling confident use of the measurement data for cable force estimation in the case of cable BA-M11.Furthermore, when the validated mode orders and frequency values are used with both the string formula and Yu's proposed formulas, the results show that Yu's formulas tend to provide more accurate estimations compared to the string theory, with average differences in cable force estimation of approximately 4.33% and 2.97% relative to the lift-off force.The contribution of this research lies in the utilization of numerical verification to correct inaccuracies in accelerometer-measured mode orders and frequency values. Subsequently, armed with validated mode orders and frequency values, Yu's proposed formulas demonstrate superior accuracy in predicting cable forces compared to the string theory when both are compared with lift-off test data.

The presence of unknown complex boundary conditions usually imposes difficulties in estimating the cable forces in cable-stayed bridges when using conventional model-based force identification methodologies. Therefore, there exists a need for new methodologies that can overcome these challenges while achieving acceptable force identification accuracy. This paper presents an innovative method to estimate the forces within stay cables with complex boundary conditions. The proposed approach transforms the cable force estimation problem from the common procedure of constructing and solving the equation of motion of the cable to a simpler problem of finding the zero-amplitude points of its mode shapes. Ultimately, the presented methodology yields accurate cable force estimations regardless of the complexity of the boundary conditions. An equivalent segmental model whose length is given by the distance between these points is used next to find an estimate of the cable tension force. A stay cable under axial force and constrained by end rotational springs is employed to analytically investigate the force identification accuracy of the proposed method. It is observed that with mode orders lower than 18, the proposed method achieves a maximum relative error less than 5% regardless of the end-restraint condition. Therefore, the proposed method has great potential for practical application because of its theoretical simplicity, accuracy, and feasibility.

Cable force estimation is essential for security assessment of cable-stayed bridges. Cable force estimation methods based on the relationship between cable force and frequency have been extensively studied and used during both construction phase and service phase. However, the effect induced by inclination angle of the cable is not included in the establishment of frequency-cable force relationship as horizontal cable model is normally employed. This study aims to investigate the influence of the inclination angle on vibration based cable force estimation and provide practical formulas accordingly. Firstly numerical examples of fixed-fixed and hinged-hinged cables are simulated to illustrate the necessity of considering the inclination angle effect on the modal parameters and cable force estimation for inclined cables with small sag. Then practical formulas considering the inclination angle effect to estimate the cable force of fixed-fixed and hinged-hinged cables via the fundamental frequency are established accordingly. For the inclined cables with unknown boundary conditions, the coefficients reflecting boundary condition are predicted via the practical formulas for fixed-fixed and hinged-hinged cables. And the cable force considering the influence of inclination angle and unknown boundary conditions is obtained by iteration method. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed method.

Noncontact cable force estimation with unmanned aerial vehicle and computer vision

The portable and rapid cable force measurement for cable-supported structures, such as a cable-stayed bridge and a suspension bridge, has an important and practical significance in the evaluation of initial damage and the recovery of transportation networks after severe events such as earthquake and typhoon. Extracting natural frequencies of cable is considered as an essential issue in the cable force estimation. Therefore, in this study, a nontarget vision-based approach is proposed for identifying the natural frequencies of cable using handheld shooting of smartphone camera. The boundary of cable is selected as a target to be tracked in the region of interest (ROI) of video image sequence captured by smartphone camera, and the natural frequencies of cable are identified according to its dynamic displacement responses in frequency domain. The moving average is adopted to eliminate the noise associated with shaking of smartphone camera during measurement. A laboratory scale cable model test is carried out to evaluate the proposed approach. The results demonstrate the feasibility of vision-based cable force estimation using smartphone camera

R.H. Bridge Fisabilillah (Bridge I) is a cable-stayed bridge, included in the series of Barelang Bridges (Batam-Rempang-Galang) which were built from 1992 to 1998. With the service life of the bridge reaching 25 years, it is necessary to check the health condition of the bridge structure. Bridge cables are one of the most important elements of a cable-stayed bridge. These cable elements dominantly experience tensile forces when transmitting loads from the decks to the bridge pylons. Cable force inspection methods can be carried out through the direct measurement method (e.g. static test using a lift-off method) and indirect measurement (e.g. dynamic test using accelerometer sensors, electromagnetic/EM sensors, and so on). This study aims to compare the cable tensile force based on the static test (lift-off method) in 2017 against the dynamic test (accelerometer sensors) in 2022. Evaluation of the cable tensile force based on the dynamic test was carried out using the taut string theory and beam string theory approaches. From the study, the two empirical approaches yielded insignificantly different results where a difference in the mean difference of -1,71% was found with a maximum difference of 28,15%. The study also shows an increase in cable force capacity to a maximum of 47,20% UTS (ultimate tensile strength) based on the taut string theory and a maximum of 53.37% UTS based on the beam string theory. This value is greater when compared to the results of the cable force based on the static test (lift-off) in 2017, which was a maximum of 41.64% UTS. It is recommended to carry out further and more comprehensive studies to determine the effect of changes in cable force distribution on the behavior of the structure on the R.H. Fisabilillah Bridge.

The stayed-cable is an important component of cable-stayed bridges, with cable force being a focal point during construction and bridge operation. The advancement of camera and image processing technology has facilitated the integration of computer vision technology in structural inspection and monitoring. This paper focuses on enhancing cable force measurement methods and addressing the limitations of traditional testing techniques by conducting experimental research on cable force estimation using video recording. The proposed approach involves capturing video footage of the target on the cable with a smartphone. Subsequently, a combination of techniques such as the background subtraction method, image morphology processing, and Hough transform image processing technology are employed to detect the precise center coordinates and ultimately obtain the accurate displacement-time curve of the cable's vibration. In addition, the graphic Circularity Coefficient (CC) has been introduced to assess its effectiveness in post-motion-blur image processing for circular targets. The fundamental frequency of the cable is determined by the fast Fourier transformation, and the relationship between the cable force and the fundamental frequency is used to estimate the cable force. The experimental results are compared with data from accelerometers and force gauges, demonstrating that the frequency measurement error is below 1.2% and the cable force test error is less than 3%. In the process of acquiring the cable's fundamental frequency, the test directly employs the pixel as the displacement unit, eliminating the need for image calibration. The innovative use of the CC in processing motion-blurred targets ensured accurate recognition of target coordinates. The experimental findings highlight the method's simplicity, speed, and accuracy.

Currently, due to the rapid development and popularization of smartphones, the usage of ubiquitous smartphones has attracted growing interest in the field of structural health monitoring (SHM). The portable and rapid cable force measurement for cable-supported structures, such as a cable-stayed bridge and a suspension bridge, has an important and practical significance in the evaluation of initial damage and the recovery of transportation networks. The extraction of dynamic characteristics (natural frequencies) of cable is considered as an essential issue in the cable force estimation. Therefore, in this study, a vision-based approach is proposed for identifying the natural frequencies of cable using handheld shooting of smartphone camera. The boundary of cable is selected as a target to be tracked in the region of interest (ROI) of video image sequence captured by smartphone camera, and the dynamic characteristics of cable are identified according to its dynamic displacement responses in frequency domain. The moving average is adopted to eliminate the noise associated with the shaking of smartphone camera during measurement. A laboratory scale cable model test and a pedestrian cable-stayed bridge test are carried out to evaluate the proposed approach. The results demonstrate the feasibility of using smartphone camera for cable force estimation.

Long-span cable-stayed bridges have gained increasing popularity due to their appealing aesthetics, increased stiffness compared to the suspension bridges, and relatively small size of bridge components. For cable-stayed bridges with steel girders in particular, fatigue has been one of the most important failure modes and the prediction of the remaining fatigue lives has been the primary focus in recent years. To enable the evaluation of fatigue performance, a reasonable baseline finite element model reflecting the real structural behaviour is indispensable. This thesis describes the establishment of baseline finite element models of cable-stayed bridges for fatigue analysis considering the mean stress effect. Ting Kau Bridge in Hong Kong is chosen as a real life example to demonstrate the practical application of the methods developed.
\nFirstly, the initial cable forces of the finite element model of the cable-stayed bridge are calibrated to ensure that the initial geometry of the finite element model under permanent loading agrees with that specified on the as-built drawings within reasonable tolerance. The traditional cable force adjustment method is often a trial-and-error process which is empirical, time-consuming and occasionally difficult in convergence. An optimization method based on the Kriging surrogate model is therefore developed in order to establish the relation between the deck geometry and he initial cable forces of the cable-stayed bridge. The efficiency of the proposed approach is further verified using Ting Kau Bridge against the traditional method. 
\nFurthermore, the initial finite element model developed on the basis of engineering blueprints is updated by modifying the uncertain parameters so as to achieve a refined model using the Kriging predictor. An objective function can be formulated in terms of the discrepancy between the theoretical and measured responses captured by the Wind and Structural Health Monitoring System (WASHMS) installed on the bridge. A novel feature of the proposed method is that it enables the simultaneous use of static load testing data and the dynamic information, which is not feasible by using the conventional sensitivity-based approaches. 
\nFinally, the fatigue lives of selected critical components of Ting Kau Bridge are worked out with different methods including the deterministic and probabilistic approaches based on the baseline finite element model. In the deterministic approach, the equivalent annual standard fatigue vehicle spectrum is built based on the WASHMS measurements to estimate the damage accumulation and predict fatigue lives. In the probabilistic approach, a probabilistic loading model is proposed to simulate the vehicles running along the bridge by using the available information collected by the weigh-in-motion system. Combined with the traffic load model, a statistical approach is employed to obtain the stress time-histories, and the probability density curve of the fatigue life can be obtained. One significant advantage of the proposed method is that the diurnal variation of traffic flow within different time intervals can be accounted for properly. The results of these approaches are compared with one another. As the mean stress effect is significant in the estimation of fatigue lives, the baseline calibration of cable-stayed bridges is considered essential.

The emergency of asymmetric hybrid-girder cable-stayed bridges with different shapes of main single tower brings numerous challenges to the analysis of cable forces of completed bridges.In this paper, based on an asymmetric hybrid-girder cable-stayed bridge with fish-shuttle-shaped single tower in China, the finite element software Midas Civil was adopted to build the model of the cable-stayed bridge.On the basis of the designed cable forces, the influence matrix method was used to optimize the cable forces of the completed cable-stayed bridge, and the reasonable completed bridge state was determined.The results indicated that the error of cable forces of the entire bridge was controlled within 10%, namely, the cable-stayed bridge could reach a reasonable completed bridge state by adjusting the cable forces in a small range.In addition, it was practical to adopt influence matric method to optimize the completed bridge cable forces of asymmetric hybrid-girder cable-stayed bridge with fish-shuttle-shaped single tower, and the optimized cable forces was reasonable.The tendency of the bending moment of main girder under optimized cable forces was the same as that under designed cable forces, and the maximum positive and negative bending moments decreased by 49.84% and 27.35%, respectively.Compared with the designed cable forces, the maximum positive and negative displacements of the main girder and the maximum deviation displacement of the main tower along the bridge under optimized cable forces decreased by 76.33%, 39.20%, and 6.59%, respectively, leading to a smoother alignment of the main girder and tower.Compared with the designed cable forces, the support reaction forces were significantly reduced under the optimized cable forces, with a maximum reduction of 41.61%, which provided a larger pressure reserve to support and no negative reaction forces occurred.The optimized cable forces using the influence matrix method had obvious improvement effects on the bending moment, deformation and bearing force of the overall structure, which met the requirements of reasonable completed bridge state for cable-stayed bridges.The research results can provide reference for the optimization of cable forces in similar completed cable-stayed bridges.

Due to its easy operation and wide applicability, the ambient vibration method is commonly adopted to determine the cable force by first identifying the cable frequencies from the vibration signals. With given vibration length and flexural rigidity, an analytical or empirical formula is then used with these cable frequencies to calculate the cable force. It is, however, usually difficult to decide the two required parameters, especially the vibration length due to uncertain boundary constraints. To tackle this problem, a new concept of combining the modal frequencies and mode shape ratios is fully explored in this study for developing an accurate method merely based on ambient vibration measurements. A simply supported beam model with an axial tension is adopted and the effective vibration length of cable is then independently determined based on the mode shape ratios identified from the synchronized measurements. With the effective vibration length obtained and the identified modal frequencies, the cable force and flexural rigidity can then be solved using simple linear regression techniques. The feasibility and accuracy of the proposed method is extensively verified with demonstrative numerical examples and actual applications to different cable-stayed bridges. Furthermore, several important issues in engineering practice such as the number of sensors and selection of modes are also thoroughly investigated.

Cable-stayed bridges have commonly been built for crossing large-span obstacles, such as rivers, valleys, and existing structures. Obtaining an optimum design for a cable-stayed bridge is challenging, due to the large number of design variables and design constraints that are typically nonlinear and usually conflict with each other. Therefore, it is a reasonable alternative to turn the large and complex optimization problem into two sub-problems, i.e., optimizing the internal force distribution by adjusting the cable prestressing forces, and optimizing the other sizing or geometrical parameters. However, conventional methods are lacking in efficiency when dealing with the problem of optimization of cable forces in the first sub-problem, under the circumstance that iteration between the two sub-problems is required. To address this, this paper presents a surrogate-model-assisted method to construct a cable forces predictor ahead of the structural optimization process, so that cable forces can be effectively predicted rather than optimized in each iterative round. Additionally, B-spline interpolation curve is adopted for variable condensation when sampling for the surrogate model. Finally, the structure optimization in the second sub-problem is performed by leveraging an optimization program based on particle swarm optimization method. The performance of the proposed framework is tested with a practical engineering application. Results show that the proposed method showcases good efficiency and accuracy. The theoretical raw material consumption of the towers and the cables is 32% lower than the original design.

A flexibility-based distributed computing strategy (DCS) for structural health monitoring (SHM) has recently been proposed which is suitable for implementation on a network of densely distributed smart sensors. In that approach, a hierarchical strategy is proposed in which adjacent smart sensors are grouped together to form sensor communities. Structural health monitoring is done without relying on central data acquisition and processing. The main purpose of this paper is to experimentally verify this flexibility-based DCS approach. The damage locating vector method that forms foundation of the DCS approach is reviewed. An overview of the DCS approach is presented. This flexibility-based approach is then experimentally verified employing a 5.6 m long three-dimensional truss structure. To simulate damage in the structure, the original truss members are replaced by ones with a reduced cross section. Both single and multiple damage scenarios are studied. Experimental results show that the DCS approach can successfully detect the damage at local elements using only locally measured information.

A flexibility-based distributed computing strategy (DCS) for structural health monitoring (SHM) has recently been proposed which is suitable for implementation on a network of densely distributed smart sensors. This approach uses a hierarchical strategy in which adjacent smart sensors are grouped together to form sensor communities. A flexibility-based damage detection method is employed to evaluate the condition of the local elements within the communities by utilizing only locally measured information. The damage detection results in these communities are then communicated with the surrounding communities and sent back to a central station. Structural health monitoring can be done without relying on central data acquisition and processing. The main purpose of this paper is to experimentally verify this flexibility-based DCS approach using wired sensors; such verification is essential prior to implementation on a smart sensor platform. The damage locating vector method that forms foundation of the DCS approach is briefly reviewed, followed by an overview of the DCS approach. This flexibility-based approach is then experimentally verified employing a 5.6 m long three-dimensional truss structure. To simulate damage in the structure, the original truss members are replaced by ones with a reduced cross section. Both single and multiple damage scenarios are studied. Experimental results show that the DCS approach can successfully detect the damage at local elements using only locally measured information.

Since cables are the critical structural components for ensuring the overall structural integrity and safety of cable‐stayed bridges, the portable and rapid cable force measurement has a very important practical significance in the bridge health monitoring. In this paper, a method of simultaneous and continuous estimation of the forces of multiple cables using a self‐developed microwave interferometric radar was proposed. In the proposed approach, first, the time‐varying modal frequencies are identified by the Hilbert transform algorithm from the displacements of multiple cables, which are monitored by the self‐developed radar, and then, the time‐varying cable forces are obtained by the vibration method. In addition, for the problem of multiple cables located in the same range bin when distance test or the test angle is nonideal, a single‐channel blind source separation algorithm based on vibrational mode decomposition (VMD) and time‐frequency analysis was proposed to separate the cable signals. Furthermore, a method based on the spectrum correlation coefficients was proposed to determine the number of decomposition layers in VMD. The “Nanjing Eye” cable‐stayed footbridge was used to conduct field measurement to validate the proposed method. Field measurement results show a good agreement with reference measurements, which demonstrate that the proposed method can perform well in an actual project for portable and rapid cable force estimation.