Cable Bending Stiffness Research Articles

Damping effects of a stay cable attached with a viscous damper and a high damping rubber damper considering cable bending stiffness and damper support flexibility

To mitigate cable oscillations in cable-stayed bridges, a common approach involves using a strategically positioned viscous damper near the cable’s anchorage and bridge deck. However, for longer cables, this method may be insufficient due to installation constraints. In such cases, supplementing the damping system with a high-damping rubber (HDR) damper near the cable’s anchorage point on the bridge tower becomes imperative to enhance the cable’s damping ratio. Conventional designs often overlook crucial factors like viscous damper support stiffness and stay cable bending stiffness when integrating dampers into cable-stayed structures. This study presents findings on achieving an effective damping ratio in stay cables by using both a viscous damper and an HDR damper, considering the influence of viscous damper support stiffness and stay cable bending stiffness. The results indicate that the combined deployment of these dampers achieves a damping efficiency approximately equivalent to the sum of their individual effects. Importantly, decreased viscous damper support stiffness significantly affects the damping effectiveness, leading to a rapid decline in the stay cable’s damping ratio. While stay cable bending stiffness also influences the damping ratio, its impact is relatively less pronounced than that of viscous damper support stiffness. The study outcomes enable a more accurate prediction of the achievable damping ratio for a stay cable with additional components, considering both damper support stiffness and stay cable bending stiffness. Furthermore, the study explores parameters of both the viscous damper and HDR damper, such as the viscous coefficient, loss coefficient, HDR damper stiffness, and damper placement, evaluating their influence on the first damping ratio of the stay cable. The survey results provide valuable insights for determining optimal parameters for both dampers, maximizing the damping efficiency of cable-stayed bridges.

Joint Identification of Cable Force and Bending Stiffness Using Vehicle-Induced Cable–Beam Vibration Responses

Joint Identification of Cable Force and Bending Stiffness Using Vehicle-Induced Cable–Beam Vibration Responses

Analytical Estimation and Experimental Validation of the Bending Stiffness of the Transmission Line Conductors

The bending stiffness of transmission line conductors can vary significantly, ranging from maximum stiffness when behaving monolithically to minimum stiffness when wires behave loosely. This large range makes it challenging to estimate stiffness accurately at intermittent bending stages. To address this issue, a mathematical model that accounts for both frictional forces between wires in the same layer and the clenching effects of helical wires from preceding layers is proposed in this paper. The proposed model estimates cable bending stiffness as a function of axial load and curvature for multilayered strands by considering slip caused by wire behavior. To evaluate the bending stiffness, experiments were conducted on Panther and Moose Indian Power Transmission line conductors. The proposed slip model considers Coulomb frictional effects and clenching effects caused by Hertzian contact forces, filling the void in the estimation procedure. Additionally, the model considers the wire stretch effect, a parameter not previously accounted for in cable research. The predicted numerical results of the proposed model were found to vary within a maximum of 7% from the experimental tests. The proposed mathematical model thus offers a more accurate and comprehensive way of estimating the bending stiffness of transmission line conductors, addressing the existing limitations in the estimation procedure.

Damping in stay cable with damper: Practical universal damping curve and full-scale measurement

This paper includes two main parts which are the development of a universal damping curve for cables of cable-stayed bridges with dampers and a full-scale measurement of cable damping. In the first part, the universal damping curve is derived for cables with viscous dampers or High Damping Rubber (HDR) dampers. The proposed damping curve is a single curve which incorporates several cable-damper parameters like cable bending stiffness, boundary conditions at cable ends (hinged, fixed or restraint ends), damper stiffness, damper support stiffness and added negative stiffness. A numerical approach by a finite difference method (FDM) is also developed to verify the proposed universal damping curve. In addition to FDM, the accuracy of the proposed curve was confirmed with damping formulations in literature over ten different damper types. The results showed a good agreement of estimated damping between proposed damping curve and others. Additionally, a significant improvement in estimating cable damping was observed through two available full-scale tests, which compared the results between proposed curve and current curve. The second part in this study presents a full-scale measurement of cable vibration and a real application of the proposed damping curve for the identification of damping in cables of Shinminato Bridge (cable lengths from 100 to 200 m). Both HDR and viscous dampers were used for the cable vibration control of the bridge. The results of damping analysis indicated that the HDR damper effectiveness was higher than 0.7 whereas this effectiveness was smaller than 0.4 for the viscous damper.

How the Material Characteristics of Optical Fibers and Soil Influence the Measurement Results of Distributed Acoustic Sensing.

Fiber optic distributed acoustic sensing (DAS) technology is widely used in security surveillance and geophysical survey applications. The response of the DAS system to external vibrations varies with different types of fiber optic cable connections. The mechanism of mutual influence between the cable's characteristics and DAS measurement results remains unclear. This study proposed a dynamic model of the interaction between the optical cable and the soil, analyzed the impact of the dynamic parameters of the optical cable and soil on the sensitivity of the DAS system, and validated the theoretical analysis through experiments. The findings suggest that augmenting the cable's bending stiffness 5.5-fold and increasing its unit mass 4.2-fold result in a discernible reduction of the system's response to roughly 0.15 times of its initial magnitude. Cables with lower unit mass and bending stiffness are more sensitive to vibration signals. This research provides a foundation for optimizing vibration-enhanced fiber optic cables and broadening the potential usage scenarios for DAS systems.

A nonlinear finite element framework for static and dynamic analysis of structural cables with deviating supports

Structural cables are important load-bearing members in many light-weight and slender structures. The mechanical behaviour of tension cables is complex as it is dominated by geometric nonlinearities, complex support conditions and uncertainties arising from the composition of the cable cross section. Numerous models to describe static and dynamic cable behaviour were developed, differing in the model assumptions and the simplifications made. Most common are geometrical linearisation, dimensional reduction, and the disregard of localised support and stiffness effects. Some models take a combination of some geometric nonlinearities, cable bending stiffness and certain boundary effects into account. This paper first presents a literature review aimed at systematically categorising cable force identification models. Based on this, the paper proposes a novel numerical cable modelling framework that allows for accurate predictions of static and dynamic cable response, e.g. for the use in the inverse system identification techniques of Structural Health Monitoring. The saddled cable model is based on a nonlinear Finite Element formulation to account for the geometric and mechanical effects of deviating supports on structural cables even for very large sag cables. After validation with analytical solutions, the models show that the effect of self-weight distribution and cable bending stiffness considerably affects the identified internal force distribution, contact length on the saddle and cable form. Further, the model is used to predict the tension force based on the identified natural frequencies of an external tendon installed on a bridge. Compared with the actual tension force, the saddled cable model provided very high prediction accuracy, even when compared with other cable models. Therefore, it is shown to be very accurate and can be used to determine force-response relationships used in system identification techniques when increased accuracy is needed for such complex cable arrangements.

Evaluation of the Cable Force by Frequency Method for the Hybrid Boundary between the Ear Plate and the Anchor Plate

For the quick and accurate determination of the cable force under the hybrid boundary between the ear plate and the anchor plate, this paper proposes a method for identifying the boundary conditions and bending stiffness of the cables based on the frequency method. First, the influence of the parameters, namely, cable boundary conditions, bending stiffness, inclination angle, and linear density, on the cable force was analyzed. Next, the actual bending stiffness of the cable under the boundary conditions of the upper hinged support and the lower solid support and the solid support at both ends were determined, and the cable force in the two states was calculated. Finally, the hydraulic jack tension and the cable force in the two states were compared and analyzed to define the actual boundary conditions and bending stiffness of the cable. The results show that the boundary conditions, bending stiffness, and linear density of the cable considerably influence the cable force, whereas the inclination angle has a negligible effect on the cable force. The bending stiffness under the two boundary conditions of the cable were both 0.12Eimax, and the calculated value of the cable force under the solid support boundary condition at both ends was consistent with the value of the hydraulic jack tension.

Combined effect of cable sag and bending stiffness on in-plane modal behavior of two horizontal shallow cables connected by an elastic cross-tie

Cross-tie solution is commonly employed to mitigate stay cable vibrations, especially the long ones. Therefore, cable sag could have a sizable impact on the dynamic response of the formed cable network and should not be neglected. Besides, the reduced cable effective length due to the installation of cross-ties and the larger diameter of longer cables would render the cable bending stiffness to have a more pronounced effect on the dynamic behavior of cable networks consisting long cables. Though the mechanism and effectiveness of cable networks have been studied by numerous researchers, majority of these work employed the taut cable assumption in modeling cable behavior. Only a few recent studies investigated the influence of either cable bending stiffness or cable sag separately on the modal response of cable networks, whereas the evaluation on their combined effect has yet to be fulfilled. In the current study, an analytical model of a flexible two-cable network proposed earlier by the authors is refined by including the sag effect. The combined effect of cable sag and bending stiffness on the dynamic behavior of cable networks with different configurations are examined, using the linear theory of shallow cables. Besides, the effect of cross-tie axial stiffness and installation location, as well as the length ratio and the bending stiffness ratio between the consisting cables, on the occurrence and characteristics of modal cross-over are discussed. Results show that while cable bending stiffness would influence all network modes, sag would only influence the symmetric modes. Both of them have a stiffening effect on the affected modes with that of the latter being more sizable, in particular, on the high order modes. Besides, modal cross-over in cable networks is observed to occur not only between two global modes, but also a global mode and a local mode. Enhance the stiffness of a cable network is found to advance the occurrence of modal cross-over at a smaller sag.

In-plane modal behavior of a single shallow flexible cable with an intermediate transverse elastic support

Cross-tie solution has been successfully applied in field to mitigate undesirable bridge stay cable vibrations, especially for cables on modern long-span cable-stayed bridges which typically have long length and relative thick cross section. In evaluating the effectiveness of the cross-tie solution and analyzing dynamic response of the formed cable network, the taut cable assumption has been used in most existing studies. Nevertheless, cable sag and bending stiffness could have non-negligible influence in networks consisting of long cables with relatively large diameter. Since a transverse cross-tie provides an elastic constraint to a vulnerable cable by connecting it with its neighbors, to fully comprehend the impact of cable sag and bending stiffness on the dynamic behavior of shallow-cable networks, it is imperative to scrutinize their effect on the behavior of a shallow cable with a transverse elastic constraint within the span. In the current study, by using the linear theory of shallow cables, an analytical model for the in-plane dynamics of a single flexible low-sag cable with an intermediate transverse elastic support is proposed. The impact of the intermediate support stiffness, location, cable bending stiffness, and sag on the in-plane modal response of the studied cable system are examined. Results show that the presence of an intermediate transverse elastic support breaks the symmetry and anti-symmetry of the original cable mode shapes, which renders the influence of sag extends to all cable modes. Further, the level of such a mode shape distortion would determine whether or not modal cross-over would occur. For any given intermediate support location, it is found that there always exists a specific level of intermediate support stiffness of which the cable sag corresponding to the occurrence of modal cross-over would be independent of cable bending stiffness. By reducing or increasing the intermediate support stiffness to be lower or higher than this specific level, the increase of cable bending stiffness would either delay modal cross-over to occur at a larger cable sag or advance its occurrence at a smaller cable sag. Reducing the intermediate support stiffness would render a local mode to evolve into a global mode, and the presence of cable bending stiffness and sag is found to allow this mode evolution to occur when a more flexible intermediate support is used.

Analytical investigation of the main cable bending stiffness effect on free flexural vibration of suspension bridges with a 3D cable system

This paper proposed a dimensionless continuum model of suspension bridges to comprehensively study the bending stiffness effect of the 3D cable system on the structural free flexural vibration, which has seldom been investigated in previous studies. In this model, coupled differential equations including the cable bending resistance term are derived via the Galerkin method. The equations are then nondimensionalized from a set of key design parameters, including the lateral inclination of hangers. The precision of the model is validated with the finite element (FE) method through an example. Then, parametric comparative studies of the effect under dimensionless parameters are conducted between 3D and 2D cable system suspension bridges. The results show that, the bending stiffness effects of the 3D cable system are more evident on the high order modes and antisymmetric modes of suspension bridges, which is similar with those effects of the 2D cable system. The hanger lateral inclination restrains the effect on the structural frequencies following a relationship of linear negative correlation; when the inclination angle varies from 30 to 60 degrees, the effect ratio of 3D to 2D cable system correspondingly varies from 0.33 to 0.67. Under a given hanger lateral inclination (48.6 degrees) with other identical conditions, the variation trend of the effects on the high order frequencies are similar between 3D and 2D cable systems; the effect value of the 3D cable system is on average 0.625 times that value of the 2D cable system.

Theoretical Investigation on the Impact of Two HDR Dampers on First Modal Damping Ratio of Stay Cable

Stay cables are one of the vital components of a cable-stayed bridge. Due to their flexible nature, stay cables are vulnerable to external excitation and often vibrate with large amplitude under wind action which leads to the fatigue failure of the cables. To suppress such kind of large amplitude vibration by improving the damping ratio of the cable various dampers such as magnetorheological damper, friction damper; oil damper; or high damping rubber (HDR) damper are utilized and gained popularity over time. This paper focuses on improving the damping ratio of stay cables using a combination of two HDR dampers. First, the theoretical model is formulated considering cable bending stiffness to evaluate the damping effect of cable-HDR dampers system. Then, the impact of various design parameters of HDR dampers on cable damping considering the cable stiffness is performed. The comparative analysis of results shows that the considered parameters such as loss factor, spring factor, and installation location of dampers have much effect on the stay cables damping ratio. Finally, the optimal parameters of the two HDR dampers are proposed for damper design.

Effect of the main cable bending stiffness on flexural and torsional vibrations of suspension bridges: Analytical approach

In the design and analysis of suspension bridges, the main cables are conventionally treated as flexible ropes transmitting only tensile loads. Large-span suspension bridges used in new railways require the main cables to be designed with larger cross-sections and higher bending stiffness. The latter can affect the dynamic behavior of the entire bridge. To this end, the analytical expression of the main cable's shape is established in this paper, considering the bending stiffness under the dead loads. According to the D'Alembert principle, the partial differential equations of vibration of the cable in the suspension bridge are derived and solved. Finally, frequencies and mode shapes of the flexural and torsional vibrations of the bridge are calculated considering the main cable bending stiffness. To verify the feasibility of the proposed method, a case study of the Wufengshan Yangtze River Bridge is investigated. The analytical results are compared with the finite element solutions obtained by the ANSYS software, with good agreement achieved. By using the proposed method, the effect of the main cable bending stiffness on the flexural and torsional vibrations of suspension bridges is analyzed. The results show that the cable bending stiffness has a larger effect on flexural vibrations than torsional vibrations. The bridge's frequencies become higher with the increase of cable bending stiffness. Besides, as the order of vibration mode or the ratio of the bending stiffness of cables to the beam increases, the effect of main cable bending stiffness gets greater.

In-plane modal responses of two-cable networks considering cable bending stiffness effect

Cross-tie solution has proven to be effective in mitigating cable vibrations on cable-stayed bridges. In the formed cable network, the consisting cables are divided into shorter segments by cross-ties and thus significantly reduced their effective length. While bending stiffness plays an important role in the behavior of shorter cables, its effect has not been considered in any existing cable network analytical models. To assess the impact of cable bending stiffness on the in-plane dynamic response of cable networks, in the current study, an existing analytical model of a two-cable network is refined by taking into account the cable bending stiffness in the formulation. A set of closed-form solutions are derived to clearly reveal the impact of cable bending stiffness on the modal response of two-cable networks having different configurations. The proposed analytical model is validated by numerical simulations and the modal analysis results are compared with those of the corresponding taut-cable networks. It is found that the cable bending stiffness would influence all the modes of a cable network by increasing their modal frequencies, with the stiffening effect on the high order global and local modes being more considerable. If neglected, an underestimation of network fundamental frequency up to 7.1% could occur. Further, the cable bending stiffness has more effect on local modes than global modes. Some local modes could evolve into global modes. Nevertheless, the modal order is not affected by the presence of cable bending stiffness.

Cable Force Identification Based on Bending Waves in Substructures

A method is proposed to study the dynamic characteristics of cable structures from the perspective of traveling waves based on the modified Timoshenko beam axial tension model. Considering the propagation characteristics of the bending wave in a beam structure, once the frequency response of the three measuring points is measured, the wave component coefficients can be obtained by the least squares method, and then the cable force and bending stiffness can be identified with the aim of minimizing the fitting residual. The accuracy of this method is verified by a numerical simulation experiment of the cable vibration. Compared with the traditional frequency method, this method focuses on the cable force identification of the substructure, so the effect of the shock absorber is invalid. Moreover, the cable force of each position of the cable can be calculated reversely by static analysis with the identified cable force of the substructure, which breaks the concept that the cable force is a single value. Furthermore, the cable force can be identified at each frequency sampling point, reducing the impact of the external disturbance.

Effects of bending stiffness and support excitation of the cable on cable rain-wind induced inclined vibration

The main objective of this paper is to investigate the responses of the inclined cable due to rain-wind induced vibration (RWIV) considering the bending stiffness and support excitation of the cable. The single-degree-of-freedom (SDOF) model is employed to determine the aerodynamic forces. The 3D model of a cable subjected to RWIV is developed using the linear theory of the cable oscillation and the central difference algorithm in which the influences of wind speed change according to the height above the ground, bending stiffness, and support excitation of the cable are considered. The numerical results showed that the cable displacement calculated by considering cable bending stiffness in RWIV is slightly smaller than in the case of neglecting it. And, the cable diameter had a nonlinear relationship with cable displacement, where when both diameter and mass per unit length of cable increase cable displacement will decrease. In addition, the periodic oscillation of cable supports extremely increases the amplitude of RWIV if its frequency is nearby that of the cable.
 Keywords:
 3D model; inclined cable; rain-wind induced vibration; rivulet; analytical model; vibration.

Cable Tension Estimation for Suspen-Dome Structures Based on Numerical Method

Suspen-dome structures have been widely adopted due to their superiority over traditional structures. Cable condition is crucial to the safety of suspen-dome structures, especially the magnitude of cable force. However, thus far, investigations on estimating the tension in this type of structure remain lacking. In this study, double-element method was initially developed to simulate cables, and its accuracy was validated. Tension compensation method was adopted to ensure that the cable force is equal to the designed value. Then, transient analysis was conducted based on an actual suspen-dome structure. The dynamic responses derived based on the overall structures and the simplified models were compared, and the reliability of the simplified models was validated. Finally, parametrical analysis was conducted to investigate the influence of support stiffness and cable-bending stiffness on the fundamental frequency of cables. A cable tension estimation program, which was suitable for suspen-dome structures, was updated. Results indicate that vibration method can maintain good accuracy when used in suspen-dome structures.

Research on characteristic function for cable inverse analysis based on dynamic stiffness theory and its application

Parameter identification of cables in the operation period is a significant issue related to the state of serviceability and safety evaluation of cables. A generalized characteristic function is established based on the dynamic stiffness theory to identify cable parameters in the study. The characteristic function exhibits an analytical form. The function relatively fits actual engineering cables due to comprehensively considering cable bending stiffness, sag, inclination, etc., and thus it is a relatively accurate inverse-analysis model of cable parameters. The variation of bending stiffness and cable force with different geometric parameters and the influence of geometric parameters on the identification accuracy of the two parameters are investigated based on numerical analysis. And a parameter identification method based on the H-I ridges of inverse-analysis characteristic function is proposed. The proposed method identifies cable parameters using the multi-order frequencies of cables simultaneously, and the order of modal frequencies and the fundamental or lower-order frequency are not required in the application of the proposed method. Finally, the results of tension tests on two real cables with lengths of 20 m and 168 m verify the applicability of the inverse-analysis characteristic function and the parameter identification method proposed in the study.

Vibration-based cable condition assessment: A novel application of neural networks

Vibration-based cable tension estimation methods demand complex computations especially when usage of comprehensive cable models is required. Avoiding mathematical calculations, this paper proposes a simple novel framework to estimate the cable tension based on Artificial Neural Networks (ANNs). Employing a comprehensive cable model, a set of data including cable length, cable mass per unit length, cable axial stiffness, cable bending stiffness, cable tension and the corresponding cable natural frequencies is generated for training, validation, and testing of the ANNs. The acquired ANNs are then used to estimate the cable tensions in new Ironton-Russell Bridge and the results are compared against the cable tensions directly measured by lift-off test. It will be shown that for new Ironton-Russell Bridge, using cable length, cable mass per unit length, cable axial stiffness, and first two cable natural frequencies as input features to ANNs, the cable tensions can be accurately estimated.

A coupled finite difference mooring dynamics model for floating offshore wind turbine analysis

This study develops a coupled nonlinear hydrodynamic model of a mooring system consisting of multiple cables for analyses of floating offshore wind turbines (FOWTs). The model is based on a finite difference model of submerged cables which considers cable-seabed interaction, current effect, cable bending and torsional stiffness. The implementation of the model proposes a parallelization scheme for solving the cable responses to improve the computational efficiency. The developed program is then coupled with a spar type FOWT and verified using an experimentally validated open source mooring simulation program. Furthermore, the model is used to study the impact of nonlinear dynamics of the mooring system on FOWT responses in the presence of current. Both static responses of a spar FOWT under current load and dynamic responses of the spar FOWT under wind, wave and current loads are investigated. Responses are compared where varied mooring models are used including the linear model, quasi-static model and nonlinear mooring models without and with current effect on cables considered. The results show that the current effect on cables can have a considerable impact on the restoring effect of the mooring system and hence the spar and cable responses. The current effect on mooring cables needs to be properly considered in the FOWT analysis.

Finite element modeling of temperature load effects on the vibration of local modes in multi-cable structures

Understanding thermal effects on the vibration of local (cable-dominant) modes in multi-cable structures is a complicated task. The main difficulty lies in the modification by temperature change of cable tensions, which are then undetermined. This paper applies a finite element procedure to investigate the effects of thermal loads on the linear dynamics of prestressed self-weighted multi-cable structures. Provided that boundary conditions are carefully handled, the discretization of cables with nonlinear curved beam elements can properly represent the thermoelastic behavior of cables as well as their linearized dynamics. A three-step procedure that aims to replace applied pretension forces with displacement continuity conditions is used. Despite an increase in the computational cost related to beam rotational degrees of freedom, such an approach has several advantages. Nonlinear beam finite elements are usually available in commercial codes. The overall method follows a thermoelastic geometrically non-linear analysis and hereby includes the main sources of non-linearities in multi-cable structures. The effects of cable bending stiffness, which can be significant, are also naturally accounted for. The accuracy of the numerical approach is assessed thanks to an analytical model for the vibration of a single inclined cable under temperature change. Then, the effects of thermal loads are investigated for two cable bridges, highlighting how natural frequencies can be affected by temperature. Although counterintuitive, a reverse relative change of natural frequency may occur for certain local modes. This phenomenon can be explained by two distinct mechanisms, one related to the physics intrinsic to cables and the other related to the thermal deflection of the superstructure. Numerical results show that cables cannot be isolated from the rest of the structure and the importance of modeling the whole structure for a quantitative analysis of temperature effects on the dynamics of cable bridges.