Cable Sag Effect Research Articles

Experimental Investigations on effectiveness of viscous and viscous inertial dampers on reducing the multimodal vibrations of stay cables

The additional damping ratios provided by viscous dampers (VDs) and viscous inertial damper (VID), were tested in the laboratory to estimate the effectiveness of these dampers on mitigating the multimodal vibrations of stay cable. First, two plate eddy current dampers (PECDs), in which the damping coefficients can be conveniently adjusted, were specially designed and manufactured. The mechanical property of these PECDs was obtained by tests and numerical simulations. Second, the natural frequencies of a test stay cable were carefully determined by tests and numerical analysis, and the effect of cable sag was carefully analyzed. Third, a series of tests including 12 test cases were carried out to validate the effectiveness of a single VD, a single VID, and two VDs, in which PECD is a typical VD, an additional mass is attached on the test stay cable near the PECD and the output is similar to that of a VID. The amplitude and frequency of the vibration have little effect on the mechanical properties of PECD. The inherent damping ratios of the first 25 modes are in the range from 0.068% to 0.22%. It appears that low efficiency to simultaneously enhance the damping ratios of the first 25 modes can be found by adding a single VD and a single VID with the installation position of 4.75%l. However, two VDs installing on 2.75%l and 4.75%l has a significant improvement on the damping ratios of the 15th∼25th modes, and should be considered to a reasonable countermeasure to simultaneously reduce the low-order rain-wind induced vibrations and high-order vortex-induced vibration by raising the modal damping ratios.

Optimization of the cable forces of completed cable-stayed bridges with differential evolution method

Long-span cable-stayed bridges have been massively erected worldwide in recent decades, and reasonable cable forces play an essential role in achieving the desired configurations of completed bridges. In the current research, a differential evolution (DE) algorithm is adopted to optimize the cable forces considering the counterweight. An asymmetrical cable-stayed bridge is taken as a case study, and the numerical model of the bridge is established with the OpenSees platform. The cable sag effect and geometric nonlinearity are considered in the numerical model. The weighted total bending energy of the girder and pylons is taken as the objective function of the optimization problem. In addition, the constraints of the cable forces, the range and magnitude of the counterweight, and the bearing reaction forces at the transition and auxiliary piers are considered. The numerical results reveal that the proposed framework is efficient and stable for optimizing the cable forces, and several optimum solutions can be obtained. In addition, the performance of the proposed optimum algorithm is less sensitive to the values of the initial parameters (feasible regions and population size). Moreover, the proposed algorithm automatically determines the control parameters (scaling factor and crossover rate) of the DE method, which makes it easy to use in practical engineering. In addition, the proposed algorithm can be used to optimize the cable forces in the final stage and the construction stages.

A shallow cable with a tuned mass-high damping rubber damper

Tuned mass-high damping rubber damper (TM-HDR-D), as a high-efficiency damper, has great practical feasibility and good application prospect for cable vibration mitigation. To investigate the effect of TM-HDR-D on the cable dynamics, a refined model which considers the cable sag effect is established via complex modal method. Then, based on this model, the influence of cable sag and the stiffness, location, loss factor, and mass of the TM-HDR-D on the cable dynamics is systematically investigated. Meanwhile, the multi-mode damping effect of TM-HDR-D is also analyzed. Results show that TM-HDR-D can significantly increase the additional damping of the system. The cable sag only affects the first-order modal damping, and the effect of TM-HDR-D is significantly affected by its frequency, loss factor, and installation position. Improving the mass of the damper can improve the damping effect. Reasonable selection of damper tuning frequency can achieve multi-order modal damping enhancement.

Qualification of the Ernst formula for modeling the sag effect of super-long stay cables in the long-span railway cable-stayed bridges

It is believed that the cable sag effect becomes more substantial with ever-increasing main span of the railway cable-stayed bridges, which have much heavier stay cables and larger live-to-dead load ratio than the highway cable-stayed bridges. The qualification of the Ernst formula to simulate the sag effect of super-long stay cables should be examined for design use. An explicit secant modulus formula has been proposed to conveniently assess the reliability of the Ernst formula. A finite element model for a railway cable-stayed bridge with the main span of 1316 m is established based on ANSYS software package, in which the stay cable has been consecutively modeled by the Ernst formula, the explicit secant modulus formula, and the multi-straight bar element model. The induced stresses and displacements in the cable, main girder and pylon obtained based on those cable models are compared to demonstrate their validity in dealing with the sag effect of railway cable-stayed bridge. The comparisons show that the relative error of the explicit secant modulus formula is less than 1 % against the benchmark solutions given by the multi-straight bar approach. The relative error of the stretching stiffness of stay cable obtained by the Ernst formula is within 8 %, and that of the displacements and internal forces in the girder and pylon is not more than 5 %. Due to its soften property, the Ernst formula is slightly conservative for structural analysis of the long-span railway cable-stayed bridges.

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.

Longitudinal damage of cable-stayed bridges subjected to near-fault ground motion pulses

This paper investigated the nonlinear seismic performance of an existing cable-stayed bridge longitudinally subjected to a set of simulated near-fault ground motion pulses. An elaborated non-linear finite element model of the bridge was established which particularly considered cable sag effect, material nonlinearity of the tower and the deck. Through non-linear dynamic response analyses, seismic responses of the tower, deck and cables were evaluated at the yield and ultimate state of the structure. In particular, the yield and ultimate state of the structure were defined in the text based on damage levels of the structure and structural integrity requirement. It is revealed that the pulse period (Tp), by determining the relative contribution of multiple modes of the structure, strongly affected the damage process of the structure. As Tp was close to the period of first vertical vibration of the deck, the responses of the deck and cables were largely excited so that the deck might yield prior to yield of the tower, the cables failed prior to the ultimate state of the tower, and the deck suffered most of the damage despite of the yielding of the tower.

Exact dynamic analysis of multi-segment cable systems

Cable-supported structure is a common system widely used in engineering, with the rapid development of the infrastructure and transportation industry, its structural form has become more and more complicated. Multi-segment cable system, which consists of a suspension cable and several lateral supports, such as the cable-damper system, the main cable of a suspension bridge, etc., is one of the important cable support systems. Due to the low structural rigidity and damping, its dynamic problems have always been the focus of engineering. To obtain a more general conclusion, a unified dynamic model considering the effect of cable sag, additional cable force, flexural stiffness, and the support stiffness of lateral components are considered in this paper for the first time. However, the exact analysis of this model is difficult, in view of this, the dynamic stiffness method is employed in this paper to investigate the dynamic characteristic of the multi-segment cable system. The accuracy of the proposed method is verified by comparing with finite element solutions and experimental results. Results show that the position of lateral supports has a strong coupling effect and influence the lower-order mode more significantly than higher-order modes, and the maximum frequency value can be reached by installing the supports equidistantly.

Impact of cable sag on the efficiency of an inertial mass damper in controlling stay cable vibrations

Passive negative stiffness dampers (NSDs) that possess superior energy dissipation abilities, have been proved to be more efficient than commonly adopted passive viscous dampers in controlling stay cable vibrations. Recently, inertial mass dampers (IMDs) have attracted extensive attentions since their properties are similar to NSDs. It has been theoretically predicted that superior supplemental damping can be generated for a taut cable with an IMD. This paper aims to theoretically investigate the impact of the cable sag on the efficiency of an IMD in controlling stay cable vibrations, and experimentally validate superior vibration mitigation performance of the IMD. Both the numerical and asymptotic solutions were obtained for an inclined sag cable with an IMD installed close to the cable end. Based on the asymptotic solution, the cable attainable maximum modal damping ratio and the corresponding optimal damping coefficient of the IMD were derived for a given inertial mass. An electromagnetic IMD (EIMD) with adjustable inertial mass was developed to investigate the effects of inertial mass and cable sag on the vibration mitigation performance of two model cables with different sags through series of first modal free vibration tests. The results show that the sag generally reduces the attainable first modal damping ratio of the cable with a passive viscous damper, while tends to increase the cable maximum attainable modal damping ratio provided by the IMD. The cable sag also decreases the optimum damping coefficient of the IMD when the inertial mass is less than its optimal value. The theoretically predicted first modal damping ratio of the cable with an IMD, taking into account the sag generally, agrees well with that identified from experimental results, while it will be significantly overestimated with a taut-cable model, especially for the cable with large sag.

Nonlinear Analysis of Spatial Cable of Long-Span Cable-Stayed Bridge considering Rigid Connection

To determine the cable sag effect and solve the rigid connection problem at cable ends of a long-span cable-stayed bridge, a two-node spatial catenary cable element with arbitrary rigid arms is developed. Using the finite rotation formula of the space vector and a differential method, the incremental relation between displacement and force at both ends of the rigid arm is given. Then, explicit expression of the tangent stiffness matrix of the element with arbitrary rigid arms is derived based on the catenary equations. Two numerical examples are provided to verify the validity of the new element. A long-span cable-stayed bridge application model is established, and the cables are simulated using three methods. The results show that the rigid end effect has influence on displacements, bending moments and rigidity and should not be ignored. The catenary cable element with arbitrary rigid arms can be used to simulate the geometric nonlinear mechanical behavior of the cables and can well solve the rigid connection problem.

Design formulas for vibration control of sagged cables using passive MR dampers

In this paper, a method for analyzing the damping performance of stay cables incorporating magnetorheological (MR) dampers in the passive control mode is developed taking into account the cable sag and inclination, the damper coefficient, stiffness and mass, and the stiffness of damper support. Both numerical and asymptotic solutions are obtained from complex modal analysis. With the asymptotic solution, analytical formulas that evaluate the equivalent damping ratio of the sagged cable-damper system in consideration of all the above parameters are derived. The main thrust of the present study is to develop an general design formula and a universal curve for the optimal design of MR dampers for adjustable passive control of sagged cables. Two sag-affecting coefficients are derived to reflect the effects of cable sag on the maximum attainable damping ratio and the optimal damper coefficient. For the cable configurations commonly used in cable-stayed bridges, the sag-affecting coefficients are directly expressed in terms of the sag-extensibility parameter to facilitate the control design. A case study on adjustable passive vibration control of the longest cable (536 m) on Stonecutters Bridge is carried out to demonstrate the influence of the sag for the damper design, and to figure out the necessity of adjustability of damper coefficients for achieving maximum damping ratio for different vibration modes.

Design formulas for vibration control of taut cables using passive MR dampers

In this paper, a method for analyzing the damping performance of stay cables incorporating magnetorheological (MR) dampers in the passive control mode is developed taking into account the cable sag and inclination, the damper coefficient, stiffness and mass, and the stiffness of damper support. Both numerical and asymptotic solutions are obtained from complex modal analysis. With the asymptotic solution, analytical formulas that evaluate the equivalent damping ratio of the sagged cable-damper system in consideration of all the above parameters are derived. The main thrust of the present study is to develop an general design formula and a universal curve for the optimal design of MR dampers for adjustable passive control of sagged cables. Two sag-affecting coefficients are derived to reflect the effects of cable sag on the maximum attainable damping ratio and the optimal damper coefficient. For the cable configurations commonly used in cable-stayed bridges, the sag-affecting coefficients are directly expressed in terms of the sag-extensibility parameter to facilitate the control design. A case study on adjustable passive vibration control of the longest cable (536 m) on Stonecutters Bridge is carried out to demonstrate the influence of the sag for the damper design, and to figure out the necessity of adjustability of damper coefficients for achieving maximum damping ratio for different vibration modes.

Ultimate Behavior of Steel Cable-Stayed Bridges During Construction

This paper presents an investigation of the ultimate behavior of steel cable-stayed bridges under construction. In general, cable-stayed bridges are subjected to quite large compressive forces induced by stayed cables, and may become unstable due to these excessive compressive forces, especially while under construction because of the characteristics of the general construction method. To investigate the ultimate behavior of steel cable-stayed bridges under construction, a three-step analysis, consisting of initial shape analysis, construction stage analysis and external load analysis, is proposed and used with considering various geometric and material nonlinearities of the structure, such as the cable-sag effect, beam-column effect of the girder and mast, large displacement effect and gradual yield effect. By performing extensive analytical study, the general ultimate behavior and mode are found and described in detail. Also, the effects of various geometric parameters on the ultimate behavior and load carrying capacity are studied, such as the cable-arrangement types, girder-mast flexural stiffness ratio and sectional area of stay cables.

Multi-point earthquake response of the Bosphorus Bridge to site-specific ground motions

The study presents the earthquake performance of the Bosphorus Bridge under multi-point earthquake excitation considering the spatially varying site-specific earthquake motions. The elaborate FE model of the bridge is firstly established depending on the new considerations of the used FEM software specifications, such as cable-sag effect, rigid link and gap elements. The modal analysis showed that singular modes of the deck and the tower were relatively effective in the dynamic behavior of the bridge due to higher total mass participation mass ratio of 80%. The parameters and requirements to be considered in simulation process are determined to generate the spatially varying site-specific ground motions. Total number of twelve simulated ground motions are defined for the multi-support earthquake analysis (Mp-sup). In order to easily implement multi-point earthquake excitation to the bridge, the practice-oriented procedure is summarized. The results demonstrated that the Mp-sup led to high increase in sectional forces of the critical components of the bridge, especially tower base section and tensile force of the main and back stay cables. A close relationship between the dynamic response and the behavior of the bridge under the Mp-sup was also obtained. Consequently, the outcomes from this study underscored the importance of the utilization of the multi-point earthquake analysis and the necessity of considering specifically generated earthquake motions for suspension bridges.

Dynamic Modeling and Feedback Linearization Control of Wheeled Mobile Cable-Driven Parallel Robot Considering Cable Sag

Cable robots are one of the subcategories of parallel robots used particularly for the applications where the large workspace is required. This type of robots can cover larger workspace compared with serial robots or parallel robots with rigid links. The main issues of cable robots are the cable flexibility due to the need to long cable length as well as inappropriate cable tension distribution in the proximity of the workspace boundaries. By combining a cable robot with a wheeled mobile robot, an unlimited area of workspace can be reached with shorter cable length. Moreover, as the end-effector does not need to come close to its boundary in the mobile platform frame to reach any desired point in the workspace, the tensions are distributed appropriately. The cable sagging in the cable robots is another issue that increases the system error and deviation of the end-effector from the desired trajectory. Therefore, this paper presents the dynamic modeling and control of a cable-suspended parallel robot with a wheeled mobile platform considering the weight of the cable. To this end, the dynamic equations of the wheeled mobile cable robot are obtained using Gibbs–Appel method. Then, the equations related to the weight and sagging of the cable are added to the equations of motion. Control of the system using feedback linearization method is presented. Furthermore, investigating the effect of cable sag on the tension and the end-effector pose of the closed-loop system is demonstrated. Finally, verification is carried out by comparing the simulation and experimental results.

Structural stability of cable-stayed bridges during construction

This paper presents an investigation of the stability characteristics of steel cable-stayed bridges during construction. In general, cable-stayed bridges are subjected to quite large compressive forces induced by stayed cables, and may become unstable during their construction stage, due to the excessive compressive forces induced by added construction loads. To solve the structural instability problems of the bridges under construction, a nonlinear analysis program was developed based on the theory of nonlinear finite element analysis. The complex stability characteristics of cable-stayed bridges during construction were investigated through a series of rigorous geometric nonlinear analyses, including various structural nonlinearities such as cable-sag effect, beam-column effect of girder and mast, large displacement effect, and girder-mast-cable interaction. To consider the construction characteristics of the cable-stayed bridges, a three-step analysis method is proposed, and used in the present study. In addition, the effects of various cable-arrangement types and girder-mast stiffness ratios on the stability characteristics were extensively investigated. Typical buckling modes can be classified into two categories, depending on the location of critical member or members.

Nonlinear aerostatic stability analysis of Hutong cable-stayed rail-cum-road bridge

To investigate the nonlinear aerostatic stability of the Hutong cable-stayed rail-cum-road bridge with ultra-kilometer main span, a FEM bridge model is established. The tri-component wind loads and geometric nonlinearity are taken into consideration and discussed for the influence of nonlinear parameters and factors on bridge resistant capacity of aerostatic instability. The results show that the effect of initial wind attack-angle is significant for the aerostatic stability analysis of the bridge. The geometric nonlinearities of the bridge are of considerable importance in the analysis, especially the effect of cable sag. The instable mechanism of the Hutong Bridge with a steel truss girder is the spatial combination of vertical bending and torsion with large lateral bending displacement. The design wind velocity is much lower than the static instability wind velocity, and the structural aerostatic resistance capacity can meet the requirement.

Ultimate behavior of steel cable-stayed bridges - II. Parametric study -

This paper presents the characteristics of the ultimate behavior of steel cable-stayed bridges through considering various geometric parameters. Steel cable-stayed bridges show complex ultimate behavior, because of their geometric characteristics and various nonlinearities. In this study, the patterns of the ultimate behavior of steel-cable stayed bridges under the critical live load case are classified. In addition, the effects of various geometric parameters on the ultimate behavior, such as cable-arrangement type, height of the girder and mast, and area of the stay cables, are studied. For rational analytical research, the analysis method suggested in the previous paper, Ultimate behavior of steel cable-stayed bridges-I. Rational ultimate analysis method (Kim et al., 2016), is mainly used. Using the analysis method, the main geometric and material nonlinearities, such as the cable sag effect, beam-column effect of the girder and mast, large-displacement effect, girder-mast-cable interaction, and gradual yield effect of steel members, are reflected and considered in the analytical research. After the analytical study, the characteristics of the change of ultimate mode and load carrying capacity are investigated, with respect to the change of various geometric parameters.

Ultimate behavior of steel cable-stayed bridges - I. Rational ultimate analysis method -

This paper presents an investigation on the ultimate behavior of steel cable-stayed bridges. In general, various nonlinear factors affect the global behavior of cable-stayed bridges, such as material nonlinearities, cable-sag effect, beam-column effect, large displacement effect, and girder-mast-cable interaction. These effects also affect the ultimate behavior of cable-stayed bridges. Therefore, a rational analysis method should be performed to study the ultimate behavior of cable-stayed bridges. Because of various nonlinearities, the analysis should be based on the theory of nonlinear finite element analysis. Moreover, rational ultimate analysis can reflect characteristics of the design and construction of cable-stayed bridges. In this study, a rational ultimate analysis method for steel cable-stayed bridges is developed and proposed based on the theory of nonlinear finite element analysis. A two-step analysis method is proposed and used in this study. Through this analysis scheme, the structural state under dead load is considered before the live load analysis. The developed program is used to study the ultimate behavior of steel cable-stayed bridges under vertically applied live load cases. Analytical study is used to investigate governing ultimate modes under the considered live load cases. By comparing the analysis results under each live load case, the critical load case is determined. The effects of geometric nonlinearities and material nonlinearities on the ultimate behavior of steel cable-stayed bridges are studied by performing geometric nonlinear analysis, as well as ultimate analysis.

Force Distribution With Pose-Dependent Force Boundaries for Redundantly Actuated Cable-Driven Parallel Robots

This paper addresses the force distribution of redundantly actuated cable-driven parallel robots (CDPRs). A new and efficient method is proposed for the determination of the lower-boundary of cable forces, including the pose-dependent lower-boundaries. In addition, the effect of cable sag is considered in the calculation of the force distribution to improve the computational accuracy. Simulations are made on a 6DOF CDPR driven by eight cables to demonstrate the validity of the proposed method. Results indicate that the pose-dependent lower-boundary method is more efficient than the fixed lower-boundary method in terms of minimizing the motor size and reducing energy consumption.

THE INFLUENCE OF CABLE SAG ON THE DYNAMIC BEHAVIOUR OF CABLE-STAYED SUSPENSION BRIDGE WITH VARIABLE SUSPENSION TO MAIN SPAN RATIO

The demand of long span bridge is increasing with infrastructure magnification. To achieve maximum central span in bridges is a motivating rational challenge. The bridge with more central span can be achieved using high strength materials and innovative forms of the bridges. The cable-stayed bridge has better structural stiffness and suspension bridge has ability to offer longer span thus combination of above two structural systems could achieve very long span cable-stayed suspension hybrid bridge. To distinguish behaviour and check the feasibility of this innovative form of hybrid bridge, 1400m central span and 312m side span cable-stayed suspension hybrid bridge is considered for analysis. The suspension portion length in central span is also playing important role in behaviour of the entire bridge. Bridge behaviour is presented for variable length of suspension portion in form of suspension portion to main span ratio. The main cable sag in central span is playing important role on behavior of the entire bridge. It directly influences the inclination angles of the main cables, the height of pylon and thus forces in pylon. The axial force in main cable is directly depending on the sag of main cable. The effects of main cable sag is studied by considering dimensionless parameter as sag to main span ratio as 1/9, 1/10 and 1/11. Paper also discusses results of nonlinear static analysis and modal analysis carried out using SAP2000 v14.0.0. The time period of bridge is used to present the behavior of bridge.