Simulations on cross-ties for vibration control of long span cablestayed

Transcription

1 ibe 00 Nottingham University Press Proeedings of the International Conferene on Computing in Civil and Building Engineering W Tizani (Editor) Simulations on ross-ties for vibration ontrol of long span ablestayed bridges Limin Sun, Hongwei Huang, Yonglong He and Yagang Zhou State Key Laboratory of Disaster Redution in Civil Engineering, Tongji University, Shanghai, China Abstrat Wind or wind/rain indued vibration of long ables is an issue seriously onerned for able supported strutures, espeially long span able-stayed bridges. Using ross-ties and dampers is one of the onsiderable ountermeasures for vibration ontrol. However, the mehanism and design method of ross-ties and dampers have not been established yet. This paper proposed to use the ombination of ross-ties and dampers installed between the ross ties and the stay ables to inrease both the natural frequenies and strutural damping of the ables for suppressing vibrations. The numerial simulations on the ross-tied-stay ables system were arried out to evaluate the effetiveness of ross-ties. The FEM model is a half of one able plane in the mid span of a long able stay bridge with the mid span 400 m, and inludes 4 flexible type ross-ties with dampers. The simulation results showed that the effetiveness of able dampers was improved when the stay ables were onneted eah other by the ross-ties; and the ross-tie dampers ould effetively inrease the able damping. The damping evaluation equations for a taut able with an arbitrary number of Kelvin dampers were derived. The formula was expeted to be able to evaluate the damping of ross-ties-stay able system in the further study. Keywords: long stay able, vibration mitigation, ross-ties, dampers, simulation Introdution Cable-stayed bridges have beome very popular over the last forty years beause of their eonomy, strutural effiieny and aesthetis. Stay ables are ritial strutural omponents in able-stayed bridges. Owing to their large flexibility, small mass and low inherent damping, and stay ables are suseptible to large-amplitude vibration aused by wind, rain and support motion. Continued osillations may result in fatigue and orrosion of strands, reduing the life of ables (Watson and Stafford, 988). Using ross ties is one of the onsiderable ountermeasures for vibration mitigation of long stay ables; however, the analysis method for design of ross ties has not been established yet. One mehanism of the ross ties is to inrease the frequenies of stay ables, another mehanism is to inrease the damping. Ehsan and Sanlan (990) studied the redistribution of osillation energy in the able system, whih onsists of stay ables onneted by ross-ties, with the omponent mode synthesis approah and showed the effiieny of ross-ties for the vibration mitigation of stay ables. Caraoglia and Jones (005) simulated the frequenies harateristis of the able system by a general analytial proedure. However, the above investigations did not take into onsideration the damping of the able system. Sun et al (004) onduted a full-sale experiment on the vibration mitigation of

2 stay able and showed that the damping of stay able is very low. Yamaguhi and Nagahawatta (995) investigated the strutural damping of able system by applying the energy method based on the measured damping in a test of single able and showed that there is more or less a damping-inrease when using ross ties. Bosh and Park (005) simulated the performane of stay ables with ross ties and able dampers and found that the benefits of a ombined use of able dampers and ross ties are not neessarily the sum of benefits from the dampers and ross ties separately. Zhou and Sun (006, 007) presented a three-element Maxwell model and a Kelvin model with arbitrary number of dampers for the analysis of onentrated damping in a able system, and as well as the analysis of the whole able system. Sun et al (007a, b) onduted experiments on a saled model of able-ross tie system and investigated the effets of tensioning method, initial tension of ross ties, type and stiffness of ross ties on the in-plane vibration mitigation of the able system. In the present study, numerial simulations on the ross-tied-stay ables system were arried out to evaluate the effetiveness of ross-ties with dampers. The analytial model is a half of one able plane in the mid span of a long able stay bridge with the mid span 400 m, whih inludes 38 stay ables and their ross-ties. The effetiveness of ross-ties and their dampers at the onnetions with stay ables were disussed. An analytial model was also derived for optimal design of parameters of ross-ties. Numerial simulations. Cable-stayed bridge with 400m mid span Numerial simulations on the ross-tied-stay ables system were arried out using a FEM model of half of one able plane in the mid span of a long able stay bridge with the mid span 400 m (Figure ). The bridge is a proposed design sheme, whih inludes 38 stay ables and the longest able is 745m. The able vibration ontrol ountermeasure by using able dampers was investigated, and it was found that even the installation height of dampers is 5.33m the maximum available damping for the ables is in a range of (logarithmi derement). Considering a damper effiient fator 0.5, the able damping an not satisfy the required speifiation of Therefore the installation of ross-ties is neessary. Figure, Cable-stayed bridge with 400m mid span. Effet of ross-tie Zhou et al (006 and 007) have disussed the effetiveness of number, distribution, tensioning method and initial tension of ross-ties, then two types of ross-ties, namely the stiff and flexible types, were suggested. In this paper, four flexible type ross-ties with dampers were adopted (Figure ). In the FEM model, it is assumed that the ross ties possess enough initial stress so that it will not loosen when the stay ables vibrate. The mode shapes of st, nd, 3 rd, 0 th, 0 th, and 30 th modes are shown in Figure 3. It is found that the lower modes are global ones but the higher modes are more like

3 loal ones. With the installation of the ross-ties, the fundamental natural frequeny of the stay ableross-ties system is inreased from 0.53Hz ( st freq. of the longest stay able) to 0.60Hz. The effet of ross-ties on the able dampers installed near to the anhorages of stay ables (girder side) was first studied, and in this ase there is no damper installed at between ross-ties and stay ables. It is found that the effetiveness of able dampers is hanged beause of a able network is formed after onneting the stay ables eah other by ross-ties. Figure 4 shows that the damping for the three lowest modes, the logarithmi derement an be inreased to 0.09, and 0.067, respetively. The simulation also shows that the able damping an be further improved by inreasing the damping oeffiient of able dampers. Tower F Cross tie damper model F F3 F4 C Figure. FEM model of ross-tied-stay ables system C38 (a) st mode (b) nd mode () 3 rd mode (d) 0 th mode (e) 0 th mode (f) 30 th mode Figure 3, Modes of stay ables-ross-ties system.3 Effet of ross-tie damper The ross-tie dampers that installed between ross-ties and stay ables are shown in Figure. The damping oeffiient of eah damper assumed to be the same, and there was no able damper installed near the anhorages of stay ables. The ross-tie dampers are installed on the ross-tie F, and 5 ases are onsidered in the simulation (Table ). Figure 5 plots the logarithmi derement of able system, δ, orresponding to the st mode, and it an be seen that δ inreases with inreasing number of dampers. Figure 6 shows δ of the first 3 modes when ross-tie dampers (Case 5) were installed, and it is observed that the optimal value of δ for every mode of vibration is about the same, around 0.4, whih an satisfy the engineering design requirement. Figure 7 plots the logarithmi derement of able system, δ, orresponding to the st mode of vibration, when installing dampers at the ross-tie F, F, F3 and F4 respetively. It an be seen that

4 the dampers are more effetive when installed at the middle two ross-ties F or F3, as the modal displaements are larger at this two positions for this partiular mode. able damping (log derement). δ 0.5 first-order mode seond-order mode 0.0 3third-order mode damping oef. C/C opt Figure 4, Effet of ross-tie on able damper able damping (log derement). δ Case 工况 Case 工况 Case 工况 3 Case 工况 4 Case 工况 damping oef. of ross tie damper (KN s/m) Figure 5, Effet of ross-tie damper ( st mode) Table. Cases in simulation on ross-tie damper effetiveness Case number Cross-tie damper positions Case C37 Case C37,C35 Case 3 C37,C35,C33 Case 4 C37,C35,C33,C3,C9,C7,C5 Case 5 C37,C35,C33,C3,C9,C7,C5,C3,C,C9,C7 able damping (log derement). δ mode 一阶 mode 二阶 mode 三阶 damping oef. of ross tie damper (KN s/m) Figure 6. Logarithmi derement of st, nd and 3 rd modes (Case 5) abledamping (log derement). δ Analytial model of able-damper system 工况 Case F 工况 Case F 工况 Case F3 3 工况 Case F damping oef. of ross tie damper (KN s/m) Figure 7. Logarithmi derement of st mode (Cross-ties damper installed at F, F, F3 and F4 respetively) In Setion, the effetiveness of the ross-ties with dampers was disussed based on FEM analysis. Although some trends were found, it is still expeted to have a losed form to optimize the parameters of the ross-ties theoretially. In this setion, an analytial method was proposed to evaluate the damping of a stay able with an arbitrary number of Kelvin dampers whih onsist of visous dampers and linear springs. The inherent stiffness of ross-ties and the dampers loated at the position of ross-ties were represented as Kelvin dampers.

5 Figure 8, Taut able with an arbitrary number of Kelvin dampers The problem under onsideration is illustrated in Figure 8, where L is the length of the able, m is the mass per unit length, and T is the able fore. The loation of Kelvin dampers are denoted as X, X, L Xk, and the nondimensional length of eah orresponding able segments are denoted as l, l, L l k +. The damping oeffiient and the stiffness of eah Kelvin damper are denoted as,, L k and k, k L,, kk respetively. Assuming that the able fore is unhanged under linear osillations, and the bending stiffness and inherent damping are negligible, the linear osillations of eah segment of string an be desribed by the following partial differential equation, Y( X, t) Y( X, t),,,, K,, 0, () X t i i T = m Xi < X < Xi i= k+ X0 = Xk+ = L where Yi ( X, t ) is the transverse displaement from stati position. The boundary onditions of zero displaement at the ends of the string must be satisfied as follows, Y(0, t) = 0; Y( L, t) = 0 () and the displaement of the string must be ontinuous aross eah string segment, that is, Y( X, t) = Y ( X +, t), i =,, K, k. (3) i i i+ i At the loation of the joint there is a disontinuity in the slope of the string, providing a transverse fore mathing the fore in the damper and the spring, as given in the following equation, Yi+ ( X, t) Yi( X, t) Yi+ ( X, t) H = i + ky i i+ ( Xi+, t), i=,, K, k, (4) X X X i+ X t i Xi+ After introduing nondimensional length and time X Y x=, y =, τ = ω0t (5) L L where ω0 = π H / m/ L, the nondimensional equations of equations (), (3) and (4) are as follows, yi( x, τ) mlω0 yi( x, τ) =, xi < x< xi, i=,, L, k+ (6) x T τ y ( x, t) = y ( x +, t) (7) i i i+ i yi+ ( x, t) yi( x, t) yi+ ( x, τ ) T = L i ω0 + kilyi+ ( x, τ ) x x x i+ x τ i xi+ The transverse displaement of free vibrations of the string an be expressed in the following form, y( xt, ) = wxe ( ) jωτ (9) where ω is a omplex natural eigenfrequeny and wx ( ) is the orresponding omplex mode shape. (8)

6 Substituting the assumed solution (9) into equation (6) yields the following ordinary differential equation, wi ( x) x + λ wi x = xi < x< xi i= k+ ( ) 0,,,, L, (0) where the wave number λ has been introdued as λ = πω. Define the following nondimensional damping oeffiient η and stiffness oeffiient μ, η = ; k( l+ l) L μ = () Tm T For able system with one damper installed at l and one ross-tie at l, the omplex frequeny solution an be obtained as follows, where ( jη+ D)sin ( λl ) tan( λ) = + ( jη + D)sin( λl )os( λl ) () ' μ/ λl k + ot ( λl ) D = (3) ' ' ' + ot ( λlk) + μ/ λlk ot( λlk) ot( λl) and l = l, l = l + l, l = l + l, l = l. (4) ' ' k 3 k 3 Aording to the relation between eigenfrequeny ω n and modal damping ratio ζ n, i.e. ωn = ωn ( ζn + iζn), and onsidering λ = πω, the damping harateristis of the able-damperspring system ould be omputed by iteration. 4 Conluding remarks The ountermeasures for suppressing able vibration of a trial-designed able-stayed bridge with a enter span 400m were investigated. It was found that using only able dampers is hard to satisfy the required damping, therefore the ross-ties have to be adopted. The flexible type ross-ties with dampers were designed for the bridge and FEM analyses were arried out to evaluate their effetiveness. The simulations showed that: ) the effetiveness of able dampers was improved when the stay ables were onneted eah other by the ross-ties; and ) the ross-tie dampers ould effetively inrease the able damping. The alloation and damping oeffiients of the ross-tie dampers need to be optimized. The damping evaluation equations for a taut able with an arbitrary number of Kelvin dampers were derived. The formula was expeted to be able to evaluate the damping of ross-ties-stay able system in the further study. Aknowledgements The authors would like to aknowledge the support of the National Nature Siene Foundation of China (Grant No.: ) and National High-teh R&D Program (863 Program Grant No.: 006AAZ0).

7 Referenes H.R. BOSCH, S.W. PARK, 005, Effetiveness of external dampers and rossties in mitigation of stay able vibrations, Sixth International Symposium on Cable Dynamis. L. CARACOGLIA, N.P. JONES, 005, In-plane dynami behavior of able networks. Part : formulation and basi solutions, Journal of Sound and Vibration, 79(3-5): p F. EHSAN, R.H. SCANLAN, 990, Damping stay ables with ties, 5th US-Japan Bridge Workshop, p L.M. SUN, C. SHI, H.J. ZHOU, 004, A full-sale experiment on vibration mitigation of stay able, IABSE Symposium Shanghai 004. Shanghai: China Communiations Press. L.M. SUN, Y.G. ZHOU, 007a, Experimental study on vibration mitigation of long stay ables using ross ties, IABSE007 L.M. SUN, Y.G. ZHOU, H.W. HUANG, 007b, "Experiment and damping evaluation on stay ables onneted by ross ties", Seventh International Symposium on Cable Dynamis. S.C. WATSON, D. STAFFORD, 988, Cables in Trouble, Civil Engineering, 58(4): p H. YAMAGUCHI, H.D. NAGAHAWATTA, 995, Damping effets of able ross ties in able-stayed bridges, Journal of Wind Engineering and Industrial Aerodynamis, 54: p Y.G. ZHOU, L.M. SUN, 006, Complex modal analysis of a taut able with three-element Maxwell damper, Tongji Daxue Xuebao/Journal of Tongji University, 34(): p. 7-. Y.G. ZHOU, L.M. SUN, 007, "Kelvin model for analysis of a stay able with ross ties", Seventh International Symposium on Cable Dynamis.