Motion-Induced and Parametric Excitations of Stay Cables: A Case Study

Transcription

1 Motion-Induced and Paraetric Excitations of Stay Cables: A Case Study Authors: Stoyan Stoyanoff, Rowan Willias Davies and Irwin, Inc., 09 bd. de Broont, Broont, Quebec, JL K7, Stoyan.Stoyanoff@rwdi.co Brad A. Pridha, Rowan Willias Davies and Irwin, Inc., 650 Woodlawn Road West, Guelph, Ontario, NK B8, Brad.Pridha@rwdi.co ABSTRACT This paper presents an overview of existing theoretical ethods for the analysis of otion-induced and paraetric excitations of stay-cables. Application of the ethods is illustrated with a practical exaple of assessent of stay vibrations due to wind buffeting on the proposed Ironton-Russell Bridge, in Ohio. The results obtained fro this analysis are copared against direct nuerical (finite eleent) siulations. The selected schee for cable vibration itigation is presented, together with a discussion of its effectiveness. INTRODUCTION Aong the any challenges facing wind engineers in the design of cable-supported bridges, is the assessent and itigation of stay-cable vibrations. Stay-cables vibrate due to their inherently low daping, lightness, and flexibility, when excited by direct wind loads (buffeting and vortex shedding), as well as anchorage otions due to the bridge s response to operational loads (wind, traffic, pedestrians etc.). Full-scale observations of cable-stayed bridge vibrations confir that indirect, or otion-induced vibrations can result in significant cable displaceent aplitudes [Caetano, Macdonald]. These otions are a consequence of siilarity between global odal frequencies of the bridge, and local odal frequencies of the cables. In the case of a priary resonance, where there is a atching (or near-atching) of a global odal frequency of the bridge with the fundaental odal frequency of the cable, significant sway (out-of-plane) otions of the cable are possible. A second, and potentially ore critical case, known as paraetric resonance, can occur when a bridge odal frequency is twice the stay frequency. Theoretical results indicate that this can lead to large aplitude instabilities [Lilien and Pinto Da Costa], although this phenoenon has not been conclusively identified on any existing bridges. Both priary and paraetric resonance can result in displaceents beyond those generally peritted for noral bridge operation, since they can cause probles associated with structural integrity and fatigue, or invoke discofort in users of the bridge. It is therefore iportant that a detailed assessent of otion-induced vibrations

2 is conducted when designing a bridge, and there, if necessary, itigation easures be ipleented to prevent excessive cable otions. In this paper, theoretical ethods for the prediction of otion-induced stay-cable vibrations due to wind loading are reviewed. Application of these ethods is illustrated with data fro the actual cable study of the proposed Ironton-Russell Bridge in Ohio. A coparison between the analytical forulation and direct nuerical siulations of the stay-cable otions is presented. The paper presents the cable vibration itigation schee selected for the Ironton-Russell Bridge and concludes with a discussion of its effectiveness at suppressing cable vibrations. MOTION-INDUCED AND PARAMETRIC EXCITATION OF CABLES Motion-induced cable vibrations, and cable-bridge interaction phenoena are highly coplex, and as a result, generally not well understood. Recently there have been a nuber of notable developents in the literature [Macdonald, Georgakis, Caetano, Gattulli], focusing on explanation of soe of the phenoena. However, the results fro these resent studies have yet to lend theselves to a widespread practical ipleentation. This is due, in part, to the coputational requireent for solution of the non-linear equation of otion, which typically restricts the analysis to a single cable. In practice, siplifying assuptions to the non-linear cable behavior can be ade, resulting in ethods that are ore easily applied to the assessent of ultiple cables during the design [SETRA, Virolgeux]. In this paper, these siplified ethods are adopted for the analysis of cables on the Ironton-Russell Bridge. Details of the selected ethodology are provided below. Analysis Paraeters Figure illustrates the geoetry of a oving stay-cable of length l, subjected to average tension T, anchorage otions δ, and an inclination angle Θ with respect to the deck. In this paper, only the response of the cable to wind is considered, however, the FIGURE GEOMETRY OF THE MOTION-INDUCED EXCITATION OF A CABLE STAY

3 results could be extended to other excitations if estiates of the overall bridge responses are available. Associated with the th deck/tower ode is a set of anchorage odal displaceents (estiated fro a finite eleent odel), which are transfored into a coponent along the chord axis (X ), and a coponent perpendicular to the chord axis, in the plane of cable vibration (Y ). These coponents are coputed using the noralized ode shape coordinates for each anchorage (δx, δy, δz, taken fro the odal analysis), and the generalized aplitude coordinates coputed fro the buffeting analysis (a ), as follows: ( t b t b = ), and, ( ) X x x a Y = y y a, () where the values x and y are the ode shape coponents along the chord, and perpendicular to the chord in the plane of cable otions, respectively. These values are coputed via projections onto the chord axis, and onto the plane perpendicular to the chord axis, respectively. The superscripts t and b represent top (tower level) and botto (deck level). The quantities X and Y are the net end displaceents of the cable, scaled according to the generalized coordinate a, that could be either estiated theoretically via wind buffeting analysis, directly easured fro an aeroelastic odel test, or recorded fro an existing bridge. A key paraeter for the quantification of the cable response is the ratio of the th bridge (deck/tower) frequency (Ω ) to the fundaental (lowest) odal frequency of the cable (ω, obtained fro the linear free-vibration theory): r = Ω, () ω Motion-induced and paraetric excitation aplitudes are ost severe when the bridge odal frequency is equal to, or twice, the cable s fundaental frequency (i.e., r =, or r = ). Since there is an uncertainty regarding the true values of the estiated frequencies, judgent should be applied in practice when coputing (). The authors are proposing as a conservative approach, to round values of r close to.0 (or.0) within a ± 0% deviation. This is considered consistent with the expected accuracy of the frequency predictions. The total daping of a stay (ξ T ) coprises aerodynaic (ξ A ) and structural daping (ξ S = c/µω ): βρudcd + ξt = ξa + ξs = µω c, (3) where ρ is the density of air, U is the ean wind speed for the analysis (corresponding to the value used for estiation of the generalized coordinates a ), D is the cable diaeter, C D is the drag coefficient, c is the structural viscous daping constant for the cable, and β is a constant equal to 0.5 for vertical (in-plane) otion, and equal to for sway (out-of-

4 plane) otion. Full-scale easureents of stay cable properties indicated very low structural daping values, ξ S 0.05% [Stoyanoff et al] which was retained for the current study. Motion-Induced Excitations due to Vertical and Lateral Anchorage Displaceents In this paper, the ethodology presented by Virlogeux is used for the estiation of resonant cable responses. A siilar ethod was proposed by SETRA, however, based on direct nuerical coparisons by the authors, it was deterined that Virlogeux s forulation provides ore realistic estiates (particularly for the case of shorter stays). In this case, a haronic excitation function of the following for is assued for the n th cable: ( ) y () t = Y cos Ω t. (4) b n Overlooking variations in tension and longitudinal displaceents, the axiu response for each ode is given by: r n rn Ak, n = Yn Hξ k, 3 k T,, (5) π k where, HξT, rn, = k k r k ξ r n T, k n + k, (6) is the echanical adittance function. It is clear fro (6) that the displaceent is a axiu when r n is unity. The total response of the n th cable for the th bridge ode is given by the suation: A n = K A, k = k n, (7) where K is the nuber of cable odes considered (K = 0 in this study) The total response of the n th cable for all M bridge (deck/tower) odes considered in the analysis is then calculated as: A n = M A = n. (8)

5 Paraetric Excitation by Along-the-Cable-Chord Anchorage Displaceents Considering now an along-the cable-chord haronic excitation of the cable of the following for: ( ) x () t = X cos Ω t, (9) n and recognizing that the variation in cable tension T can no longer be neglected, results in the non-linear Mathieu-Hill equation [Nayfeh and Mook] governing the transverse response (y(t)): ESnX ESng π ESn &&() yt + ωξ T yt &( ) + ω + cos ( t) y() t cos ( ) y () t Ω Tl + 3π Θ + ω Tl 4 T ESn Xg = cos π T l ( Θ) cos( Ω t), (0) where S n is the cross-sectional area of the n th cable, and E is Young s Modulus. The second coefficient of y(t) is the so-called paraetric excitation ter, due to the presence of the input excitation function. Two zones of instability govern the response of a cable under this excitation regie. Zone corresponds to r n =, and Zone corresponds to r n =. The liit aplitudes for each region are given by [SETRA]: A Zone n = ω l rn ESnX + Tl 3π ESn ω 4 T ( ξt nrn ), and, () A Zone n ( ) ESnXgcos Θ = ESnX Tl rn π ω Tl. () APPLICATION TO THE PROPOSED IRONTON-RUSSELL BRIDGE The ethods presented in the previous sections were applied to the stay cables of the proposed Ironton-Russell Bridge over the Ohio River. This bridge consists of a single tower and a two-deck with 34 back span, and 36 ain span stays. The bridge is approxiately syetric about the center of the deck. The length of the longest stay is approxiately 960 ft, and estiated fundaental stay frequencies vary between 0.44 Hz and.6 Hz.

6 Early in the design, a preliinary assessent of the cables indicated the possibility for otion-induced and paraetric excitations, due to the proxiity of several deck/tower (global) odal frequencies, to that of the stay cables (Figure ). x st Cable frequency r = r = st Cable frequency Tower Motions Longitudinal Lateral Deck Motions Lateral Torsion Vertical Frequency (Hz) FIGURE DECK/TOWER AND STAY CABLE MODAL FREQUENCY COMPARSION IRONTON-RUSSELL BRIDGE. As indicated in Figure, several deck/tower odes lay in close proxiity to once, and twice- the stay frequencies (i.e., r n and r n ). A particular concern during the assessent was the frequency proxiity of bridge ode (0.447 Hz lateral tower ode) to several of the frequencies of the longest stays. Since this was a lateral tower ode, it was recognized that itigation could not be accoplished through the use of crossties, and, if necessary, a suppleentary daper solution would be required. Paraetric Excitation Assessent Figure 3 is a graphical illustration of the assessent of the potential for paraetric instabilities to develop in cables R/L07 of the bridge, during the design wind speed stor of 69 ph. The reduced aplitude of the cable (a = ESX /Tl) is plotted for 30- deck/tower odal anchorage displaceents (corresponding to the first 30 odes of the deck/tower), together with the instability regions coputed using the total daping (ξ T ) for the cable. Zone is uch broader than Zone, and requires saller reduced aplitudes to be reached. However, the failure of any of the reduced aplitudes to reach the instability regions indicates that paraetric excitation instability is unlikely to occur for this cable.

7 .5 Zone Frequency Ratio.5 Cable Data Zone Reduced Aplitude - a FIGURE 3 GRAPHICAL ILLUSTRATION OF PARAMETRIC INSTABILITY ASSESSMENT FOR CABLES R/L07. The assessent was conducted for all 35 cables on one of the cable planes, and it was deterined that paraetric instabilities are unlikely to occur on the Ironton-Russell Bridge. It should be noted that for the bridge odes higher that about 0.7 Hz, the predicted buffeting responses were very low. Motion-Induced Stay Vibration Aplitudes The otion-induced response aplitudes of each cable were estiated for the design wind speed using (6) through (9). The daping for each cable was estiated using (4), the value of β was selected according to the type of otion induced in the stay, by the deck/tower (i.e., in-plane or out-of-plane). Figure 4 plots the predicted axiu displaceent aplitude of each cable at both the design wind speed of 69 ph, and the service wind speed 46 ph. The values shown are with respect to the equilibriu position of the cable (i.e., ultiply by two to arrive at peak-to-peak aplitudes). Also shown on the plot is the acceptable aplitude criterion applied to each cable, which was selected as 0.5D, or half of the cable diaeter (D peak-to-peak), at the service wind speed. The aplitude was selected based on concerns for huan cofort on the bridge, and has been previously adopted on other bridge projects [Bosch]. Except for six of the shortest cables, the peak aplitudes exceed the recoended aplitude criterion for both the design and service wind events. Table lists the axiu odal contributions (A n ) to the total response of cables R/L0, R/L0, R/L33, R/L34, and R/L35, for the design wind speed. Modes (lateral tower otion) and 3 (vertical deck otion) of the deck/tower were found to be the priary contributors to the response of these cables.

8 5 Design Wind Speed - 69 ph Service Wind Speed - 46 ph Aplitude Criterion - 0.5D Response Aplitude (ft) Cable FIGURE 4 PREDICTED MAXIMUM CABLE REPONSE AMPLITUDES FROM LATERAL ANCHORAGE DISPLACEMENTS. Cable Peak total response (ft) Maxiu odal response (ft) Bridge ode causing ax response R/L st Lateral tower ode R/L0 3.6 R/L R/L R/L st Lateral tower ode 3 nd vertical ode 3 nd vertical ode st Lateral tower ode 3 nd vertical ode st Lateral tower ode 3 nd vertical ode st Lateral tower ode TABLE PREDICTED MAXIMUM MODAL RESPONSES OF TEN OF THE LONGEST CABLES ON THE BRIDGE (69 MPH WINDS). Coparison with Direct Nuerical Siulations A finite eleent odel of cable R35 was developed using the SAP000 Non-linear software, and copared with the analytical results. The tension of the cable was siulated using initial strains, and the non-linear geoetric stiffness atrix was evaluated for estiation of eigenvalues. A sinusoidal excitation function with a frequency of 0.44 Hz was used as input to the anchorages. The siulation predicted a peak cable displaceent of.4 ft, which was alost exactly that calculated using the analytical ethod (see Table ). The nuerical approach was retained for further analysis of vibration itigation scheas.

9 MITIGATION OF STAY CABLE VIBRATIONS Based on the otion-induced and paraetric excitation vibration assessents, it was proposed a itigation schee consisting of both cable crossties, and external lateral daping devices, required to reduce the anticipated cable vibrations to the required level of 0.5D, at the service wind speed 46 ph. Cross-Tie Syste The accepted crosstie schea will increase vertical stiffness of the back span and ain span cable fans, connecting all cables with the deck. Pretension of the ties was designed to sustain the axiu cable dynaic displaceents, avoiding any slackening, which could result in failures of the ties during extree wind events. A finite eleent odel of the ain span and back span cable fans was developed using the SAP000 Non-linear software. Figure 5 shows the first in-plane ode shapes for the tied back span and ain span cable arrays. Table lists a coparison between the in-plane and out-of-plane fundaental odal frequencies of the un-tied and tied cable arrays. st in-plane ode back span:.5 Hz st in-plane ode ain span:. Hz FIGURE 5 FIRST IN-PLANE MODES FOR THE BACK SPAN AND MAIN SPAN TIED CABLE ARRAYS Main Span Array Back Span Array Mode Un-tied Tied Relative Un-tied Tied Relative frequency frequency variation frequency frequency variation (Hz) (Hz) (%) (Hz) (Hz) (%) First out-of-plane First in-plane TABLE COMPARISON BEWTEEN UN-TIED AND TIED CABLE ARRAY FUNDAMENTAL FREQUENCIES. The installation of crossties can have a significant affect on the in-plane odal frequencies of the syste, whereas, the affect on out-of-plane frequencies is negligible. As a result, the crossties could not be expected to reduce out-of-plane otions due to lateral otions of the tower, and suppleental dapers were deeed necessary on the five longest cables with fundaental frequencies in the range of Hz.

10 External Lateral Dapers The external lateral daping devices (ELDs) were designed to provide the axiu possible aount of daping to the cables, given the constraints on their placeent along the deck. Figure 6 shows graphics of Motioneering s design of the ELDs. Five of the devices are proposed for each side of the deck, on the cables listed in Table. The ELD units are expected to contribute approxiately 4% additional equivalent odal daping for the first ode of each of these cables. FIGURE 6 PROPOSED EXTERNAL LATERAL DAMPING DEVICE FOR SUPPRESSION OF OUT-OF-PLANE CABLE VIBRAITONS. Effectiveness of Proposed Mitigation Schee An equivalent finite eleent odel of cable longest cable on the ain span (R35) was developed consisting of spring eleents representing the crossties, and a dashpot representing the external lateral daper. The first and second out-of-plane, and first inplane frequencies of the reduced syste were tuned to within % of the full fan odel (Figure 5). A 40-inute tie history of wind buffeting loads and responses [Stoyanoff], estiated fro a siulation of the design windstor, was used as input to the cables and their anchorages. Siulations of the reduced odel were run for three scenarios: ) no itigation, ) ELD only (no cross ties), 3) ELD and crossties. Results fro the siulations are shown in Table 3. Response Predicted peak displaceent: unitigated (ft) Predicted peak displaceent: External Lateral Dapers only (ft) Predicted peak displaceent: External Lateral Dapers and Crossties (ft) Lateral Vertical Total TABLE 3 PREDICTED CABLE DISPLACEMENTS OF CABLE R35 (69 MPH WINDS).

11 The ELDs are expected to reduce out-of-plane displaceents to approxiately onequarter, and the crossties are expected to decrease in-plane displaceents to at least onetenth of their un-itigated aplitudes. Considering that the cable pipe diaeter is approxiately 9 in, the resulting aplitude would be reduced fro approxiately 4D to D, for the design windstor. For the serviceability windstor (46 ph), axiu cable displaceents were estiated to be approxiately 45 50% of the values listed in Table 3 (i.e, peak displaceents of approxiately ft). These values are less than, or approxiately equal to, the 0.5D criterion. Since it was recognized that the serviceability wind speed will occur ore frequently, and that the bridge would likely be closed for service during the design wind speed event, the proposed itigation schea was deeed acceptable for the design. CONCLUSIONS Analytical techniques for estiation of otion-induced and paraetric excitation of stay cables were presented, and applied to the proposed Ironton-Russell Bridge over Ohio River. The analytical ethods predictions were in a good agreeent with these obtained via direct nuerical siulations on the longest cables of the bridge. Results fro the cable vibration assessent indicated the requireent for a vibration itigation schee consisting of both crosstie cables and external lateral daping devices. These devices were designed, and their perforance verified, with nuerical siulations of the ain span and back span cable arrays. Siulation results indicated significant reductions in in-plane and out-of-plane displaceents of the longest cables exited by bridge buffeting. For the proposed itigation schee, the axiu cable displaceents during serviceability wind events are expected to be less than one cable diaeter, peakto-peak. ACKNOWLEDGEMENTS The authors wish to acknowledge Michael Baker Jr. Inc., structural engineers, for the bridge data, and the bridge owners, the Ohio Departent of Transportation. Motineering Inc., who designed the lateral dapers and crosstie anchoring are also acknowledged. The conceptual design of crosstie collars and anchoring is credited to Genesis Structures, Kansas City, USA. REFERENCES [] Caetano, E., Dynaics of Cable-stayed Bridges: Experiental Assessent of Cable-Structure Interaction. Ph.D. Thesis, University of Porto, Portugal, 000. [] Macdonald, J.H.G., Identification of the Dynaic Behaviour of a Cable-stayed Bridge Fro Full-scale Testing During and After Construction. Ph.D. Thesis, University of Bristol, Departent of Civil Engineering, 00. [3] Lilien, J.L., and Pinto Da Costa, A., Vibration aplitudes caused by paraetric excitation of cablestayed structures, Journal of Sound and Vibration, Vol. 74, 994, pp [4] Georgakis, C., Non-linear Dynaics of Cable Stays and Cable-Structure Interaction, Ph.D. Thesis, University of Bristol, Departent of Civil Engineering, 00.

12 [5] Gattulli, V., Lepidi, M., Macdonald, J.H.G. and Taylor, C., One-to-two global-local interaction in a cable-stayed bea observed through analytical, finite eleent and experiental ethods, International Journal of Non-linear Dynaics, Vol. 40, 005, pp [6] SETRA-LCPC, Cable Stays, Recoendation of French Interinisterial Coission on Prestressing, June 00. [7] Virlogeux, M., Cable vibrations in cable-stayed bridges, in Larsen, A. & Esdahl, S. (eds) Bridge Aerodynaics, Balkea, Rotterda, 998, pp [8] Stoyanoff, S., Theryo, T. and Garcia, P. Full dynaic tests of the stay cables of the Sunshine Skyway Bridge, Proceedings of the 6 th International Syposiu on Cable Dynaics, Charleston, South Carolina, 005. [9] Nayfeh, A. and Mook, D. Nonlinear Oscillations, John Wiley and Sons, 979. [0] Stoyanoff, S. A unified approach for 3D stability and tie doain response analysis with application of quasi-steady theory, Journal of Wind Engineering and Industrial Aerodynaics, v.89, 00, pp , 00. [] Bosch, H. Wind induced vibrations of stay cables, Missouri Departent of Transportation Research Report No , February, 005.