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Fully coupled buffeting analysis of Tsing Ma suspension bridge
Journal of Wind Engineering and Industrial Aerodynamics 85 (2000) 97}117
Fully coupled bu!eting analysis of Tsing Ma suspension bridge Y.L. Xu!,*, D.K. Sun", J.M. Ko!, J.H. Lin" !Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong "Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian, People's Republic of China Received 17 February 1999; received in revised form 19 April 1999; accepted 17 November 1999
Abstract A fully coupled bu!eting analysis of the Tsing Ma suspension bridge in Hong Kong is carried out in this paper. The features of the bridge, its natural frequencies and mode shapes from "nite element analysis are "rst presented. Followed is an introduction to the formulation for fully coupled bu!eting analysis of cable-supported bridges, which is featured mainly by a combination of "nite element approach and pseudo-excitation method in the frequency domain. The bu!eting response of bridge deck is then computed and compared with that from the conventional method. After the comparison, the e!ects from aeroelastic forces, multimodes and intermodes of vibration on deck response are examined. Finally, the bu!eting response of the whole bridge is investigated with aerodynamic forces on not only bridge deck but also bridge towers and main cables. The results show that aeroelastic forces signi"cantly a!ect the torsional response of the deck. The multimode responses are larger than single-mode responses in the vertical and torsional motions of the deck. The signi"cant intermode responses may exist between the vibration modes of similar mode shapes or similar modal components. The bu!eting of towers and main cables a!ects the lateral vibration of bridge deck and, in turn, the vibration of the bridge deck impacts the bu!eting of the bridge towers and main cables. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Bu!eting analysis; Tsing Ma bridge; Multimodes; Intermodes; Component interaction
* Corresponding author. Tel.: 852-2766-6050; fax: 852-2334-6389. E-mail address: [email protected] (Y.L. Xu) 0167-6105/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 1 3 3 - 6
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1. Introduction Modern long-span cable-supported bridges tend to have low and closely spaced natural frequencies. The contributions from multimodes of vibration and intermodes of vibration to bu!eting response may have to be included. To consider the multimode bu!eting response of a bridge deck, Lin and Yang [1] proposed a general linear theory for the computation of cross-spectra of the deck response to turbulent wind in the time domain. Jain et al. [2] considered both multimode and intermode bu!eting responses using random vibration-based mode superposition method in the frequency domain. Katsuchi et al. [3] re"ned the frequency domain approach in [2] and applied it to the Akashi}Kaikyo Bridge, which is the longest suspension bridge built in the world so far. The wind-induced dynamic responses of bridge deck, towers and cables are traditionally determined separately to simplify the problem. Wind-induced dynamic forces on towers and cables are thus not considered in the aforementioned work. However, with respect to #utter instability of cable-stayed bridges, Ogawa et al. [4] pointed out that the ignorance of interaction between bridge deck, towers and cables may positively or negatively a!ect the prediction of #utter instability of a bridge. Thus, the #utter derivatives of the suspension deck section near midspan are sometime measured with cable models to partially consider the interaction between deck and cables [3]. Davenport [5] also mentioned several possible mechanisms of interaction between bridge deck and cables and advocated the prediction of aerodynamic response of the deck, towers and cables as a system [6]. In this connection, a fully coupled three-dimensional bu!eting analysis of a longspan cable-supported bridge, featured mainly by a combination of "nite element approach and a pseudo-excitation method in the frequency domain, was presented by the writers [7]. Aeroelastic forces on bridge deck are changed into nodal forces to form aeroelastic damping and sti!ness matrices that are similar to those used in the "nite element-based #utter analysis [8}10]. The aerodynamic forces on the bridge deck, towers and cables are converted into node forces to obtain a loading vector. After the system equation of motion is assembled, the pseudo-excitation method is applied and the formulation is intentionally programmed so that a modern personal computer can be used to execute the bu!eting analysis of a bridge of hundreds of degrees of freedom in one or two hours. A fully coupled bu!eting analysis is now carried out for the Tsing Ma suspension bridge in Hong Kong. The features of the bridge, its natural frequencies and mode shapes from "nite element analysis are "rst presented. A brief introduction to the formulation for fully coupled bu!eting analysis of cable-supported bridges is then followed. The bu!eting responses of bridge deck are computed and compared with those from the conventional method. After the comparison, the e!ects from aeroelastic forces, multimodes and intermodes of vibration on deck response are examined. Finally, the bu!eting response of the bridge is investigated with aerodynamic forces on not only the bridge deck but also the bridge towers and main cables.
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2. Features and dynamic properties of Tsing Ma bridge Hong Kong's new airport and port development are located on Lantau Island. The key section of the transportation between the new facilities and the existing commercial centres of Hong Kong Island and Kowloon is the Lantau Fixed Crossing, in which the Tsing Ma long suspension bridge is the central structure carrying a dual three-lane highway on the upper level of the bridge deck and two railway tracks and two carriage ways on the lower level within the deck [11]. The Tsing Ma bridge, stretching from Tsing Yi Island to Ma Wan Island, has a main span of 1377 m between the Tsing Yi tower in the east and the Ma Wan tower in the west (see Fig. 1). The height of the towers is 206 m, measured from the base level to the tower saddle. The two main cables of 36 m apart in the north and south are accommodated by the four saddles located at the top of the tower legs in the main span. On the Tsing Yi side, the main cables are extended from the tower saddles to the main anchorage through the splay saddles, forming a 300 m Tsing Yi side span. On the Ma Wan side, the main cables extended from the Ma Wan tower are held "rst by the saddles on Pier M2 at a horizontal distance of 355.5 m from the Ma Wan tower and then by the main anchorage through splay saddles at the Ma Wan abutment. The bridge deck is a hybrid steel structure continuing between the two main anchorages. It is suspended by suspenders in the main span and the Ma Wan side span. On the Tsing Yi side the deck is supported by three piers rather than suspenders. This arrangement and the di!erence of span between the Ma Wan side and the Tsing Yi side introduce some asymmetry with respect to the midspan of the bridge. A three-dimensional dynamic "nite element model has been established for the Tsing Ma bridge after the completion of deck welding connections [12]. Threedimensional Timoshenko beam elements with rigid arms were used to model the two bridge towers. The cables and suspenders were modelled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel bridge deck was represented by a single beam with equivalent cross-sectional properties determined from a "nite element analysis using detailed sectional models. The connections between bridge components and the supports of the bridge were properly modelled. The modal analysis of the Tsing Ma bridge shows that the natural frequencies of the bridge are spaced very closely. The "rst 20 natural frequencies include the "rst 12 lateral modes, the "rst six vertical modes, and the "rst two torsional modes. They range from 0.068 to 0.380 Hz only. Table 1 lists the "rst 20 natural frequencies and the
Fig. 1. Con"guration of Tsing Ma bridge.
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Table 1 The First 20 Natural Frequencies and Mode Shapes of Tsing Ma bridge Mode number (1)
Mode direction (2)
Natural freq. (Hz) (3)
Nature of mode shape (4)
1 2 3 4 5
L L L L L
0.068 0.158 0.210 0.230 0.232
6 7
L L#T
0.240 0.285
8
L
0.333
9
L
0.340
10
L
0.365
11 12
L L
0.367 0.380
1 2
V V
0.117 0.137
3 4 5 6 1
V V V V T#L
0.189 0.245 0.294 0.325 0.271
2
T
0.311
Deck and cables in main span move in phase (a half-wave) Deck and cables in main span move in phase (one wave) Cables in main span move out of phase (a half-wave) Cables in main span move out of phase (one wave) Deck and cables in main span move out of phase (a halfwave) Deck and cables in main span move out of phase (one wave) Deck and cables in main span move in phase (one and a halfwaves) and couple with torsion Deck and cables in Ma Wan span move in phase (a halfwave) Cables in main span move out of phase (one and a halfwaves) Deck and cables in main span move out of phase (one and a half-waves) Cables in Ma Wan span move out of phase (a half-wave) Cables and deck in Tsing Yi span move in phase (a halfwave) Deck and cables in main span move in one wave Deck and cables in main span move in a half-wave, approximately Deck and cables in main span move in one and a half-waves Deck and cables in main span move in two waves Deck and cables in Ma Wan span move in a half-wave Deck and cables in main span move in two and a half-waves Deck and cable in main span rotate in a half-wave and couple with the seventh lateral mode Deck and cables in main span rotate in one wave
Note: V"vertical; T"torsional; L"lateral.
nature of corresponding mode shapes. These natural frequencies and mode shapes have been veri"ed by the "eld measurement. Among the "rst 20 natural frequencies, the lowest frequency of the bridge is 0.068 Hz, corresponding to the "rst lateral mode of a half-wave with the bridge deck and cables moving in phase in the main span. The "rst vertical mode of the bridge is almost antisymmetric in the main span at a natural frequency of 0.117 Hz in one wave while the second vertical mode is almost symmetric in the main span at a natural frequency of 0.137 Hz in a half-wave approximately. The "rst torsional vibrational mode occurs at a natural frequency of 0.271 Hz in a halfwave in the main span. This mode contains the lateral component that is similar to the seventh lateral mode, whereas the seventh lateral mode contains the torsional component that is similar to the "rst torsional mode. The more information on the structural properties, the "nite element modelling, the natural frequencies and mode shapes of the bridge can be found in Ref. [12].
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3. Formulation for fully coupled bu4eting analysis The following is a brief introduction to the formulation for fully coupled bu!eting analysis [7], which is featured mainly by a combination of "nite element approach and pseudo-excitation method in the frequency domain. The advantages of this combined approach are to readily handle the bridge deck with signi"cantly varying structural properties and mean wind speed along the deck, to make good use of the "nite element bridge models already made for the static and eigenvalue analyses, to naturally include intermode and multimode responses, and to determine wind-induced responses of the bridge deck, towers and cables simultaneously. This approach is however relatively time-consuming compared with the Scanlan's method [2], but the required computing time is quite acceptable if the formulation is intentionally programmed and the modern personal computer is used. The equation of motion of the whole bridge for the fully-coupled bu!eting analysis can be expressed as [7] MYG (t)#CYQ (t)#KY (t)"RP (t),
(1)
in which Y (t) is the total nodal displacement vector of N dimensions including the bridge deck, towers, cables, and other components; M is the N]N total mass matrix; C is the N]N total damping matrix which consists of both aeroelastic damping matrix C!% and structural damping matrix C4 ; K is the N]N total sti!ness matrix 4 4 containing the aeroelastic sti!ness matrix K!% and the structural sti!ness matrix K4 ; 4 4 P is the total aerodynamic loading vector of m dimensions (in general, m;N) for the bridge deck, towers, cables, and other components; and R is the N]m matrix consisting of 0 and 1, which expands the m-dimensional loading vector into the N-dimensional loading vector. The system aeroelastic sti!ness matrix K!% and the system aeroelastic damp4 ing matrix C!% are assembled from the element aeroelastic sti!ness matrices K!% i 4 and the element aeroelastic damping matrices C!% in the same way as the system i structural sti!ness matrix K4 and the system structural damping matrix C4 are 4 4 assembled from the element structural sti!ness matrices K4 and the element structural i damping matrices C4, respectively. The total aerodynamic loading vector can be i obtained by n P"+ T P" , i i,% i
(2)
where T is the co-ordinate transformation matrix for the ith element with the i dimensions equal to the dimension of the system nodal force vector times 12 for a beam element or six for a cable element; P" can be either the aerodynamic force i,% vector of the deck element or the tower element or the cable element in local co-ordinate; and n is the total number of the elements subject to wind loading. The spectral density function matrix of the nodal bu!eting forces acting on the whole
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bridge in the global co-ordinate system is
C
D
S% 1 1 (u) S% 1 2 (u) 2 S% 1 n (u) 11 11 11 S% 2 1 (u) S% 2 2 (u) 2 S% 2 n (u) 11 11 S (u)"T 1 1 TT, 11 F F } F S% n 1 (u) 11
S% n 2 (u) 11
S% n n (u) 11
(3)
where T"[T , T ,2T ,2T ]; the superscript T means the transposition of 1 2 i n a matrix; and S% i j (u) is the cross-spectral density function matrix of the nodal 11 bu!eting forces acting on the ith and jth elements. To reduce the computation e!orts for determining bu!eting response and at the same time to include multimode and intermode e!ects, a pseudo-excitation algorithm is suggested to determine the spectral density function matrix for the bu!eting response. The pseudo-excitation method was proposed by Lin [13] to deal with dynamic response of structures subject to random seismic excitation. In this method, the determination of random response of a structure is converted to the determination of response of the structure under a series of harmonic loads, i.e., the so-called pseudo-excitation. The advantages of this method include the less computation e!ort required, the retention of all cross-correlation terms between normal modes, and the feasibility for the determination of random internal force responses. The basic equations involving in the present bu!eting analysis are listed below, and the detailed derivation can be found in Ref. [7]. The spectral density function matrix S (u) is usually a symmetric matrix, and thus PP this excitation spectral matrix can be decomposed as m (4) S (u)"LHDLT" + d LHLT, PP kk k k k/1 in which L is the lower triangular matrix; D is the diagonal matrix; L is the kth k column of L; and d is the kth diagonal element of D. kk The pseudo-excitations can then be constituted as follows: f "L exp(iu t) (k"1, 2,2, m). k k
(5)
For each pseudo-excitation (harmonic excitation) vector, a pseudo-displacement response vector, Y (u), can be easily and traditionally determined by k Y "H(u)Rf , k k
(6)
H(u)"[!u2M#iuC#K ]~1
(7)
The spectral density function matrix of the displacement response of the bridge can be then proved to be S
YY
m (u)" + d YHYT. kk k k k/1
(8)
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Because the above equation involves one summation only, much less computation e!orts are needed to calculate the response spectral density matrix. Further reduction of computation time can be also achieved if the mode reduction technique is used. Let Q be the modal matrix containing the "rst r mode shapes, and introduce a linear NCr transformation. "Q z , (9) NC1 NCr rC1 in which z is the r]1 generalised displacement response vector. Eq. (1) of N dimensions can be reduced to the equation of r dimensions. Y
M zK (t)#C z5 (t)#K z(t)"P (t). r r r r The pseudo-excitations are then computed according to
(10)
f "QTRL exp(iu t) (k"1, 2,2, m). (11) rk k For each pseudo-excitation vector, the pseudo-displacement response vector, Y (u), is k now determined by Y "QH (u)Rf (k"1, 2,2, m). (12) k r rk The spectral density function matrix for the displacement response attributed to the "rst r modes of vibration is computed according to Eq. (8). The standard deviations of the displacement, velocity, and acceleration of the node can be readily computed according to the random vibration theory, after the auto spectral density functions for each node are determined. Di!erent from the SRSS method (the square root of the sum of the squares of the model response), the pseudo-excitation method retains the cross-correlation terms between the "rst r normal modes. Thus, it is a kind of CQC methods (the complete quadratic combination). This CQC method also makes it possible to use a personal computer to execute the bu!eting analysis of the system of hundreds of degrees of freedom in one or two hours.
4. Aeroelastic and aerodynamic parameters In this section, the aerodynamic and aeroelastic parameters used for the bu!eting analysis of the Tsing Ma bridge are provided. Fig. 2 shows a typical deck section of the Tsing Ma bridge with global axis sign convention. The x-axis is in the longitudinal direction of the bridge, the y-axis is in the lateral direction, and the z-axis is in the vertical direction. The #utter derivatives of the bridge deck corresponding to the 03 wind incidence without tra$c and trains are used, but only the eight #utter derivatives HH and i AH(i"1, 2, 3, 4) were measured from the wind tunnel tests [14]. By "tting the wind i tunnel test results, Figs. 3 and 4 show the eight #utter derivatives related to vertical and torsional motions, respectively, against the reduced wind speed. The reference width of the deck used in the tests was half the width of the deck, i.e., B/2"20.5 m. It is seen that all HH curves are basically negative within the reduced wind speed range i
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Fig. 2. Typical section of bridge deck.
Fig. 3. Flutter derivatives for vertical motion.
concerned. Particularly, the negative HH and HH indicate the positive aerodynamic 1 2 damping, which will enhance the stability and reduce the vertical motion of the bridge deck. Among the AH curves, the AH curve exhibits the positive values in the reduced i 1 velocity ranging from 6 and 12, indicating the possibility of negative aerodynamic damping in the torsional motion. The negative AH curve, on the other hand, indicates 4 that the aeroelastic force from vertical motion may generate positive torsional sti!ness. Fig. 5 shows the aerodynamic force coe$cients of the bridge deck measured from wind tunnel tests [14]. The drag, lift, and moment coe$cients are 0.135, 0.090 and 0.063, respectively, at the wind incidence of 03 with respect to the deck width of
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Fig. 4. Flutter derivatives for torsional motion.
Fig. 5. Drag, lift, and moment coe$cient curves [14].
41 m. The "rst derivatives of the drag, lift, and moment coe$cients with respect to wind incidence at the wind incidence of 03 are !0.253, 1.324, and 0.278, respectively. The aerodynamic admittance function of the bridge is not considered due to the lack of information. For the two bridge towers, the drag coe$cient is taken as 1.5 with respect to the tower width of 9.25 m. The lift and moment coe$cients for the towers as well as the "rst derivatives of the drag, lift and moment coe$cients are not available to the writers. For the main cables, force coe$cients critically depend on the Reynold's number and cable surface roughness. Conservatively, the drag coe$cient of 1.0 is chosen for the cable diameter of 1.1 m. In the spectral density functions of gusty wind, the exponential decay coe$cients C , C , and C are taken as 16, 10 and 0, respectively [15]. The friction velocity of x y z the wind, uH, is chosen as 1.15 m/s in consideration of the open terrain surrounding the bridge. The variation of mean wind speed with height follows a power law,
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;(z)"; (z/z )a with an exponent a of 0.19 for the open terrain. The reference mean r r wind speed, ; , is taken as 25 m/s at the reference height of 60 m, i.e., the bridge deck r level near the tower. This mean wind speed is not the design mean wind speed of the bridge at the deck level, but it is a relatively frequently occurring mean wind speed. At this wind speed level, the vortex shedding response of the bridge deck can be neglected. Furthermore, this mean wind speed level guarantees the range of frequencies for the computation of bu!eting response within the experimental range for #utter derivatives. To maintain a high spectral resolution in the interested frequency range and to avoid a poor estimate of response from numerical integration, a frequency interval about 0.0004 Hz is used within the range from 0.043 to 0.48 Hz. This frequency range covers well the "rst 20 modes of vibration of the bridge. The structural damping ratio is taken as 0.008 for all modes of vibration.
5. Comparison of deck response In this section, the displacement and acceleration responses of the bridge deck along the span are computed using the new formulation and compared with those from the Scanlan's method modi"ed by Jain et al. [2]. To make this comparison possible, only the bridge deck is subjected to bu!eting loading and no bu!eting loading is applied to the towers and cables although the present approach can take all bu!eting loads into consideration. The "rst 20 modes of vibration listed in Table 1 are included in the computation. Each mode of vibration contains three components, but for most of vibrational modes only one component is dominant. The results from both methods encompass multimodes and intermodes contributions as well as aeroelastic e!ects. Figs. 6}8 show variations of standard deviation of lateral, vertical, and angular displacement responses of the deck along the span. It is seen that the displacement responses and their variations along the bridge span obtained from the two methods
Fig. 6. Variations of deck lateral displacement response along span.
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Fig. 7. Variations of deck vertical displacement response along span.
Fig. 8. Variations of deck angular displacement response along span.
are fairly similar. The new formulation gives slightly larger values around the midspan of the bridge than those from the Scanlan's method [2]. This is because in the "nite element-based new formulation, the actual variations of mean wind speed and turbulence along the span are taken into consideration while in the Scanlan's method the mean wind speed and turbulence remain unchanged for the whole bridge [2], although the modi"ed Scanlan's method can also take into account of the varying mean wind speed and turbulence, as done just recently by Katsuchi et al. in Ref. [3]. It is seen that the maximum standard deviation of lateral displacement response of the bridge deck occurs near the midspan and reaches about 0.23 m. The shape of lateral displacement response curve is similar to the "rst lateral mode of vibration of the deck. The maximum standard deviation of vertical displacement response occurs about the quarter of the main span from either bridge tower and attains about 0.11 m.
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The pattern of vertical displacement response curve in the main span does not follow either the "rst vertical mode or the second vertical mode of vibration. The variation of angular displacement response along the main span appears a half wave and the maximum response comes about 0.0018 rad at the midspan. The displacement response curves from single mode of vibration, which are also displayed in Figs. 6}8, will be discussed later. Apart from displacement responses, the lateral, vertical, and angular acceleration responses of the bridge deck from the new formulation are also compared with those from the Scanlan's method [2], and the comparison is quite satisfactory. Fig. 9 shows an example in which the variations of deck vertical acceleration response along the span from the two methods are compared. The satisfactory comparison veri"es the accuracy of the new formulation and the new computer program.
6. Aeroelastic e4ects It is a common belief that self-excited forces (aeroelastic forces) would a!ect bu!eting response of long-span bridges. To understand such e!ects on the Tsing Ma bridge, the response spectra of the bridge deck with and without #utter derivatives are computed using the new formulation. Fig. 10 shows the power spectra of lateral displacement response of the bridge deck at the midspan with and without aeroelastic e!ects from the "rst 20 modes of vibration. The response spectrum from the "rst lateral mode of vibration only is also displayed in Fig. 10. The response peaks appearing in the spectrum without aeroelastic forces clearly identify the "rst, second, and "fth natural frequencies of the bridge in the lateral direction (see Table 1). The third and fourth natural frequencies of the bridge in the lateral direction correspond to cable vibration rather than deck vibration and thus they do not appear in the spectrum. Because lateral #utter derivatives are not considered in the computation, the e!ects of aeroelastic forces are very small
Fig. 9. Variations of deck vertical acceleration response along span.
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Fig. 10. Power spectra of lateral displacement of deck at midspan.
and the spectra with and without aeroelastic forces are similar to each other. However, a preliminary study recently carried out by Katsuchi et al. [3] indicated that the lateral #utter derivative PH would provide a damping e!ect for the lateral bu!eting 1 response of the bridge deck, and the measurement of this parameter together with other lateral #utter derivatives would be an important task for future wind tunnel test programs. It is also seen from Fig. 10 that due to the similarity of mode components between the seventh lateral mode and the "rst torsional mode, the spectrum with aeroelastic forces is slightly di!erent from the spectrum without aeroelastic forces within the frequency range from 0.26 to 0.30 Hz. This indicates that the torsional #utter derivatives indirectly a!ect the lateral bu!eting response of the bridge deck. By comparing the spectrum from the "rst mode of vibration with the spectrum from the "rst 20 modes of vibration, it is seen that in the lateral displacement response, the "rst lateral mode of vibration is dominant, which is consistent with the results shown in Fig. 6. The spectrum without aeroelastic forces in Fig. 11 clearly demonstrates the "rst, second, third, fourth, and sixth natural frequencies of the bridge in the vertical direction. The "fth natural frequency corresponds to the vertical vibration of the Ma Wan side span rather than the main span, and thus it does not appear in the spectrum. Because the "rst vertical mode of the deck is almost antisymmetric, the second vertical mode of vibration becomes dominant for the vertical response of the bridge deck around the midspan. Since both the "rst vertical and the second vertical modes are important, the vertical displacement response of the bridge deck is dominated by these two modes, as seen in Fig. 7. Furthermore, by comparing the spectra with aeroelastic forces and without aeroelastic forces, it is seen that the positive aeroelastic damping reduces the peak response amplitudes signi"cantly, in particular in the lower modes of vibration. The e!ects of aeroelastic sti!ness are not signi"cant since the peak frequencies do not have any clear shift. The aeroelastic coupling of vertical motion with torsional motion leads to an additional small peak appearing on the spectrum near the frequency of 0.29 Hz.
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Fig. 11. Power spectra of vertical displacement of deck at midspan.
Fig. 12. Power spectra of angular displacement of deck at midspan.
Appearing in Fig. 12 are the spectra of angular displacement response of the deck at the midspan. It is seen that the aeroelastic coupling between the vertical and torsional motion signi"cantly a!ects the torsional vibration of the bridge deck. Without aeroelastic forces, two resonance peaks occur at the "rst and second natural frequencies in torsion while another small peak between them is due to the similarity of mode components between the "rst torsional mode and the seventh lateral mode. There is also a very small peak at a frequency of 0.068 Hz due to the torsional component in the "rst lateral mode of vibration. After introducing aeroelastic forces, the spectrum of angular displacement response of the deck at the midspan is signi"cantly changed. There are three new resonance peaks occurring at the "rst three natural frequencies in vertical direction. The largest resonance peak corresponds to the second vertical mode of vibration, indicating that the aeroelastic coupling between the second vertical and "rst torsional modes of vibration is considerable. In fact, due to the aeroelastic
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coupling between the second vertical and the "rst torsional modes, the angular displacement response of the bridge deck is similar to either the "rst torsional mode or the second vertical mode, as shown in Fig. 8. Furthermore, the magnitudes of resonance peaks at the original "rst and second torsional frequencies are slightly reduced, and the frequencies are shifted. If only the "rst torsional mode is used to compute the torsional response spectrum in consideration of aeroelastic forces, the resonance peak reduces slightly but the "rst torsional natural frequency remains unchanged. This further con"rms the aeroelastic coupling between vertical and torsional modes of vibration. Displayed in Fig. 13 are the vertical acceleration response spectra of the bridge deck at the midspan. Compared with the vertical displacement response spectrum in Fig. 10, it is clear that the e!ects of higher modes of vibration are much more signi"cant on acceleration response than displacement response since acceleration resonance peaks in higher modes are compatible with those in lower modes. This is also why the pattern of the vertical acceleration response along the span (see Fig. 9) is much complicated that that of the vertical displacement response (see Fig. 7). Similar phenomena are also observed for the torsional and lateral displacement and acceleration responses of the bridge deck. From the above discussion, it is concluded that the aeroelastic damping considerably reduces the vertical response of the deck. The aeroelastic coupling between the second vertical and the "rst torsional modes of vibration signi"cantly a!ects the torsional vibration of the bridge deck.
7. Multimode and intermode e4ects The spectral analysis of the deck response has already demonstrated that the vertical and torsional displacement responses may not be dominated by a single mode of vibration. To further discuss this matter, the lateral displacement response
Fig. 13. Power spectra of vertical acceleration of deck at midspan.
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attributed to the "rst lateral mode only, the vertical displacement response due to the second vertical mode only, and the angular displacement response arising from the "rst torsional mode only are computed and displayed in Figs. 6}8, respectively, together with the same quantities from the "rst 20 modes of vibration. It is seen that the "rst lateral mode of vibration absolutely dominates the total lateral displacement response of the bridge deck. For the vertical and angular displacement responses, signi"cant di!erences appear in the responses between the single mode of vibration and multimodes of vibration. In particular, for acceleration response, the e!ects from multimodes of vibration become much more signi"cant, as shown in Fig. 9 for the vertical acceleration response. These results are consistent with the spectral analysis. As a result, the multimode approach should be used to determine bu!eting response of the Tsing Ma long suspension Bridge. To investigate e!ects of intermodes, a number of pairs of vibrational modes are selected from lateral, vertical, and torsional motions, respectively, as well as their combination. For each pair of vibrational modes, the deck response along the span from each single mode is computed "rst and then the deck response from the two modes are computed using both the SRSS method and the CQC method (the complete quadratic combination). The CQC method used here is actually the pseudo-excitation method as introduced in Section 3. By comparing the responses from both the methods, inter-mode e!ects can be estimated. Fig. 14 shows the deck vertical displacement responses along the span from the "rst vertical mode only, the second vertical mode only, as well as from both modes using either the SRSS method and the CQC method. The natural frequencies of the "rst and second vertical modes are close. The "rst one is 0.117 Hz and the second one is 0.137 Hz, but the "rst mode shape is almost antisymmetric and the second mode shape is almost symmetric. The SRSS method gives almost the same response as the CQC method. This means that the contribution from the intermodes is almost zero. The intermode e!ects from the "rst and second lateral modes of vibration and from the "rst and second torsional modes of vibration are also found negligible.
Fig. 14. Inter-mode e!ects from "rst and second vertical modes.
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Fig. 15. Inter-mode e!ects from "rst torsional and seventh lateral modes.
The investigation then moves to the "rst torsional mode and the seventh lateral mode. Fig. 15 shows the lateral displacement responses along the span from the lateral component of the "rst torsional mode, the lateral component of the seventh lateral mode, as well as from both components. The lateral component of the "rst torsional mode is similar in shape to the lateral component of the seventh lateral mode of one and a half-waves in the man span. In this case, the SRSS method gives larger lateral displacement response than the CQC method. This is because the lateral displacement response from the "rst torsional mode is out of phase with that from the seventh lateral mode and consequently, the intermode contribution is negative and the CQC method renders smaller response than the SRSS method. For these two modes but the torsional components, it is found that the SRSS method gives slightly smaller response than the CQC method. This is because the torsional component from the "rst torsional mode is in phase with the torsional component of the seventh lateral mode of vibration. Since the "rst mode of vibration is dominant in the lateral response, the intermode e!ects from the seventh lateral and the "rst torsional modes in the direction can still be neglected. The investigation also goes to the typical aeroelastically coupled modes, i.e., the "rst torsional mode and the second vertical mode. This is because the mode shape of the second vertical mode is similar to the "rst torsional mode, and these two modes are aeroelastically coupled. Fig. 16 shows the angular displacement responses along the span from the torsional component of the "rst torsional mode and the torsional component from the second vertical mode. Since the torsional component from the second vertical mode is almost zero, the angular response from this component is almost zero. As a result, the SRSS method yields the same response as the torsional component of the "rst torsional mode. However, the CQC method produces much larger torsional response than the SRSS method. This is because the CQC method includes the aeroelastic coupling between the two modes. This is consistent with the results from the spectral analysis. No other signi"cant aeroelastically coupled mode combinations were found in this study.
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It is thus concluded that for the Tsing Ma bridge, multimode e!ects are signi"cant for the vertical and torsional displacement responses but not for the lateral displacement response. Intermode e!ects are negligible for the two modes from either lateral motion or vertical motion or torsional motion but intermode e!ects are considerable for the aeroelastically coupled vertical and torsional modes of similar mode shapes and for the torsional and lateral modes of similar mode components.
8. Interaction between deck, towers and cables In this section, apart from the bu!eting forces and aeroelastic forces on the bridge deck the bu!eting forces on the towers and the cables are also included in the computation. The results are denoted by the term `full bridgea. The full-bridge results
Fig. 16. Inter-mode e!ects from "rst torsional and second vertical modes.
Fig. 17. Interactive e!ects on deck lateral displacement response.
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Fig. 18. Interactive e!ects on deck vertical displacement response.
Fig. 19. Interactive e!ects on deck angular displacement response.
are then compared with those with the forces on the bridge deck only, the bridge tower only, and the main cable only. In this way, the interaction between the bridge deck, towers, and main cables can be examined. Fig. 17 shows the variations of deck lateral displacement response along the span for the full bridge and for the deck only. It is clear that due to the consideration of bu!eting forces on the main cables and towers, the lateral displacement response of the bridge deck increases about 15% at the midspan. The further computation of the bridge with the forces on the bridge deck and towers only shows that the 15% increase of the lateral displacement response of the bridge deck is almost due to the aerodynamic forces on the main cables. Figs. 18 and 19 show, respectively, the variations of deck vertical and angular displacement responses along the span for the full bridge and for the deck only. Clearly, the bu!eting forces on the main cables and towers do
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not a!ect the vertical and torsional displacement responses of the bridge deck. This may be because the lift and moment coe$cients and their derivatives for the bridge towers are not considered and the time delay of bu!eting forces on the front cable and the back cable is also not taken into consideration in the computation. For the dynamic response of the bridge tower in the full-bridge case, the computed results show that the tower has not only alongwind response but also crosswind and torsional responses. The maximum alongwind displacement of the Tsing Yi tower occurs at the top of the tower about 0.003 m. The maximum crosswind displacement also occurs at the top of tower about 0.009 m. The maximum angular displacement response occurs at about 70 m high with 0.00014 rad. From the computation of the dynamic response of the main cable in the full-bridge case and in the main cable case, it is found that the lateral and vertical displacement responses of the main cable are signi"cantly a!ected by the bridge deck motion. The maximum lateral displacement response of the front cable is 0.24 m in the full-bridge case while the maximum vertical displacement response of the front cable is 0.13 m in the full-bridge case. The longitudinal motion of the main cable is also induced by the bridge deck vibration.
9. Conclusions The fully coupled bu!eting analysis of the Tsing Ma suspension bridge in Hong Kong has been carried out in this paper in the frequency domain. The bu!eting response of the bridge deck only obtained from the suggested approach is in good agreement with that computed using the Scanlan's method. The aeroelastic damping was found to reduce the vertical response of the bridge deck, and the aeroelastic e!ects on the torsional vibration of the Tsing Ma bridge are signi"cant. The multimode e!ects are considerable on the vertical motion and torsional motion but not on the lateral motion of the bridge deck. Intermode e!ects can be neglected for the two modes in either lateral motion or vertical motion or torsional motion but intermode e!ects should be considered for the aeroelastically coupled vertical and torsional modes of similar mode shapes. Finally, the results from the present approach reveals that the bu!eting of bridge deck considerably impacts the bu!eting of towers and main cables whereas the bu!eting of towers and main cables moderately a!ects the lateral vibration of bridge deck. It is the writers' wish that the numerical results presented in this paper will be validated through a comparison with wind tunnel test results and "eld measurement data in near future.
Acknowledgements The writers are grateful for the "nancial supports from the Hong Kong Research Grant Council through a UGC grant (CERG 5027/98E) for the "rst and third writer. The support from the Lantau Fixed Crossing Project Management O$ce, Highways Department of Hong Kong, to allow the writers to access the relevant materials for academic purpose only is particularly appreciated.
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References [1] Y.K. Lin, J.N. Yang, Multimode bridge response to wind excitations, J. Eng. Mech. ASCE 109 (2) (1983) 586}603. [2] A. Jain, N.P. Jones, R.H. Scanlan, Coupled #utter and bu!eting analysis of long-span bridges, J. Struct. Eng. ASCE 122 (7) (1996) 716}725. [3] H. Katsuchi, N.P. Jones, R.H. Scanlan, Multimode coupled #utter and bu!eting analysis of the Akashi-Kaikyo bridge, J. Struct. Eng. 125 (1) (1999) 60}70. [4] K. Ogawa, H. Shimodoi, H. Ishizaki, Aeroelastic stability of a cable-stayed bridge with girder, tower and cables, J. Wind Eng. Ind. Aerodyn. 41}44 (1992) 1227}1238. [5] A.G. Davenport, A simple representation of the dynamics of a massive stay cable in wind, AFPC Conference, Deauville, France, 1994, pp. 427}437. [6] A.G. Davenport, The consistent safety of long span bridges against wind with special reference to Le Pont De Normandie, AFPC Conference, Deauville, France, 1994, pp. 3}14. [7] Y.L. Xu, D.K. Sun, J.M. Ko, J.H. Lin, Bu!eting analysis of long span bridges: a new algorithm, Comput. Struct. 68 (1998) 303}313. [8] T.J.A. Agar, Dynamic instability of suspension bridges, Comput. Struct. 41 (6) (1991) 1321}1328. [9] A. Namini, P. Albrecht, H. Bosch, Finite element-based #utter analysis of cable-suspension bridges, J. Struct. Engi. 118 (6) (1992) 1509}1526. [10] M.S. Pfeil, R.C. Batista, Aeroelastic stability analysis of cable-stayed bridges, J. Struct. Eng. ASCE 121 (12) (1995) 1784}1794. [11] A.S. Beard, J.S. Young, Aspect of the design of the Tsing Ma bridge, Proceedings of the International Conferences on Bridge into 21st Century, Impressions Design and Print Ltd., Hong Kong, 1995, pp. 93}100. [12] Y.L. Xu, J.M. Ko, W.S. Zhang, Vibration studies of Tsing Ma suspension bridge, J. Bridge Eng. ASCE 2 (4) (1997) 49}156. [13] J.H. Lin, A fast CQC algorithm of PSD matrices for random seismic responses, Comput. Struct. 44 (3) (1992) 683}687. [14] C.K. Lau, K.Y. Wong, Aerodynamic stability of Tsing Ma Bridge, Proceedings of the Fourth International Kerensky Conference Hong Kong, Sept., 1997, pp. 131}138. [15] E. Simiu, R.H. Scanlan, Wind E!ects on Structures, Wiley, New York, 1996.
Fully coupled bu!eting analysis of Tsing Ma suspension bridge Y.L. Xu!,*, D.K. Sun", J.M. Ko!, J.H. Lin" !Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong "Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian, People's Republic of China Received 17 February 1999; received in revised form 19 April 1999; accepted 17 November 1999
Abstract A fully coupled bu!eting analysis of the Tsing Ma suspension bridge in Hong Kong is carried out in this paper. The features of the bridge, its natural frequencies and mode shapes from "nite element analysis are "rst presented. Followed is an introduction to the formulation for fully coupled bu!eting analysis of cable-supported bridges, which is featured mainly by a combination of "nite element approach and pseudo-excitation method in the frequency domain. The bu!eting response of bridge deck is then computed and compared with that from the conventional method. After the comparison, the e!ects from aeroelastic forces, multimodes and intermodes of vibration on deck response are examined. Finally, the bu!eting response of the whole bridge is investigated with aerodynamic forces on not only bridge deck but also bridge towers and main cables. The results show that aeroelastic forces signi"cantly a!ect the torsional response of the deck. The multimode responses are larger than single-mode responses in the vertical and torsional motions of the deck. The signi"cant intermode responses may exist between the vibration modes of similar mode shapes or similar modal components. The bu!eting of towers and main cables a!ects the lateral vibration of bridge deck and, in turn, the vibration of the bridge deck impacts the bu!eting of the bridge towers and main cables. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Bu!eting analysis; Tsing Ma bridge; Multimodes; Intermodes; Component interaction
* Corresponding author. Tel.: 852-2766-6050; fax: 852-2334-6389. E-mail address: [email protected] (Y.L. Xu) 0167-6105/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 1 3 3 - 6
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1. Introduction Modern long-span cable-supported bridges tend to have low and closely spaced natural frequencies. The contributions from multimodes of vibration and intermodes of vibration to bu!eting response may have to be included. To consider the multimode bu!eting response of a bridge deck, Lin and Yang [1] proposed a general linear theory for the computation of cross-spectra of the deck response to turbulent wind in the time domain. Jain et al. [2] considered both multimode and intermode bu!eting responses using random vibration-based mode superposition method in the frequency domain. Katsuchi et al. [3] re"ned the frequency domain approach in [2] and applied it to the Akashi}Kaikyo Bridge, which is the longest suspension bridge built in the world so far. The wind-induced dynamic responses of bridge deck, towers and cables are traditionally determined separately to simplify the problem. Wind-induced dynamic forces on towers and cables are thus not considered in the aforementioned work. However, with respect to #utter instability of cable-stayed bridges, Ogawa et al. [4] pointed out that the ignorance of interaction between bridge deck, towers and cables may positively or negatively a!ect the prediction of #utter instability of a bridge. Thus, the #utter derivatives of the suspension deck section near midspan are sometime measured with cable models to partially consider the interaction between deck and cables [3]. Davenport [5] also mentioned several possible mechanisms of interaction between bridge deck and cables and advocated the prediction of aerodynamic response of the deck, towers and cables as a system [6]. In this connection, a fully coupled three-dimensional bu!eting analysis of a longspan cable-supported bridge, featured mainly by a combination of "nite element approach and a pseudo-excitation method in the frequency domain, was presented by the writers [7]. Aeroelastic forces on bridge deck are changed into nodal forces to form aeroelastic damping and sti!ness matrices that are similar to those used in the "nite element-based #utter analysis [8}10]. The aerodynamic forces on the bridge deck, towers and cables are converted into node forces to obtain a loading vector. After the system equation of motion is assembled, the pseudo-excitation method is applied and the formulation is intentionally programmed so that a modern personal computer can be used to execute the bu!eting analysis of a bridge of hundreds of degrees of freedom in one or two hours. A fully coupled bu!eting analysis is now carried out for the Tsing Ma suspension bridge in Hong Kong. The features of the bridge, its natural frequencies and mode shapes from "nite element analysis are "rst presented. A brief introduction to the formulation for fully coupled bu!eting analysis of cable-supported bridges is then followed. The bu!eting responses of bridge deck are computed and compared with those from the conventional method. After the comparison, the e!ects from aeroelastic forces, multimodes and intermodes of vibration on deck response are examined. Finally, the bu!eting response of the bridge is investigated with aerodynamic forces on not only the bridge deck but also the bridge towers and main cables.
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2. Features and dynamic properties of Tsing Ma bridge Hong Kong's new airport and port development are located on Lantau Island. The key section of the transportation between the new facilities and the existing commercial centres of Hong Kong Island and Kowloon is the Lantau Fixed Crossing, in which the Tsing Ma long suspension bridge is the central structure carrying a dual three-lane highway on the upper level of the bridge deck and two railway tracks and two carriage ways on the lower level within the deck [11]. The Tsing Ma bridge, stretching from Tsing Yi Island to Ma Wan Island, has a main span of 1377 m between the Tsing Yi tower in the east and the Ma Wan tower in the west (see Fig. 1). The height of the towers is 206 m, measured from the base level to the tower saddle. The two main cables of 36 m apart in the north and south are accommodated by the four saddles located at the top of the tower legs in the main span. On the Tsing Yi side, the main cables are extended from the tower saddles to the main anchorage through the splay saddles, forming a 300 m Tsing Yi side span. On the Ma Wan side, the main cables extended from the Ma Wan tower are held "rst by the saddles on Pier M2 at a horizontal distance of 355.5 m from the Ma Wan tower and then by the main anchorage through splay saddles at the Ma Wan abutment. The bridge deck is a hybrid steel structure continuing between the two main anchorages. It is suspended by suspenders in the main span and the Ma Wan side span. On the Tsing Yi side the deck is supported by three piers rather than suspenders. This arrangement and the di!erence of span between the Ma Wan side and the Tsing Yi side introduce some asymmetry with respect to the midspan of the bridge. A three-dimensional dynamic "nite element model has been established for the Tsing Ma bridge after the completion of deck welding connections [12]. Threedimensional Timoshenko beam elements with rigid arms were used to model the two bridge towers. The cables and suspenders were modelled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel bridge deck was represented by a single beam with equivalent cross-sectional properties determined from a "nite element analysis using detailed sectional models. The connections between bridge components and the supports of the bridge were properly modelled. The modal analysis of the Tsing Ma bridge shows that the natural frequencies of the bridge are spaced very closely. The "rst 20 natural frequencies include the "rst 12 lateral modes, the "rst six vertical modes, and the "rst two torsional modes. They range from 0.068 to 0.380 Hz only. Table 1 lists the "rst 20 natural frequencies and the
Fig. 1. Con"guration of Tsing Ma bridge.
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Table 1 The First 20 Natural Frequencies and Mode Shapes of Tsing Ma bridge Mode number (1)
Mode direction (2)
Natural freq. (Hz) (3)
Nature of mode shape (4)
1 2 3 4 5
L L L L L
0.068 0.158 0.210 0.230 0.232
6 7
L L#T
0.240 0.285
8
L
0.333
9
L
0.340
10
L
0.365
11 12
L L
0.367 0.380
1 2
V V
0.117 0.137
3 4 5 6 1
V V V V T#L
0.189 0.245 0.294 0.325 0.271
2
T
0.311
Deck and cables in main span move in phase (a half-wave) Deck and cables in main span move in phase (one wave) Cables in main span move out of phase (a half-wave) Cables in main span move out of phase (one wave) Deck and cables in main span move out of phase (a halfwave) Deck and cables in main span move out of phase (one wave) Deck and cables in main span move in phase (one and a halfwaves) and couple with torsion Deck and cables in Ma Wan span move in phase (a halfwave) Cables in main span move out of phase (one and a halfwaves) Deck and cables in main span move out of phase (one and a half-waves) Cables in Ma Wan span move out of phase (a half-wave) Cables and deck in Tsing Yi span move in phase (a halfwave) Deck and cables in main span move in one wave Deck and cables in main span move in a half-wave, approximately Deck and cables in main span move in one and a half-waves Deck and cables in main span move in two waves Deck and cables in Ma Wan span move in a half-wave Deck and cables in main span move in two and a half-waves Deck and cable in main span rotate in a half-wave and couple with the seventh lateral mode Deck and cables in main span rotate in one wave
Note: V"vertical; T"torsional; L"lateral.
nature of corresponding mode shapes. These natural frequencies and mode shapes have been veri"ed by the "eld measurement. Among the "rst 20 natural frequencies, the lowest frequency of the bridge is 0.068 Hz, corresponding to the "rst lateral mode of a half-wave with the bridge deck and cables moving in phase in the main span. The "rst vertical mode of the bridge is almost antisymmetric in the main span at a natural frequency of 0.117 Hz in one wave while the second vertical mode is almost symmetric in the main span at a natural frequency of 0.137 Hz in a half-wave approximately. The "rst torsional vibrational mode occurs at a natural frequency of 0.271 Hz in a halfwave in the main span. This mode contains the lateral component that is similar to the seventh lateral mode, whereas the seventh lateral mode contains the torsional component that is similar to the "rst torsional mode. The more information on the structural properties, the "nite element modelling, the natural frequencies and mode shapes of the bridge can be found in Ref. [12].
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3. Formulation for fully coupled bu4eting analysis The following is a brief introduction to the formulation for fully coupled bu!eting analysis [7], which is featured mainly by a combination of "nite element approach and pseudo-excitation method in the frequency domain. The advantages of this combined approach are to readily handle the bridge deck with signi"cantly varying structural properties and mean wind speed along the deck, to make good use of the "nite element bridge models already made for the static and eigenvalue analyses, to naturally include intermode and multimode responses, and to determine wind-induced responses of the bridge deck, towers and cables simultaneously. This approach is however relatively time-consuming compared with the Scanlan's method [2], but the required computing time is quite acceptable if the formulation is intentionally programmed and the modern personal computer is used. The equation of motion of the whole bridge for the fully-coupled bu!eting analysis can be expressed as [7] MYG (t)#CYQ (t)#KY (t)"RP (t),
(1)
in which Y (t) is the total nodal displacement vector of N dimensions including the bridge deck, towers, cables, and other components; M is the N]N total mass matrix; C is the N]N total damping matrix which consists of both aeroelastic damping matrix C!% and structural damping matrix C4 ; K is the N]N total sti!ness matrix 4 4 containing the aeroelastic sti!ness matrix K!% and the structural sti!ness matrix K4 ; 4 4 P is the total aerodynamic loading vector of m dimensions (in general, m;N) for the bridge deck, towers, cables, and other components; and R is the N]m matrix consisting of 0 and 1, which expands the m-dimensional loading vector into the N-dimensional loading vector. The system aeroelastic sti!ness matrix K!% and the system aeroelastic damp4 ing matrix C!% are assembled from the element aeroelastic sti!ness matrices K!% i 4 and the element aeroelastic damping matrices C!% in the same way as the system i structural sti!ness matrix K4 and the system structural damping matrix C4 are 4 4 assembled from the element structural sti!ness matrices K4 and the element structural i damping matrices C4, respectively. The total aerodynamic loading vector can be i obtained by n P"+ T P" , i i,% i
(2)
where T is the co-ordinate transformation matrix for the ith element with the i dimensions equal to the dimension of the system nodal force vector times 12 for a beam element or six for a cable element; P" can be either the aerodynamic force i,% vector of the deck element or the tower element or the cable element in local co-ordinate; and n is the total number of the elements subject to wind loading. The spectral density function matrix of the nodal bu!eting forces acting on the whole
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bridge in the global co-ordinate system is
C
D
S% 1 1 (u) S% 1 2 (u) 2 S% 1 n (u) 11 11 11 S% 2 1 (u) S% 2 2 (u) 2 S% 2 n (u) 11 11 S (u)"T 1 1 TT, 11 F F } F S% n 1 (u) 11
S% n 2 (u) 11
S% n n (u) 11
(3)
where T"[T , T ,2T ,2T ]; the superscript T means the transposition of 1 2 i n a matrix; and S% i j (u) is the cross-spectral density function matrix of the nodal 11 bu!eting forces acting on the ith and jth elements. To reduce the computation e!orts for determining bu!eting response and at the same time to include multimode and intermode e!ects, a pseudo-excitation algorithm is suggested to determine the spectral density function matrix for the bu!eting response. The pseudo-excitation method was proposed by Lin [13] to deal with dynamic response of structures subject to random seismic excitation. In this method, the determination of random response of a structure is converted to the determination of response of the structure under a series of harmonic loads, i.e., the so-called pseudo-excitation. The advantages of this method include the less computation e!ort required, the retention of all cross-correlation terms between normal modes, and the feasibility for the determination of random internal force responses. The basic equations involving in the present bu!eting analysis are listed below, and the detailed derivation can be found in Ref. [7]. The spectral density function matrix S (u) is usually a symmetric matrix, and thus PP this excitation spectral matrix can be decomposed as m (4) S (u)"LHDLT" + d LHLT, PP kk k k k/1 in which L is the lower triangular matrix; D is the diagonal matrix; L is the kth k column of L; and d is the kth diagonal element of D. kk The pseudo-excitations can then be constituted as follows: f "L exp(iu t) (k"1, 2,2, m). k k
(5)
For each pseudo-excitation (harmonic excitation) vector, a pseudo-displacement response vector, Y (u), can be easily and traditionally determined by k Y "H(u)Rf , k k
(6)
H(u)"[!u2M#iuC#K ]~1
(7)
The spectral density function matrix of the displacement response of the bridge can be then proved to be S
YY
m (u)" + d YHYT. kk k k k/1
(8)
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Because the above equation involves one summation only, much less computation e!orts are needed to calculate the response spectral density matrix. Further reduction of computation time can be also achieved if the mode reduction technique is used. Let Q be the modal matrix containing the "rst r mode shapes, and introduce a linear NCr transformation. "Q z , (9) NC1 NCr rC1 in which z is the r]1 generalised displacement response vector. Eq. (1) of N dimensions can be reduced to the equation of r dimensions. Y
M zK (t)#C z5 (t)#K z(t)"P (t). r r r r The pseudo-excitations are then computed according to
(10)
f "QTRL exp(iu t) (k"1, 2,2, m). (11) rk k For each pseudo-excitation vector, the pseudo-displacement response vector, Y (u), is k now determined by Y "QH (u)Rf (k"1, 2,2, m). (12) k r rk The spectral density function matrix for the displacement response attributed to the "rst r modes of vibration is computed according to Eq. (8). The standard deviations of the displacement, velocity, and acceleration of the node can be readily computed according to the random vibration theory, after the auto spectral density functions for each node are determined. Di!erent from the SRSS method (the square root of the sum of the squares of the model response), the pseudo-excitation method retains the cross-correlation terms between the "rst r normal modes. Thus, it is a kind of CQC methods (the complete quadratic combination). This CQC method also makes it possible to use a personal computer to execute the bu!eting analysis of the system of hundreds of degrees of freedom in one or two hours.
4. Aeroelastic and aerodynamic parameters In this section, the aerodynamic and aeroelastic parameters used for the bu!eting analysis of the Tsing Ma bridge are provided. Fig. 2 shows a typical deck section of the Tsing Ma bridge with global axis sign convention. The x-axis is in the longitudinal direction of the bridge, the y-axis is in the lateral direction, and the z-axis is in the vertical direction. The #utter derivatives of the bridge deck corresponding to the 03 wind incidence without tra$c and trains are used, but only the eight #utter derivatives HH and i AH(i"1, 2, 3, 4) were measured from the wind tunnel tests [14]. By "tting the wind i tunnel test results, Figs. 3 and 4 show the eight #utter derivatives related to vertical and torsional motions, respectively, against the reduced wind speed. The reference width of the deck used in the tests was half the width of the deck, i.e., B/2"20.5 m. It is seen that all HH curves are basically negative within the reduced wind speed range i
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Fig. 2. Typical section of bridge deck.
Fig. 3. Flutter derivatives for vertical motion.
concerned. Particularly, the negative HH and HH indicate the positive aerodynamic 1 2 damping, which will enhance the stability and reduce the vertical motion of the bridge deck. Among the AH curves, the AH curve exhibits the positive values in the reduced i 1 velocity ranging from 6 and 12, indicating the possibility of negative aerodynamic damping in the torsional motion. The negative AH curve, on the other hand, indicates 4 that the aeroelastic force from vertical motion may generate positive torsional sti!ness. Fig. 5 shows the aerodynamic force coe$cients of the bridge deck measured from wind tunnel tests [14]. The drag, lift, and moment coe$cients are 0.135, 0.090 and 0.063, respectively, at the wind incidence of 03 with respect to the deck width of
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Fig. 4. Flutter derivatives for torsional motion.
Fig. 5. Drag, lift, and moment coe$cient curves [14].
41 m. The "rst derivatives of the drag, lift, and moment coe$cients with respect to wind incidence at the wind incidence of 03 are !0.253, 1.324, and 0.278, respectively. The aerodynamic admittance function of the bridge is not considered due to the lack of information. For the two bridge towers, the drag coe$cient is taken as 1.5 with respect to the tower width of 9.25 m. The lift and moment coe$cients for the towers as well as the "rst derivatives of the drag, lift and moment coe$cients are not available to the writers. For the main cables, force coe$cients critically depend on the Reynold's number and cable surface roughness. Conservatively, the drag coe$cient of 1.0 is chosen for the cable diameter of 1.1 m. In the spectral density functions of gusty wind, the exponential decay coe$cients C , C , and C are taken as 16, 10 and 0, respectively [15]. The friction velocity of x y z the wind, uH, is chosen as 1.15 m/s in consideration of the open terrain surrounding the bridge. The variation of mean wind speed with height follows a power law,
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;(z)"; (z/z )a with an exponent a of 0.19 for the open terrain. The reference mean r r wind speed, ; , is taken as 25 m/s at the reference height of 60 m, i.e., the bridge deck r level near the tower. This mean wind speed is not the design mean wind speed of the bridge at the deck level, but it is a relatively frequently occurring mean wind speed. At this wind speed level, the vortex shedding response of the bridge deck can be neglected. Furthermore, this mean wind speed level guarantees the range of frequencies for the computation of bu!eting response within the experimental range for #utter derivatives. To maintain a high spectral resolution in the interested frequency range and to avoid a poor estimate of response from numerical integration, a frequency interval about 0.0004 Hz is used within the range from 0.043 to 0.48 Hz. This frequency range covers well the "rst 20 modes of vibration of the bridge. The structural damping ratio is taken as 0.008 for all modes of vibration.
5. Comparison of deck response In this section, the displacement and acceleration responses of the bridge deck along the span are computed using the new formulation and compared with those from the Scanlan's method modi"ed by Jain et al. [2]. To make this comparison possible, only the bridge deck is subjected to bu!eting loading and no bu!eting loading is applied to the towers and cables although the present approach can take all bu!eting loads into consideration. The "rst 20 modes of vibration listed in Table 1 are included in the computation. Each mode of vibration contains three components, but for most of vibrational modes only one component is dominant. The results from both methods encompass multimodes and intermodes contributions as well as aeroelastic e!ects. Figs. 6}8 show variations of standard deviation of lateral, vertical, and angular displacement responses of the deck along the span. It is seen that the displacement responses and their variations along the bridge span obtained from the two methods
Fig. 6. Variations of deck lateral displacement response along span.
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Fig. 7. Variations of deck vertical displacement response along span.
Fig. 8. Variations of deck angular displacement response along span.
are fairly similar. The new formulation gives slightly larger values around the midspan of the bridge than those from the Scanlan's method [2]. This is because in the "nite element-based new formulation, the actual variations of mean wind speed and turbulence along the span are taken into consideration while in the Scanlan's method the mean wind speed and turbulence remain unchanged for the whole bridge [2], although the modi"ed Scanlan's method can also take into account of the varying mean wind speed and turbulence, as done just recently by Katsuchi et al. in Ref. [3]. It is seen that the maximum standard deviation of lateral displacement response of the bridge deck occurs near the midspan and reaches about 0.23 m. The shape of lateral displacement response curve is similar to the "rst lateral mode of vibration of the deck. The maximum standard deviation of vertical displacement response occurs about the quarter of the main span from either bridge tower and attains about 0.11 m.
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The pattern of vertical displacement response curve in the main span does not follow either the "rst vertical mode or the second vertical mode of vibration. The variation of angular displacement response along the main span appears a half wave and the maximum response comes about 0.0018 rad at the midspan. The displacement response curves from single mode of vibration, which are also displayed in Figs. 6}8, will be discussed later. Apart from displacement responses, the lateral, vertical, and angular acceleration responses of the bridge deck from the new formulation are also compared with those from the Scanlan's method [2], and the comparison is quite satisfactory. Fig. 9 shows an example in which the variations of deck vertical acceleration response along the span from the two methods are compared. The satisfactory comparison veri"es the accuracy of the new formulation and the new computer program.
6. Aeroelastic e4ects It is a common belief that self-excited forces (aeroelastic forces) would a!ect bu!eting response of long-span bridges. To understand such e!ects on the Tsing Ma bridge, the response spectra of the bridge deck with and without #utter derivatives are computed using the new formulation. Fig. 10 shows the power spectra of lateral displacement response of the bridge deck at the midspan with and without aeroelastic e!ects from the "rst 20 modes of vibration. The response spectrum from the "rst lateral mode of vibration only is also displayed in Fig. 10. The response peaks appearing in the spectrum without aeroelastic forces clearly identify the "rst, second, and "fth natural frequencies of the bridge in the lateral direction (see Table 1). The third and fourth natural frequencies of the bridge in the lateral direction correspond to cable vibration rather than deck vibration and thus they do not appear in the spectrum. Because lateral #utter derivatives are not considered in the computation, the e!ects of aeroelastic forces are very small
Fig. 9. Variations of deck vertical acceleration response along span.
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Fig. 10. Power spectra of lateral displacement of deck at midspan.
and the spectra with and without aeroelastic forces are similar to each other. However, a preliminary study recently carried out by Katsuchi et al. [3] indicated that the lateral #utter derivative PH would provide a damping e!ect for the lateral bu!eting 1 response of the bridge deck, and the measurement of this parameter together with other lateral #utter derivatives would be an important task for future wind tunnel test programs. It is also seen from Fig. 10 that due to the similarity of mode components between the seventh lateral mode and the "rst torsional mode, the spectrum with aeroelastic forces is slightly di!erent from the spectrum without aeroelastic forces within the frequency range from 0.26 to 0.30 Hz. This indicates that the torsional #utter derivatives indirectly a!ect the lateral bu!eting response of the bridge deck. By comparing the spectrum from the "rst mode of vibration with the spectrum from the "rst 20 modes of vibration, it is seen that in the lateral displacement response, the "rst lateral mode of vibration is dominant, which is consistent with the results shown in Fig. 6. The spectrum without aeroelastic forces in Fig. 11 clearly demonstrates the "rst, second, third, fourth, and sixth natural frequencies of the bridge in the vertical direction. The "fth natural frequency corresponds to the vertical vibration of the Ma Wan side span rather than the main span, and thus it does not appear in the spectrum. Because the "rst vertical mode of the deck is almost antisymmetric, the second vertical mode of vibration becomes dominant for the vertical response of the bridge deck around the midspan. Since both the "rst vertical and the second vertical modes are important, the vertical displacement response of the bridge deck is dominated by these two modes, as seen in Fig. 7. Furthermore, by comparing the spectra with aeroelastic forces and without aeroelastic forces, it is seen that the positive aeroelastic damping reduces the peak response amplitudes signi"cantly, in particular in the lower modes of vibration. The e!ects of aeroelastic sti!ness are not signi"cant since the peak frequencies do not have any clear shift. The aeroelastic coupling of vertical motion with torsional motion leads to an additional small peak appearing on the spectrum near the frequency of 0.29 Hz.
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Fig. 11. Power spectra of vertical displacement of deck at midspan.
Fig. 12. Power spectra of angular displacement of deck at midspan.
Appearing in Fig. 12 are the spectra of angular displacement response of the deck at the midspan. It is seen that the aeroelastic coupling between the vertical and torsional motion signi"cantly a!ects the torsional vibration of the bridge deck. Without aeroelastic forces, two resonance peaks occur at the "rst and second natural frequencies in torsion while another small peak between them is due to the similarity of mode components between the "rst torsional mode and the seventh lateral mode. There is also a very small peak at a frequency of 0.068 Hz due to the torsional component in the "rst lateral mode of vibration. After introducing aeroelastic forces, the spectrum of angular displacement response of the deck at the midspan is signi"cantly changed. There are three new resonance peaks occurring at the "rst three natural frequencies in vertical direction. The largest resonance peak corresponds to the second vertical mode of vibration, indicating that the aeroelastic coupling between the second vertical and "rst torsional modes of vibration is considerable. In fact, due to the aeroelastic
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coupling between the second vertical and the "rst torsional modes, the angular displacement response of the bridge deck is similar to either the "rst torsional mode or the second vertical mode, as shown in Fig. 8. Furthermore, the magnitudes of resonance peaks at the original "rst and second torsional frequencies are slightly reduced, and the frequencies are shifted. If only the "rst torsional mode is used to compute the torsional response spectrum in consideration of aeroelastic forces, the resonance peak reduces slightly but the "rst torsional natural frequency remains unchanged. This further con"rms the aeroelastic coupling between vertical and torsional modes of vibration. Displayed in Fig. 13 are the vertical acceleration response spectra of the bridge deck at the midspan. Compared with the vertical displacement response spectrum in Fig. 10, it is clear that the e!ects of higher modes of vibration are much more signi"cant on acceleration response than displacement response since acceleration resonance peaks in higher modes are compatible with those in lower modes. This is also why the pattern of the vertical acceleration response along the span (see Fig. 9) is much complicated that that of the vertical displacement response (see Fig. 7). Similar phenomena are also observed for the torsional and lateral displacement and acceleration responses of the bridge deck. From the above discussion, it is concluded that the aeroelastic damping considerably reduces the vertical response of the deck. The aeroelastic coupling between the second vertical and the "rst torsional modes of vibration signi"cantly a!ects the torsional vibration of the bridge deck.
7. Multimode and intermode e4ects The spectral analysis of the deck response has already demonstrated that the vertical and torsional displacement responses may not be dominated by a single mode of vibration. To further discuss this matter, the lateral displacement response
Fig. 13. Power spectra of vertical acceleration of deck at midspan.
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attributed to the "rst lateral mode only, the vertical displacement response due to the second vertical mode only, and the angular displacement response arising from the "rst torsional mode only are computed and displayed in Figs. 6}8, respectively, together with the same quantities from the "rst 20 modes of vibration. It is seen that the "rst lateral mode of vibration absolutely dominates the total lateral displacement response of the bridge deck. For the vertical and angular displacement responses, signi"cant di!erences appear in the responses between the single mode of vibration and multimodes of vibration. In particular, for acceleration response, the e!ects from multimodes of vibration become much more signi"cant, as shown in Fig. 9 for the vertical acceleration response. These results are consistent with the spectral analysis. As a result, the multimode approach should be used to determine bu!eting response of the Tsing Ma long suspension Bridge. To investigate e!ects of intermodes, a number of pairs of vibrational modes are selected from lateral, vertical, and torsional motions, respectively, as well as their combination. For each pair of vibrational modes, the deck response along the span from each single mode is computed "rst and then the deck response from the two modes are computed using both the SRSS method and the CQC method (the complete quadratic combination). The CQC method used here is actually the pseudo-excitation method as introduced in Section 3. By comparing the responses from both the methods, inter-mode e!ects can be estimated. Fig. 14 shows the deck vertical displacement responses along the span from the "rst vertical mode only, the second vertical mode only, as well as from both modes using either the SRSS method and the CQC method. The natural frequencies of the "rst and second vertical modes are close. The "rst one is 0.117 Hz and the second one is 0.137 Hz, but the "rst mode shape is almost antisymmetric and the second mode shape is almost symmetric. The SRSS method gives almost the same response as the CQC method. This means that the contribution from the intermodes is almost zero. The intermode e!ects from the "rst and second lateral modes of vibration and from the "rst and second torsional modes of vibration are also found negligible.
Fig. 14. Inter-mode e!ects from "rst and second vertical modes.
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Fig. 15. Inter-mode e!ects from "rst torsional and seventh lateral modes.
The investigation then moves to the "rst torsional mode and the seventh lateral mode. Fig. 15 shows the lateral displacement responses along the span from the lateral component of the "rst torsional mode, the lateral component of the seventh lateral mode, as well as from both components. The lateral component of the "rst torsional mode is similar in shape to the lateral component of the seventh lateral mode of one and a half-waves in the man span. In this case, the SRSS method gives larger lateral displacement response than the CQC method. This is because the lateral displacement response from the "rst torsional mode is out of phase with that from the seventh lateral mode and consequently, the intermode contribution is negative and the CQC method renders smaller response than the SRSS method. For these two modes but the torsional components, it is found that the SRSS method gives slightly smaller response than the CQC method. This is because the torsional component from the "rst torsional mode is in phase with the torsional component of the seventh lateral mode of vibration. Since the "rst mode of vibration is dominant in the lateral response, the intermode e!ects from the seventh lateral and the "rst torsional modes in the direction can still be neglected. The investigation also goes to the typical aeroelastically coupled modes, i.e., the "rst torsional mode and the second vertical mode. This is because the mode shape of the second vertical mode is similar to the "rst torsional mode, and these two modes are aeroelastically coupled. Fig. 16 shows the angular displacement responses along the span from the torsional component of the "rst torsional mode and the torsional component from the second vertical mode. Since the torsional component from the second vertical mode is almost zero, the angular response from this component is almost zero. As a result, the SRSS method yields the same response as the torsional component of the "rst torsional mode. However, the CQC method produces much larger torsional response than the SRSS method. This is because the CQC method includes the aeroelastic coupling between the two modes. This is consistent with the results from the spectral analysis. No other signi"cant aeroelastically coupled mode combinations were found in this study.
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It is thus concluded that for the Tsing Ma bridge, multimode e!ects are signi"cant for the vertical and torsional displacement responses but not for the lateral displacement response. Intermode e!ects are negligible for the two modes from either lateral motion or vertical motion or torsional motion but intermode e!ects are considerable for the aeroelastically coupled vertical and torsional modes of similar mode shapes and for the torsional and lateral modes of similar mode components.
8. Interaction between deck, towers and cables In this section, apart from the bu!eting forces and aeroelastic forces on the bridge deck the bu!eting forces on the towers and the cables are also included in the computation. The results are denoted by the term `full bridgea. The full-bridge results
Fig. 16. Inter-mode e!ects from "rst torsional and second vertical modes.
Fig. 17. Interactive e!ects on deck lateral displacement response.
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Fig. 18. Interactive e!ects on deck vertical displacement response.
Fig. 19. Interactive e!ects on deck angular displacement response.
are then compared with those with the forces on the bridge deck only, the bridge tower only, and the main cable only. In this way, the interaction between the bridge deck, towers, and main cables can be examined. Fig. 17 shows the variations of deck lateral displacement response along the span for the full bridge and for the deck only. It is clear that due to the consideration of bu!eting forces on the main cables and towers, the lateral displacement response of the bridge deck increases about 15% at the midspan. The further computation of the bridge with the forces on the bridge deck and towers only shows that the 15% increase of the lateral displacement response of the bridge deck is almost due to the aerodynamic forces on the main cables. Figs. 18 and 19 show, respectively, the variations of deck vertical and angular displacement responses along the span for the full bridge and for the deck only. Clearly, the bu!eting forces on the main cables and towers do
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not a!ect the vertical and torsional displacement responses of the bridge deck. This may be because the lift and moment coe$cients and their derivatives for the bridge towers are not considered and the time delay of bu!eting forces on the front cable and the back cable is also not taken into consideration in the computation. For the dynamic response of the bridge tower in the full-bridge case, the computed results show that the tower has not only alongwind response but also crosswind and torsional responses. The maximum alongwind displacement of the Tsing Yi tower occurs at the top of the tower about 0.003 m. The maximum crosswind displacement also occurs at the top of tower about 0.009 m. The maximum angular displacement response occurs at about 70 m high with 0.00014 rad. From the computation of the dynamic response of the main cable in the full-bridge case and in the main cable case, it is found that the lateral and vertical displacement responses of the main cable are signi"cantly a!ected by the bridge deck motion. The maximum lateral displacement response of the front cable is 0.24 m in the full-bridge case while the maximum vertical displacement response of the front cable is 0.13 m in the full-bridge case. The longitudinal motion of the main cable is also induced by the bridge deck vibration.
9. Conclusions The fully coupled bu!eting analysis of the Tsing Ma suspension bridge in Hong Kong has been carried out in this paper in the frequency domain. The bu!eting response of the bridge deck only obtained from the suggested approach is in good agreement with that computed using the Scanlan's method. The aeroelastic damping was found to reduce the vertical response of the bridge deck, and the aeroelastic e!ects on the torsional vibration of the Tsing Ma bridge are signi"cant. The multimode e!ects are considerable on the vertical motion and torsional motion but not on the lateral motion of the bridge deck. Intermode e!ects can be neglected for the two modes in either lateral motion or vertical motion or torsional motion but intermode e!ects should be considered for the aeroelastically coupled vertical and torsional modes of similar mode shapes. Finally, the results from the present approach reveals that the bu!eting of bridge deck considerably impacts the bu!eting of towers and main cables whereas the bu!eting of towers and main cables moderately a!ects the lateral vibration of bridge deck. It is the writers' wish that the numerical results presented in this paper will be validated through a comparison with wind tunnel test results and "eld measurement data in near future.
Acknowledgements The writers are grateful for the "nancial supports from the Hong Kong Research Grant Council through a UGC grant (CERG 5027/98E) for the "rst and third writer. The support from the Lantau Fixed Crossing Project Management O$ce, Highways Department of Hong Kong, to allow the writers to access the relevant materials for academic purpose only is particularly appreciated.
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References [1] Y.K. Lin, J.N. Yang, Multimode bridge response to wind excitations, J. Eng. Mech. ASCE 109 (2) (1983) 586}603. [2] A. Jain, N.P. Jones, R.H. Scanlan, Coupled #utter and bu!eting analysis of long-span bridges, J. Struct. Eng. ASCE 122 (7) (1996) 716}725. [3] H. Katsuchi, N.P. Jones, R.H. Scanlan, Multimode coupled #utter and bu!eting analysis of the Akashi-Kaikyo bridge, J. Struct. Eng. 125 (1) (1999) 60}70. [4] K. Ogawa, H. Shimodoi, H. Ishizaki, Aeroelastic stability of a cable-stayed bridge with girder, tower and cables, J. Wind Eng. Ind. Aerodyn. 41}44 (1992) 1227}1238. [5] A.G. Davenport, A simple representation of the dynamics of a massive stay cable in wind, AFPC Conference, Deauville, France, 1994, pp. 427}437. [6] A.G. Davenport, The consistent safety of long span bridges against wind with special reference to Le Pont De Normandie, AFPC Conference, Deauville, France, 1994, pp. 3}14. [7] Y.L. Xu, D.K. Sun, J.M. Ko, J.H. Lin, Bu!eting analysis of long span bridges: a new algorithm, Comput. Struct. 68 (1998) 303}313. [8] T.J.A. Agar, Dynamic instability of suspension bridges, Comput. Struct. 41 (6) (1991) 1321}1328. [9] A. Namini, P. Albrecht, H. Bosch, Finite element-based #utter analysis of cable-suspension bridges, J. Struct. Engi. 118 (6) (1992) 1509}1526. [10] M.S. Pfeil, R.C. Batista, Aeroelastic stability analysis of cable-stayed bridges, J. Struct. Eng. ASCE 121 (12) (1995) 1784}1794. [11] A.S. Beard, J.S. Young, Aspect of the design of the Tsing Ma bridge, Proceedings of the International Conferences on Bridge into 21st Century, Impressions Design and Print Ltd., Hong Kong, 1995, pp. 93}100. [12] Y.L. Xu, J.M. Ko, W.S. Zhang, Vibration studies of Tsing Ma suspension bridge, J. Bridge Eng. ASCE 2 (4) (1997) 49}156. [13] J.H. Lin, A fast CQC algorithm of PSD matrices for random seismic responses, Comput. Struct. 44 (3) (1992) 683}687. [14] C.K. Lau, K.Y. Wong, Aerodynamic stability of Tsing Ma Bridge, Proceedings of the Fourth International Kerensky Conference Hong Kong, Sept., 1997, pp. 131}138. [15] E. Simiu, R.H. Scanlan, Wind E!ects on Structures, Wiley, New York, 1996.