Field observations of rain-wind-induced cable vibration in cable-stayed Dongting Lake Bridge

ARTICLE IN PRESS

Journal of Wind Engineering and Industrial Aerodynamics 95 (2007) 303–328 www.elsevier.com/locate/jweia

Field observations of rain-wind-induced cable vibration in cable-stayed Dongting Lake Bridge Y.Q. Nia,, X.Y. Wanga,1, Z.Q. Chenb, J.M. Koa a

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b School of Civil Engineering, Hunan University, Changsha, Hunan 410082, PR China Received 15 March 2004; received in revised form 1 July 2006; accepted 18 July 2006 Available online 7 September 2006

Abstract The cable-stayed Dongting Lake Bridge has been observed several times to exhibit large-amplitude cable oscillation under simultaneous action of rain and wind. To investigate excitation mechanism and response characteristic of the rain-wind-induced stay vibration, a series of field measurements lasting 45 days have been conducted on the Dongting Lake Bridge by deploying accelerometers, anemometers and rain gauge for continuous monitoring. This paper presents the measurement results of rain and wind excitations as well as dynamic response of a typical stay in three rain-wind excitation events. The measurement data show that under specific combination of rain and wind, the maximum acceleration response of the cable reaches 10 g and the maximum displacement response (peak-to-peak) is around 0.7 m. It is revealed that the large-amplitude rain-wind-induced stay oscillation occurs in the bridge when the mean wind velocity at deck level ranges from 6 to 14 m/s, the wind attack angle (relative yaw angle) ranges from 101 to 501, and the rainfall is light to moderate (less than 8 mm/h). For the observed cable, the overall dominant mode of cable vibration during rain-wind excitations is the third mode. However, in the evolution process of this kind of vibration, the dominant mode may differ for different response segments. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cable-stayed bridge; Cable vibration; Rain-wind excitation; Field measurement

Corresponding author. Tel.: +852 2766 6004; fax: +852 2334 6389.

E-mail address: [email protected] (Y.Q. Ni). On leave from School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China. 1

0167-6105/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2006.07.001

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1. Introduction Owing to large flexibility, relatively small mass and extremely low inherent damping, cables in cable-supported bridges are susceptible to vibration caused by various excitation mechanisms. For example, incidences of large-amplitude cable oscillation in cable-stayed bridges under specific combinations of rain and wind have been reported worldwide (Hikami and Shiraishi, 1987; Pacheco and Fujino, 1993; Matsumoto et al., 1995a; Poston, 1998; Irwin et al., 1999; Persoon and Noorlander, 1999; Main et al., 2001; Chen and Tanaka, 2002; Ni et al., 2002). This kind of large-amplitude vibration may cause reduced life of the cables and their connections due to fatigue and breakdown of protections against corrosion (Pacheco and Fujino, 1993; Poston, 1998), and invoke the risk of losing public confidence to the bridges (Persoon and Noorlander, 1999). In the past decade, the mitigation of rain-wind-induced cable vibration by means of aerodynamic, mechanical and structural means has been extensively studied. Although a lot of research efforts have been made, the excitation mechanism of rainwind-induced cable vibration is still an imperfectly understood phenomenon (Matsumoto et al., 1995b; Verwiebe and Ruscheweyh, 1998). The research on rain-wind excitation mechanisms has been conducted by wind tunnel testing (Hikami and Shiraishi, 1987; Ohshima and Nanjo, 1987; Matsumoto et al., 1990, 1992; Kinoshita et al., 1991; Flamand, 1995; Honda et al., 1995; Bosdogianni and Olivari, 1996), analytical and numerical modeling (Yamaguchi, 1990; Geurts et al., 1998; Verwiebe, 1998; Ruscheweyh, 1999; Wang and Xu, 1999; Peil et al., 2003), and field observation (Matsumoto et al., 1989, 2003; Main and Jones, 1999; Persoon and Noorlander, 1999; Main et al., 2001; Schwarzkopf and Sedlacek, 2003). The proposed excitation mechanisms include water rivulet formation on upper surface of a cable, axial flow in a wake of cable, low-frequency vortex shedding along the cable axis, and vortex-induced vibration at high reduced wind speed. In order to verify the validity of these proposed excitation mechanisms, understanding the excitation and response features of rain-wind-induced cable vibration actually occurring in the field becomes extremely important. Although visible observations of the rain-wind-induced vibration have been reported for numerous cable-stayed bridges, in situ measurement data of this kind of cable vibration at full-scale conditions are seldom available. The Dongting Lake Bridge is a three-tower prestressed concrete cable-stayed bridge recently built in Hunan, China. It has a total length of 880 m, consisting of two main spans of 310 m each and two side spans of 130 m each. The central tower is 125.7 m high and the side towers are 99.3 m high each. The bridge deck is 23.4 m wide with four traffic lanes, with the clearance height of 25.0 m above water level. It is supported by 222 cables of size ranging from 28 to 201 m in length and 99 to 159 mm in diameter, which are coated into smooth surface with polyethylene (PE). Shortly after opening to traffic in the end of 2000, the bridge has been observed several times to exhibit severe cable vibration under low wind speed and light-to-moderate rain. The frequent occurrence of such rain-wind-induced vibration has agitated the administrative authority and management engineers who finally adopted MR-based damping technology for cable vibration mitigation (Chen et al., 2004). It also provides a good test-bed for in situ measurement of excitation and response characteristics of cable vibration under the rain-wind excitation conditions and other excitation conditions. For this purpose, a typical cable of 122 m long was selected for monitoring, and the MR dampers were intentionally dismantled from the cable to cater for vibration. An instrumentation system, consisting of 15 accelerometers, one displacement

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transducer, two tri-axial anemometers, one rain gauge, and one data acquisition and processing system, has been used for continuous 45-day measurements in the bridge site. This paper presents the measurement data during three typical rain-wind excitation events and analysis results of cable response at different time segments. The measurement databased analyses on rain-wind excitation mechanism (mainly on the mechanism of vortexinduced cable vibration at high reduced wind velocity) and on wind-induced and trafficinduced cable oscillation will be reported separately. 2. Description of measured cable and instrumentation system Fig. 1 shows the elevation and plan of the Dongting Lake Bridge. The bridge axis direction is NNW201. The previous observations on the Dongting Lake Bridge indicated that rain-wind-induced cable vibration in this bridge occurred commonly under a combined action of yawed north wind and rain, and behaved as large-amplitude oscillation at all stays declining in the direction of the wind flow and small or invisible oscillation at the cables inclining in the direction of the wind flow. Hence, the cable A12, as shown in Fig. 1, was selected for response measurement. Fig. 2 illustrates the cable A12 after the dampers are dismantled. The main parameters of the cable A12 are as follows: length L ¼ 121.9 m, inclination angle a ¼ 35:21, diameter D ¼ 119 mm, initial tension T ¼ 3150 KN, mass per unit length m ¼ 51.8 kg/m, elastic modulus E ¼ 2.0  105 MPa. The first four in-plane modal frequencies of the cable are 1.07, 2.14, 3.20 and 4.23 Hz, respectively. Referring to Fig. 3, the instrumentation system includes the following primary components:

    

One three-axis ultrasonic anemometer at the top of south side tower, 2 m high above the tower top (Fig. 3(a)). It is situated at an elevation of 102 m; One three-axis ultrasonic anemometer at deck level near by the cable A12, 4 m stretching out from the deck edge with a horizontal cantilever (Fig. 3(b)). It is situated at an elevation of 26 m; One rain gauge at deck level near by the cable A12 (Fig. 3(c)); Four uniaxial accelerometers on the locations of L/6 and L/20 from the lower anchorage for cable in-plane and out-of-plane acceleration measurement (Fig. 3(d)); Eight uniaxial accelerometers on neighboring MR-damped cables A10, A11, A13 and A14 for in-plane and out-of-plane acceleration measurement; Anemometer Cable A12 Yueyang city

Huarong 130 m

310 m

310 m

130 m

Elevation Wind

N

θ

Bridge axis

Plan

Fig. 1. Dongting Lake Bridge and deployment of sensors.

Cable A12 Anemometer on deck level Rain gauge

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Fig. 2. Cable A12 with dampers being dismantled.

Fig. 3. Sensor installation for field measurement: (a) anemometer at tower top; (b) anemometer at deck level; (c) rain gauge at deck level; and (d) accelerometers on cable A12.

 

Two uniaxial accelerometers at south side tower near by the upper anchorage of cables A11 and A13 for tower longitudinal acceleration measurement; One uniaxial accelerometer at the deck for vertical acceleration measurement;

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One vibration transducer at the deck for vertical displacement measurement; One data acquisition and processing system in the bridge site.

This system was settled in situ in mid-March 2003. Continuous field measurements were conducted from 24 March to 11 May 2003. 3. Correlation of wind velocities at different heights Wind velocity and direction were measured by two three-axis ultrasonic anemometers at the tower top and deck level, respectively. Fig. 4 shows a sample of wind velocity records and 3-min mean wind velocities at the two heights. It is seen that the characteristic of wind velocity is basically identical at the tower top and at deck level. A statistical analysis indicates that the ratio of mean wind velocities at the tower top and at deck level is around 1.55. The correlation of wind velocities at the two heights can be expressed approximately as  a z1 Uðz1 Þ ¼ Uðz2 Þ , (1) z2 where z1 and z2 denote height from the ground; U(z1) and U(z2) are wind velocity at the heights z1 and z2, respectively; a is an exponential coefficient. The wind fetch around the bridge is almost uniform. With the measurement data, a is estimated by curve fitting to be 0.2723. In the following, only the wind velocity and direction data measured at deck level are presented and analyzed. The orientation of the anemometers was calibrated along the bridge axis, namely, when wind direction has an angle y with respect to the bridge axis (refer to Fig. 1), the direction output of the anemometers is also y. Considering the inclination angle of stay cable and orientation angle between stay axis and bridge axis in plan view, the wind direction will be represented by the relative yaw angle bn defined as (Matsumoto et al., 1995b; Main et al., 2001) bn ¼ sin1 ðsin b cos aÞ,

(a)

(b)

:30 21

:00

:00

:30

:30

21

20

20

19

:00

:30

0 19

:30 21

:00

:30

Time (1 April 2003)

21

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:30

:00

:30

:00 20

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19

:00

:30

18

18

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:00

0

5

:00

5

10

18

10

15

:30

15

Tower top Deck level

20

:00

20

25

18

Tower top Deck level

17

Wind velocity (m/s)

25

17

3-minutes mean wind velocity (m/s)

(2)

Time (1 April 2003)

Fig. 4. Measured wind velocities at tower top and deck level: (a) records from anemometers; and (b) 3-min mean wind velocities.

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Wind *





Fig. 5. Definition of yaw and inclination angles.

where b is the horizontal yaw angle and a the inclination angle of stay cable. The relative yaw angle bn defines relative angle between the wind direction and the cable axis as shown in Fig. 5. It is equal to zero when the wind direction is perpendicular to the cable axis and equal to 90o when the wind direction is parallel to the cable axis. A positive bn corresponds to the cable declining in the direction of the wind flow, while a negative bn corresponds to the cable inclining in the direction of the wind flow. 4. Measurement results under rain-wind excitations Three typical rain-wind excitation events were observed on 1–2 April, 18–19 April and 28–29 April 2003, respectively. In the first rain-wind excitation event, large-amplitude stay cable vibration occurred with intermission from 4:00 pm of 1 April to 3:00 am of 2 April under a combined action of wind and rain. Fig. 6 shows the time history of in-plane and out-of-plane acceleration responses of the cable A12 at L/6 location during the first rainwind excitation event. In this event, the maximum cable acceleration response approached to 10 g. Fig. 7 illustrates the corresponding wind velocity, wind direction and rainfall, respectively. By comparing Figs. 6 and 7, it is observed that when the rain stopped occasionally (19:10–20:10 and 21:50–22:20 of 1 April) or the wind direction changed dramatically (0:05–0:25 of 2 April), the cable ceased to vibrate or reduced its response suddenly. The cable vibration amplitude varied significantly with the weather condition and large-amplitude oscillation could not keep a stationary level for a long time. Fig. 8 shows the power spectral density of cable in-plane and out-of-plane acceleration responses. It is evident that the dominant mode of both in-plane and out-of-plane responses for the cable in this rain-wind excitation event is the third mode. The second and fourth modes also participate in the cable vibration, but the fundamental mode (first mode) has no significant contribution to the rain-wind-excited cable response. In the second rain-wind excitation event, large-amplitude stay cable vibration occurred discontinuously from 8:00 am of 18 April to 7:00 am of 19 April under simultaneous wind and rain. Fig. 9 shows a time segment of in-plane and out-of-plane acceleration responses of the cable A12 at L/6 location during the second rain-wind excitation event. The largest cable acceleration response was found about 10 g. Fig. 10 illustrates the corresponding wind velocity, wind direction and rainfall, respectively. It is seen that when the wind velocity was low (much less than 10 m/s) or the wind direction deviated considerably, the

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0

30 0:

:3

0 :3

:3

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18

0:

1:

30

0 :3

0 23

:3

0

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:3

22

21

0

:3 20

0

:3 19

:3 18

Time (18:30 1 April to 1:30 2 April 2003)

(a)

23

-8 -10

0

-6

22

-4

:3

0 -2

0

2

21

4

:3

6

10 8 6 4 2 0 -2 -4 -6 -8 -10

0

Out-of-plane acceleration (g)

In-plane acceleration (g)

8

20

10

309

Time (18:30 1 April to 1:30 2 April 2003)

(b)

Fig. 6. Time history of cable response in first rain-wind excitation event: (a) in-plane acceleration; and (b) out-ofplane acceleration.

90 Wind direction β* (degree)

10

5

(a)

70 60 50 40 30 20 10 0 30 1:

30 0:

0

0

:3 23

:3 22

0 :3

0 21

:3

0 20

18

(b)

Time (18:30 1 April to 1:30 2 April 2003)

:3

0 :3

30 1:

0:

30

0 :3

0 23

0

:3 22

:3

0 :3

21

20

:3

0

-10 19

18

:3

0

0

80

19

Wind velocity (m/s)

15

Time (18:30 1 April to 1:30 2 April 2003)

6

Rainfall (mm/h)

5 4 3 2 1

(c)

0 1: 3

0 0: 3

0

30

23 :3

22 :

30 21 :

30 20 :

30 19 :

18 :

30

0

Time (18:30 1 April to 1:30 2 April 2003)

Fig. 7. Wind velocity, wind direction and rainfall in first rain-wind excitation event: (a) wind velocity at deck level; (b) wind direction (b*); and (c) rainfall.

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×103 PSD 0f out-of-plane acceleration (g2/Hz)

PSD of in-plane acceleration (g2/Hz)

310

12 10 8 6 4 2 0 0

(a)

1

2

3 4 5 6 7 Frequency (Hz)

8

4000 3500 3000 2500 2000 1500 1000 500 0

9 10

0 (b)

1

2

3 4 5 6 7 Frequency (Hz)

8

9 10

Fig. 8. Power spectral density of cable response in first rain-wind excitation event: (a) in-plane acceleration; and (b) out-of-plane acceleration.

10 5 0 -5 -10 -15 10:10

(a)

15 Out-of-plane acceleration (g)

In-plane acceleration (g)

15

10:40 11:10 11:40 Time (18 April 2003)

10 5 0 -5 -10 -15 10:10

12:10 (b)

10:40 11:10 11:40 Time (18 April 2003)

12:10

Fig. 9. Time history of cable response in second rain-wind excitation event: (a) in-plane acceleration; and (b) outof-plane acceleration.

cable exhibited very small response. It is worth mentioning that even in the case of small amount of rainfall, large cable vibration might be generated with proper wind velocity and wind direction. Fig. 11 shows the power spectral density of cable in-plane and out-of-plane acceleration responses. It is indicated again that the dominant mode of both in-plane and out-of-plane responses for this cable under rain-wind excitation is the third mode. In the third rain-wind excitation event, large-amplitude stay cable vibration occurred from 11:00 pm of 28 April to 3:00 am of 29 April with combined wind and rain. Fig. 12 shows a time segment of in-plane and out-of-plane acceleration responses of the cable A12 at L/6 location during the third rain-wind excitation event. The maximum cable acceleration response exceeded 10 g. Fig. 13 illustrates the corresponding wind velocity, wind direction and rainfall, respectively. It is evident that when the wind velocity was lower than a certain value or the wind direction deviated beyond a certain range, the cable ceased to vibrate.

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90 Wind direction β* (degree)

Wind velocity (m/s)

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10:40

11:10

11:40

30

0

-30 10:10

12:10

Time (18 April 2003)

(a)

60

10:40

11:10

11:40

12:10

Time (10:10 ~12:10, 18 April 2003)

(b)

Rainfall (mm/h)

40

30

20

10

0 10:10

(c)

10:40 11:10 11:40 Time (18 April 2003)

12:10

8000

PSD of out-of-plane acceleration (g2/Hz)

PSD of in-plane acceleration (g2/Hz)

Fig. 10. Wind velocity, wind direction and rainfall in second rain-wind excitation event: (a) wind velocity at deck level; (b) wind direction (b*); and (c) rainfall.

6000 4000 2000 0 0

(a)

1

2

3 4 5 6 7 Frequency (Hz)

8

9 10

1500

1000

500

0 0

(b)

1

2

3 4 5 6 7 8 Frequency (Hz)

9 10

Fig. 11. Power spectral density of cable response in second rain-wind excitation event: (a) in-plane acceleration; and (b) out-of-plane acceleration.

Fig. 14 shows the power spectral density of cable in-plane and out-of-plane accelerations. The dominant mode of cable in-plane and out-of-plane responses in this event is the third mode again. It is, therefore, concluded that rain-wind-induced vibrations

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15 Out-of-plane acceleration (g)

In-plane acceleration (g)

15 10 5 0 -5 -10 -15 0:20 (a)

0:50 1:20 Time (29 April 2003)

10 5 0 -5 -10 -15 0:20

1:50 (b)

0:50 1:20 Time (29 April 2003)

1:50

90

0:50 1:20 Time (29 April 2003)

Rainfall (mm/h)

(a)

20 18 16 14 12 10 8 6 4 2 0 0:20

Wind direction β* (degree)

Wind velocity (m/s)

Fig. 12. Time history of cable response in third rain-wind excitation event: (a) in-plane acceleration; and (b) outof-plane acceleration.

(c)

10 9 8 7 6 5 4 3 2 1 0 0:20

60

30

0

-30 0:20

1:50 (b)

0:50

1:20

0:50 1:20 Time (29 April 2003)

1:50

1:50

Time (29 April 2003)

Fig. 13. Wind velocity, wind direction and rainfall in third rain-wind excitation event: (a) wind velocity at deck level; (b) wind direction (b*); and (c) rainfall.

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PSD of in-plane acceleration (g2/Hz)

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9000

6000

3000

0 0

1

2

3

4

5

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7

8

2500 2000 1500 1000 500 0

9 10

Frequency (Hz)

(a)

313

0

1

2

3

4

5

6

7

8

9 10

Frequency (Hz)

(b)

RMS of out-of-plane acceleration (g)

Fig. 14. Power spectral density of cable response in third rain-wind excitation event: (a) in-plane acceleration; and (b) out-of-plane acceleration.

RMS in-plane acceleration (g)

5 4 3 2 1 0 0 (a)

3 6 9 12 1 min mean wind velocity (m/s)

5 4 3 2 1 0

15

0 (b)

3 6 9 12 1 min mean wind velocity (m/s)

15

Fig. 15. Cable RMS response versus 1-min mean wind velocity: (a) in-plane acceleration; and (b) out-of-plane acceleration.

3 2 1 0 5 4 3 2 1 0 -90

(a)

RMS of out-in-plane acceleration (g)

RMS in-plane acceleration (g)

4

-60

-30

0

30

60

1 min mean wind direction β* (degree)

5 4 3 2 1 0 -90

90

(b)

-60

-30

0

30

60

90

1 min mean wind direction β* (degree)

Fig. 16. Cable RMS response versus 1-min mean wind direction (b*): (a) in-plane acceleration; and (b) out-ofplane acceleration.

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occurring in the cable A12 are dominated by the third mode. However, it is worth mentioning that for different cables (even adjacent cables with similar configurations) in a cable-stayed bridge, their dominant modes in rain-wind-induced vibrations may differ with each other (Ni et al., 2002). A statistical analysis has been conducted with the measurement data from all three rainwind excitation events. For the convenience of comparison with measurement data obtained by other investigators (Main et al., 2001; Main and Jones, 2001), the recorded cable acceleration signals were divided at 1-min intervals and the root-mean-square (RMS) response of the cable for each interval is obtained against the corresponding 1-min mean wind velocity, 1-min mean wind direction and amount of rainfall, respectively. Figs. 15–17 plot the cable RMS response versus mean wind velocity, mean wind direction (bn ) and rainfall, respectively. It is evident from the figures that the critical mean wind velocity for RMS of out-in-plane acceleration (g)

RMS in-plane acceleration (g)

5 4 3 2 1 0 0 (a)

10

20

30

Rainfall (mm/h)

40

50

5 4 3 2 1 0 0

(b)

10

20

30

40

Rain (mm/h)

Fig. 17. Cable RMS response versus rainfall: (a) in-plane acceleration; and (b) out-of-plane acceleration.

Fig. 18. Cable RMS in-plane acceleration versus reduced wind velocity.

50

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1-minute RMS of in-plane acceleration (g)

5

y=2x

4

3

2

1

0 0

0.5 1.5 1 2 1-minute RMS of out-of-plane acceleration (g)

2.5

Fig. 19. Cable RMS in-plane acceleration versus RMS out-of-plane acceleration.

Table 1 Measured and design values of turbulence intensity Time

April 1–2

April 18–19

April 28–29

Location

16:51–21:36 16:51–21:36 22:10–02:48 22:10–02:48 7:31–9:15 9:17–12:29 12:30–11:57 21:32–03:22

Deck (20pzp30) Tower (100pzp150) Deck (20pzp30) Tower (100pzp150) Deck (20pzp30) Deck (20pzp30) Deck (20pzp30) Deck (20pzp30)

Measured value

Design value

Iu

Iv

Iw

Iu

Iv

Iw

0.0916 0.0655 0.0938 0.0744 0.0948 0.1060 0.1178 0.1119

0.0855 0.0528 0.0988 0.0767 0.0912 0.0874 0.0945 0.0909

0.0778 0.0747 0.0824 0.0792 0.0681 0.0655 0.0601 0.0561

0.13 0.10 0.13 0.10 0.13 0.13 0.13 0.13

0.11 0.09 0.11 0.09 0.11 0.11 0.11 0.11

0.07 0.05 0.07 0.05 0.07 0.07 0.07 0.07

rain-wind-induced cable vibrations is 6–14 m/s and the critical mean wind direction (relative yaw angle bn ) ranges from 101 to 501. The rain-wind-induced cable vibrations with large amplitude occurred in light to moderate rain (less than 8 mm/h). During the 45-day field measurements, no large-amplitude cable vibration was observed in the case of heavy rain as wind velocity was often reduced to 1–2 m/s when heavy rain came down. Fig. 18 illustrates RMS in-plane acceleration response versus reduced wind velocity (V/fD) where V is the wind velocity, f the vibration frequency and D the cable diameter. It is seen that the large-amplitude responses correspond to the reduced wind velocity ranging from 15 to 35. Fig. 19 shows the relation of RMS in-plane acceleration to RMS out-of-plane acceleration of the cable under rain-wind excitations. It is found that the in-plane acceleration response amplitude is approximately two times the out-of-plane acceleration response amplitude. Table 1 provides the measured turbulence intensities at the tower top and the deck level during the three rain-wind excitation events, where Iu indicates the turbulence intensity in

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the longitudinal direction, Iv the turbulence intensity in the lateral direction, and Iw the turbulence intensity in the vertical direction. For comparison, the corresponding design values are also listed. 5. Analysis of response segments As afore illustrated, the cable vibration response during a rain-wind excitation event does not perform as monotone increasing and then decreasing. Instead, it consists of numerous segments each of which accommodates rising and descending of the oscillation. For example, the recorded cable response in the first rain-wind excitation event includes more than 20 segments. In order to understand response properties of the cable at different segments, an analysis is conducted on three typical response segments obtained from the first rain-wind excitation event. 5.1. Response segment 1 Fig. 20(a) illustrates time histories of the 1600-s response segment 1 starting at 6:40 pm of 1 April 2003, while Fig. 20(b) shows the 3-s records corresponding to large-amplitude quasi-stationary response within this segment. The maximum in-plane acceleration of the cable in this segment is about 1.5 g and the corresponding wind velocity recorded at deck level is around 7–8 m/s. Fig. 21 shows the power spectral density of in-plane and out-ofplane acceleration responses at different time intervals (each interval is 200 s). It is evident that the dominant frequency of this response segment is 1.07 Hz, which corresponds to the first modal frequency of the cable. It is also observed from the spectral diagrams that the cable vibration is first tuned to a dominant frequency, and then its amplitude increases rapidly with the effect of wind and rain. The acceleration response is mainly attributed to the first three modes. The response segment 1 is one of very few cases where the first mode is the dominant mode of cable vibration. Based on the measured acceleration response data, displacement response modal components of the cable in its quasi-steady state are estimated. Assume that the acceleration response in quasi-steady state can be approximated in the form of n components as a¼

n X

Ai sinðoi t þ fi Þ,

(3)

i¼1

where a is the cable acceleration response; Ai the ith modal component of acceleration response; oi the angular frequency of the ith mode; and fi denotes the phase angle of the ith mode. Because rain-wind-induced cable vibration is dominated by only first few modes, the acceleration response a can be well approximated with a combination of the first 5–10 modes. With the known modal frequencies oi, the parameters Ai and fi in Eq. (3) can be obtained by curve fitting to the measured data. Then modal components of the displacement response are estimated from the identified acceleration modal components. Fig. 22 shows a comparison of the measured in-plane acceleration response and the response reproduced by the identified first five modal components for the 3-s quasi-

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2

2

1

In-plane Out-of-plane Acceleration (g)

Acceleration (g)

In-plane Out-of-plane

0

1

0

-1

-1

-2 0

400

800

1200

Time (s)

(a)

-2 800

1600

(b)

801

802

803

Time (s)

(a)

200 150 100 50 0 10 Fre 8 6 que ncy 4 2 0 0 (Hz )

1600 1200 800 400 (s) Time

PSD of out-of-plane acceleration (g2/Hz)

PSD of in-plane acceleration (g2/Hz)

Fig. 20. Time history of cable acceleration response (segment 1): (a) signals in 1600 s; and (b) signals in 3 s.

(b)

30 25 20 15 10 5 0 10 Fre 8 6 que 4 ncy 2 0 0 (Hz )

1200 800 ) 400 e (s Tim

1600

Fig. 21. Power spectral density of cable acceleration response (segment 1): (a) in-plane acceleration; and (b) outof-plane acceleration.

stationary response. A good agreement between the measured and simulated responses is observed. After modal components of the displacement response at the measurement point are obtained by the above method, the displacement response at arbitrary location of the cable in quasi-steady state can be estimated by modal combination with the aid of theoretical mode shapes. Fig. 23 illustrates the cable in-plane displacement response and its modal components at measurement point (L/6) and at mid-span for the 3-s quasi-stationary response. When the cable vibration is dominated by the first mode, the maximum displacement response is expected to occur at mid-span. Therefore, from the plotted midspan response curve we may evaluate the cable maximum displacement response for segment 1. It is seen from Fig. 23 that the first mode contributes the overwhelming majority of displacement response. The cable displacement response amplitudes (peak-topeak) are estimated to be 0.34 m at L/6 location and 0.66 m at mid-span, which correspond to 2.9 times and 5.5 times diameter of the cable. Fig. 24 plots the locus diagrams of acceleration and displacement responses at L/6 location. The displacement locus is

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318

10 Measured Fitted In plane acceleration (g)

5

0

-5

-10

-15 800

801

802

803

Time (s)

0.2

0.1

0

-0.1

-0.2 800

(a)

Total displacement Mode 1 component Mode 2 component Mode 3 component Mode 4 component Mode 5 component

801

802 Time (s)

0.2 In-plane diplacement at L/6 (m)

In-plane diplacement at L/6 (m)

Fig. 22. Comparison between measured and fitted in-plane acceleration (segment 1).

0.1

0

-0.1

-0.2 800

803

(b)

Total displacement Mode 1 component Mode 2 component Mode 3 component Mode 4 component Mode 5 component

801

802

803

Time (s)

Fig. 23. Estimated cable displacement response (segment 1): (a) at L/6 location; and (b) at L/2 location.

approximately a single ellipse, while the acceleration locus is much more complicated due to the exaggerated contribution of higher modes. Due to the negative aerodynamic damping generated by combined wind and rain, the cable oscillation can be accelerated to very large amplitude within a few minutes. The net damping ratio of the cable in a mode can be expressed as the sum of mechanical damping ratio and aerodynamic damping ratio (Main and Jones, 2001): xnet ¼ xmech þ xaero ,

(4)

where xmech and xaero are the mechanical damping ratio and the aerodynamic damping ratio, respectively. Under specific combinations of wind and rain, xaero becomes possibly negative damping. As long as the negative aerodynamic damping exceeds the mechanical damping, the oscillation amplitude increases dramatically. The mechanical damping of the

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0.2 In-plane displacement (m)

1.5 In-plane acceleration (g)

319

1 0.5 0 -0.5 -1

(a)

-0.5 0 0.5 Out-of-plane acceleration (g)

0

-0.1

-0.2 -0.08

-1.5 -1

0.1

1 (b)

-0.04 0 0.04 Out-of-plane displacement (m)

0.08

Fig. 24. Vibration locus of cable response at L/6 location (segment 1): (a) acceleration; and (b) displacement.

Table 2 Measured mechanical damping for in-plane vibration modes Mode no.

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

Damping ratio (%)

0.178

0.157

0.122

0.097

0.108

0.104

0.076

0.080

0.079

0.086

cable A12 has been measured by ambient vibration and forced vibration tests (Ko et al., 2002). Table 2 shows the identified modal damping ratios for the first ten in-plane vibration modes of the cable. It is very difficult to accurately evaluate aerodynamic damping based on acceleration response measurement during rain-wind excitation. In this study, net negative damping ratio in a mode is roughly estimated from the identified dominant modal displacement components at different instances. For response segment 1, the first modal displacement component for each 3-s response is estimated at intervals of 50 s. As shown in Fig. 25, the displacement response amplitude increases continuously from 400 to 850 s. By fitting the identified displacement amplitudes with an exponentially increasing function, the net negative damping ratio for the first mode is estimated to be 0.0548%. 5.2. Response segment 2 Fig. 26(a) illustrates time histories of the 1600-s response segment 2 starting at 8:10 pm of 1 April 2003, while Fig. 26(b) shows the 3-s records corresponding to large-amplitude quasi-stationary response within this segment. The cable largest in-plane acceleration response in this segment is nearly 2 g and the corresponding wind velocity recorded at deck level is about 8 m/s. Fig. 27 shows the power spectral density of in-plane and out-of-plane acceleration responses at different time intervals. It is found that the dominant frequency of this response segment is 2.14 Hz, which corresponds to the second modal frequency of the cable. Similarly, it is observed that the cable vibration is first tuned to a dominant frequency, and then increases dramatically in the dominant mode. Other modes have

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Displacement amplitude of mode 1 (m)

320

0.2 Displacement amplitude Fitted curve (0.0548%)

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0 400 450 500 550 600 650 700 750 800 850 Time (s) Fig. 25. Estimation of net damping ratio of first mode (segment 1).

3

3 In-plane Out-of-plane

1 0 -1

1 0 -1 -2

-2

-3 1200

-3 0 (a)

In-plane Out-of-plane

2 Acceleration (g)

Acceleration (g)

2

400

800 Time (s)

1200

1600 (b)

1201

1202

1203

Time (s)

Fig. 26. Time history of cable acceleration response (segment 2): (a) signals in 1600 s; and (b) signals in 3 s.

contributions to the cable vibration only when the vibration amplitude is considerably large. The response is mainly attributed to the first five modes. The displacement modal components for response segment 2 are obtained by the same method as aforementioned. When the cable vibration is dominated by the second mode, the maximum displacement response should be at the location of quarter-span. Fig. 28 shows the cable in-plane displacement response and its modal components at measurement point (L/6) and at the location of L/4 away from the lower end for the 3-s quasi-stationary period. It is seen from Fig. 28 that although the second mode is dominant, the first mode also contributes significantly to the cable response. The peak-to-peak displacement response amplitudes are 0.20 m at L/6 location and 0.25 m at L/4 location, which correspond to 1.7 times and 2.0 times diameter of the cable, respectively. Fig. 29 plots the locus diagrams of acceleration and displacement responses at L/6 location. In this case,

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600 400 200 0 10 8 Fre 6 4 que 2 ncy (Hz 0 0 )

(a)

400

1600 1200 800 ) e (s Tim

PSD of out-of-plane acceleration (g2/Hz)

PSD of in-plane acceleration (g2/Hz)

Y.Q. Ni et al. / J. Wind Eng. Ind. Aerodyn. 95 (2007) 303–328

321

200 150 100 50 0 10

8 6 4 que 2 ncy (Hz 0 0 )

Fre

(b)

400

1200 (s) ime

1600

800

T

0.2 0.15

In-plane displacement at L/4 (m)

In-plane displacement at L/6 (m)

Fig. 27. Power spectral density of cable acceleration response (segment 2): (a) in-plane acceleration; and (b) outof-plane acceleration.

Total displacement Mode 1 component Mode 2 component Mode 3 component

0.1 0.05 0 -0.05 -0.1 -0.15 1200

(a)

1201 1202 Time (s)

1203

0.25 0.2 0.15

Total displacement Mode 1 component Mode 2 component Mode 3 component

0.1 0.05 0 -0.05 -0.1 -0.15 1200

(b)

1201 1202 Time (s)

1203

Fig. 28. Estimated cable displacement response (segment 2): (a) at L/6 location; and (b) at L/4 location.

0.15

1 0.5 0 -0.5 -1 -1.5 -2 -1.5

(a)

In-plane displacement (m)

In-plane acceleration (g)

2 1.5

-0.5 0.5 -1 0 1 Out-of-plane acceleration (g)

1.5

0.1 0.05 0 -0.05 -0.1

-0.05

(b)

0 Out-of-plane displacement (m)

0.05

Fig. 29. Vibration locus of cable response at L/6 location (segment 2): (a) acceleration; and (b) displacement.

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Displacement amplitude of mode 2 (m)

0.08 Displacement amplitude Fitted curve (0.016%) 0.06

0.04

0.02

0 850

950 1050 1150 1250 850 Time (s)

950 1050 1150

Fig. 30. Estimation of net damping ratio of second mode (segment 2).

-10

-10

In-plane Out-of-plane

10

10 5 Acceleration (g)

0 -5 -10 10 5 0

-10 10 5

(b)

44 3

44 2

44 0

00

00

36

00

32

00

28

00

Time (s)

24

00

20

00

16

12

80 0

-5 -10 40 0

-5

1

0

-10 0

(a)

0 -5

44

Acceleration (g)

5

In-plane Out-of-plane

Time (s)

Fig. 31. Time history of cable acceleration response (segment 3): (a) signals in 3600 s; and (b) signals in 3 s.

both the displacement and acceleration loci are twin-ellipse formation because of simultaneous participation of the first and second modal components. The net negative damping ratio for the second mode is roughly estimated from the identified dominant modal displacement components. Fig. 30 plots the identified second modal components from 850 to 1150 s at intervals of 50 s. By fitting the identified displacement amplitudes with an exponential function, the net negative damping ratio for the second mode is estimated to be 0.016%. 5.3. Response segment 3 Fig. 31(a) illustrates time histories of the 3600-s response segment 3 starting at 10:20 pm of 1 April 2003, while Fig. 31(b) shows the 3-s records corresponding to large-amplitude

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10 8 6 4 2 0 10

3000 2500 2000 1500 1000 500 0 10

2

0

0

)

e (s

0

(b)

4 (Hz )

80

6 ncy

00 32 0 0 28 00 24 0 0 20 00 16 0 0 12

8 que

Fre

0

T

ime

(s)

323

40

0

0

00 32 0 0 28 00 24 0 0 20 00 16 0 0 12

80

0

0

8 Fre 6 que 4 ncy 2 (Hz )

40

(a)

×103

PSD of out-of-planeacceleration (g2/Hz)

PSD of in-plane acceleration (g2/Hz)

Y.Q. Ni et al. / J. Wind Eng. Ind. Aerodyn. 95 (2007) 303–328

Tim

Fig. 32. Power spectral density of cable acceleration response (segment 3): (a) in-plane acceleration; and (b) outof-plane acceleration.

In-plane displacement at L/6 (m)

0.25

Total displacement Mode 1 component Mode 2 component Mode 3 component Mede 4 component Mode 5 component

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 440

441

442

443

Time (s) Fig. 33. Estimated cable displacement response at L/6 (segment 3).

quasi-stationary response within this segment. The maximum in-plane acceleration of the cable in this segment approaches to 10 g and the corresponding wind velocity recorded at deck level is about 9 m/s. Fig. 32 shows the power spectral density of in-plane and out-of-plane acceleration responses at different time intervals. It is found that the dominant frequency of this response segment is 3.21 Hz, which corresponds to the third modal frequency of the cable. The first five modes participate in the cable vibration. Fig. 33 shows the estimated in-plane displacement response and its modal components at measurement point (L/6) for the 3-s quasi-stationary response. When the cable vibration is dominated by the third mode, the maximum displacement response is expected to occur just at the measurement point, and therefore only the response at L/6 location is plotted here. It is seen from Fig. 33 that with the third mode being dominant, the first, second, fourth and fifth modes also contribute to the cable response. The peak-to-peak

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In-plane displacement at L/6 (m)

In-plane acceleration at L/6 (g)

324

10 8 6 4 2 0 -2 -4 -6 -8 -10 -8

(a)

-4

0

4

8

Out-of-plane acceleration at L/6 (g)

0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.15

(b)

-0.1 -0.05

0

0.05

0.1

0.15

Out-of-plane displacement at L/6 (m)

Fig. 34. Vibration locus of cable response at L/6 location (segment 3): (a) acceleration; and (b) displacement.

Displacement amplitude of Mode 3 (m)

0.15 Displacement amplitude Fitted curve (0.069%) 0.1

0.05

0 200

250

300

350

400

450

500

Time (s) Fig. 35. Estimation of net damping ratio of third mode (segment 3).

displacement response amplitude at L/6 location is 0.36 m, which is about three times diameter of the cable. It is noted that the displacement response amplitude for segment 3 is only about half of that for segment 1, even though the acceleration response amplitude for segment 3 is five times as large as that for segment 1. Fortunately, cable vibration dominated by the first mode occurs infrequently under rain-wind excitations. Fig. 34 plots the locus diagrams of acceleration and displacement responses at L/6 location. In this case, both the displacement and acceleration loci are multiple-ellipse formation due to participation of several modes. The net negative damping ratio for the third mode is roughly estimated from the identified dominant modal displacement components. Fig. 35 plots the identified third modal components from 200 to 500 s at intervals of 50 s. By fitting the identified displacement amplitudes with an exponential function, the net negative damping ratio for

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the third mode is estimated to be 0.0691%, which is the largest net negative damping ratio among the three cases.

6. Modal participation The analysis given in the above section shows that the dominant mode of the cable A12 during rain-wind-induced vibration may be the first, second and third modes. To investigate the occurrence frequency of these modes as dominant mode, the response records from all three rain-wind excitation events are divided into 360, 3-s data samples at intervals of 50 s. Each sample is then used to identify the displacement modal components and the mode which contributes largest modal displacement is regarded as dominant mode of this response sample. Fig. 36 shows a histogram of number of the first four modes occurring as dominant mode. It is obvious that about two-thirds of the 360 data samples exhibit the third mode as dominant mode, while one-third of the samples exhibit the second mode as dominant mode. To evaluate the degree of modal participation and estimate whether a single dominant mode or more modes are participating during the rain-wind excitation events, the Modal Contribution Index (MCI) is calculated by use of the 360 data samples. The MCI is defined as (Main et al., 2001). An MCI ¼ qffiffiffiffiffiffiffiffiffiffiffiffi P 2 i Ai

(5)

Number of occurrences

where An is the displacement component of the dominant mode and Ai is the displacement component of the ith participation mode. The MCI approaches unity when the response is a result of pure single-mode vibration, and the MCI decreases when the participation of other modes becomes more significant. Fig. 37 shows the histogram of MCI. It is known from the figure that in the three rain-wind excitation events, only a small portion of the cable responses are resulting from nearly single-mode vibration and most of the responses manifest themselves as multi-mode vibration with participation of the dominant mode and other modes.

250

221

200 150

126

100 50 11

2

0 1

2

3 Mode order

Fig. 36. Histogram of dominant mode.

4

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Number of ocurrences

120 97

100 80

69

60

51 42

40 20

47

28 6

0 0.650.7

0.70.75

0.750.8

0.80.85

0.850.9

0.90.95

0.95-1

Range of MCI Fig. 37. Histogram of modal contribution index (MCI).

7. Conclusions Continuous 45-day field measurements were conducted on the cable-stayed Dongting Lake Bridge to measure the cable vibration in rain-wind excitation conditions and to monitor dynamic interaction among the cables, deck and towers under wind loading and traffic loading. The measurement data and preliminary analysis results of cable response, wind velocity, wind direction and rainfall during three typical rain-wind excitation events are presented in this paper. This study reaches the following conclusions: (i) The critical mean wind velocity at deck level to produce rain-wind-induced cable vibration is 6–14 m/s and the critical mean wind direction (relative yaw angle) ranges from 10o to 50o. The rainwind-induced cable vibration with large amplitude occurs in light-to-moderate rain (less than 8 mm/h). The cable ceases to vibrate as wind velocity and wind direction are beyond the above range; (ii) The cable in-plane acceleration response amplitudes are approximately two times the out-of-plane acceleration response amplitudes during rainwind-induced vibrations; (iii) For the observed cable, the overall dominant mode of rainwind-induced vibration is the third mode in all the rain-wind excitation events. In the process of vibration evolution, however, the dominant mode may differ for different response segments; (iv) The cable maximum acceleration response may reach 10 g during rain-wind-induced vibration. However, the displacement response amplitude corresponding to the maximum acceleration response is not as large as expected; (v) During rainwind-induced cable vibrations, only a few segments of the response manifest themselves as nearly single-mode vibration and the majority of the response involves participation of the dominant mode as well as other low-order modes. Acknowledgements The work described in this paper was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. PolyU 5051/02E) and partially by a grant from the Hong Kong Polytechnic University through the Area of Strategic Development Program (Research Centre for Urban Hazards Mitigation). The support from National Natural Science Foundation of China (Grant no. 50178013) is also acknowledged.

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