On wind–rain induced vibration of cables of cable-stayed bridges based on quasi-steady assumption

ARTICLE IN PRESS J. Wind Eng. Ind. Aerodyn. 97 (2009) 381–391

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On wind–rain induced vibration of cables of cable-stayed bridges based on quasi-steady assumption Ming Gu  State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China

a r t i c l e in fo

abstract

Article history: Accepted 18 May 2009 Available online 1 July 2009

Wind–rain induced vibration of cables of cable-stayed bridges is presently a problem of great concern. Similar to the classical galloping theory, this paper adopts quasi-steady assumption to study wind–rain induced vibration of cables. A wind tunnel test was first made to measure wind pressures and thus wind forces acting on a 3-D cable model and the upper artificial rivulet model. A new theoretical model for instability of a 2-D sectional rigid model with a moving artificial rivulet is then established and the instability criterion is proposed. The instability criterion is verified through wind tunnel test on a 2-D rigid sectional cable model with a moving artificial rivulet. Finally, theoretical models of wind–rain induced vibration of 3-D sectional cables and 3-D continuous cables are, respectively, developed based on the measured mean wind forces mentioned above, and the vibration characteristics are investigated as well as an explanation of the mechanism of wind–rain induced vibration of stay cables is made. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Cable of cable-stayed bridge Wind–rain induced vibration Quasi-steady assumption Wind tunnel test Theoretical model Aerodynamic instability Mechanism

1. Introduction Excessive and unanticipated vibration of cables in cable-stayed bridges under the simultaneous occurrence of wind and rain has been reported widely in the past two decades (Hikami and Shiraishi, 1988; Ohshima and Nanjo, 1987; Matsumoto et al., 2003; Matsumoto, 1998; Pacheco and Fujino, 1993; Main and Jones, 1999; Persoon and Noorlander, 1999; Gu et al., 1998; Shi et al., 2003). The author also observed wind–rain induced cable vibration from cable-stayed bridges built in Shanghai (Gu et al., 1998), and in Nanjing (Shi et al., 2003), which have main span of 602 and 628 m, respectively. Wind–rain induced cable vibration is presently a great concern to bridge engineering and wind engineering communities. As is well known, wind–rain induced vibration of cables is a complex problem. Field measurements (Hikami and Shiraishi, 1988; Ohshima and Nanjo, 1987; Matsumoto, 1998; Main and Jones, 1999; Persoon and Noorlander, 1999; Shi et al., 2003), wind tunnel simulation tests (Matsumoto et al., 1995; Matsumoto et al., 1992; Flamand, 1995; Bosdogianni and Olivari, 1996; Gu et al., 2002; Gu and Du, 2005b; Cosentino et al., 2003a), and theoretical analyses (Yamaguchi, 1990; Gu and Lu, 2001; Wilde and Witkowski, 2003; Geurts and Staalduinen, 1999; Xu and Wang, 2003; Cao et al., 2003; Cosentino et al., 2003b; Yuscheweyh, 1999; Zhou and Xu, 2007) were conducted to investigate the character-

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istics and the mechanism of the complex phenomenon. In wind tunnel simulation tests, there are mainly two kinds of approaches for simulating rivulet on cable section model: one is to spray water appropriately onto the surface of cable models to form rivulet (Matsumoto et al., 1995; Cosentino et al., 2003a), mainly for investigating cable’s vibration characteristics, and the other is to stick artificial rivulet model on the cable model surface (Gu and Du, 2005a; Yamaguchi, 1990; Gu and Lu, 2001), mainly for vibration characteristics and wind forces. In the theoretical studies on rigid sectional cable, quasi-steady and unsteady aerodynamic force models were both applied to describe the motion of cables. Matsumoto et al. (2005) suggested an unsteady aerodynamic forces, which were expressed by aerodynamic derivatives, Hi ði ¼ 124Þ, to study the aerodynamic damping and the instability of cables. Moreover, Matsumoto et al. (2001) discovered that stay cables might vibrate at high reduced wind speeds, such as 20, 40, 60 and 80, under wind–rain condition or only wind condition, and raised an explanation of axial flow and axial vortexes as another possible mechanism of wind–rain induced vibration. Even so, quasi-steady aerodynamic forces were adopted by most of the researchers to theoretically investigate the vibration characteristics and mechanism of rigid cables (Yamaguchi, 1990; Gu and Lu, 2001; Wilde and Witkowski, 2003; Geurts and Staalduinen, 1999; Xu and Wang, 2003; Cao et al., 2003; Cosentino et al., 2003b; Gu et al., 2009). Through the studies, it is now believed that the water rivulet formed on surface of cables and its motion play important roles in rain–wind induced vibration. But unfortunately, the mechanism of the complex phenomenon has not been clearly recognized up to now. As for

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3-D continuous cables, theoretical studies on wind–rain induced vibration and control were performed based on the above results of 2-D cables (Gu and Huang, 2008; Gu et al., 2009). In this paper, similar to the classical galloping theory, the quasi-steady assumption is adopted to study wind–rain induced vibration of cables of cable-stayed bridges. A wind tunnel test was first made to measure the wind pressures and then mean wind forces acting on a 3-D sectional cable model and the attached upper artificial rivulet model, which are the base of the new theoretical studies on wind–rain induced vibration of cables in the present paper. A new theoretical model for instability of a 2-D rigid sectional cable model with a moving artificial rivulet is established and the instability criterion is proposed. Finally, theoretical models for wind–rain induced vibration of 3-D sectional cables and 3-D continuous cables are, respectively, developed and the vibration characteristics are investigated and a possible mechanism is suggested.

2. Wind tunnel test for wind pressures and forces acting on cable and rivulet models To establish the theoretical models of wind–rain induced vibration of 3-D cables, a wind tunnel test on a 3-D sectional cable model with upper and lower artificial rivulet models was performed for obtaining wind forces acting on the models. 2.1. Cable model and test condition As is known, stay cables in cable stayed bridges usually have diameters from 10 to 18 cm; and, moreover, the width of water rivulet is about 1–2 cm (Gu et al., 1998; Gu and Du, 2005b). This

means that if the cable model and the artificial rivulet model for the wind pressure test had the same sizes with the actual ones, the too small artificial rivulet model would make the measurement of wind pressures acting on the rivulet very difficult. Therefore the sizes of cable model and accordingly the artificial rivulet model for the wind tunnel test were both amplified for the test, about two or three times the actual ones. On the other hand, in order to assure the Re number of the model is the same with the actual cable, the test wind speed should be 12213 of the actual wind speed at which the rain–wind induced vibration of real stay cables usually appears. The diameter of the cable was thus 350 mm and the model length was 3.5 m. The testing wind speed ranged from 3 to 10 m/s; the Re number was accordingly from 7:04  104 to 2:35  105 . Two kinds of artificial upper rivulets, which are schematically shown in Fig. 1, were adopted in the test. On the model 176 pressure taps were arranged in four sections. The section B where 63 pressure taps were arranged is shown in Fig. 2. Especially, to understand detailed information of wind pressures on the rivulet, the pressure taps were arranged in three rows on the rivulet model (see Fig. 2b). The test was carried out in smooth flow in TJ-3 BLWT in Tongji University, which has a working section of 15 m in width and 2 m in height. The cable model could be easily adjusted to the required wind angles and cable’s inclined angles on a specially designed experimental set up. Fig. 3 shows the photograph of the test set up and the model. Fig. 4 illustrates the 3-D rigid cable model described by inclined cable angle a and wind angle b. In the test, the cable inclined angle was fixed at 301 and the wind directions were 01, 251, 301, 351, 401 and 451. Furthermore, in order to measure the wind pressures under the condition of the rivulet being at different locations, which is denoted by yu (see Fig. 2a), the cable model could rotate around its axis to make the rivulet at 47.4

R2

R2

50

50

17

8.5

37.6

Fig. 1. Sizes of artificial rivulets (unit: mm).

u

upper rivulet 8.5

5.3

cy1

.6

24

5.3

37.6

9x3

37

B-B

2x7.5

Fig. 2. Arrangement of pressure taps. (a) Arrangement of pressure taps on cable model. (b) Arrangement of pressure taps on upper rivulet.

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383

mean pressure coefficient RMS pressure coefficient 1 0 0

45

90

135

180

225

270

315

360

225

270

315

360

-1 -2 -3 θcyl (°) mean pressure coefficient RMS pressure coefficient

Fig. 3. Photo of test setup and model.

1 0

cable model

0

45

90

135

180

-1 -2

Wind -3

α

θcyl (°) Fig. 5. Wind pressure coefficients. (a) yu ¼ 01 and (b) yu ¼ 461.

β

Fig. 4. Space state of 3-D rigid sectional cable model.

different positions. In the test, the increment interval of the upper rivulet was 21 for the upper rivulet ranging from 301 to 821, while 41 for the upper rivulet in the other range. A DSM3000 electronic pressure scan valve system was used to measure the wind pressures acting on the cable and rivulet models. The transfer function method was adopted to modify the distortion of fluctuating wind pressures passing through the measuring tubes. The test results show that the mean and fluctuating wind pressures on the four sections are almost the same, so only the pressures on the section B are presented hereafter.

2.2. Main testing results As is expected, the wind pressures acting on the cable for the two kinds of different artificial rivulets have the similar trend, which has also been pointed out by Gu and Lu (2001). So only the wind pressure results of the cable model with the smaller artificial rivulet under the conditions of 301 cable inclined angle and 351 wind direction angle are shown here. More detailed testing results can be found in Gu and Du (2005a). Fig. 5 shows the mean and fluctuating wind pressure coefficients of the cable model for the upper rivulet at two typical positions: yu ¼ 01 and 461. The Re number is about 110 000. When the upper rivulet is located at 01, the standing point goes down to 3401 (i.e., 201) for the present 3-D cable model; and accordingly the mean wind pressure coefficient at the standing point is somewhat smaller than that of a 2-D circular cylinder without a rivulet. When the upper rivulet ranges from 101 to 461, the absolute values of the mean wind pressures at the tap

positions just behind the upper rivulet suddenly become much larger than those in the front of the upper rivulet; while those at the other tap positions are almost unvaried. When the upper rivulet position is located in the range of 481–901, which means that the rivulet is behind the separation point, the effects of the rivulet on the wind pressure distribution become much less significant. From Fig. 5 it can also be seen that the RMS fluctuating wind pressure coefficients at most of the pressure taps are small and can hardly affected by the rivulet. Figs. 6 and 7 show the variations of aerodynamic force coefficients of the cable–rivulet model and the attached rivulet model with the position of the rivulet, respectively. It can be seen from Fig. 6 that for the cable–rivulet model the drag force increases violently but the lift force decreases violently when yu is from 441 to 491. The violent variations of the drag force and lift force may be the key aerodynamic reason of wind–rain induced vibration of cables based on the viewpoint of classical galloping. Furthermore, from Fig. 7 it can be found that the ranges of violent increase and decrease of the aerodynamic forces on the rivulet model with yu are also between 441 and 491, but the variation trends of the aerodynamic forces with yu are entirely contrary to those of the cable–rivulet model. When the aerodynamic forces on the cable–rivulet model increase (or decrease) with yu , the forces on the rivulet model decrease (or increase). In addition, the testing results show that the only upper rivulet but not the lower rivulet has effects on the wind forces. In fact, in previous studies, high frequency force balance (HFFB) technique was used to measure quasi-steady aerodynamic forces acting on cable–rivulet models (Yamaguchi, 1990; Gu and Lu, 2001), which can be compared with the present results (Fig. 6). The comparison indicates that the aerodynamic forces obtained by the different techniques, i.e., wind pressure measurement technique and HFFB technique, have similar variation trends. But on the other hand, the different shape and size of the rivulet can

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3.1. Basic assumptions

1.2 1.0

CD

The phenomenon of instability of a 2-D cable with a moving rivulet is very complex. In order to simplify the analysis, some appropriate assumptions are made as follows:

CL

(1) In order to only consider the effects of motion of the rivulet on the unstable motion of the cables, the axial flow and axial vortexes along the cable will not be taken into account in the analysis. (2) Only upper rivulet will be considered in the following analysis. (3) Quasi-steady assumption will be applied in the present analysis.

0.8 0.6 0.4 0.2 0.0 0

10

20

30

40

50

60

70

80

90

u Fig. 6. Aerodynamic force coefficients of cable.

3.2. Equations governing motion of cable and rivulet

2.0 1.5

cd

1.0 0.5 0.0 -0.5 -1.0

0

10

20

30

40

50

60

70

80

90

cl

Based on the analysis of forces acting on the sectional cable, the equations governing the motion of the cable model in y direction (across-wind direction) and the motion of the rivulet in tangential direction on the cable model can be derived. Similar to the derivation process of classical galloping, the nondimensional equations governing the motions of the cable and rivulet can be further written as (Gu and Huang, 2008)

Z00 þ 2zZ0 þ Z ¼ 

-1.5



u Fig. 7. Aerodynamic force coefficients of rivulet.

lead to different values of the aerodynamic forces but do not change the variation trends. Furthermore, the present experiment, i.e., the individual measurement of wind pressures on the cable–rivulet model and the attached rivulet model, has obviously advantages over the previous measurements by using high force balance technique. The former method can be used to individually obtain the aerodynamic forces acting on not only the cable–rivulet model but also the alone rivulet model, which can be further adopted to individually derive the motion equations of the cable model and the rivulet model; whereas the latter method can be used to only obtain the whole forces acting the cable–rivulet model.

3. Instability of 2-D cable with movable rivulet (Gu and Huang, 2008) The above section presents the wind tunnel test on a 3-D sectional cable model with upper and lower rivulet models and the wind forces acting on the models from the test. The results will be adopted later to develop theoretical models of wind–rain induced vibration of 3-D sectional cables and 3-D continuous cables. This section presents briefly theoretical and experimental studies on aerodynamic instability of a 2-D cable with movable attachment based on quasi-steady aerodynamic force assumption. The study on the instability of 2-D cable model with a movable rivulet model and the derivation of the criterion for instability of the system are similar to the study on the classical galloping instability and the Den Hartog criterion for galloping instability.

C y ðy; aÞ

(1)

g 

Pt cosðb0 þ gÞ þ F 1 ¼ (2) mRo2 Ro pffiffiffiffiffiffiffiffiffiffiffi where t ¼ ot; Z ¼ y=R; o ¼ K=M is the natural circular frequency of the cable, ranging generally from 2 to 18; z is the damping ratio of the cable system, which is generally smaller than 0.5% (Shi et al., 2003); d ¼ m=M is the mass ratio between the rivulet and the cable; and R is the radius of the cable;  ¼ pR2 r=M; mr ¼ Ro=U; Z0 and g0 denote the derivatives of Z and g, respectively, with respect to the non-dimensional time, t; C y ðy; aÞ ¼ 1= cos a½C D ðyÞ tan a þ C L ðyÞ; g is the vibration angle of the rivulet; F1 is the tangential component of the surface tension between the rivulet and the cable surface; Pt is the tangential projection of the aerodynamic force acting on the cable. In order to derive the criterion for the unstable motion of the cable, F y ðyÞ is expanded into Taylor’s series at a ¼ g ¼ 0, and the items higher than the first order are neglected. Then applying Lyapnov stability criterion to Eq. (1), the criterion for the unstable motion of the cable finally derived as    A 1 @C L ðy; b0 Þ 1 @C L ðy; b0 Þ þ D ¼ 2z þ  C D ðyÞ þ o0 (3) pmr B @y @y pm2r

g00 þ Z00 þ

-2.0

1

pm2r

2

So far, one can find the unstable balance angle, b0, corresponding to the unstable motion of cable using Eq. (3). In Eq. (3), A/B is the amplitude ratio of the cable and rivulet. In fact, the value of A/B is difficult to be precisely predicted for a real cable, so criterions for two limit conditions of A/B are further derived as follows. (1) A=B ! 1: A=B ! 1 means B ! 0, that is to say, the rivulet is motionless. This situation corresponds to galloping. Thus the range of the unstable balance angle can be found from   1 @C L ðy; b0 Þ o0 (4) C D ðyÞ þ 2z þ  pmr @y The above equation is the same as Den Hartog galloping criterion.

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385

(2) A=B ! 0: A=B ! 0 means that the non-dimensional amplitude of cable is much smaller than that of the rivulet. The criterion for this condition thus can be derived from Eq. (3) as @C L ðy; b0 Þ o0 @y

(5)

3.3. Numerical computation The parameters of the cable for the computation can be found in (Gu and Huang (2008). Fig. 8 shows the variation range of the unstable balance angle of the rivulet with A/B. The curves A and A0 in Fig. 8 are computed with Eq. (3), and the zone between them corresponds to unstable zone for different values of A/B. Accordingly, the curves B and B0 are computed with Eq. (4); and the curves C and C0 are obtained with Eq. (5). The zone between the curves B and B0 corresponds to the unstable zone of galloping of the cable; and that between the curves C and C0 corresponds to the upper limitation of the unstable state, as mentioned above. The unstable zone between A and A0 vary gradually from the upper boundaries (C and C0 ) to lower boundaries (B and B0 ) with the increase of A/B. This reveals that the range of unstable balance angle of rivulet of a cable with a movable upper rivulet can be much wider than that of unstable galloping of a cable with a fixed rivulet. This is consistent with the result reported by Gu and Lu (2001). 3.4. Experimental verification In order to verify the above theoretical model, a new 2-D aeroelastic cable model with a movable rivulet model was tested in TJ-1 Boundary Layer Wind Tunnel in Tongji University. The testing results were then compared with the theoretical ones. Different from the cable model with a fixed upper artificial rivulet in other studies, the new cable model system had a movable artificial upper rivulet model (see Fig. 9). The cable model of weight of 2.3 kg was a hollow PVC tube with an outside diameter of 110 mm and a wall-thickness of 3 mm and a length of 1.2 m; the movable artificial rivulet of weight of 0.096 kg was a hollow aluminum tube with an outside diameter of 6 mm and a wall-thickness of 1 mm and a length of 1.26 m. As is indicated above, the position of rivulet on the cable surface, rather than shape and size of the rivulet, is the most important factor for the aerodynamic force. The frequencies of the cable model and the movable artificial rivulet were 1.47 and 1.4 Hz, respectively. The damping of the rivulet was difficult to be precisely measured but

75 70

C

very small, about 2–4%. In addition, the artificial rivulet was required not to touch the cable model during its vibration process and the interval between the artificial rivulet and cable model was as small as possible. Finally the interval was carefully adjusted to be about 0.4–0.5 mm. The unstable balance angle was first investigated for the condition that the artificial rivulet was fixed on the cable model. The testing results show that the range of the unstable balance angle of the rivulet is from 381 to about 461, which is just classical galloping phenomenon. This meets the above expectation of Eq. (4). When the rivulet model was movable, the artificial rivulet could arrive at a certain balance angle from its initial position at a certain wind speed. Several groups of cable models with typical damping ratios (0.1%, 0.53%, 1.06% and 1.7%) and initial angles of the rivulet (101, 201, 301 and 401) were tested for the comparison between the theoretical and testing results. The parameters for the computation were the same as those for the wind tunnel test, but the values of A/B are assumed to be 4, 10 and 20. All the comparisons show the same tendency. Fig. 10 indicates one of the comparison groups (z ¼ 0.017; initial angle ¼ 101; wind speed (5–11) m/s; A/B ¼ 4), together with the unstable boundaries computed with Eq. (3). From the figure it can be seen that the unstable balance angles of the rivulet for different wind speeds are within the upper and lower boundaries.

4. Theoretical model for wind–rain induced vibration of 3-D sectional cable The instability criterion of a 2-D rigid cable model has been proposed above in terms of the quasi-steady assumption. Here the vibration characteristics and possible mechanism of 3-D cables are studied based on the wind forces acting on the 3-D rigid cable model and rivulet model, which have been presented in Section 2. 4.1. Aerodynamic forces on cable and rivulet models

65 A

60 0 (°)

Fig. 9. Schematic diagram for experimental setup.

55 50 45

B

40

B'

35

A' C'

30 1

10 A/B

Fig. 8. Unstable balance angle vs. A/B (computation).

100

The coordinate system of the 3-D cable model and the directions of aerodynamic forces are schematically indicated in Fig. 11. The space stage of the 3-D cable is described by the inclined angle a and the wind direction angle b, the included angle f between the relative wind speed U rel and the horizontal plan. The x coordinate is parallel to the ground and perpendicular to the axial plan of the cable model; and the y coordinate is also perpendicular to the model’s axis. The axial plan of the cable model is along the model’s axis and perpendicular to the ground. The initial rivulet angle and the rivulet instantaneous position are denoted by b0 and g, respectively. The aerodynamic forces can be obtained through integrating the wind pressures over all the taps

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48

Zone where model vibrated unstably (from test)

47 46 45 44

Upper boundary of unstable balance angel (from computation)

0 (°)

43 42 41 40 39 38 37 Lower boundary of unstable balance angel (from computation)

36 35 5

6

7

8 U/m/s

9

10

11

Fig. 10. Comparison of theoretical and testing results.

0

0

02

C L ðf Þ ¼  138:187 þ 364:113f  355:520f 03

þ 152:456f  24:222f

CL



CD

z

φ′

(9)

Furthermore, the formulas of aerodynamic force coefficients of the rivulet are fitted as well. 0 For 0:337  f  1:174 (radian),

0 U λφ y· Urel

04

0

0

02

03

cd ðf Þ ¼  556:828 þ 3077:682f  6752:521f þ 7350:843f 04

05

 3976:649f þ 856:480f 0

0

04

0

02

03

C D ðf Þ ¼ 414:612  5874:960f þ 36063:162f  125707:500f 04

06

(11)

and for 1:174of  1:907 (radian), 0

02

03

cd ðf Þ ¼ 11668:679  54592:670f þ 109115:881f  120775:568f 04

05

06

þ 79945:337f  31642:867f þ 6933:343f  648:674f

07

(12) 0

0

02

03

cl ðf Þ ¼  479:655 þ 1562:689f  2030:702f þ 1319:811f 04

05

 429:234f þ 55:884f

(13)

4.2. Theoretical formulation (Gu et al., 2009)

05

þ 274498:563f  389948:057f 06

þ 360930:20381f 07

08

09

 210205:891f þ 69997:4332f  10166:924f 0

02

(6)

03

C L ðf Þ ¼ 89:308  1241:370f þ 7479:664f  25870:496f 04

05

þ 56997:146f  83166:040f 06

07

þ 80355:099f  49500:190f 08

09

þ 17608:969f  2751:762f

(7)

0

and for 1:174of  1:907 (radian), 0

02

C D ðf Þ ¼  59:281 þ 146:406f  130:364f 03

05

0

0

presented in Section 2. The variations of the mean lift and drag coefficients, CL and CD, of the cable model for the artificial upper rivulet at different positions are presented in Figs. 6 and 7. The formulas of the mean lift and drag coefficients for the conditions of cable inclined angle of 301 and wind angle of 351 covering a much wider range than the unstable angles are fitted as a function 0 of f ¼ f þ g þ b0 (see Fig. 11). The formulas of the mean lift and drag coefficients are 0 for 0:337  f  1:174 (radian),

0

03

 123178:026f þ 54728:168f  10075:677f

Fig. 11. Coordinate system of the model and directions of aerodynamic forces.

0

02

cl ðf Þ ¼  5080:425 þ 34688:718f  98080:616f þ 146999:950f

y

0

(10)

04

þ 50:418f  7:164f

(8)

The assumptions in Section 3 are still adopted here. Furthermore, it should be indicated that the axial flow and axial vortexes are not taken into account as well even if the present cable model is three dimensional. The forces acting on the cable are: the elastic force of cable itself; the structural damping force; the inertial force of cable; the aerodynamic forces; the gravity force; the forces caused by the rivulet, such as the friction force (damping force) between the cable surface and rivulet, which are much smaller than the above forces acting on the cable and thus are omitted in the following cable’s motion equation. On the other hand, when the water rivulet moves around the surface of the vibrating cable, the forces acting on the rivulet are: the friction force (damping force) between the water rivulet and the cable’s surface, F 0 þ cr Ry_ (Gu et al., 2009); the inertial forces caused by the cable’s motion

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and the rivulet motion itself; the rivulet’s gravity force and the aerodynamic forces. It is worthy of noticing that the damping force between the water rivulet and the cable’s surface F 0 þ cr Ry_ defined in the present paper is somewhat different from that in literature (Pacheco and Fujino, 1993), which includes two part: constant damping force F 0 and linear damping force cr Ry_ (Huang and Ding, 1994). In the cable’s motion equation this force is neglected, as mentioned above; but in the rivulet motion equation this force plays an important role. Based on the discussions above, the equation governing the cable motion in y direction (across-wind direction) can be written as y€ þ 2zy oy y_ þ o2y y ¼ F y =M

(14)

where y is the dynamic displacement of the cable; zy and oy are the damping ratio and circular frequency of the cable, respectively; Fy is the wind force acting on the cable which may lead to the wind–rain induced cable vibration. According to the characteristics of forces acting on the cable model discussed above, the aerodynamic force Fy(t) can be written as F y ðtÞ ¼ 12rDU 2rel ½C L ðf Þ cosðfÞ þ C D ðf Þ sinðfÞ 0

0

(15)

where CL and CD are the lift and drag coefficients of the cable, respectively, which are from Eqs. (6) and (9). Moreover, the motion equation of the rivulet in tangential direction can be written as (Zhou and Xu, 2007) mRg€ þ F 0 þ cr Rg_ ¼ f t ðzÞ þ my€ cosðg þ b0 Þ  mg cos a cosðg þ b0 Þ (16) where m and R are the mass per unit length and characteristic size of the rivulet, respectively; F0 and cr are the constant damping force and linear damping coefficient between the cable surface and the rivulet; ft(z) is the aerodynamic force on the rivulet f t ðzÞ ¼ 12rU 2rel B½cd sinðfÞ þ cl cosðfÞ

(17)

cd and cl are the drag and lift coefficients of the rivulet, respectively, the fitted formulas being from Eqs. (10)–(13). 4.3. Numerical example and possible mechanism The cable parameters in Gu and Du (2005b) are adopted in the present computation. The vibration characteristics of the cable and the rivulet at different wind velocities are first computed 0.24

F0 = 5% of gravity F0 = 10% of gravity F0 = 20% of gravity F0 = 40% of gravity (Gu and Du, 2005b)

Cable Amplitude (m)

0.20 0.16 0.12

387

according to Eqs. (14)–(17). The wind velocities for the computations are U0 ¼ 7.0, 7.2, 7.3, 7.4, 7.5, 7.6, 7.8, 8.0, 8.3, 8.5, 8.8, 9.0, 9.1, 10.0, 11.0 and 12.0 m/s. Fig. 12 presents the variation of vibration amplitudes of the cable with typical wind velocities, together with the experimental results from Gu and Du (2005b). From the figure it can be clearly seen that the cable exhibits velocity- and amplitude-restricted response characteristics, which has also been pointed out (Gu et al., 2002; Gu and Du, 2005b). Moreover, the onset velocities and amplitudes from the present computation and the wind tunnel test are comparable. Especially when the constant damping coefficient is taken as 0.04 N/m, the amplitudes are much closer to the testing results in Gu and Du (2005b). The detailed computations provide the time histories of vibration amplitudes of the cable and rivulet, which are shown in Fig. 13. When the wind velocity is low the upper water rivulet cannot form on the cable’s surface and the cable thus cannot vibrate. When the wind velocity ranges between 7.3 and 9.0 m/s, the upper water rivulet can form on the surface of the cable, between the upper and lower limits of ‘‘dangerous zone’’ (‘‘dangerous zone’’ will be explained in the following text), and accordingly the cable has large vibration amplitudes. Furthermore, when the wind velocity increases to 9.1 m/s, the mean balance position of rivulet suddenly increases to a position out of the ‘‘dangerous zone’’, leading to a stop of the cable vibration. More detailed discussions on the results in the figure will be made in the following text. The study performed by the author Gu and Lu (2001) suggested an explanation of mechanism of wind–rain induced vibration of a 2-D cable. The suggested mechanism for a 2-D cable seems still to be able to explain the mechanism of wind–rain induced vibration of the 3-D cable. From the present detailed computations the author finds that there might exist an ‘‘unstable zone’’ of the initial rivulet position and a ‘‘dangerous zone’’ of the instantaneous rivulet position. The ‘‘unstable zone’’ means that when the rivulet is initially located at this zone, the forces acting on the rivulet could make the rivulet move to the ‘‘dangerous zone’’. And the ‘‘dangerous zone’’ is the negative slope zone of the aerodynamic force coefficient, i.e., C D þ dC L =dyo0. When the rivulet’s instantaneous position reaches the ‘‘dangerous zone’’, the vibrating cable can absorb energy through the aerodynamic force, which is similar to the phenomenon of galloping, and as a result, the vibration amplitudes of rivulet and cable will increase till very large amplitudes take place. For the present model the ‘‘dangerous zone’’ of yu ¼ g þ b0 is from about 441 to 491 (see Section 2). As is indicated above, for the conditions of cable inclination angle of 301 and wind yaw angle of 351, l ¼ 19:3 . Therefore, when a new angle c is defined to be the sum of g, b0 and l, that is, c ¼ g þ b0 þ l, the ‘‘dangerous zone’’ of c ranges between 631 and 681, which can be seen in Fig. 13. Obviously, the ‘‘unstable zone’’ is generally much wider than the corresponding ‘‘dangerous zone’’.

5. Theoretical model for wind–rain induced vibration of 3-D continuous cable

0.08

5.1. Theoretical formulation 0.04 0.00 7

8

9

10

U0 (m/s) Fig. 12. Comparison between responses of cable from present computation and test (Gu and Du, 2005b).

The 3-D continuous cable here represents a real cable. For a long real cable, the wind speeds acting on different parts of the cable are certainly different due to the mean wind profile. Therefore, mean wind speed profile should be taken into account of the theoretical formulation, whose exponent is here assumed to be 0.16. The quasi-steady assumption and the only upper rivulet assumption are still adopted here; the assumption of no axial flow and axial vortexes is also acceptable. The forces acting on the

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0.08

76 Cable Displacement (m)

0.06

ψ (°)

72 68 64 60

Dangerous zone 0

10

20

30

0.02 0.00 -0.02 -0.04 -0.06 -0.08

40

50 60 Time (s)

70

80

90 100

78

0

10

20

30

40

50 60 Time (s)

70

80

90 100

0

10

20

30

40

50 60 Time (s)

70

80

90 100

0.04

76 Cable Displacement (m)

74 72 70

ψ (°)

0.04

68 66

Dangerous zone

64 62

0.02 0.00 -0.02 -0.04

60 -0.06

58 0

10

20

30

40

50 60 Time (s)

70

80

90 100

80

0.000 Cable Displacement (m)

76

ψ (°)

72 68 Dangerous zone

64 60 56

-0.005 -0.010 -0.015 -0.020 -0.025

0

20

40

60 Time (s)

80

100

0

20

40 60 Time (s)

80

100

Fig. 13. Cable’s vibration amplitudes for different rivulet positions. (a) U0 ¼ 8.0 m/s, (b) U0 ¼ 8.8 m/s, (c) U0 ¼ 9.1 m/s.

cable model and the rivulet model are similar to those described in Section 4, but the force values are different at different heights due to different wind velocities. Fig. 14 is the schematic diagram of 3-D continuous cable under wind and rain action. The equations governing the motions of a 3-D continuous cable can be written as 2

2

2

A1

@ u @ v @u @v @ u @u þ A2 2 þ A3 þ A4 þ F x ðy; tÞ ¼ M 2 þ C @x @x @t @x2 @x @t

(18)

A5

@2 v @2 u @v @u @2 v @v þ A2 2 þ A6 þ A4 þ F y ðy; tÞ ¼ M 2 þ C 2 @x @x @t @x @x @t

(19)

where u and v are the displacements of a node in horizontal and vertical directions (in xy coordinate system) in the cable plane, respectively (see Fig. 14); M and C are the mass per unit length and damping coefficient of the cable, respectively, H EA A1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ; 1 þ y2x ð1 þ y2x Þ2 A3 ¼ 

3EAyx @2 y ; ð1 þ y2x Þ3 @x2

A4 ¼

H EAy2x ; A5 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 þ y2x ð1 þ y2x Þ2

A2 ¼

EAyx ð1 þ y2x Þ2

;

EAð1  2y2x Þ @2 y ; ð1 þ y2x Þ3 @x2 A6 ¼

EAð2yx  y3x Þ @2 y ; ð1 þ y2x Þ3 @x2

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389

from the wind forces acting on a rigid sectional cable model in Section 4. 5.2. Case study and discussion on possible mechanism Cable A20 in No. 2 Nanjing Bridge over Yangtze River is taken as an example. The cable has a length of 330.4 m and a diameter of 11.4 cm. The first natural frequency of the cable is 0.42 Hz. The first mode damping ratio is about 0.1%. Wind–rain induced vibrations of the cable have been found in field observations

49.0 Fig. 14. Schematic diagram of 3-D continuous cable under wind and rain action.

48.0

0.35 γ + β0 (°)

47.0

Cable Amplitude (m)

0.30 0.25

46.0 45.0

0.20 0.15

44.0

0.10 43.0

0.05

0.0

0.1

0.2

0.3

0.4

0.5 y/R

0.6

0.7

0.0

0.1

0.2

0.3

0.4

0.5 y/R

0.6

0.7

0.8

0.9

1

40 0.0

0.1

0.2

0.3

0.4

0.5 y/R

0.6

0.7

0.8

0.9

1

0.00

0.8

0.9

1

-0.05 55

Fig. 15. Maximum amplitude of cable vs. wind speed.

50

H is the horizontal component of cable tension under action of gravity; E and A are the modulus of elasticity and across section area of cable, respectively; Fx(z,t) and Fy(z,t) are the aerodynamic forces

γ + β0 (°)

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 Ud (m/s)

40

F x ðy; tÞ ¼ 12rDU 2rel ðyÞ½C L ðf ðyÞÞ cosðfðyÞÞ 0

0

þ C D ðf ðyÞÞ sinðfðyÞÞ sin a

(20) 35

F y ðy; tÞ ¼  12rDU 2rel ðyÞ½C L ðf ðyÞÞ cosðfðyÞÞ 0

0

þ C D ðf ðyÞÞ sinðfðyÞÞ cos a

(21)

CL and CD are the lift and drag coefficients (Eqs. (6)–(9)). The motion equation of the rivulet in tangential direction can be written as @2 g @g ¼ f t ðyÞ þ mðy€ þ x€ Þ cosðg þ b0 Þ þ F 0 þ cr R t t2  mg cos a cosðg þ b0 Þ

0 U 2rel ðyÞB½cd ðf Þ sinðfðyÞÞ

f t ðyÞ ¼ r

0

þ cl ðf Þ cosðfðyÞÞ

65 60

(22)

where m and R are the mass per unit length and characteristic size of the rivulet, respectively; F0 and cr are the constant damping force and linear damping coefficient between the cable surface and the rivulet; ft(z) is the aerodynamic force on the rivulet: 1 2

70

(23)

cd and cl are the drag and lift coefficients of the rivulet, respectively, (Eqs. (10)–(13)). As can be seen from Eqs. (20), (21) and (23), the wind forces acting on the 3-D continuous cable and rivulet vary with height due to the mean wind speed profile, which are obviously different

γ + β0 (°)

mR

45

55 50 45

Fig. 16. Rivulet positions on cable for different wind speed at bridge deck height. (a) Ud ¼ 6.35 m/s (t ¼ 1000 s), (b) Ud ¼ 7.0 m/s (t ¼ 400 s) and (c) Ud ¼ 8.5 m/s (t ¼ 1000 s).

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Cable Amplitude (m)

0.16 0.12 0.08 0.04 0.00 0.0

0.1

0.2

0.3

0.4

0.5 z/L

0.6

0.7

0.8

0.9

1.0

Moreover, most of the previous studies were focused on rigid sectional cable models to study the wind–rain induced vibration phenomenon. In these studies, the cable models had only one frequency; while in the present study, the effects of higher modes of cable can be taken into account. Fig. 17 shows some typical results of vibration mode of the cable at two time instances when Ud ¼ 6.9 m/s. At the beginning of the cable’s vibration, the cable vibrates in its first mode; and with the increase of time, the higher modes may gradually participate in the vibration and the cable’s vibrating shape becomes obviously different from the first mode shape. This seems to be the reason why cables in wind–rain induced vibration commonly exhibit ‘‘beat’’ phenomenon, and finally the participation of more higher-modes may gradually mitigate the wind–rain induced vibration, which was also found in field observations by the author and colleagues for several times (Gu et al., 1998; Shi et al., 2003).

0.03

Cable Amplitude (m)

6. Concluding remarks

0.00

-0.03 0.0

0.1

0.2

0.3

0.4

0.5 z/L

0.6

0.7

0.8

0.9

1.0

Fig. 17. Participation of modes of cable at two typical moments (Ud ¼ 6.9 m/s). (a) t ¼ 600 s and (b) t ¼ 1000 s.

(Shi et al., 2003). Typical computation results of the cable’s vibration amplitude in the cable plane under the given condition with wind speed are presented in Fig. 15. The largest vibration amplitude of the cable in normal direction, which is composed of the amplitude components in u and v directions, is about 0.33 m, which matches the field observations to some extent; and moreover, the range of wind speed at bridge deck height at which the wind–rain induced vibration takes place is from 6.4 to about 8 m/s. As is mentioned above, when the rivulet’s instantaneous position moves from ‘‘unstable zone’’ to ‘‘dangerous zone’’, the aerodynamic force could provide energy to the vibrating cable, which is similar to classical galloping to certain extent. As a result, the vibration amplitudes of rivulet and cable will increase till very large amplitudes take place. From the present study, it seems the explanation can be further extended to 3-D continuous cables. Fig. 16 shows the rivulet positions on the continuous cable with wind speed. In the figures, two horizontal solid lines mean the boundaries of the so called ‘‘dangerous zone’’. When the wind speed Ud at bridge deck height is 6.35 m/s, the upper rivulet can form only on 40% of the cable near the top of tower, and the cable’s amplitude is small. When Ud ¼ 6.6 m/s, the rivulet covers the range of around 60% of the cable within the ‘‘dangerous zone’’, and the cable’s peak amplitude is about 0.2 m. Especially when Ud ¼ 7.0 m/s, the range of the rivulet within the ‘‘dangerous zone’’ on the cable exceeds 90%, the cable’s vibration amplitude reaches up to 0.33 m. Furthermore, when the wind speed becomes higher, such as 8.5 m/s, the rivulet can only form on the low part of the cable because of too high wind speed at the upper part, and as a result, the vibration almost vanishes.

This paper studies wind–rain induced vibration of cables of cable-stayed bridges based on quasi-steady assumption. Through the wind tunnel test on a 3-D sectional cable model, the wind pressures and mean wind forces acting on the cable model and upper artificial rivulet model are obtained. A new theoretical model for instability of a 2-D sectional rigid model with a moving artificial rivulet is established and the instability criterion is proposed. The instability criterion was then experimentally verified. In the theoretical models of wind–rain induced vibration of 3-D sectional and continuous stay cables, the equations governing motions of cable and rivulet are both developed. For 3-D continuous cables, mean wind profile and higher modes of cable are taken into account. The vibration characteristics of the cables are numerically investigated and a possible mechanism of wind–rain induced vibration of 3-D continuous stay cables is proposed. Based on the studies, the following conclusions can be made: (1) The wind pressures and forces acting on the 3-D sectional cable model and artificial rivulet model are obtained, which are the base of the theoretical models of wind–rain induced vibration of 3-D sectional and continuous cables developed in this study. Violent variations of the wind forces acting on the cable model at some rivulet’s positions can be found, which may be the key aerodynamic reason of wind–rain induced vibration of cables from a quasi-steady viewpoint. (2) The instability criterion for a 2-D sectional cable with a movable rivulet, which simulates the instability of wind–rain vibration of cable, seems to be able to judge the instability of cables. The unstable range of the rivulet due to the rivulet’s motion is much wider than the classical galloping unstable range. A wind tunnel test on a new cable system with a movable artificial rivulet model has proved the theoretical model. (3) The present theoretical models for wind–rain induced vibration of 3-D cables seem to be able to estimate the vibration responses and the onset wind speed of cables. The responses computed with the models match the responses of the aeroelastic cable model under the simulated wind and rain conditions. A new explanation of the mechanism of wind–rain induced vibration of real continuous cables is made according to the theoretical model and the computation results. (4) Quasi-steady assumption seems to be appropriate to describe the complex problem of wind–rain induced vibration of cables of cable-stayed bridges.

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Acknowledgement This project is financially supported by National Natural Science Foundation of China (50621062), which the author gratefully appreciates. References Bosdogianni, A., Olivari, D., 1996. Wind- and rain-induced oscillations of cables of stayed bridges. J. Wind Eng. Ind. Aerodyn. 64, 171–185. Cao, D.Q., Tucker, R.W., Wang, C., 2003. A stochastic approach to cable dynamics with moving rivulets. J. Sound Vib. 268, 291–304. Cosentino, N., Flamand, O., Ceccoli, C., 2003a. Rain–wind induced vibration of inclined stay cables, part 1: experimental investigation and physical explanation. Wind Struct. 6 (6), 471–484. Cosentino, N., Flamand, O., Ceccoli, C., 2003b. Rain–wind induced vibration of inclined stay cables, part 1: mechanical modeling and parameter characterization. Wind Struct. 6 (6), 485–498. Flamand, O., 1995. Rain–wind induced vibration of cables. J. Wind Eng. Ind. Aerodyn. 57, 353–362. Geurts, C.P.W., Staalduinen, P.C.V., 1999. Estimation of the effects of rain–wind induced vibration in the design stage of inclined stay cables. In: Larsen, A., Larose, G.L., Livesey, F.M. (Eds.), Proceedings of Tenth International Conference on Wind Engineering, vol. 2, Copenhagen, Denmark, 21–24 June, pp. 885–892. Gu, M., Du, X.Q., 2005a. Testing study on wind pressure distributions of stayed cables with an artificial rivulet. ACTA Aerodyn. Sin. 23, 419–424 (in Chinese). Gu, M., Du, X.Q., 2005b. Experimental investigation of rain–wind-induced vibration of cables in cable-stayed bridges and its mitigation. J. Wind Eng. Ind. Aerodyn. 93, 79–95. Gu, M., Huang, L., 2008. Theoretical and experimental studies on aerodynamic instability of two-dimensional circular cylinder with moving attachment. J. Fluid Struct. 24, 200–211. Gu, M., Lu, Q., 2001. Theoretical analysis of wind–rain induced vibration of cables of cable-stayed bridges. J. Wind Eng. 89, 125–128. Gu, M., Liu, C.J., lou, G.Q., Lin, Z.X., Xiang, H.F., 1998. Rain–wind induced vibration of cables on cable-stayed bridges and its control. Shanghai J. Mech. 4, 281–288 (in Chinese). Gu, M., Liu, C.J., Xu, Y.L., Xiang, H.F., 2002. Response characteristics of wind excited cable with artificial rivulet. Appl. Math. Mech. 23 (10), 1176–1187. Gu, M., Li, S.Y., Du, X.Q., 2009. Experimental and theoretical simulations on wind–rain induced vibration of 3-D rigid stay cables. J. Sound Vib. 320, 184–200. Hikami, Y., Shiraishi, N., 1988. Rain–wind induced vibrations of cables in cable stayed bridges. J. Wind Eng. Ind. Aerodyn. 29, 409–418.

391

Huang, Z.Q., Ding, E.J., 1994. Surface Infiltration and Its Phase Transformation. Higher Education Press (in Chinese). Main, J.A., Jones, N.P., 1999. Full-scale measurements of stay cable vibration. In: Larsen, A., Larose, G.L., Liversey, F.M. (Eds.), Proceedings of the Tenth International Conference on Wind Engineering, Copenhagen, Denmark, 21–24 June, pp. 963–970. Matsumoto, M., 1998. Observed behavior of prototype cable vibration and its generation mechanism. In: Larsen, A., Esdahl, S. (Eds.), Bridge Aerodynamics. Balkema, Rotterdam, pp. 189–211. Matsumoto, M., Shirashi, N., Shirato, H., 1992. Rain–wind induced vibration of cables of cable-stayed bridges. J. Wind Eng. Ind. Aerodyn. 41–42, 2011–2022. Matsumoto, M., Saitoh, T., Kitazawa, M., Shirato, H., Nishizaki, T., 1995. Response characteristics of rain–wind induced vibration of stay-cable of cable-stayed bridges. J. Wind Eng. Ind. Aerodyn. 57, 323–333. Matsumoto, M., Yagi, T., Shigemura, Y., Tsusgima, D., 2001. Vortex-induced cable vibration of the cable-stayed bridges at high reduced wind velocity. J. Wind Eng. Ind. Aerodyn. 89, 633–647. Matsumoto, M., Yagi, T., Sakai, S., Ohya, J., Okada, T., 2003. Aerodynamic force/ aerodynamic response characteristics of inclined/yawed cable. In: Proceedings of the Eleventh International Conference on Wind Engineering, Texas, USA, June 21–23, pp. 2003. Matsumoto, M., et al., 2005. Motion-effect of water rivulet on rain–wind induced vibration of inclined stay-cables. In: Proceedings of the Sixth International Symposium on Cable Dynamics, pp. 255–262. Ohshima, K., Nanjo, M., 1987. Aerodynamics stability of the cables of a cable-stayed bridge subject to rain (a case study of the Aji River Bridge). In: Proceedings of US–Japan Joint Seminar on Natural Resources, pp. 324–336. Pacheco, B.M., Fujino, Y., 1993. Keeping cables calm. Civ. Eng. ASCE 63 (10), 56–58. Persoon, A.J., Noorlander, K., 1999. Full scale measurements on the Erasmus Bridge after rain/wind induced cable vibration. In: Larsen, A., Larose, G.L., Livesey, F.M., (Eds.), Proceedings of the Tenth International Conference on Wind Engineering, vol. 2, Copenhagen, Denmark, 21–24 June, 1999, pp. 1019–1026. Shi, J.J., Gu, M., et al., 2003. Wind induced vibration of cables of Nanjing No. 2 Bridge over Yangtze river, Research Report, Department of Bridge Engineering, Tongji University (in Chinese). Wilde, K., Witkowski, W., 2003. Simple model of rain–wind-induced vibrations of stayed cables. J. Wind Eng. Ind. Aerodyn. 91, 873–891. Xu, Y.L., Wang, L.Y., 2003. Analytical study of wind–rain-induced cable vibration. J. Wind Eng. Ind. Aerodyn. 91, 27–40. Yamaguchi, H., 1990. Analytical study on growth mechanism of rain vibration of cables. J. Wind Eng. Ind. Aerodyn. 33, 73–80. Yuscheweyh, H.P., 1999. The mechanism of rain–wind induced vibration. In: Larsen, A., Larose, G.L., Livesey, F.M., (Eds.), Proceedings of the Tenth International Conference on Wind Engineering, vol. 2, Copenhagen, Denmark, 21–24 June, pp. 1041–1047. Zhou, H.J., Xu, Y.L., 2007. Wind–rain-induced vibration and control of stay cables in a cable-stayed bridge. J. Struct. Control Health Monit. 14, 1013–1033.