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Publication Number: FHWA-HRT-05-083
Date: August 2007
Appendix C. Wind-Induced Cable Vibrations
MECHANICS OF WIND-INDUCED VIBRATIONS
There are many possible types of wind-induced vibrations of cables:
Some of these are more critical or probable than others, but they are all listed here for completeness. They are discussed in turn in the following sections.
Vortex Excitation of an Isolated Cable and Groups of Cables
Vortex excitation of a single isolated cable is caused by the alternate shedding of vortices from the two sides of the cable when the wind is approximately at right angles to the cable axis. The vortices are shed from one side of the cable at a frequency, n, that is proportional to the wind velocity U and inversely proportional to the cable diameter D. Thus as shown in equation 33,
where S is a nondimensional parameter, the Strouhal number, that remains constant over extended ranges of wind velocity. For circular cross-section cables in the Reynolds number range 104 to about 3×105, S is about 0.2.
Each time a vortex is shed it gives rise to a force at right angles to the wind direction. The alternate shedding thus causes an oscillating across-wind force. If the frequency of the oscillating force matches the frequency Nr of the rth natural mode of vibration of the cable, then oscillations of the cable in that mode will be excited. The wind velocity UVS at which this matching of vortex shedding frequency n to natural frequency Nr occurs can be deduced from equation 33 and is shown in equation 34:
Thus, as an example of a typical situation for a long stay cable, if the cable natural frequency Nr were 2 Hz and its diameter were D = 0.15 m (5.9 inches) then, using a Strouhal number S = 0.2, it can be determined that the vortex shedding excitation of the rth mode will occur at a wind speed of UVS = 1.5 m/s (3.4 mi/h). This is clearly a very low wind speed, showing that vortex shedding can begin in the lower modes at very modest speeds. For higher modes the wind speed will be higher.
The amplitude of the cable oscillations depends on the mass and damping of the cable. An approximate formula for the maximum amplitude y0 as a fraction of the diameter is shown in equation 35:
The lift coefficient CL has some dependence on oscillation amplitude as well as Reynolds number, but a rough value suitable for order of magnitude estimates is CL ≈ 0.3.
It can be seen from this relationship that increasing the mass and damping of the cables reduces oscillation amplitudes. The parameter (mζ/D2) is called the Scruton number. Higher values of Scruton number will tend to suppress vortex excitation and, as will be seen later, other types of wind-induced oscillation also tend to be mitigated by increasing the Scruton number.
It is difficult to give a precise estimate of the damping expected to occur in the cables of cablestayed bridges; however, cable damping ratios can range anywhere from 0.0005 to 0.01 (0.05 to 1.0 percent of critical). The lower end of this range is typical of very long cable stays before cement grouting, while the upper end of this range is more typical of shorter cable stays with grouting and some end damping.
For a bundled steel cable stay with a damping ratio of ζ = 0.005, the Scruton number (mζ/D2) has a value in the range of about 7 to 12 depending on the sheathing material, on whether grouting is used, and if so, on how much of the cable system mass consists of grouting. The value of (UVS/nD) ≈ 5, so the above expression leads to y0/D ≈ 0.008 × 0.3 × (1/7) × 25 = 0.0084 for the lower end of the range of (mζ/D2). The lower end of the range of (mζ/D2) would correspond to a typical cable on a cable-stayed bridge prior to grouting. The predicted amplitude of oscillation is small–of order 1 percent of the cable diameter–and it would drop to about y0/D ≈ 0.0049 (i.e., about one–half of 1 percent of the cable diameter) for the higher value of (mζ/D2) = 12 that corresponds to a grouted cable with 0.005 damping ratio.
If the damping ratio of the stay cables is extremely low (e.g., 0.001), as has been observed on some cable-stayed bridges before grouting is applied, then the amplitude could conceivably increase to about y0/D = 0.044 (i.e., about 4 percent of the cable diameter). Typically, 4 percent of the cable diameter would amount to not more than a few millimetres (i.e., still a small amplitude). Over many cycles at this amplitude it may be possible for fatigue problems to arise, but these larger oscillations are expected to be primarily a construction phase phenomenon when low values of (mζ/D2) occur on ungrouted and lightly damped cables. Therefore the time period involved is less likely to be long enough for fatigue problems to develop.
From the above discussion, it is clear that the classical vortex type of excitation of a single isolated cylindrical shape is unlikely to lead to serious oscillations of typical bridge cables. The predicted amplitudes are small even for lightly damped stay cables prior to grouting.
When one cable is close to other cables, especially when it lies in their wakes, the interactions become very complex especially at close spacings (e.g., 2 to 6 diameters). The vortex shedding behavior is modified, occurring at slightly different wind speeds, and leading to amplitudes that can be several times larger than for the isolated cable. However, even with this further magnification of the vortex response because of interaction effects, the amplitudes still do not reach magnitudes sufficient to explain the vibrations observed on some bridges.
Therefore, a general conclusion is that vortex shedding from the cables themselves is unlikely to be the root cause of cable vibration problems on bridges. There are other more serious forms of wind-induced oscillations, as explained below, that are more likely candidates for causing fatigue problems. Almost any small amount of damping that is added to the cables will be sufficient to effectively suppress vortex excitation.
Wake Galloping for Groups of Cables
When a cable lies in the wake of another cable the wind forces on it depend on its position in the wake. When the cable is near the center of the wake the wind velocity is low and it can move upwind against a lower drag force. If it is in the outer part of the wake, it experiences a stronger drag force and will tend to be blown downwind. Also, because of the shear flow in the wake of the upwind cable, the downwind cable will experience an across-wind force tending to pull it away from the wake center, the magnitude of this across wind force being a function of the distance from the wake center. These variations in drag and across wind forces can lead to the cable undergoing oscillations which involve both along-wind and across-wind components (i.e., the cable moves around an elliptical orbit). Over each complete orbit it can be shown that there is a net transfer of energy from the wind into the cable motion. For smaller spacings of the cables, say 2 to 6 diameters, the downwind cable moves around a roughly circular orbit. For larger spacings the orbit becomes more elongated into an ellipse with its major axis roughly aligned with the wind direction.
This type of instability is called wake galloping. The wind speeds involved are typically substantially higher than those for the onset of vortex excitation. It can cause oscillations much larger in amplitude than those seen in vortex excitation. For example, oscillation amplitudes of order 20 cable diameters have been observed on bundled power conductors for cable spacings in the 10–20 diameter range. In some cases adjacent cables clashed with each other. This type of wake galloping could potentially occur on the cable arrays on cable-stayed bridges or for grouped hangers. Less severe forms of galloping, but still problematic, can occur at smaller spacings in the 2 to 6 diameter range.
As for vortex excitation of the isolated cable, the Scruton number (mζ/D2) is an important guide as to the likelihood of there being a problem due to wake galloping effects. Cooper has proposed an approximate global stability criterion, based on earlier work by Connors.(15,20) This criterion gives the wind velocity UCRIT above which instability can be expected because of wake galloping effects. It is given in terms of (mζ/D2) as shown in equation 36:
where c is a constant. For close cable spacings (e.g., 2 to 6 diameters), the value of the constant cappears to be about 25 but for spacings in the 10 to 20 diameter range it goes up to about 80. This relationship shows that increasing the Scruton number (mζ/D2) or natural frequency Nr will make the cable array stable up to a higher wind velocity.
Thus, if for example (mζ/D2) = 10,D = 152 mm (6 inches), and Nr = 1 Hz, then for the spacing in the range 2 to 6 diameters we find UCRIT = 43.5 km/h (27 mi/h). This is quite low and is a speed common enough to have the potential to cause fatigue problems. However, Nr may be increased by installing crossties to the cables to shorten the effective length of cable for the vibration mode of concern. If crossties were used at two locations along this cable, dividing it into three equal lengths, the frequency Nr would be tripled resulting in UCRIT = 129 km/h (80 mi/h), which is high enough to have a much smaller probability of occurring. Added to the stiffening effect of the spacers is the additional damping that they most likely cause. For a cable, significant damping occurs at the points where it is clamped, such as at its ends or at spacers placed along its length.
It should be noted that the values of c in equation 36 quoted above were for wind normal to the axis of the cable. For cable-stayed bridges, wind normal to the axis of the cable typically is not possible, at least for wind directions where wake interference can occur. The angle is typically in the range of 25° to 60° rather than 90°. Therefore, it is probable that for stay cable arrays the values of c would be higher than those quoted above. Most cables on cable-stayed bridges are separated by more than 6 diameters. Therefore, it is probably conservative to assume a c-value of 80 when estimating the critical velocity for wake galloping. More research is needed in this area to better define wake galloping stability boundaries for inclined cables.
The oscillations caused by wake galloping are known to have caused fatigue of the outer strands of bridge hangers at end clamps on suspension and arch bridges. Fatigue problems of this type have yet to be encountered on cable-stayed bridges, but could potentially occur on cross cables. Therefore, it is good practice to avoid sharp corners where the cross cables enter the clamps linking them to the main cables or the deck. Bushings of rubber or other visco-elastic materials at the clamps can help reduce fatigue and can be a source of extra damping.
Galloping of Dry Single Cables
Single cables of circular cross sections do not gallop when they are aligned normal to the wind. However, when the wind velocity has a component along the span, it is no longer normal to the cable axis and for cables inclined to the wind, an instability with the same characteristics as galloping has been observed. In figure 33, the data of Saito et al. are shown plotted in the form of UCRIT ⁄(f D) versus Sc.(13) The data came from a series of wind tunnel experiments on a section of bridge cable mounted on a spring suspension system. Also plotted are curves calculated from equation 15 for several values of c in the range of 25 to 55. It can be seen that all the data points except one lie above the curve for c = 40 and that this value could be used to predict the onset of single inclined cable galloping.
Another possible mechanism of single inclined cable galloping which has not received a lot of attention in the literature is the notion that the wind “sees” an elliptical cross section of cable, for the typical wind directions where single cable galloping has been seen. Elliptical sections with ellipticity of about 2.5 or greater have a lift coefficient with a region of negative slope at angles of attack between 10° to 20°. An ellipticity of 2.5 would correspond to an angle of inclination of the cable of approximately 25°, which can occur in the outermost cables of long-span bridges. (Ellipticity is defined as the maximum width divided by the minimum width—e.g., a circle has an ellipticity of 1.0.) The negative slope of the lift coefficient may result in galloping instability if the level of structural damping in the cables is very low. The ellipticity range and angle range where galloping occurs is likely to be sensitive to surface roughness and Reynolds number.
Figure 33. Graph. Galloping of inclined cables.
There is a need for further experimental studies to confirm the results of Saito et al. and to extend the range of conditions studied.(13) Saito‘s results were nearly all at very low damping. There is a particular need to investigate if galloping of an inclined cable is indeed possible at damping ratios of 0.005 and higher.
The conclusion of recent testing (see appendix D) was that instability occurs at very low damping levels; however, if enough damping is added these instabilities disappear. It is expected that if enough cable system damping is supplied to mitigate rain/wind-induced vibrations, these vibrations should also be suppressed.
It has been observed on several bridges that the combination of rain and wind will cause cable vibrations. Hikami and Shiraishi described this phenomenon as it was observed on the Meikonishi cable-stayed bridge on cables of about 140-mm (5.5-inch) diameter.(11) This welldocumented case is a good illustration of the phenomenon. Oscillation single amplitudes of more than 254 mm (10 inches) developed. In other cases, amplitudes in excess of 1 m (3 ft) have been observed. The cables, consisting of parallel wires inside a PE pipe, had masses of 37.2 and 52.1 kg/m (25 and 35 lb/ft) before and after cement grouting, respectively. The damping ratio was reported to be in the range of 0.0011 to 0.0046 depending on cable length, vibration mode, and construction situation. It is probable that the lower values of damping corresponded to the ungrouted case. With this assumption, the Scruton number for the ungrouted cables was as low as (mζ/D2) = 1.7. The cable lengths were in the range of 64 to 198 m (210 to 650 ft).
The oscillations were seen in the wind speed range of 29 to 48 km/h (18 to 30 mi/h) and the modes of vibration affected by oscillations all had frequencies in the 1- to 3-Hz range and were any one of the first four modes. Based on wind tunnel tests that reproduced the oscillations, it was established that rivulets of water running down the upper and lower surfaces of the cable in rainy weather were the essential component of this aeroelastic instability. The water rivulets changed the effective shape of the cable. Furthermore they moved as the cable oscillated, causing cyclical changes in the aerodynamic forces which led, in a not fully understood way, to the wind feeding energy into oscillations. The wind directions causing the excitation were at about 45° to the plane of the cables with the affected cables being those sloping downwards in the direction of the wind. The particular range of wind velocities that caused the oscillations appears to be that which maintained the upper rivulet within a critical zone on the upper surface of the cable. A lower velocity simply allowed the water rivulet to drain down to the bottom surface and a higher velocity pushed it too far up onto the upper surface for it to be in the critical zone.
As with vortex excitation and galloping, any increase in the Scruton number (mζ/D2) is beneficial in reducing the cable‘s susceptibility to rain/wind-induced vibrations. It is noteworthy that many of the rain/wind-induced vibrations that have been observed on cable-stayed bridges have occurred during construction when both the damping and mass of the cable system are likely to have been lower than in the completed state, resulting in a low Scruton number. The routing of the completed cables adds both mass and probably damping, and often sleeves of visco-elastic material are added to the cable end connections which further raises the damping. The available circumstantial evidence indicates that the rain/wind type of vibration primarily arises as a result of some cables with exceptionally low damping, down in the ζ = 0.001 range.
Since many bridges have been built without experiencing problems from rain/wind-induced vibration of cables it appears probable that in many cases the level of damping naturally present is sufficient to avoid the problem. The rig test data of Saito et al., obtained using realistic cable mass and damping values, are useful in helping to define the boundary of instability for rain/wind oscillations.(13) Based on their results it appears that rain/wind oscillations can be avoided provided that the Scruton number is greater than 10 (equation 37):
This criterion can be used to assess how much damping a cable needs to avoid rain wind oscillation problems. Recent full-scale data have generally supported this criterion.(21)
Since the rain/wind oscillations are due to the formation of rivulets on the cable surface, it is probable that the instability is sensitive to the surface roughness or to small protrusions on the surface and to the type of sheathing material. One approach to solving the rain/wind problem is to have small protrusions running parallel to the cable axis or coiled around its surface. For example, Matsumoto et al. indicate that they found axially aligned protrusions of about 4.8 mm (0.2 inch) height and 11 mm (0.4 inch) width at 30-degree intervals around the perimeter of 153- mm (6-inch) diameter cables were successful in suppressing oscillations.(12) This method has been used on the Higashi-Kobe Bridge and has proven effective. However, for longer main spans, the additional drag force on the cables introduced by the protrusions can become a substantial part of the overall wind loads.
Flamand has used helical fillets 1.6 mm (0.06 inch) high and 2.4 mm (0.09 inch) wide with a pitch length of 0.6 m (2 ft) on the cables of the Normandie Bridge.(8) This technique has proven successful, with a minimal increase in drag coefficient. Work by Miyata and Yamada has shown that lumped surface roughness elements, typically of order 1 percent of cable diameter, can be used to introduce aerodynamic stability in rain/wind conditions with no appreciable increase in drag force.(22)
Examples of these techniques are illustrated in figure 34.
Figure 34. Drawing. Aerodynamic devices.
Galloping of Cables with Ice Accumulations
The accumulation of ice on a cable in an ice or freezing rain storm can lead to an effective change in shape of the cable to one that is aerodynamically unstable. This has caused large amplitude oscillations of long power conductor cables and could potentially occur on bridge cables. However, we are not aware of this being a common problem on bridges. In the power industry special dampers such as the Stockbridge damper have been employed to mitigate this problem. For bridges, the general measure of ensuring that cables do not have excessively low damping is probably sufficient to avoid most problems from this source.
Galloping of Cables in the Wakes of Other Structural Components
The wakes of bridge components such as towers or arches have velocity gradients and turbulence in them, and if a cable becomes impacted by these disturbed flows they can, in principle, experience galloping oscillations. It might be difficult to distinguish oscillations caused by this mechanism from those due to other causes such as buffeting, galloping in the wakes of other cables, or rain/wind oscillations. There do not appear to be any reports of cases where galloping in the wake of another structural component, such as a bridge tower, has been specifically identified. As with other types of instability, ensuring the cable has as high a damping as possible would be a good general preventive measure against this type of galloping.
Cable Oscillations Caused by Aerodynamic Excitation of Other Bridge Components
The natural modes of vibration of a long-span bridge and its cables, treated as a single system, are numerous. Many of these modes involve substantial cable motions accompanying relatively small motions of other major components such as the deck. It is conceivable, therefore, that the deck could be excited, by vortex shedding, for example, into very small oscillations which are of little significance for the deck but which involve concomitant motions of one or more of the cables at much larger amplitude. To the observer this type of response to wind could well appear like pure cable oscillations if the deck motions were too small to notice, and yet the source of the excitation would, in this case, be wind action on the deck.
Pinto da Costa et al. have shown analytically that small amplitudes of anchorage oscillation can lead to large cable responses if the exciting frequency is near the natural frequency of the lower modes of the cables.(23) Anchorage displacement amplitudes as low as 38.1 mm (1.5 inches) are shown to cause steady-state cable displacements of over 1.8 m (6 ft) for a 442-m (1,450-ft) stay cable with a critical damping ratio below 0.1 percent, typical of bridge stay cables during erection. Anchorage motions with frequencies equal to or double the first natural frequency of vibration of the cables are most likely to excite the cables. This type of excitation appears not to have been identified specifically in full-scale observations of bridges. On the other hand, it is a subtle effect that could easily be confused with other forms of cable excitation. It is a subject requiring further research.
Motions Caused by Wind Turbulence Buffeting
Flexible structures such as long bridges and their cable systems undergo substantial motions in strong winds simply because of the random buffeting action of wind turbulence. Very long cables will have their lower modes of vibration excited by this effect, but it is not an aeroelastic instability. Even very aerodynamically stable structures will be seen to move in strong winds if they are flexible. Buffeting motions are not typically a problem for bridge cables. However, they may be mistakenly identified as the beginnings of an aeroelastic instability. The buffeting motions increase gradually with wind speed, rather than in the sudden fashion associated with an instability.
Motion Caused by Fluctuating Cable Tensions
Fluctuating forces caused by turbulence produces fluctuating tension in the stays, which induces fluctuating forces at the anchorage points (lifting on the deck and pulling on the tower). Davenport has noted that the fluctuating axial tension in cable stays produced by drag is an additional excitation mechanism for the cables.(24) Denoting the change in drag per unit length of the cable by ΔFD, the magnitude of the fluctuating tension as a fraction of the fluctuating lateral load on the stays is shown in equation 38:
A strong multiplier effect is indicated in equation 38 through the term T/(mL) and the term (1 + (ke/kg) – (ω/ ωo)2) which has resonance characteristics at ω = ωo.
It is important to note the relationship between anchorage displacements and cable tension. A combination of fluctuating tension in the stay cables and oscillation of the anchorage points will likely have the effect of feeding energy into the cables, amplifying the motion. Ensuring adequate damping levels for the stay cables will reduce cable motion considerably.
From the above discussion it is clear that there are a number of causes of aerodynamic excitation of cables, so there are several possible approaches to developing mitigating measures. Some of these are not always practical for implementation but they are all listed below for completeness.
A number of bridges that have reported cable vibration problems are listed in table 10. Also listed in table 10 are remedies used to solve the cable vibration problem. All of the remedies listed in the table use some of the measures discussed above, and have apparently been effective in mitigating the cable vibration problem. These methods are illustrated in figures 35–37.
Figure 35. Drawing. Cable crossties.
Figure 36. Drawing. Viscous damping.
Figure 37. Drawing. Material damping.
Topics: research, infrastructure, structures
Keywords: research, structures, cable-stayed bridge, cables, vibrations, wind, rain, dampers, crossties
TRT Terms: research, infrastructure, Facilities, Structures