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Chapter 4. Design Guidelines, Wind-Induced Vibration of Stay Cables

NEW CABLE-STAYED BRIDGES

General

A sufficiently detailed cable vibration analysis (including modal analysis of the cable system) must be performed as part of the bridge design to identify the potential for cable vibration. The following factors must be examined: (1) the dynamic properties of the cables, (2) dynamics of the structural system, (3) geometry of the cable layout, (4) cable spacing, (5) exposure conditions, and (6) estimated Scruton numbers (S_c).

Mitigation of Rain/Wind Mechanism

At a minimum, providing an effective surface treatment for cable pipes to mitigate rain/wind-induced vibrations is highly recommended. One common method is the use of double-helical beads. The effectiveness of the surface treatment must be based on the tests applicable to the specific system, provided by the manufacturer.

Additional Mitigation

Depending on the outcome of the vibration study (item 1), the provision of at least one of the following major cable vibration mitigation measures (in addition to surface treatment) is recommended:

• Additional damping (using external dampers).
• Cable crossties.

Minimum Scruton Number

Following are minimum desired Scruton numbers (S_c):

• mζ / ρD² > 10   for regular cable arrangements.
• mζ /ρD² > 5     for cable pipes with effective surface treatment suppressing rain/wind-induced vibrations (see note below).

Note: Limited tests on cables with double-helix surface treatments have suggested that mζ/ρD² > 5 may be acceptable.⁽¹⁹⁾ However, such reductions should be made only for regularly spaced, single cable arrangements. In general, it is recommended to keep the Scruton number as high as possible by providing external dampers and/or crossties. For unusual geometry or double-stay arrangements where parallel stays are placed within close proximity to one another, careful case-by- case evaluation of these limits is recommended.

External Dampers

Manufacturer warranties should be provided for all damping devices. Most dampers used in bridges are proprietary items and design details should be provided by the manufacturer.

A damper can be tuned to yield optimal damping in any one selected mode (see figure 8). For other modes the level of damping will be less than this optimal value. Rain/wind-induced vibrations occur predominantly in mode 2. Therefore, if a damper is to be tuned to a particular mode to mitigate rain/wind-induced vibrations, it appears logical to select mode 2.

There are many types and designs of dampers, and linear dampers have been shown to be effective through their widespread use in the past. However, recent analytical studies show that nonlinear dampers can be used to provide a more optimal condition than linear dampers as these are effective over a larger range of modes. In particular, the damping performance of square-root dampers (β = 0.5) is independent of the mode number and is only affected by the amplitude of vibration.

With some dampers (such as dashpot type), an initial static friction force must be overcome before engaging of the viscous element. Field experiments have shown the presence of this stickmove- stick-move behavior associated with such dampers. This may effectively provide a fixed node instead of the intended damping for the cable at low-amplitude oscillations and should be considered. The visco-elastic type dampers where an elastomeric element is permanently engaged between the cable and the supporting elements, theoretically, are free of such initial frictional thresholds. On the other hand, there are also damper designs that rely on friction as the energy dissipation mechanism, and the static friction threshold for such dampers may be higher than for the other types.

Another factor needing consideration is the directionality of the damper. The cable vibrations observed in the field indicate both vertical and horizontal components of motion. Some damper designs are antisymmetric and provide damping against cable motion in any direction. Other dampers (e.g., dashpot types) provide damping against motion only along the axis of the damper. It is possible to arrange two or more such dampers so that the combination is effective in all directions. As the majority of the observed motion caused by rain/wind-induced vibrations is in the vertical direction, it may be sufficient to provide damping against only the vertical motion. However, this has not been clearly established. It is recommended that damping be made effective against cable movement in any direction.

Damper mounting details may transfer lateral forces caused by damper action onto components of the cable anchorage. Such forces must be considered in the design of the cable anchorages.

Cable Crossties

If used, provide clear and mandatory specifications for cable crossties. Experience shows that crossties, when properly detailed and installed, can be an effective method for suppressing undesirable levels of cable vibrations. Reported failures of crossties have been generally traced to improper details and material selection.

The use of crossties creates local modes, which must be considered in design. The frequency of the first plateau of local modes (see chapter 3) should be kept as high as possible. Symmetric configurations of the restrainers with respect to intermediate-length cables is preferred to increase the frequency interval (lower limit, in particular) corresponding to local modes since they minimize the longest segment length.

Cable crossties must be provided with initial tension sufficient to prevent slack of the crossties during design wind events. The level of tension depends on the dynamic properties of the cable system and the design wind event. The initial crosstie tensions must be established based on rational engineering analysis. Also, the tie to cable connection must be carefully designed and detailed for the transfer of the design forces.

User Tolerance Limits

A preliminary survey (see appendix I) on sensitivity of bridge users to stay cable vibrations has indicated that the comfort criteria for cable displacement can be described using the following maximums (within a 0.5 to 2.0 Hz range):

• 0.5 D (preferred).
• 1.0 D (recommended).
• 2.0 D (not to exceed).

While this aspect may need further study, the above can be used as a guide when such displacements can be computed and/or needed as input for the design of such elements as dampers and crossties. The displacement limits need not be considered for extreme events.

RETROFIT OF EXISTING BRIDGES

If an existing bridge is found or suspected to exhibit episodes of excessive stay cable vibration, an initial field survey and inspection of the cable system should be performed to assemble the following information:

• Eyewitness accounts, video footage of episodes.
• Condition of the stay cable anchorages and related components, noting any visible damage and/or loose, displaced components.

A brief field instrumentation and measurement program can be used to obtain such parameters as the existing damping levels of the cables. Instrumentation of cables to record the vibration episodes, wind direction, wind velocity, and rain intensity during their occurrences could also provide some confirmation of the nature of cable vibrations.

The mitigation methods available for retrofit of existing bridges follow closely those provided for the new bridges. However, the application of surface treatment may be difficult, impractical, or cost-prohibitive on existing structures. The addition of crossties and/or dampers is recommended.

A split pipe with surface modifications can be installed over the existing cable pipe if this is practical and cost-effective. In many of the older bridges for cables using PE pipes, ultraviolet (UV) protection to cable pipes is provided by wrapping the PE pipe with Tedlar tape. These cables require periodic rewrapping as part of routine maintenance. The newer high-density polyethylene (HDPE) cable pipes are manufactured with a coextruded outer shell that provides the needed UV resistance, thus providing a split pipe as a secondary outer pipe has the added benefit of eliminating the need for future Tedlar taping for the UV protection.

In addition, any damaged cable anchorage hardware must be properly retrofitted or replaced. It is recommended that the original cable supplier be contacted to ensure the replacement of cable anchorage components and that adding mitigative devices is compatible with the original design of the stay anchorage area.

WORKED EXAMPLES

Example 1

The following application shows how the information presented in this report can be used to assess the vulnerability of a given cable and provide mitigation measures.

Properties

Assume the following has been established based on the site meteorological data:

As the live-load cable tension under normal service is only a small fraction of the dead load, the cable vibration evaluations typically ignore the live load.

Under dead load, the modal frequencies of the cable are computed to be:

• f₁ = 0.875 Hz
• f₂ = 1.750 Hz
• f₃ = 2.625 Hz

Rain/Wind

Design for S_c = mζ /ρD² ≥ 10, as established in the design guidelines. Note that typical damping inherent in cables are in the range of 0.1 to 0.2 percent.

For S_c = mζ /ρD² ≥ 10, using ρ = 1.225 kg/m³ (0.0765 lb/ft³), the minimum damping required for S_c ≥ 10 is ζ ≥ 0.005 (0.5 percent).

Galloping

(Note that the present investigation has shown that galloping is not a major issue for normally spaced cable stays. However, this is included in this example to demonstrate its application.)

The critical wind velocity for onset of galloping was given in equation 6.

The numerical value of the constant c is typically taken as:

As galloping leads to divergent oscillations, it is considered a stability issue and the critical wind speed for onset of galloping is desired to be over 209 km/h (130 mi/h).

For U_CRIT = cf D √S_c ≥ 209 km/h (130 mi/h (190.7 ft/s)),

(13)

Assuming S_c ≥ 10:

Design Options

Option A

If the cable geometry falls within normal cable spacing and ignoring dry inclined galloping*:

• Galloping: The requirement for mitigation of wake galloping is automatically met (f₁ = 0.875 Hz > f = 0.82 Hz). Thus cable crossties are not required to raise the natural frequency of the cable.
• Rain/wind: For rain/wind, two options exist— (1) damping, and (2) crossties.

(1) Provide damping such that ζ ≥ 0.005 (0.5 percent, corresponding to S_c = 10). Assuming ζ ≥ 0.005 is to be achieved in the 1^st mode and the damper is to be located 3.6 m (11.7 ft) from the lower anchorage of the cable:

(14)

* Note that the wind tunnel tests conducted in the current study indicate that this can be ignored for normally spaced cables with damping ratios exceeding about 0.3 percent.

From figure 8 and equation 11:

(15)

Therefore:

(16)

(17)

Rain/wind-induced vibrations generally involve the first three modes of cable vibration, mode 2 being the most predominant.

Provide ζ₁ = 0.005 (1^st mode damping ratio, 0.5 percent).

This requires a damper with a damping constant:

(18)

Compute damping ratio in 2^nd and 3^rd modes:

2^nd mode (i = 2):

(19)

From figure 8:

(20)

3^rd mode (i = 3):

(21)

From figure 8:

(22)

Therefore, a damper with a damping coefficient of c = 2,850 lbf-s/ft at 11’ 8" from the lower end anchorage provides the following damping ratios for the first three modes:

• Mode 1 ζ ₁ = 0.0050 (0.50 percent) and S_c = 10.0.
• Mode 2 ζ₂ = 0.0077 (0.73 percent) and S_c = 15.2.
• Mode 3 ζ ₃ = 0.0120 (1.17 percent) and S_c = 23.7.

These are considered sufficient to mitigate rain/wind-induced vibrations based on the Scruton number criterion. Note that the above computation ignores the natural damping present in the cable. Hence the actual Scruton numbers, including the inherent damping, will be larger than those computed, making the design somewhat conservative.

(2) Provide cable crossties.

Note that by providing two cable crossties at the 1/3 points, cable modes 1 and 2 are eliminated, raising the natural frequency of the cable net to about 2.63 Hz. The cable crossties also raisedamping of the cable system considerably, as demonstrated by the preliminary vibrationmeasurements taken from the Leonard P. Zakim Bunker Hill Bridge (chapter 3)

Option B

If the cables are spaced closer together, using:

(23)

Providing two cable crossties at the 1/3 points would raise the natural frequency to 2.63 Hz. This is out of range for rain/wind-induced vibrations. It is generally believed that additional dampers can be neglected as the cable crossties are more than likely to raise the damping ratio beyond the 0.5 percent needed for mitigating rain/wind-induced vibrations.

Summary

Thus, the cable vibration mitigation following recommendations of the current study would consist of:

• For normal cable arrangements: Either a damper with c = 41.58 kN-s/m (2,850 lbf-s/ft) mounted 3.6 m (11.7 ft) from the end anchorage or cable crossties at 1/3 points.
• For closely spaced cable arrangements: Cable crossties at 1/3 points. The designer can consider providing additional damping depending on the details of the project, careful analysis of the specifics, and engineering judgment and expertise.

In all cases, an effective surface treatment, such as the double-helix spiral beads provided by leading cable manufacturers, is recommended for normal cable spacings. Limited tests show that the damper size may be reduced by as much as 50 percent with an effective surface treatment. Note that the elimination of dry inclined cable galloping as a design consideration for normal cable arrangements makes it possible to provide a mitigation design using only dampers, without the use of crossties.

Example 2

Example 1 demonstrated the use of the design guide in sizing the dampers so that the minimum Scruton number criteria is met for the 1^st and higher modes. Example 2 is based on data provided in table 4 (as shown in table 8), and is an illustration of providing maximum possible damping in mode 1.

Cable AS16

Substituting into equation 12:

(24)

Then using equation 11:

(25)

Mode 1:

(26)

Mode 2:

(27)

Mode 3:

(28)

Cable AS23

Equation 11:

(29)

Mode 1:

(30)

Mode 2:

(31)

Mode 3:

(32)

Note that the optimum value of ζ/(l /L) for linear dampers near the end of the stay is approximately 0.50, as is evident from figure 8. For cables AS16 and AS23, the damping provided in the 1^st mode is very close to this optimum, with damping in higher modes falling below the optimal level. The damper coefficients for the dampers used here are very large compared to the value obtained in example 1, and the Scruton numbers for modes 1–3 are much greater than the minimum of 10 recommended for the suppression of rain/wind-induced vibrations.

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