Aerodynamic Analysis of Cable-Suspended Bridges

The wind-induced failure on November 7, 1940, of the Tacoma Narrows Bridge in the state of Washington shocked the engineering profession. Many were surprised to learn that failure of bridges as a result of wind action was not unprecedented. During the slightly more than  12 decades prior to the Tacoma Narrows failure, 10 other bridges were severely damaged or destroyed by wind action (Table 15.12). As can be seen from Table 12a, wind-induced failures have occurred in bridges with spans as short as 245 ft up to 2800 ft. Other ‘‘modern’’ cable-suspended bridges have been observed to have undesirable oscillations due to wind (Table 15.12b).

Required Information on Wind at Bridge Site

Prior to undertaking any studies of wind instability for a bridge, engineers should investigate the wind environment at the site of the structure. Required information includes the character  of strong wind activity at the site over a period of years. Data are generally obtainable from local weather records and from meteorological records of the U.S. Weather Bureau. However, caution should be used, because these records may have been attained at a point some distance from the site, such as the local airport or federal building. Engineers should also be aware of differences in terrain features between the wind instrumentation site and the structure site that may have an important bearing on data interpretation. Data required are wind velocity, direction, and frequency. From these data, it is possible to predict high wind speeds, expected wind direction and probability of occurrence.
The aerodynamic forces that wind applies to a bridge depend on the velocity and direction of the wind and on the size, shape, and motion of the bridge. Whether resonance will occur under wind forces depends on the same factors. The amplitude of oscillation that may build up depends on the strength of the wind forces (including their variation with amplitude of bridge oscillation), the energy-storage capacity of the structure, the structural damping, and the duration of a wind capable of exciting motion.

The wind velocity and direction, including vertical angle, can be determined by extended observations at the site. They can be approximated with reasonable conservatism on the basis of a few local observations and extended study of more general data. The choice of the wind conditions for which a given bridge should be designed may always be largely a matter of judgment.
At the start of aerodynamic analysis, the size and shape of the bridge are known. Its energy-storage capacity and its motion, consisting essentially of natural modes of vibration, are determined completely by its mass, mass distribution, and elastic properties and can be computed by reliable methods.
The only unknown element is that factor relating the wind to the bridge section and its motion. This factor cannot, at present, be generalized but is subject to reliable determination in each case. Properties of the bridge, including its elastic forces and its mass and motions (determining its inertial forces), can be computed and reduced to model scale. Then, wind conditions bracketing all probable conditions at the site can be imposed on a section model.
The motions of such a dynamic section model in the properly scaled wind should duplicate reliably the motions of a convenient unit length of the bridge. The wind forces and the rate at which they can build up energy of oscillation respond to the changing amplitude of the motion. The rate of energy change can be measured and plotted against amplitude. Thus, the section-model test measures the one unknown factor, which can then be applied by calculation to the variable amplitude of motion along the bridge to predict the full behavior of the structure under the specific wind conditions of the test. These predictions are not precise but are about as accurate as some other features of the structural analysis.

Criteria for Aerodynamic Design

Because the factor relating bridge movement to wind conditions depends on specific site and bridge conditions, detailed criteria for the design of favorable bridge sections cannot be written until a large mass of data applicable to the structure being designed has been accumulated.
But, in general, the following criteria for suspension bridges may be used:
• A truss-stiffened section is more favorable than a girder-stiffened section.
• Deck slots and other devices that tend to break up the uniformity of wind action are likely to be favorable.
• The use of two planes of lateral system to form a four-sided stiffening truss is desirable because it can favorably affect torsional motion. Such a design strongly inhibits flutter and also raises the critical velocity of a pure torsional motion.
• For a given bridge section, a high natural frequency of vibration is usually favorable:
For short to moderate spans, a useful increase in frequency, if needed, can be attained by increased truss stiffness. (Although not closely defined, moderate spans may be regarded as including lengths from about 1,000 to about 1,800 ft.)

For long spans, it is not economically feasible to obtain any material increase in natural frequency of vertical modes above that inherent in the span and sag of the cable.
The possibility should be considered that for longer spans in the future, with their unavoidably low natural frequencies, oscillations due to unfavorable aerodynamic characteristics of the cross section may be more prevalent than for bridges of moderate span.
• At most bridge sites, the wind may be broken up; that is, it may be nonuniform across the site, unsteady, and turbulent. So a condition that could cause serious oscillation does not continue long enough to build up an objectionable amplitude. However, bear in mind:
There are undoubtedly sites where the winds from some directions are unusually steady and uniform.
There are bridge sections on which any wind, over a wide range of velocity, will continue to build up some mode of oscillation.
• An increase in stiffness arising from increased weight increases the energy-storage capacity of the structure without increasing the rate at which the wind can contribute energy. The effect is an increase in the time required to build up an objectionable amplitude. This may have a beneficial effect much greater than is suggested by the percentage increase in weight, because of the sharply reduced probability that the wind will continue unchanged for the greater length of time. Increased stiffness may give added structural damping and other favorable results.
Although more specific design criteria than the above cannot be given, it is possible to design a suspension bridge with a high degree of security against aerodynamic forces. This involves calculation of natural modes of motion of the proposed structure, performance of dynamic-section-model tests to determine the factors affecting behavior, and application of these factors to the prototype by suitable analysis.
Most long-span bridges built since the Tacoma bridge failure have followed the above procedures and incorporated special provisions in the design for aerodynamic effects. Designers of these bridges usually have favored stiffening trusses over girders. The second Tacoma Narrows, Forth Road, and Mackinac Straits Bridges, for example, incorporate deep stiffening trusses with both top and bottom bracing, constituting a torsion space truss. The Forth Road and Mackinac Straights Bridges have slotted decks. The Severn Bridge, however, has a streamlined, closed-box stiffening girder and inclined suspenders. Some designs incorporate longitudinal cable stays, tower stays, or even transverse diagonal stays (Deer Isle Bridge). Some have unloaded backstays. Others endeavor to increase structural damping by frictional or viscous means. All have included dynamic-model studies as part of the design.

Wind-Induced Oscillation Theories

Several theories have been advanced as models for mathematical analysis to develop an understanding of the process of wind excitation. Among these are the following.
Negative-Slope Theory. When a bridge is moving downward while a horizontal wind is blowing (Fig. 15.66a), the resultant wind is angled upward (positive angle of attack) relative to the bridge. If the lift coefficient CL , as measured in static tests, shows a variation with wind angle  such as that illustrated by curve A in Fig. 15.66b, then, for moderate amplitudes, there is a wind force acting downward on the bridge while the bridge is moving downward. The bridge will therefore move to a greater amplitude than it would without this wind force. The motion will, however, be halted and reversed by the action of the elastic forces. Then, the vertical component of the wind also reverses. The angle of attack becomes
negative, and the lift becomes positive, tending to increase the amplitude of the rebound.
With increasing velocity, the amplitude will increase indefinitely or until the bridge is de stroyed. A similar, though more complicated situation, would apply for torsional or twisting motion of the bridge.

Vortex Theory. This attributes aerodynamic excitation to the action of periodic forces having a certain degree of resonance with a natural mode of vibration of the bridge. Vortices, which form around the trailing edge of the airfoil (bridge deck), are shed on alternating sides, giving rise to periodic forces and oscillations transverse to the deck.
Flutter Theory. The phenomenon of flutter, as developed for airfoils of aircraft and applied to suspension-bridge decks, relates to the fact that the airfoil (bridge deck) is supported so that it can move elastically in a vertical direction and in torsion, about a longitudinal axis.
Wind causes a lift that acts eccentrically. This causes a twisting moment, which, in turn, alters the angle of attack and increases the lift. The chain reaction becomes catastrophic if the vertical and torsional motions can take place at the same coupled frequency and in appropriate phase relation.
F. Bleich presented tables for calculation of flutter speed vF for a given bridge, based on flat-plate airfoil flutter theory. These tables are applicable principally to trusses. But the tables  are difficult to apply, and there is some uncertainty as to their range of validity.
A. Selberg has presented the following formula for flutter speed:

Selberg has also published charts, based on tests, from which it is possible to approximate the critical wind speed for any type of cross section in terms of the flutter speed.
Applicability of Theories. The vortex and flutter theories apply to the behavior of suspension bridges under wind action. Flutter appears dominant for truss-stiffened bridges, whereas vortex action seems to prevail for girder-stiffened bridges. There are mounting indications, however, these are, at best, estimates of aerodynamic behavior. Much work has been done and is being done, particularly in the spectrum approach and the effects of nonuniform, turbulent winds.

Design Indices

Bridge engineers have suggested several criteria for practical design purposes. O. H. Ammann, for example, developed two analytical-empirical indices that were applied in the design of the Verrazano Narrows Bridge, a vertical-stiffness index and a torsional-stiffness index.
Vertical-Stiffness Index Sv . This is based on the magnitude of the vertical deflection of the suspension system under a static downward load covering one-half the center span. The index includes a correction to allow for the effect of structural damping of the suspended structure and for the effect of different ratios of side span to center span.

Typical values of these indices are listed in Table 15.13 for several bridges.
Other indices and criteria have been published by D. B. Steinman. In connection with
these, Steinman also proposed that, unless aerodynamic stability is otherwise assured, the

Solution of these equations for the natural frequencies and modes of motion is dependent on the various possible static forms of suspension bridges involved (see Fig. 15.9). Numerous lengthy tabulations of solutions have been published.

Damping

Damping is of great importance in lessening of wind effects. It is responsible for dissipation of energy imparted to a vibrating structure by exciting forces. When damping occurs, one part of the external energy is transformed into molecular energy, and another part is transmitted to surrounding objects or the atmosphere. Damping may be internal, due to elastic hysteresis of the material or plastic yielding and friction in joints, or Coulomb (dry friction), or atmospheric, due to air resistance.

Aerodynamics of Cable-Stayed Bridges

The aerodynamic action of cable-stayed bridges is less severe than that of suspension bridges, because of increased stiffness due to the taut cables and the widespread use of torsion box decks. However, there is a trend towards the use of the composite steel-concrete superstructure girders (Fig. 15.16) for increasingly longer spans and to reduce girder dead weight. This configuration, because of the long spans and decreased mass, can be relatively more sensitive to aerodynamic effects as compared to a torsionally stiff box.

Stability Investigations

It is most important to note that the validation of stability of the completed structure for expected wind speeds at the site is mandatory. However, this does not necessarily imply that the most critical stability condition of the structure occurs when the structure is fully completed.
A more dangerous condition may occur during erection, when the joints have not been fully connected and, therefore, full stiffness of the structure has not yet been realized.
In the erection stage, the frequencies are lower than in the final condition and the ratio of torsional frequency to flexural frequency may approach unity. Various stages of the partly erected structure may be more critical than the completed bridge. The use of welded components in pylons has contributed to their susceptibility to vibration during erection.
Because no exact analytical procedures are yet available, wind-tunnel tests should be used to evaluate the aerodynamic characteristics of the cross section of a proposed deck girder, pylon, or total bridge. More importantly, the wind-tunnel tests should be used during the design process to evaluate the performance of a number of proposed cross sections for a particular project. In this manner, the wind-tunnel investigations become a part of the design decision process and not a postconstruction corrective action. If the wind-tunnel evaluations are used as an after-the-fact verification and they indicate an instability, there is the distinct risk that a redesign of a retrofit design will be required that will have undesirable ramifications on schedules and availability of funding.

Rain-Wind Induced Vibration

Well known mechanisms of cable vibration are vortex and wake galloping. Starting in approximately
the mid-1980’s, a new phenomenon of cable vibration has been observed that occurs during the simultaneous presence of rain and wind, thus, it is given the name ‘‘rainwind vibration,’’ or rain vibration.
The excitation mechanism is the formation of water rivulets, at the top and bottom, that run down the cable oscillating tangentially as the cables vibrate, thus changing the aerodynamic profile of the cable (or the enclosing HDPE pipe). The formation of the upper rivulet appears to be the more dominant factor in the origin of the rain-wind vibration.
In the current state-of-the-art, three basic methods of rain-wind vibration suppression are being considered or used:
• Rope ties interconnecting the cable stays in the plane of the stays, Fig. 15.67a
• Modification of the external surface of the enclosing HDPE pipe, Fig. 15.67b
• Providing external damping

The interconnection of stays by rope ties produces node points at the point of connection of the secondary tie to the cable stays. The purpose is to shorten the free length of the stay and modify the natural frequency of vibration of the stay. The modification of the surface may be such as protuberances that are axial, helical, elliptical or circular or grooves or dimples. The intent is to discourage the formation of the rivulets and/or its oscillations.
Various types of dampers such as viscous, hydraulic, tuned mass and rubber have also been used to suppress the vibration.
The rain-wind vibration phenomenon has been observed during construction prior to grout injection which then stabilizes after grout injection. This may be as a result of the difference in mass prior to and after grout injection (or not). It also has been noticed that the rain-wind vibration may not manifest itself until some time after completion of the bridge. This may be the results of a transition from initial smoothness of the external pipe to a roughness, sufficient to hold the rivulet, resulting from an environmental or atmospheric degradation of the surface of the pipe.
The interaction of the various parameters in the rain-wind phenomenon is not yet well understood and an optimum solution is not yet available. It should also be noted that under similar conditions of rain and wind, the hangars of arch bridges and suspenders of suspension bridges can also vibrate.