Busting the Misconceptions of Tacoma Narrows Bridge collapse
Watch the video below before you read the remaining post. It is among the most famous film clips in history, important enough to have been selected for preservation in the US National Film Registry by the Library of Congress.
The Tacoma Narrows Bridge in the U.S. state of Washington which was opened for the traffic on July 1, 1940, collapsed dramatically just after four months on November 7. The only fatality was Tubby, a cocker spaniel that drowned inside a car after it tumbled into Puget Sound over which the bridge was built. Professor Farquharson, a dog lover, tried rescuing Tubby but he snapped at the friendly hand, nipping the knuckle, a photo of which was captioned “Token of Gratitude” in the Seattle Post-Intelligencer the next day.
Above all, this dramatic disaster that attracted wide attention at the time has been one of the most fascinating topics of engineering case studies ever since. However, despite the fact that it has elicited recurring references, notably in undergraduate physics textbooks, its treatment is either at best inadequate or at worst misleading.
In the 1991 American Journal of Physics paper by Billah and Scanlan, the authors scoured five college libraries, two high-school libraries, and three public libraries, as well as three campus bookstores and two of the largest textbook-carrying stores in New York City, revealing how widespread the misconceptions were pertaining to the reason behind the collapse.
Misconception #1: Collapse due to simple Wind Induced Resonance
Within a body, the atoms are in a constant state of motion having an average frequency of vibration called the natural frequency. When a periodic external load is applied, energy is stored in the body in the form of kinetic and potential energies and transmitted via atomic vibrations. Much of this stored energy is lost as heat due to friction. However, when the frequency of the applied load is equal (or nearly equal) to the natural frequency, energy is transmitted with minimal loss and results in the phenomenon of resonance. As a result, the body starts oscillating with relatively large amplitudes.
As Billah and Scanlan quoted one of the textbooks, “The wind produced a fluctuating resultant force in resonance with the natural frequency of the structure. This caused a steady increase in amplitude until the bridge was destroyed”. However, texts were vague about just what the exciting force was and how it acquired the necessary periodicity. “Gale winds” or “gusts of wind” were assumed for the origin of the force in some texts but “gale” and “gust” do not have any well-defined periodicity. Hence, such reasons were ruled out and closer investigations were sought.
Misconception #2: Collapse due to Vortex Induced Vibrations (VIV)
A likely candidate for the origin of periodicity in the force that acted on the bridge was vortex shedding.
When a bluff (non-streamlined) body is fixed in a fluid stream, the flow separates over substantial parts of its surface, i.e., the flow lines do not follow the contours of the body but break away from it. Depending on the geometry of the body and the flow velocity, periodic vortices (known as Strouhal vortices) form and detach from the low-pressure wake of the body, thus giving rise to vortex shedding. The repeating pattern of vortices thus formed is popularly known as the von Karman vortex street.
However, if the body is not mounted rigidly, it undergoes motion transverse to the direction of fluid flow. This happens due to the fact that a vortex is a low-pressure region and hence, a force in the direction of the vortex is produced on the body. This eventually gives rise to vortex induced vibrations.
As aforementioned, the vortices are shed periodically whose frequency can be determined mathematically and if it matches the natural frequency of the body, it can undergo large amplitude oscillations (due to resonance). This was exactly the conclusion that few texts probed by Billah and Scanlan made, which again was incorrect. According to Professor Farquharson, an engineering professor at the University of Washington and one of the main researchers into the cause of the bridge collapse, the wind speed during the collapse was steady at 42 mph, which gave a vortex shedding frequency of 1 Hz (from the calculation), far from the observed 0.2 Hz frequency of the bridge during its collapse. Consequently, the possibility of VIV was ruled out as well.
Moreover, Billah and Scanlan noted that even when VIVs occur, they have a “locking” effect on the body, i.e., as the amplitude starts increasing, this modifies the local fluid flow boundary conditions that instigates self-limiting forces, eventually attenuating the structural motion to small amplitudes.
Well, vortices were indeed the culprit and here’s the proof!
In another paper in the American Journal of Physics in 2006, Green and Unruh included a frame shown below from the film of the collapse taken by Barney Elliott which gave an important clue to the bridge collapse. The disintegration of the roadway threw dust into the air which acted as tracers for the airflow over the bridge.
However, a DIFFERENT kind of vortex wake was at play, NOT the von Karman vortex street!
The formation of von Karman vortex street depends only on the geometry of the body and the flow velocity, i.e., it is independent of the body motion. However, as Billah and Scanlan noted, when for any reason, a body changes the angle of attack in a fluid stream, it sheds new vorticity in its wake. This motion-induced vorticity (flutter wake) is different from the von Karman vortex street and depends on the motion as well. Furthermore, bluff bodies in oscillatory motion shed both types of vortices but at higher amplitudes, the flutter wake predominates. But what’s the physics behind a flutter wake and how exactly did it lead it to the bridge collapse?
A digression on self-oscillation…
Alejandro Jenkins in his paper on Self-oscillation in Physics Reports defines self-oscillation as “the generation and maintenance of periodic motion by a source of power that lacks periodicity”. The power source must only provide enough energy (i.e., do positive work) above a threshold to sustain the oscillation and it is the oscillator itself that sets the frequency with which it is driven, i.e., no external rate needs to be tuned to produce periodic motion. This is in contrast to resonant systems in which the oscillation is driven by the source of power that is tuned externally.
Ever heard the loud squeal or screech when someone speaks over the microphone or when music is played over it? It happens due to positive feedback. Essentially, a signal received from the microphone is amplified and passed out of the loudspeaker. The sound from the loudspeaker is received by the microphone again, amplified further, and passed out of the loudspeaker again. Self-oscillation is at the crux of the process. Clarinet acoustics and garden hose work on the same principle of self-oscillation.
Evidence of self-oscillation during the bridge collapse
As aforementioned, the source of power for self-oscillation itself lacks periodicity which complies with the observation of Professor Farquharson that the wind speed during the collapse was steady at 42 mph and not fluctuating. Furthermore, the motion of the bridge was akin to the oscillator that decided what frequency it would be driven at.
Billah and Scalan related it to the heart of a “chicken-egg” dilemma: Did the vortices cause the motion or the motion cause the vortices? In the case of flutter, it was the latter. Let’s see how.
Green and Unruh noted the existence of drifting vortices on the deck of the bridge in the film of the collapse and sought to answer as to what caused the vortices to drift. They asserted that were the bridge to remain motionless, equal strength vortices would form on either side of the deck and would have a canceling effect leading to no drift. But as the film evidence suggested, drifting did occur and hence was a result of the deck motion. They also explained the process of drift as shown in the figure above.
Anyway, how could self-oscillation have done the damage?
An important outcome of self-oscillation that occurs due to positive feedback is the generation of negative damping. Due to negative damping, as explained by Jenkins, the oscillator keeps drawing energy from the surroundings (instead of dissipating it out) as a result of which the amplitude of oscillation keeps growing exponentially until it becomes large enough for self-destruction. But there’s more to dig into…
Was the energy provided by the wind enough to sustain self-oscillation? What were the conditions that would lead to negative damping?
Allan Larsen produced the first physical model of the collapse wherein he examined the work done by the vortices as they drifted over the bridge. His model showed that there existed a critical wind speed at which a vortex crossed the deck in exactly one time period, doing no work on the bridge. Above this speed, the work done was positive, i.e., energy was fed to the bridge. This meant that negative damping came into play resulting in the exponential growth of amplitude.
The wind speed on the day of collapse (42 mph) was much higher than the critical speed (27 mph) and hence, fed surplus energy.
And, the final piece of the puzzle!
As previously mentioned, Green and Unruh already confirmed the existence of drifting vortices on the deck of the bridge during its collapse. Additionally, the vortex formed by the cement dust during the roadway disintegration drifted along the bridge deck at a rate similar to their simulations, i.e., Case 3 in the figure above which consolidated the fact that the vortex crossed the deck in less than a period, a requirement for negative damping!
What else did such a high wind speed imply?
Wind-tunnel tests done by Professor Farquharson on a 1/50th scale Tacoma Narrows Bridge model implied that at such high wind speeds, the dominant oscillation mode was a single-degree-of-freedom torsional mode that exponentially grew in amplitude. The vertical modes exhibited self-limiting effect wherein their amplitudes died out at high speeds. This was exactly what happened. In the last 45 minutes of the life of the bridge, it exhibited a torsional flutter of high amplitude as evident from the video of the collapse.
What do these analyses suggest about how to improve bridge stability?
Leon Moisseiff, the noted New York bridge engineer who served as designer and consultant engineer for the Golden Gate Bridge was also the designer of the Tacoma Narrows Bridge. His primary motive behind the design was to build the bridge for lesser money than what was listed originally and in doing so, he compromised with the structural integrity of the bridge leading to its collapse. Anyway, for what it’s worth, the wealth of lessons that this incident taught the engineers, a few basic guidelines would be:
- To avoid failure from the von Karman vortex street, the bridge must be made rigid enough to increase its natural frequency above that of the vortex street.
- To avoid failure from self-oscillation, damping must be increased as was done via the installation of hydraulic dampers at the towers and piers of the new Tacoma Narrows Bridge.
- The design must ensure high torsional stiffness so as to shift the critical speed of self-oscillation to higher speeds.
- The Tacoma Narrows Bridge design used solid plate girders instead of perforated trusses. This led to the diversion of the wind which eventually created large vortices upon the deck motion. Perforated trusses should be used to impart a more aerodynamic design that lets wind pass through the truss.
- Slots can be introduced between the bridge decks as was done in the new Tacoma Narrows Bridge. This tends to equalize the pressure above and below the deck making it stable.
The descriptions of the bridge collapse in most physics texts have their roots in oversimplified physics of resonance and do not penetrate deeply into the wealth of evidence provided by the investigations described in this post. Such texts do nothing but deviate from the truth. Given the fact that good evidence has been available in the literature for long accounting for the exact physics behind the collapse, efforts must be made to explore them and draw correct lessons from such events. A deep insight must be sought when loose explanations exist since eventually, the truth evolves by busting misconceptions!
