Tacoma Narrows Bridge (1940)

[Kristin Moore]

Resonance hypothesis

Frequently, the bridge's spectacular destruction is used as an object lesson in the necessity to consider both aerodynamics and resonance effects in civil and structural engineering. Billah and Scanlan (1991)[1] reported that in fact, many physics textbooks (for example Resnick et al.[15] and Tipler et al.[16] ) wrongly explain that the cause of the failure of the Tacoma Narrows bridge was mechanical resonance. Resonance is the tendency of a system to oscillate at maximum amplitude at certain frequencies, known as the system's natural frequencies. At these frequencies, even small periodic driving forces can produce large amplitude vibrations, because the system stores vibrational energy. For example, a child using a swing realizes that if the pushes are properly timed, the swing can move with a very large amplitude. The driving force, in this case the agent pushing the swing, exactly replenishes the energy that the system loses if its frequency equals the natural frequency of the system.

Usually, the approach taken by those physics textbooks is to introduce a first order degree-of-freedom forced oscillator, defined by the second order differential equation

[Kristin Moore]

where m, c and k stand for the mass, damping coefficient and stiffness of the linear system and F and ω represent the amplitude and the angular frequency of the exciting force. The solution of such ordinary differential equation as a function of time t represents the displacement response of the system (given appropriate initial conditions). In the above system resonance happens when ω is approximately

[Kristin Moore]

.e. ωr is the natural (resonant) frequency of the system. The actual vibration analysis of a more complicated mechanical system—such as an airplane, a building or a bridge—is basically based on the linearization of the equation of motion for the system, which is basically a multidimensional version of equation (eq. 1). The analysis requires eigenvalue analysis and thereafter the natural frequencies of the structure are found, together with the so-called degrees of freedom of the system, which are a set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system, i.e., the bridge moves as a (linear) combination of those basic deformed positions.

Each structure has natural frequencies. For resonance to occur, it is necessary to have also periodicity in the excitation force. The most tempting candidate of the periodicity in the wind force was assumed to be the so-called vortex shedding. This is because bluff bodies (non-streamlined bodies), like bridge decks, in a fluid stream do shed wakes, whose characteristics depend on the size and shape of the body and the properties of the fluid. These wakes are accompanied by alternating low-pressure vortices on the downwind side of the body (the so-called Von Kármán vortex street). The body will in consequence try to move toward the low-pressure zone, in an oscillating movement called vortex-induced vibration. Eventually, if the frequency of vortex shedding matches the resonance frequency of the structure, the structure will begin to resonate and the structure's movement can become self-sustaining.

The frequency of the vortices in the von Kármán vortex street is called the Strouhal frequency fs, and is given by

[Kristin Moore]

Here, U stands for the flow velocity, D is a characteristic length of the bluff body and S is the dimensionless Strouhal number, which depends on the body in question. For Reynolds Numbers greater than 1000, the Strouhal number is approximately equal to 0.21. In the case of the Tacoma Narrows, D  was approximately 8 feet (2.4 m) and S was 0.20.

It was thought that the Strouhal frequency was the same one of the natural vibration frequencies of the bridge i.e. 2πfs = ω, causing resonance and therefore vortex-induced vibration.

In the case of the Tacoma Narrows Bridge, there was no resonance. According to Professor Frederick Burt Farquharson, an engineering professor at the University of Washington and one of the main researchers about the cause of the bridge collapse, the wind was steady at 42 miles per hour (68 km/h) and the frequency of the destructive mode was 12 cycles/minute (0.2 Hz).[17] This frequency was neither a natural mode of the isolated structure nor the frequency of blunt-body vortex shedding of the bridge at that wind speed (which was approximately 1 Hz). It can be concluded therefore that the vortex shedding was not the cause of the bridge collapse. The event can be understood only while considering the coupled aerodynamic and structural system that requires rigorous mathematical analysis to reveal all the degrees of freedom of the particular structure and the set of design loads imposed.

Note, however, that vortex-induced vibration is a far more complex process that involves both the external wind-initiated forces and internal self-excited forces that lock on to the motion of the structure. During lock-on, the wind forces drive the structure at or near one of its resonance frequencies, but as the amplitude increases this has the effect of changing the local fluid boundary conditions, so that this induces compensating, self-limiting forces, which restrict the motion to relatively benign amplitudes. This is clearly not a linear resonance phenomenon, even if the bluff body has itself linear behaviour, since the exciting force amplitude is a nonlinear force of the structural response.[18]

[Kristin Moore]

Vortex shedding and Von Kármán vortex street behind a circular cylinder. The first hypothesis of failure of the Tacoma Narrows Bridge was resonance.[14] This is because it was thought that the Von Kármán vortex street frequency (the so-called Strouhal frequency) was the same as the torsional natural vibration frequency. This was found to be incorrect. Actual failure was due to aeroelastic flutter[1]

[Kristin Moore]

Origin of the confusion in failure modes

It is not clear what is the original source of the confusion[clarification needed]. Billah and Scanlan cite that Lee Edson in his biography of Theodore von Kármán[19] is a source of misinformation: "The culprit in the Tacoma disaster was the Karman vortex Street."

However, the Federal Works Administration report of the investigation (of which von Kármán was part) concluded that

    It is very improbable that the resonance with alternating vortices plays an important role in the oscillations of suspension bridges. First, it was found that there is no sharp correlation between wind velocity and oscillation frequency such as is required in case of resonance with vortices whose frequency depends on the wind velocity.[20]

Nowadays, even after over half a century, it is common to find a wide range of rather weak descriptions, explanations, and speculations about the failure of the original Tacoma Narrows Bridge in popular introductory physics and mathematical books for engineers.

[Kristin Moore]

Fate of the collapsed superstructure

Efforts to salvage the bridge began almost immediately after its collapse and continued into May 1943.[21] Two review boards, one appointed by the federal government and one appointed by the state of Washington, concluded that repair of the bridge was impossible, and the entire bridge would have to be dismantled and an entirely new bridge superstructure built.[22] With steel being a valuable commodity because of the involvement of the United States in World War II, steel from the bridge cables and the suspension span was sold as scrap metal to be melted down. The salvage operation cost the state more than was returned from the sale of the material, a net loss of over $350,000.[21]

The cable anchorages, tower pedestals and most of the remaining substructure were relatively undamaged in the collapse, and were reused during construction of the replacement span that opened in 1950. The towers, which supported Gertie's main cables and road deck, suffered major damage at their bases from being deflected twelve feet towards shore as a result of the collapse of the mainspan and the sagging of the sidespans. They were dismantled, and the steel sent to recyclers.

[Kristin Moore]

"Preservation" of the collapsed roadway

The underwater remains of the highway deck of the old suspension bridge act as a large artificial reef, and these are listed on the National Register of Historic Places with reference number 92001068.[23][24]

[Kristin Moore]

A lesson for history

Othmar Ammann, a leading bridge designer and member of the Federal Works Agency Commission investigating the collapse of the Tacoma Narrows Bridge, wrote:

    The Tacoma Narrows bridge failure has given us invaluable information...It has shown [that] every new structure [that] projects into new fields of magnitude involves new problems for the solution of which neither theory nor practical experience furnish an adequate guide. It is then that we must rely largely on judgement and if, as a result, errors, or failures occur, we must accept them as a price for human progress.[25]

The Bronx Whitestone Bridge, which is of similar design to the 1940 Tacoma Narrows Bridge, was reinforced shortly after the collapse. Fourteen-foot-high (4.3 m) steel trusses were installed on both sides of the deck in 1943 to weigh down and stiffen the bridge in an effort to reduce oscillation. In 2003, the stiffening trusses were removed and aerodynamic fiberglass fairings were installed along both sides of the road deck.

[Kristin Moore]

Galloping Gertie remains found on the beach below the Tacoma Narrows Bridge (Gig Harbor side). This piece of concrete was used to separate the roadway from the sidewalk. Rusted rebar prominent in the concrete section.

This piece was from the salvage operation from 1942. The concrete roadbed was simply jackhammered to the ground below. Steel was used for wartime purposes.


Photograph was taken by Tim Babcock. www.timbabcock.net

[Kristin Moore]

Replacement bridge


Because of materials and labor shortages as a result of the involvement of the United States in World War II, it took 10 years before a replacement bridge was opened to traffic. This replacement bridge was opened to traffic on October 14, 1950, and is 5,979 feet (1,822 m) long — 40 feet (12 m) longer than Galloping Gertie. The replacement bridge also has more lanes than the original bridge, which only had two traffic lanes, plus shoulders on both sides.

Half a century later, the rebuilt bridge that was completed in 1950 was exceeding its traffic capacity, and a second, parallel suspension bridge was constructed to carry eastbound traffic. The suspension bridge that was completed in 1950 was reconfigured to solely carry westbound traffic. The new parallel bridge opened to traffic in July 2007.

[Kristin Moore]

References

Notes

   1. ^ a b c d Billah, K.; R. Scanlan (1991). "Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks" (PDF). American Journal of Physics 59 (2): 118–124. Bibcode 1991AmJPh..59..118B. doi:10.1119/1.16590. http://www.ketchum.org/billah/Billah-Scanlan.pdf.
   2. ^ Henry Petroski. Engineers of Dreams: Great Bridge Builders and the Spanning of America. New York: Knopf/Random House, 1995.
   3. ^ Leon S. Moisseiff and Frederick Lienhard. "Suspension Bridges Under the Action of Lateral Forces," with discussion. Transactions of the American Society of Civil Engineers, No. 98, 1933, pp. 1080–1095, 1096–1141
   4. ^ a b c Richard Scott. In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability. American Society of Civil Engineers (June 1, 2001) ISBN 0784405425 http://books.google.com/books?id=DnQOzYDJsm8C
   5. ^ Rita Robison. "Tacoma Narrows Bridge Collapse." In When Technology Fails, edited by Neil Schlager, pp. 18–190. Detroit: Gale Research, 1994.
   6. ^ Washington State Department of Transportation (2005). "Tacoma Narrows Bridge: Eyewitness accounts of November 7, 1940". http://www.wsdot.wa.gov/tnbhistory/people/eyewitness.htm. Retrieved 2008-08-17.
   7. ^ "Professor's Analysis". Tacoma Narrows Bridge History. WDOT. http://www.wsdot.wa.gov/TNBhistory/Connections/connections3.htm.
   8. ^ As told by Clarence C. Murton, head of the Seattle Post Intelligencer Art Dept at the time, and close colleague of the photographer.
   9. ^ "Tubby Trivia". Tacoma Narrows Bridge History. Washington State Department of Transportation. http://www.wsdot.wa.gov/TNBhistory/tubby.htm.
  10. ^ "Tacoma Narrows Bridge: Weird Facts". Washington State Department of Transportation. http://www.wsdot.wa.gov/tnbhistory/weirdfacts.htm#4. "Finally, the WSTBA reimbursed Coatsworth for the loss of his car, $450.00. They had already paid him $364.40 for the loss of his car's "contents"."
  11. ^ Halacy Jr., D. S. (1965). Father of Supersonic Flight: Theodor von Kármán. pp. 119–122.
  12. ^ "Tacoma Narrows Bridge". University of Washington Special Collections. http://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html. Retrieved 2006-11-13.
  13. ^ "Weird Facts". Tacoma Narrows Bridge History. Washington State Department of Transportation. http://www.wsdot.wa.gov/TNBhistory/weirdfacts.htm#6. ""The effects of Galloping Gertie's fall lasted long after the catastrophe. Clark Eldridge, who accepted some of the blame for the bridge's failure, learned this first-hand. In late 1941, Eldridge was working for the U.S. Navy on Guam when the United States entered World War II. Soon, the Japanese captured Eldridge. He spent the remainder of the war (three years and nine months) in a prisoner of war camp in Japan. To his amazement, one day a Japanese officer, who had once been a student in America, recognized the bridge engineer. He walked up to Eldridge and said bluntly, 'Tacoma Bridge!'""
  14. ^ "Big Tacoma Bridge Crashes 190 Feet into Puget Sound. Narrows Span, Third Longest of Type in World, Collapses in Wind. Four Escape Death.". New York Times. November 8, 1940, Friday. "Cracking in a forty-two-mile an hour wind, the $6,400,000 Tacoma narrows Bridge collapsed with a roar today and plunged into the waters of Puget Sound, 190 feet below."
  15. ^ Halliday, David; Resnick, Robert; Walker, Jearl. Fundamentals of Physics, (Chapters 21-44). John Wiley & Sons. ISBN 0-470-04474-8.
  16. ^ Tipler, Paul Allen; Mosca, Gene. Physics for Scientists and Engineers : Volume 1B: Oscillations and Waves; Thermodynamics (Physics for Scientists and Engineers). W. H. Freeman. ISBN 0-7167-0903-1. )
  17. ^ F. B. Farquharson et al. Aerodynamic stability of suspension bridges with special reference to the Tacoma Narrows Bridge. University of Washington Engineering Experimental Station, Seattle. Bulletin 116. Parts I to V. A series of reports issued since June 1949 to June 1954.
  18. ^ Billah, K.Y.R. and Scanlan, R. H. "Vortex-Induced Vibration and its Mathematical Modeling: A Bibliography", Report No.SM-89-1. Department of Civil Engineering. Princenton University. April 1989
  19. ^ Theodore von Karman with Lee Edson. The wind and Beyond.Theodore von Karman: Pioneer in Aviation and Pathfinder in Space. Little Brown and Company, Boston, 1963. Page 213
  20. ^ Steven Ross, et al. "Tacoma Narrows 1940." In Construction Disasters: Design Failures, Causes, and Prevention. McGraw Hill, 1984, pp. 216–239,.
  21. ^ a b Tacoma Narrows Bridge Aftermath – A New Beginning: 1940 – 1950
  22. ^ University of Washington Special Collections
  23. ^ "National Register Information System". National Register of Historic Places. National Park Service. 2007-01-23. http://nrhp.focus.nps.gov/natreg/docs/All_Data.html.
  24. ^ "WSDOT - Tacoma Narrows Bridge: Extreme History". Washington State Department of Transportation. http://www.wsdot.wa.gov/TNBhistory/. Retrieved 2007-10-23.
  25. ^ Othmar H. Ammann, Theodore von Kármán and Glenn B. Woodruff. The Failure of the Tacoma Narrows Bridge, a report to the administrator. Report ot the Federal Works Agency, Washingthon, 1941

[Kristin Moore]

Tacoma Narrows Bridge Disaster

Stuart Doole was a fomer postdoc working with John Hogan on the dynamics of a peicewise linear model of a suspension bridge. An early preprint is available online . Contact John Hogan for more recent work , including a model that supports the transition between purely vertical oscillations and torsional modes, a feature that was observed in the collapse of the bridge. Also Alan Champneys has collaborated with Joe McKenna now of the University of Cork on the propensity of such simple models to support solitary waves; preprint 1 , preprint 2

Part of the motivation for this work was to provide a simple dynamical-systems explanation of the bi-stability and transition between modes observed in the famous collapse of the Tacoma Narrows suspension bridge.

The rest of this page provides a mirror to the information held at Carleton University Civil Engineering Exhibits Centre about the failure of the Bridge at Tacoma Narrows in May 1940.

On November 7, 1940, at approximately 11:00 AM, the first Tacoma Narrows suspension bridge collapsed due to wind-induced vibrations. Situated on the Tacoma Narrows in Puget Sound, near the city of Tacoma, Washington, the bridge had only been open for traffic a few months.

[Kristin Moore]

Photos of the Bridge collapsing

The following images and captions were taken from the report:
Smith, Doug,
A Case Study and Analysis of the Tacoma Narrows Bridge Failure
99.497 Engineering Project,
Department of Mechanical Engineering, Carleton University, Ottawa, Canada, March 29, 1974.
Supervised by Professor G. Kardos.

The full size images are generally 500x360 (approx.) 256-level grey scale images, of about 130 KBytes each.

http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html

[Kristin Moore]

This photograph shows the twisting motion of the center span just prior to failure.