View source for Engineering:Tacoma Narrows Bridge (1940)

{{Short description|Former suspension bridge in Puget Sound, Washington, USA; collapsed in 1940}}
{{Infobox bridge
|bridge_name   = Tacoma Narrows Bridge
|other_name    = Galloping Gertie
|image         = Opening_day_of_the_Tacoma_Narrows_Bridge,_Tacoma,_Washington.jpg
|caption       = The original Tacoma Narrows Bridge on its opening day on July 1, 1940
|designer      = [[Biography:Leon Moisseiff|Leon Moisseiff]]
|design        = [[Engineering:Suspension bridge|Suspension]]
|material      = [[Engineering:Carbon steel|Carbon steel]]
|lanes         = 2
|mainspan      = {{convert|2800|ft|m|1}}
|length        = {{convert|5939|ft|m|1}}
|below         = {{convert|195|ft|m|1}}
|width         = 
|clearance     = 
|begin         = September 27, 1938
|open          = July 1, 1940
|collapsed     = November 7, 1940
|replaced_by   = Tacoma Narrows Bridge (1950)
|coordinates   = {{coord|47|16|N|122|33|W|region:US|display=inline,title}}
<!-- | map_type            = USA Washington
| map_relief          = no -->
}}
[[File:Tacoma Narrows Bridge Location.png|thumb|right|Map showing location of the bridge]]

The '''1940 Tacoma Narrows Bridge''', the first bridge at this location, was a [[Engineering:Suspension bridge|suspension bridge]] in the U.S. state of {{wipe|Washington (state)}} that spanned the Tacoma Narrows [[Earth:Strait|strait]] of Puget Sound between Tacoma and the Kitsap Peninsula. It opened to traffic on July 1, 1940, and dramatically [[Physics:Structural integrity and failure|collapsed]] into Puget Sound on November&nbsp;7 of the same year.<ref>{{Cite web|title=Tacoma Narrows Bridge collapses|url=https://www.history.com/this-day-in-history/tacoma-narrows-bridge-collapses|access-date=2020-07-12|website=HISTORY|language=en}}</ref> The bridge's collapse has been described as "spectacular" and in subsequent decades "has attracted the attention of engineers, physicists, and mathematicians".<ref>Gianni Arioli and Filippo Gazzola. [https://www.sciencedirect.com/science/article/pii/S0307904X14003424?via%3Dihub A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge]. Applied Mathematical Modelling, Jan 2015. [https://doi.org/10.1016/j.apm.2014.06.022 doi.org]</ref> Throughout its short existence, it was the world's third-longest suspension bridge by main span, behind the Golden Gate Bridge and the George Washington Bridge.

Construction began in September 1938. From the time the [[Engineering:Deck (bridge)|deck]] was built, it began to move vertically in windy conditions, so construction workers nicknamed the bridge '''Galloping Gertie'''. The motion continued after the bridge opened to the public, despite several [[Physics:Damping|damping]] measures. The bridge's main span finally collapsed in {{convert|40|mph|km/h|adj=on}} winds on the morning of November 7, 1940, as the deck oscillated in an alternating twisting motion that gradually increased in amplitude until the deck tore apart.

The portions of the bridge still standing after the collapse, including the towers and cables, were dismantled and sold as scrap metal. Efforts to replace the bridge were delayed by the United States' entry into [[Social:World War II|World War II]], but in 1950, a new Tacoma Narrows Bridge opened in the same location, using the original bridge's tower pedestals and cable anchorages. The portion of the bridge that fell into the water now serves as an artificial reef.

The bridge's collapse had a lasting effect on science and engineering. In many [[Physics:Physics|physics]] textbooks, the event is presented as an example of elementary forced [[Physics:Mechanical resonance|mechanical resonance]], but it was more complicated in reality; the bridge collapsed because moderate winds produced [[Physics:Aeroelasticity#Flutter|aeroelastic flutter]] that was self-exciting and unbounded: For any constant sustained wind speed above about {{cvt|35|mph|km/h||}}, the amplitude of the ([[Physics:Torsion (mechanics)|torsion]]al) flutter oscillation would continuously increase, with a negative damping factor, i.e., a reinforcing effect, opposite to damping.<ref name="BillahScanlan91" /> The collapse boosted research into bridge [[Physics:Aerodynamics|aerodynamics]]-[[Physics:Aeroelasticity|aeroelastics]], which has influenced the designs of all later long-span bridges.

==Design and construction==
Proposals for a bridge between Tacoma and the Kitsap Peninsula date at least to the Northern Pacific Railway's 1889 trestle proposal, but concerted efforts began in the mid-1920s. The Tacoma Chamber of Commerce began campaigning and funding studies in 1923.<ref>{{Cite journal|title = Tacoma Narrows Bridges|last = Petroski|first = Henry|date = 2009|journal =American Scientist|doi = 10.1511/2009.77.103|volume = 97|issue = 2|edition = 2|issn = 0003-0996|pages = 103–107}}</ref> Several noted bridge engineers were consulted, including Joseph B. Strauss, who went on to be chief engineer of the Golden Gate Bridge, and David B. Steinman, later the designer of the Mackinac Bridge. Steinman made several Chamber-funded visits and presented a preliminary proposal in 1929, but by 1931 the Chamber had canceled the agreement on the grounds that Steinman was not working hard enough to obtain financing. At the 1938 meeting of the structural division of the American Society of Civil Engineers, during construction of the bridge, with its designer in the audience, Steinman predicted its failure.{{citation needed|date=February 2021}}

In 1937, the Washington State legislature created the Washington State Toll Bridge Authority and appropriated $5,000 (equivalent to ${{inflation|US|5000|1931|r=-3|fmt=c}} today) to study the request by Tacoma and Pierce County for a bridge over the Narrows.<ref>Plaut, R.H. (2008). "Snap Loads and Torsional Oscillation of the original Tacoma Narrows Bridge". ''[[Physics:Journal of Sound and Vibration|Journal of Sound and Vibration]]. doi: 10.1016/j.jsv.2007.07.057''</ref>

From the start, financing of the bridge was a problem: Revenue from the proposed tolls would not be enough to cover construction costs; another expense was buying out the ferry contract from a private firm running services on the Narrows at the time. Nonetheless, there was strong support for the bridge from the United States Navy, which operated the Puget Sound Naval Shipyard in Bremerton, and from the United States Army, which ran McChord Field and Fort Lewis near Tacoma.<ref name=":1">{{Cite web|title=Tacoma Narrows Bridge History: Creating the Narrows Bridge 1937- 1940|url=https://www.wsdot.com/TNBhistory/Connections/connections2.htm|access-date=2020-07-12|website=www.wsdot.com}}</ref>

Washington State engineer Clark Eldridge<!-- state employee, not just resident in the state --> produced a preliminary tried-and-true conventional suspension bridge design, and the Washington State Toll Bridge Authority requested $11 million (equivalent to ${{inflation|US|11|1931|r=0|fmt=c}} million today) from the federal Public Works Administration (PWA). Preliminary construction plans by the Washington Department of Highways had called for a set of {{convert|25|ft|m|adj=mid|-deep}} [[Physics:Truss|truss]]es to sit beneath the roadway and stiffen it.
[[File:Tacoma Narrows Bridge opening program June 30, 1940.jpg|thumb|left|Program for the opening of the Tacoma Narrows Bridge, June 30, 1940]]
However, "Eastern consulting engineers"&nbsp;— by which Eldridge meant [[Biography:Leon Moisseiff|Leon Moisseiff]], the noted New York bridge engineer who served as designer and consultant engineer for the Golden Gate Bridge&nbsp;— petitioned the PWA and the Reconstruction Finance Corporation (RFC) to build the bridge for less. Moisseiff and Frederick Lienhard, the latter an engineer with what was then known in New York as the Port Authority, had published a paper<ref>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</ref> that was probably the most important theoretical advance in the bridge engineering field of the decade.<ref name="RichardScott">{{cite book| first=Richard |last=Scott |title=In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability. |publisher=American Society of Civil Engineers |date=June 1, 2001 |isbn=0-7844-0542-5 |url=https://books.google.com/books?id=DnQOzYDJsm8C}}</ref> Their theory of elastic distribution extended the [[Engineering:Deflection (engineering)|deflection]] theory that was originally devised by the Austrian engineer Josef Melan to horizontal bending under static wind load. They showed that the stiffness of the main cables (via the suspenders) would absorb up to one-half of the static wind pressure pushing a [[Engineering:Suspended structure|suspended structure]] laterally. This energy would then be transmitted to the anchorages and towers.<ref name="RichardScott" /> Using this theory, Moisseiff argued for stiffening the bridge with a set of {{convert|8|ft|m|adj=mid|spell=in|-deep}} plate girders rather than the {{convert|25|ft|m|adj=mid|-deep}} trusses proposed by the Washington State Toll Bridge Authority. This approach meant a slimmer, more elegant design, and also reduced the construction costs as compared with the Highway Department's design proposed by Eldridge. Moisseiff's design won out, inasmuch as the other proposal was considered to be too expensive. On June 23, 1938, the PWA approved nearly $6 million (equivalent to ${{inflation|US|6|1938|r=1|fmt=c}} million today) for the Tacoma Narrows Bridge.<ref name=":1" /> Another $1.6 million (${{inflation|US|1.6|1938|r=1|fmt=c}} million today) was to be collected from tolls to cover the estimated total $8<!-- ibid --> million cost (${{inflation|US|8|1938|r=1|fmt=c}} million today).

Following Moisseiff's design, bridge construction began on September 27, 1938. Construction took only nineteen months, at a cost of $6.4 million (${{inflation|US|6.4|1938|r=1|fmt=c}} million today), which was financed by the grant from the PWA and a loan from the RFC.

The Tacoma Narrows Bridge, with a main span of {{convert|2800|ft}}, was the third-longest suspension bridge in the world at that time, following the George Washington Bridge between New Jersey and {{wipe|New York City}}, and the Golden Gate Bridge, connecting {{wipe|San Francisco}} with Marin County to its north.<ref>Henry Petroski. Engineers of Dreams: Great Bridge Builders and the Spanning of America. New York: Alfred A. Knopf/{{wipe|Random House}}, 1995.</ref>

Because planners expected fairly light traffic volumes, the bridge was designed with two lanes, and it was just {{convert|39|ft}} wide.<ref>{{Cite web|title=BUILDING BIG: Databank: Tacoma Narrows Bridge|url=https://www.pbs.org/wgbh/buildingbig/wonder/structure/tacoma_narrows.html|access-date=2020-07-12|website=www.pbs.org}}</ref> This was quite narrow, especially in comparison with its length. With only the {{convert|8|ft|m|adj=mid|-deep}} plate girders providing additional depth, the bridge's roadway section was also shallow.

The decision to use such shallow and narrow girders proved the bridge's undoing. With such minimal girders, the deck of the bridge was insufficiently rigid and was easily moved about by winds; from the start, the bridge became infamous for its movement. A mild to moderate wind could cause alternate halves of the center [[Engineering:Span (engineering)|span]] to visibly rise and fall several feet over four- to five-second intervals. This flexibility was experienced by the builders and workmen during construction, which led some of the workers to christen the bridge "Galloping Gertie". The nickname soon stuck, and even the public (when the toll-paid traffic started) felt these motions on the day that the bridge opened on July 1, 1940.

==Attempt to control structural vibration==
Since the structure experienced considerable vertical [[Oscillation|oscillation]]s while it was still under construction, several strategies were used to reduce the motion of the bridge. They included<ref>Rita Robison. "Tacoma Narrows Bridge Collapse." In ''When Technology Fails'', edited by Neil Schlager, pp. 18–190. Detroit: [[Company:Gale (publisher)|Gale Research]], 1994.</ref>
* attachment of tie-down cables to the plate girders, which were anchored to 50-ton [[Chemistry:Concrete|concrete]] blocks on the shore. This measure proved ineffective, as the cables snapped shortly after installation.
* addition of a pair of inclined cable stays that connected the main cables to the bridge deck at mid-span. These remained in place until the collapse, but were also ineffective at reducing the oscillations.
* finally, the structure was equipped with hydraulic buffers installed between the towers and the floor system of the deck to damp longitudinal motion of the main span. The effectiveness of the hydraulic dampers was nullified, however, because the seals of the units were damaged when the bridge was sand-blasted before being painted.

The Washington State Toll Bridge Authority hired Professor Frederick Burt Farquharson, an engineering professor at the [[Organization:University of Washington|University of Washington]], to make [[Physics:Wind tunnel|wind tunnel]] tests and recommend solutions in order to reduce the oscillations of the bridge. Professor Farquharson and his students built a 1:200-scale model of the bridge and a 1:20-scale model of a section of the deck. The first studies concluded on November 2, 1940—five days before the bridge collapse on November 7. He proposed two solutions:
* To drill holes in the lateral girders and along the deck so that the air flow could circulate through them (in this way reducing [[Physics:Lift (force)|lift forces]]).
* To give a more [[Physics:Aerodynamics|aerodynamic]] shape to the transverse section of the deck by adding fairings or deflector vanes along the deck, attached to the girder fascia.
The first option was not favored, because of its irreversible nature. The second option was the chosen one, but it was not carried out, because the bridge collapsed five days after the studies were concluded.<ref name="RichardScott" />

==Collapse==
[[File:Tacoma-narrows-bridge-collapse.jpg|thumb|The main bridge span falling into the strait on November 7, 1940]]
Leonard Coatsworth, an editor at ''The News Tribune'' in Tacoma, was the last person to drive on the bridge:

{{Blockquote|Around me I could hear concrete cracking. I started back to the car to get the dog, but was thrown before I could reach it. The car itself began to slide from side to side on the roadway. I decided the bridge was breaking up and my only hope was to get back to shore.

On hands and knees most of the time, I crawled {{convert|500|yd|ft m|disp=sqbr}} or more to the towers&nbsp;... My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb&nbsp;... Towards the last, I risked rising to my feet and running a few yards at a time&nbsp;... Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.<ref name=coatsworth>{{Cite web|url=https://www.wsdot.wa.gov/tnbhistory/eyewitnesses.htm|title=Eyewitness accounts|website=Tacoma Narrows Bridge history|publisher=Washington State Department of Transportation (WSDOT)}}</ref>}}

{{anchor|Tubby the dog}}
Tubby, Coatsworth's Cocker Spaniel, was the only fatality of the Tacoma Narrows Bridge disaster; he was lost along with Coatsworth's car. Professor Farquharson<ref name="Professor Farquharson">{{Cite web|url=http://www.wsdot.wa.gov/TNBhistory/Connections/connections3.htm|title=Professor's Analysis|website=Tacoma Narrows Bridge History|publisher=Washington State Department of Transportation (WSDOT)|access-date=2008-08-15|archive-date=2008-06-27|archive-url=https://web.archive.org/web/20080627110928/http://www.wsdot.wa.gov/tnbhistory/Connections/connections3.htm|url-status=dead}}</ref> and a news photographer<ref>As told by Clarence C. Murton, head of the ''Seattle Post Intelligencer'' Art Dept at the time, and close colleague of the photographer.</ref> attempted to rescue Tubby during a lull, but the dog was too terrified to leave the car and bit one of the rescuers. Tubby died when the bridge fell and neither his body nor the car was ever recovered.<ref name="tubby">{{Cite web|url=https://wsdot.wa.gov/TNBhistory/tubby-trivia.htm|title=Tubby Trivia|website=Tacoma Narrows Bridge History|publisher=Washington State Department of Transportation}}</ref> Coatsworth had been driving Tubby back to his daughter, who owned the dog. Coatsworth received $450.00 for his car (equivalent to ${{formatnum:{{Inflation|US|450|1940|r=-2}}}} today{{Inflation-fn|US}}) and $364.40 (${{formatnum:{{Inflation|US|364.40|1940|r=-2}}}} today{{Inflation-fn|US}}) in reimbursement for the contents of his car, including Tubby.<ref name=coatsworth2>{{Cite web|url=http://www.wsdot.wa.gov/tnbhistory/weirdfacts.htm#4|title=Tacoma Narrows Bridge: Weird Facts|publisher=Washington State Department of Transportation|quote=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".|access-date=2011-01-12|archive-date=2019-03-05|archive-url=https://web.archive.org/web/20190305152848/http://www.wsdot.wa.gov/tnbhistory/weirdfacts.htm#4|url-status=dead}}</ref>

==Inquiry==
[[File:Galloping Gertie fragment 01.jpg|thumb|left|A fragment of the collapsed bridge, in the Washington State History Museum in Tacoma]]
[[Biography:Theodore von Kármán|Theodore von Kármán]], the director of the [[Organization:Guggenheim Aeronautical Laboratory|Guggenheim Aeronautical Laboratory]] and a world-renowned aerodynamicist, was a member of the board of inquiry into the collapse.<ref>{{Cite book| last = Halacy Jr. | first = D. S. | title = Father of Supersonic Flight: Theodor von Kármán | year = 1965 | pages = 119–122}}</ref> He reported that the State of Washington was unable to collect on one of the [[Finance:Insurance|insurance]] policies for the bridge because its insurance agent had fraudulently pocketed the insurance premiums. The agent, Hallett R. French, who represented the Merchant's Fire Assurance Company, was charged and tried for grand larceny for withholding the premiums for $800,000 worth of insurance (equivalent to ${{inflation|US|.800|1940|r=1|fmt=c}} million today).<ref>{{cite news|url=https://www.newspapers.com/clip/8330547/prison_minimum_set_for_exinsurance/|title=Prison Minimum Set For Ex-Insurance Executive|work=The News-Review|location=Roseburg, Oregon|date=22 May 1941|page=1|via=Newspapers.com|access-date=13 January 2017}}</ref> The bridge was insured by many other policies that covered 80% of the $5.2&nbsp;million structure's value (equivalent to ${{inflation|US|5.2|1940|r=1|fmt=c}}&nbsp;million today). Most of these were collected without incident.<ref>{{Cite web | title = Tacoma Narrows Bridge | publisher = [[Organization:University of Washington|University of Washington]] Special Collections | url = http://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html | access-date = 2006-11-13 | archive-date = 2006-09-06 | archive-url = https://web.archive.org/web/20060906001739/https://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html | url-status = dead }}</ref>

On November 28, 1940, the U.S. Navy's Hydrographic Office reported that the remains of the bridge were located at [[Earth:Geographic coordinate system|geographical coordinates]] {{Coord|47|16|N|122|33|W}}, at a depth of {{convert|180|ft|m|abbr=off|sp=us}}.

===Film of collapse===
[[File:Tacoma Narrows Bridge destruction.ogv|thumb|Footage of the old Tacoma Narrows Bridge collapsing<br />(19.1&nbsp;MB video, 02:30).]]
The collapse was filmed with two cameras by Barney Elliott and by Harbine Monroe, owners of The Camera Shop in Tacoma, including the unsuccessful attempt to rescue the dog.<ref>{{cite web |title=::: Tacoma Narrows Bridge Film Collection ::: |url=https://content.lib.washington.edu/filmarchweb/tacoma.html |website=content.lib.washington.edu |access-date=7 December 2020}}</ref> Their footage was subsequently sold to Paramount Studios, which duplicated it for newsreels in black-and-white and distributed it worldwide to movie theaters. Castle Films also received distribution rights for 8 mm home video.<ref>{{cite web |title=Tacoma Narrows Bridge: Art of the Bridges Continues |url=https://www.wsdot.wa.gov/TNBhistory/Art/art3.htm |website=www.wsdot.wa.gov |access-date=7 December 2020 |archive-date=1 June 2019 |archive-url=https://web.archive.org/web/20190601053411/http://www.wsdot.wa.gov/TNBhistory/Art/art3.htm |url-status=dead }}</ref> In 1998, ''The Tacoma Narrows Bridge Collapse'' was selected for preservation in the United States National Film Registry by the Library of Congress as being culturally, historically, or aesthetically significant. This footage is still shown to [[Engineering:Engineering|engineering]], architecture, and [[Physics:Physics|physics]] students as a [[Social:Cautionary tale|cautionary tale]].<ref>{{Cite web|quote="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 [[Social:Prisoner of war|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!'"|url=http://www.wsdot.wa.gov/TNBhistory/weirdfacts.htm#6|title=Weird Facts|website=Tacoma Narrows Bridge History|publisher=Washington State Department of Transportation|access-date=2008-08-15|archive-date=2019-03-05|archive-url=https://web.archive.org/web/20190305152848/http://www.wsdot.wa.gov/tnbhistory/weirdfacts.htm#6|url-status=dead}}</ref>

Elliott and Monroe's footage of the construction and collapse was shot on 16&nbsp;mm Kodachrome film, but most copies in circulation are in black and white because newsreels of the day copied the film onto 35&nbsp;mm black-and-white [[Engineering:Original camera negative|stock]]. There were also film-speed discrepancies between Monroe's and Elliot's footage, with Monroe filming at 24 frames per second and Elliott at 16 frames per second.<ref name="vice">{{cite news |last1=Pasternack |first1=Alex |title=The Strangest, Most Spectacular Bridge Collapse (And How We Got It Wrong) |url=https://www.vice.com/en/article/kb78w3/the-myth-of-galloping-gertie |access-date=7 December 2020 |work=Vice |date=14 December 2015 |language=en}}</ref> As a result, most copies in circulation also show the bridge oscillating approximately 50% faster than real time, due to an assumption during conversion that the film was shot at 24 frames per second rather than the actual 16 fps.<ref>{{Cite news|title=A Tacoma Narrows 'Galloping Gertie' bridge-collapse surprise, 75 years later |url=http://www.seattlepi.com/science/article/A-Tacoma-Narrows-Galloping-Gertie-bridge-6617030.php|quote=By timing the torsional oscillations, the Texas State researchers determined the bridge goes through 18 twisting cycles per minute on the existing video. Stopwatch measurements taken on November 7, 1940, however, timed the bridge cycles at 12 per minute—a significant discrepancy. The Texas State researchers were able to prove that the original 16&nbsp;mm camera that filmed the oscillations was running at the slower 16 fps, not the 24 fps assumed when the conversion to video was done. When the film frames are viewed at the slower speed, the torsional cycles match the eyewitness stopwatch measurement of 12 cycles per minute. |newspaper=Seattle Post-Intelligencer |date=7 November 2015 |access-date=11 November 2015}}</ref>

Another reel of film emerged in February 2019, taken by Arthur Leach from the Gig Harbor (westward) side of the bridge, and one of the few known images of the collapse from that side. Leach was a civil engineer who served as toll collector for the bridge, and is believed to have been the last person to cross the bridge to the west before its collapse, trying to prevent further crossings from that side as the bridge became unstable. Leach's footage (originally on black-and-white film but then recorded to video cassette by filming the projection) also includes Leach's commentary at the time of the collapse.<ref>{{cite web | url = https://www.kgw.com/article/news/lost-footage-of-wild-1940-tacoma-narrows-bridge-collapse-revealed/283-edd714a8-ce83-47ce-9702-c8eefac143c0 | title = Lost footage of wild 1940 Tacoma Narrows Bridge collapse revealed | date= February 28, 2019 | access-date = February 28, 2019 | publisher = KING-TV }}</ref>

=== Federal Works Agency commission ===
A commission formed by the Federal Works Agency studied the collapse of the bridge. It included engineers Othmar Ammann and [[Biography:Theodore von Kármán|Theodore von Kármán]]. Without drawing any definitive conclusions, the commission explored three possible failure causes:
* Aerodynamic instability by self-induced vibrations in the structure
* Eddy formations that might be periodic in nature
* Random effects of turbulence, that is the random fluctuations in velocity of the wind.

===Cause of the collapse===
[[File:Tacomanarrowssimple.png|thumb|Simplistic representation of the collapse of the Tacoma Narrows Bridge]]
The original Tacoma Narrows Bridge was the first to be built with girders of [[Engineering:Carbon steel|carbon steel]] anchored in concrete blocks; preceding designs typically had open lattice beam trusses underneath the roadbed.<ref name=":3">{{Cite web|title=Construction — UW Libraries|url=https://www.lib.washington.edu/specialcollections/collections/exhibits/tnb/construction|access-date=2020-07-13|website=www.lib.washington.edu}}</ref> This bridge was the first of its type to employ plate girders (pairs of deep [[Engineering:I-beam|I-beam]]s) to support the roadbed.<ref name=":3" /> With the earlier designs, any wind would simply pass through the truss, but in the new design the wind would be diverted above and below the structure.<ref>{{Cite web|title=The Aftermath — UW Libraries|url=https://www.lib.washington.edu/specialcollections/collections/exhibits/tnb/aftermath|access-date=2020-07-13|website=www.lib.washington.edu}}</ref> Shortly after construction finished at the end of June (opened to traffic on July 1, 1940), it was discovered that the bridge would sway and [[Physics:Buckling|buckle]] dangerously in relatively mild windy conditions that are common for the area, and worse during severe winds.<ref name=":2">{{Cite web|title=Opening and Experiments to study 'ripple' — UW Libraries|url=https://www.lib.washington.edu/specialcollections/collections/exhibits/tnb/opening|access-date=2020-07-12|website=www.lib.washington.edu}}</ref> This vibration was [[Physics:Transverse wave|transverse]], one-half of the central span rising while the other lowered. Drivers would see cars approaching from the other direction rise and fall, riding the violent energy wave through the bridge. However, at that time the mass of the bridge was considered sufficient to keep it structurally sound.

The failure of the bridge occurred when a never-before-seen twisting mode occurred, from winds at {{convert|40|mph}}. This is a so-called torsional [[Physics:Normal mode|vibration mode]] (which is different from the [[Physics:Transverse mode|transversal]] or [[Physics:Longitudinal mode|longitudinal]] vibration mode), whereby when the left side of the roadway went down, the right side would rise, and vice versa, i.e., the two halves of the bridge twisted in opposite directions, with the center line of the road remaining still (motionless). This vibration was caused by [[Physics:Aeroelasticity#Flutter|aeroelastic fluttering]].

[[File:Tacoma Bridge Animation.ogv|thumb|Full-scale, two-way Fluid Structure Interaction (FSI) model of the Tacoma Narrows Bridge exhibiting aeroelastic flutter]] Fluttering is a physical phenomenon in which several [[Physics:Degrees of freedom (mechanics)|degrees of freedom]] of a structure become coupled in an unstable oscillation driven by the wind. Here, unstable means that the forces and effects that cause the oscillation are not checked by forces and effects that limit the oscillation, so it does not self-limit but grows without bound. Eventually, the amplitude of the motion produced by the fluttering increased beyond the strength of a vital part, in this case the suspender cables. As several cables failed, the weight of the deck transferred to the adjacent cables, which became overloaded and broke in turn until almost all of the central deck fell into the water below the span.

====Resonance (due to Von Kármán vortex street) hypothesis====
[[File:Vortex-street-animation.gif|thumb|right|Vortex shedding and [[Physics:Kármán vortex street|Kármán vortex street]] behind a circular cylinder. The first hypothesis of failure of the Tacoma Narrows Bridge was resonance (due to the Kármán vortex street).<ref>{{Cite news|title=Big Tacoma Bridge Crashes 190 Feet into Puget Sound. Narrows Span, Third Longest of Type in World, Collapses in Wind. Four Escape Death. |quote=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. |newspaper={{wipe|The New York Times}} |date=November 8, 1940}}</ref>
This is because it was thought that the Kármán vortex street frequency (the so-called [[Physics:Strouhal number|Strouhal frequency]]) was the same as the [[Physics:Torsion (mechanics)|torsional]] [[Natural frequency|natural vibration frequency]]. This was found to be incorrect. The actual failure was due to [[Physics:Aeroelasticity#Flutter|aeroelastic flutter]].<ref name="BillahScanlan91">{{Cite journal|last=Billah|first=K.|author2=R. Scanlan |year=1991|title=Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks|journal=[[Physics:American Journal of Physics|American Journal of Physics]]|volume=59|issue=2|pages=118–124|url=http://www.ketchum.org/billah/Billah-Scanlan.pdf|doi=10.1119/1.16590|bibcode = 1991AmJPh..59..118B }}</ref> ]]

The bridge's spectacular destruction is often used as an object lesson in the necessity to consider both [[Physics:Aerodynamics|aerodynamics]] and [[Physics:Resonance|resonance]] effects in [[Engineering:Civil engineering|civil]] and [[Engineering:Structural engineering|structural engineering]]. Billah and Scanlan (1991)<ref name="BillahScanlan91" /> reported that, in fact, many physics textbooks (for example Resnick et al.<ref name="HallidayResnickWalker94">{{Cite book|author1=Halliday, David |author2=Resnick, Robert |author3=Walker, Jearl | title=Fundamentals of Physics, (Chapters 21-44) | publisher=John Wiley & Sons | isbn=978-0-470-04474-2 |year=2008 }}</ref> and Tipler et al.<ref name="TiplerMosca">{{Cite book|author1=Tipler, Paul Allen |author2=Mosca, Gene | title=Physics for Scientists and Engineers |volume=1B: Oscillations and Waves; Thermodynamics (Physics for Scientists and Engineers) | publisher=W. H. Freeman and Company | isbn=978-0-7167-0903-9 |year=2004 }})</ref>) wrongly explain that the cause of the failure of the Tacoma Narrows bridge was externally forced mechanical resonance. Resonance is the tendency of a system to oscillate at larger amplitudes at certain frequencies, known as the system's natural frequencies. At these frequencies, even relatively small periodic driving forces can produce large amplitude vibrations, because the system stores 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 child 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 forced oscillator, defined by the [[Differential equation#Second order|second-order differential equation]]

{{Numbered block|:|<math>m\ddot{x}(t) + c\dot{x}(t) + kx(t) = F \cos (\omega t)</math> | {{EquationRef|eq. 1}}}}

where {{mvar|m}}, {{mvar|c}} and {{mvar|k}} stand for the [[Physics:Mass|mass]], [[Physics:Damping|damping coefficient]] and [[Physics:Stiffness|stiffness]] of the [[Physics:Linear system|linear system]] and {{mvar|F}} and {{mvar|ω}} represent the amplitude and the [[Physics:Angular frequency|angular frequency]] of the exciting force. The solution of such [[Ordinary differential equation|ordinary differential equation]] as a function of time {{mvar|t}} represents the displacement response of the system (given appropriate initial conditions). In the above system resonance happens when {{mvar|ω}} is approximately <math>\omega_r = \sqrt{k/m}</math>, i.e., <math>\omega_r</math> is the natural (resonant) frequency of the system. The actual vibration analysis of a more complicated mechanical system&nbsp;— such as an airplane, a building or a bridge&nbsp;— is based on the linearization of the equation of motion for the system, which is a multidimensional version of equation ({{EquationNote|eq. 1}}). The analysis requires [[Eigenvalues and eigenvectors|eigenvalue]] analysis and thereafter the natural frequencies of the structure are found, together with the so-called ''fundamental modes'' 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 [[Physics:Vortex shedding|vortex shedding]]. This is because bluff (non-streamlined) bodies&nbsp;— like bridge decks&nbsp;— in a fluid stream produce (or "shed") [[Physics:Wake|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 [[Physics:Kármán vortex street|Kármán vortex street]] or von Kármán vortex street. The body will in consequence try to move toward the low-pressure zone, in an oscillating movement called [[Physics:Vortex-induced vibration|vortex-induced vibration]]. Eventually, if the frequency of vortex shedding matches the natural 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 <math>f_s</math>, and is given by
{{Numbered block|:|<math>\frac{f_s D}{U} = S</math> |{{EquationRef|eq. 2}}}}
Here, {{mvar|U}} stands for the flow velocity, {{mvar|D}} is a characteristic length of the [[Physics:Drag coefficient#Blunt and streamlined body flows|bluff body]] and {{mvar|S}} is the dimensionless [[Physics:Strouhal number|Strouhal number]], which depends on the body in question. For [[Reynolds number]]s greater than 1000, the Strouhal number is approximately equal to 0.21. In the case of the Tacoma Narrows, {{mvar|D}}&nbsp;was approximately {{convert|8|ft}} and {{mvar|S}} was 0.20.

It was thought that the Strouhal frequency was close enough to one of the natural vibration frequencies of the bridge, i.e., <math>2\pi f_s = \omega</math>, to cause resonance and therefore vortex-induced vibration.

In the case of the Tacoma Narrows Bridge, this appears not to have been the cause of the catastrophic damage. According to Professor Frederick Burt Farquharson, an engineering professor at the University of Washington and one of the main researchers into the cause of the bridge collapse, the wind was steady at {{convert|42|mph}} and the frequency of the destructive mode was 12 cycles/minute (0.2 [[Hertz|Hz]]).<ref>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.</ref> This frequency was neither a natural mode of the isolated structure nor the frequency of blunt-body [[Physics:Vortex shedding|vortex shedding]] of the bridge at that wind speed, which was approximately 1&nbsp;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.

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 natural 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 linear behaviour, since the exciting force amplitude is a nonlinear force of the structural response.<ref name="ReferenceA">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. [[Organization:Princeton University|Princeton University]]. April 1989</ref>

====Resonance vs. non-resonance explanations====
Billah and Scanlan<ref name="ReferenceA" /> state that Lee Edson in his biography of [[Biography:Theodore von Kármán|Theodore von Kármán]]<ref>Theodore von Karman with Lee Edson (1963). ''The wind and Beyond. Theodore von Karman: Pioneer in Aviation and Pathfinder in Space''. Boston: Little, Brown and Company. p. 213</ref> 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

{{Blockquote|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.<ref>Steven Ross, et al. "Tacoma Narrows 1940." In ''Construction Disasters: Design Failures, Causes, and Prevention''. McGraw Hill, 1984, pp. 216–239.</ref> }}

A group of physicists cited "wind-driven amplification of the torsional oscillation" as distinct from resonance:

{{Blockquote|Subsequent authors have rejected the resonance explanation, and their perspective is gradually spreading to the physics community. The user's guide for the current American Association of Physics Teachers (AAPT) DVD states the bridge collapse "was not a case of resonance." Bernard Feldman likewise concluded in a 2003 article for the Physics Teacher that for the torsional oscillation mode, there was "no resonance behavior in the amplitude as a function of the wind velocity."

An important source for both the AAPT user's guide and for Feldman was a 1991 American Journal of Physics article by K. Yusuf Billah and Robert Scanlan. According to the two engineers, the failure of the bridge was related to a wind-driven amplification of the torsional oscillation that, unlike a resonance, increases monotonically with increasing wind speed. The fluid dynamics behind that amplification is complicated, but one key element, as described by physicists Daniel Green and William Unruh, is the creation of large-scale vortices above and below the roadway, or deck, of the bridge. Nowadays, bridges are constructed to be rigid and to have mechanisms that damp oscillations. Sometimes they include a slot in the middle of the deck to alleviate pressure differences above and below the road.<ref name=":0">{{Cite journal|title = The Tacoma Narrows Bridge collapse|journal =Physics Today|date = 2015-11-01|issn = 0031-9228|pages = 64–65|volume = 68|issue = 11|doi = 10.1063/PT.3.2991|first = Donald W.|last = Olson|first2 = Steven F.|last2 = Wolf|first3 = Joseph M.|last3 = Hook|bibcode = 2015PhT....68k..64O |doi-access = free}}</ref> }}

To some degree the debate is due to the lack of a commonly accepted precise definition of resonance. Billah and Scanlan<ref name="BillahScanlan91" /> provide the following definition of resonance "In general, whenever a system capable of oscillation is acted on by a periodic series of impulses having a frequency equal to or nearly equal to one of the natural frequencies of the oscillation of the system, the system is set into oscillation with a relatively large amplitude." They then state later in their paper "Could this be called a resonant phenomenon? It would appear not to contradict the qualitative definition of resonance quoted earlier, if we now identify the source of the periodic impulses as ''self-induced'', the wind supplying the power, and the motion supplying the power-tapping mechanism. If one wishes to argue, however, that it was a case of ''externally forced linear resonance'', the mathematical distinction ... is quite clear, self-exciting systems differing strongly enough from ordinary linear resonant ones."

== Link to the Armistice Day blizzard ==
The weather system that caused the bridge collapse went on to cause the 1940 Armistice Day Blizzard that killed 145 people in the Midwestern United States:

{{Blockquote|The strong winds in the Tacoma Narrows on 7 November 1940 were related to a remarkable low-pressure system that followed a track across the country and four days later produced the Armistice Day storm, one of the greatest storms ever to strike the Great Lakes region. For example, when the storm reached Illinois, the headline on the front page of the Chicago Tribune included the words "Heaviest winds in this century smash at city."

Additional details of the film and video analysis can be found in the November 2015 issue of the Physics Teacher, which also includes further description of the Armistice Day storm and the strong winds that earlier had caused the Tacoma Narrows Bridge to oscillate, twist, and collapse into the waters below.<ref name=":0" /> }}

==Fate of the collapsed superstructure==
Efforts to salvage the bridge began almost immediately after its collapse and continued into May 1943.<ref name="WSDOT">{{cite web|url=http://www.wsdot.wa.gov/tnbhistory/Connections/connections4.htm#4|title=Tacoma Narrows Bridge: Aftermath – A New Beginning: 1940–1950|website=www.wsdot.wa.gov|access-date=2008-09-16|archive-date=2012-02-05|archive-url=https://web.archive.org/web/20120205005119/http://www.wsdot.wa.gov/TNBhistory/Connections/connections4.htm#4|url-status=dead}}</ref> 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 [[Earth:Superstructure|superstructure]] built.<ref>{{cite web|url=http://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html|title=Subject Guides & Online Exhibits – UW Libraries|website=www.lib.washington.edu|access-date=2008-08-15|archive-date=2006-09-06|archive-url=https://web.archive.org/web/20060906001739/https://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html|url-status=dead}}</ref> With steel being a valuable commodity because of the involvement of the United States in [[Social:World War II|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 (equivalent to ${{inflation|US|.35|1943|r=1|fmt=c}} million today).<ref name="WSDOT" />

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 the main cables and road deck, suffered major damage at their bases from being deflected {{convert|12|ft||}} 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.

===Preservation of the collapsed roadway===
[[File:GertieRemains 1.jpg|thumb|upright|right|Remains of the collapsed bridge in May 2008]]
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.<ref name="nris">{{NRISref|2007a}}</ref><ref>{{Cite web|url=http://www.wsdot.wa.gov/TNBhistory/|title=WSDOT – Tacoma Narrows Bridge: Extreme History|publisher=Washington State Department of Transportation|access-date=2007-10-23}}</ref>

The Harbor History Museum has a display in its main gallery regarding the 1940 bridge, its collapse, and the subsequent two bridges.

===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:

{{Blockquote|The Tacoma Narrows bridge failure has given us invaluable information&nbsp;... 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 judgment and if, as a result, errors, or failures occur, we must accept them as a price for human progress.<ref>
Othmar H. Ammann, Theodore von Kármán and Glenn B. Woodruff. [http://resolver.caltech.edu/CaltechAUTHORS:20140512-105559175 The Failure of the Tacoma Narrows Bridge, a report to the administrator]. Report to the Federal Works Agency, Washington, 1941</ref>}}

Following the incident, engineers took extra caution to incorporate aerodynamics into their designs, and [[Physics:Wind tunnel|wind tunnel]] testing of designs was eventually made mandatory.<ref>{{cite web |url=https://www.history.com/this-day-in-history/tacoma-bridge-collapses |title=Tacoma Bridge collapses |date=August 21, 2018 |website=HISTORY |publisher=A&E |access-date=November 7, 2018 |quote=After the Tacoma Narrows disaster, bridge builders took care to incorporate aerodynamics into their designs and build structures with complex frequencies. Wind-tunnel testing of bridge designs eventually became mandatory.}}</ref>

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.

A key consequence was that suspension bridges reverted to a deeper and heavier [[Physics:Truss|truss]] design, including the replacement Tacoma Narrows Bridge (1950), until the development in the 1960s of box girder bridges with an [[Physics:Airfoil|airfoil]] shape such as the Severn Bridge, which gave the necessary stiffness together with reduced torsional forces.

== Replacement bridge ==
Because of shortages in materials and labor 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 {{convert|5979|ft}} long, {{convert|40|ft|spell=in}} longer than the original bridge. 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 replacement bridge exceeded 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 carry only westbound traffic. The new parallel bridge opened to traffic in July 2007.

==See also==
* [[Engineering:Engineering disasters|Engineering disasters]]
* Humen Pearl River Bridge, suspension bridge that has shakened violently until weight limits initiated
* List of bridge failures
* [[Engineering:List of structural failures and collapses|List of structural failures and collapses]]
* Millennium Bridge, London, for an engineering error
* Silver Bridge, a bridge that collapsed in 1967 on the West Virginia–Ohio border
* Volgograd Bridge, a bridge in Russia that experienced similar problems with the wind

==References==
{{Reflist|35em}}

== Further reading ==
* {{cite web |last1=Crowell |first1=Benjamin |title=Vibrations and Waves |url=http://www.lightandmatter.com/html_books/3vw/ch02/ch02.html |website=Lightandmatter.com |archive-url=https://web.archive.org/web/20070103222329/http://www.lightandmatter.com/html_books/3vw/ch02/ch02.html |archive-date=3 January 2007 |date=2006}}
* {{cite journal|title=Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge|journal=Journal of Sound and Vibration |volume=332 |issue=15 |pages=3772–3789 |doi=10.1016/j.jsv.2013.02.011|year=2013|last1=Malík|first1=Josef|bibcode=2013JSV...332.3772M}}
* {{cite web |last1=Meador |first1=Granger |title=Two Case Studies of Bridge Design Failure |url=http://failurebydesign.info/ |website=Failure By Design |archive-url=https://web.archive.org/web/20081004045144/http://failurebydesign.info/ |archive-date=4 October 2008 |date=2008}}
* {{cite web |last1=Zasky |first1=Jason |title=Suspended Animation: The collapse of the Tacoma Narrows Bridge |url=http://failuremag.com/feature/article/suspended_animation/ |website=Failure Magazine |archive-url=https://web.archive.org/web/20140101074931/http://failuremag.com/feature/article/suspended_animation/ |archive-date=1 January 2014}}
* {{cite web|url=https://www.lib.washington.edu/specialcollections/collections/exhibits/tnb |title=Exhibit: History of the Tacoma Narrows Bridge |publisher=University of Washington Libraries|work=Special Collections}}
* {{cite web|url=https://content.lib.washington.edu/farquharsonweb/index.html |title=Tacoma Narrows Bridge Collection|publisher=University of Washington Libraries|work=Digital Collections}}
* {{cite web |title=Tacoma Narrows Bridge history |url=https://wsdot.wa.gov/TNBhistory/ |publisher=Washington State Department of Transportation}}

==External links==
* {{Structurae|id=20000074|title=Tacoma Narrows Bridge (1940)}}
* [https://web.archive.org/web/20080914163203/http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html The Tacoma Narrows Bridge Disaster, November 1940] at University of Bristol
* [https://web.archive.org/web/19970628132902/http://www.civeng.carleton.ca/Exhibits/Tacoma_Narrows/ Tacoma Narrows Bridge Failure] at Carleton University
* [https://web.archive.org/web/20040624183211/http://www.nwrain.net/~newtsuit/recoveries/narrows/narrows.htm Tacoma Narrows Bridge] at the Gig Harbor Peninsula Historical Society & Museum
* [http://www.ketchum.org/bridgecollapse.html The Tacoma Narrows Bridge] at Ketchum.org


[[Category:Engineering failures]]

{{Sourceattribution|Tacoma Narrows Bridge (1940)|1}}