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| {{About|the first Tacoma Narrows Bridge, which collapsed in 1940|the article on the current bridges|Tacoma Narrows Bridges}}
| | I am 26 years old and my name is Elliot Lehner. I life in Desulo (Italy).<br><br>my web site :: [http://realmofshadows.us/wiki/index.php/User:RaleighLitchfie dog training oklahoma city] |
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| {{Infobox bridge
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| |bridge_name = Tacoma Narrows Bridge
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| |other_name = Galloping Gertie
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| |image = Image-Tacoma Narrows Bridge1.gif
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| |image_size = 275px
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| |caption = The original Tacoma Narrows Bridge roadway twisted and vibrated violently under {{convert|40|mph|adj=on}} winds on the day of the collapse
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| |design = [[suspension bridge|Suspension]]
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| |mainspan = {{convert|2800|ft|m|1}}
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| |length = {{convert|5939|ft|m|1}}
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| |below = {{convert|195|ft|m|1}}
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| |width =
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| |clearance =
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| |open = July 1, 1940
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| |collapsed = November 7, 1940
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| |coordinates = {{Coord|47|16|00|N|122|33|00|W|display=inline,title}}
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| |extra ={{Location map | USA Washington
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| |position = right
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| |lon_dir=W
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| |lat_dir=N
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| |lat_deg = 47
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| |lat_min = 16
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| |lat_sec = 00
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| |lon_deg = 122
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| |lon_min = 33
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| |lon_sec = 00
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| |width = 250
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| [[File:Tacoma Narrows Bridge Location.png|thumb|right|Map showing location of the bridge]]
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| The '''1940 Tacoma Narrows Bridge''' was the first [[Tacoma Narrows Bridge]], a [[suspension bridge]] in the [[United States|U.S.]] state of [[Washington (U.S. state)|Washington]] that spanned the [[Tacoma Narrows]] strait of [[Puget Sound]] between [[Tacoma, Washington|Tacoma]] and the [[Kitsap Peninsula]]. It opened to traffic on July 1, 1940, and dramatically [[structural failure|collapsed]] into Puget Sound on November 7 of the same year. At the time of its construction (and its destruction), the bridge was the third longest suspension bridge in the world in terms of main span length, behind the [[Golden Gate Bridge]] and the [[George Washington Bridge]].
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| Construction on the bridge began in September 1938. From the time the deck was built, it began to move vertically in windy conditions, which led to construction workers giving the bridge the nickname '''Galloping Gertie'''. The motion was observed even when the bridge opened to the public. Several measures aimed at stopping the motion were ineffective, and the bridge's main span finally collapsed under {{convert|40|mph|km/h|adj=on}} wind conditions the morning of November 7, 1940.
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| Following the collapse, the United States' involvement in [[World War II]] delayed plans to replace the bridge. The portions of the bridge still standing after the collapse, including the towers and cables, were dismantled and sold as scrap metal. Nearly 10 years after the bridge collapsed, a [[Tacoma Narrows Bridge (1950)|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]].
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| The bridge's collapse had a lasting effect on science and engineering. In many [[physics]] textbooks, the event is presented as an example of elementary forced [[Mechanical resonance|resonance]] with the wind providing an external periodic frequency that matched the bridge's natural structural frequency, though the actual cause of failure was [[Aeroelasticity#Flutter|aeroelastic flutter]].<ref name="BillahScanlan91"/> Its failure also boosted research in the field of bridge aerodynamics-aeroelastics, the study of which has influenced the designs of all the world's great long-span bridges built since 1940.
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| ==Design and construction==
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| The desire for the construction of a bridge between Tacoma and the Kitsap Peninsula dates back to 1889 with a [[Northern Pacific Railway]] proposal for a [[trestle]], but concerted efforts began in the mid-1920s. The Tacoma Chamber of Commerce began campaigning and funding studies in 1923. Several noted bridge engineers, including [[Joseph Strauss (engineer)|Joseph B. Strauss]], who went on to be chief engineer of the [[Golden Gate Bridge]], and [[David B. Steinman]], who went on to design the [[Mackinac Bridge]], were consulted. Steinman made several Chamber-funded visits, culminating in a preliminary proposal presented in 1929, but by 1931, the Chamber decided to cancel the agreement on the grounds that Steinman was not sufficiently active in working to obtain financing. Another problem with financing the first bridge was buying out the ferry contract from a private firm running service on the Narrows at the time.
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| The Washington State legislature created the [[Washington Toll Bridge Authority|Washington State Toll Bridge Authority]] and appropriated $5,000 to study the request by Tacoma and [[Pierce County, Washington|Pierce County]] for a bridge over the Narrows.
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| From the start, financing of the bridge was a problem: revenue from the proposed tolls would not be enough to cover construction costs, but there was strong support for the bridge from the [[U.S. Navy]], which operated the [[Puget Sound Naval Shipyard]] in [[Bremerton, Washington|Bremerton]], and from the [[United States Army|U.S. Army]], which ran [[McChord Air Force Base|McChord Field]] and [[Fort Lewis]] near Tacoma.
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| 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 Toll Bridge Authority]] requested $11 million from the Federal [[Public Works Administration]] (PWA). Preliminary construction plans by the Washington Department of Highways had called for a set of 25-foot-deep (7.6 m) trusses to sit beneath the roadway and stiffen it.
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| However, according to Eldridge, "Eastern consulting engineers"—by which Eldridge meant [[Leon Moisseiff]], the noted New York bridge engineer who served as designer and consultant engineer for the [[Golden Gate Bridge]]—petitioned the PWA and the [[Reconstruction Finance Corporation]] (RFC) to build the bridge for less. Moisseiff proposed shallower supports—girders {{convert|8|ft}} deep. His approach meant a slimmer, more elegant design, and also reduced the construction costs as compared with the Highway Department's design. 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 for the Tacoma Narrows Bridge. Another $1.6 million was to be collected from tolls to cover the estimated total $8<!-- ibid --> million cost.
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| Following Moisseiff's design, bridge construction began on September 27, 1938. Construction took only nineteen months, at a cost of $6.4 million, 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 [[New York City]], and the [[Golden Gate Bridge]], connecting [[San Francisco]] with [[Marin County, California|Marin County]] to its north.<ref>Henry Petroski. Engineers of Dreams: Great Bridge Builders and the Spanning of America. New York: Knopf/Random House, 1995.</ref>
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| Moisseiff and Fred Lienhard, the latter a [[Port of New York Authority]] engineer, 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">Richard Scott. In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability. ''American Society of Civil Engineers'' (June 1, 2001) ISBN 0-7844-0542-5 http://books.google.com/books?id=DnQOzYDJsm8C</ref> Their theory of elastic distribution extended the 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 suspended structure laterally. This energy would then be transmitted to the anchorages and towers.<ref name="RichardScott"/>
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| Using this theory, Moisseiff argued for stiffening the bridge with a set of eight-foot-deep plate girders rather than the {{convert|25|ft}}-deep trusses proposed by the Washington Toll Bridge Authority. This change was a substantial contributor to the difference in the projected costs of the designs.
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| Because planners expected fairly light traffic volumes, the bridge was designed with two lanes, and it was just {{convert|39|ft}} wide. This was quite narrow, especially in comparison with its length. With only the {{convert|8|ft}}-deep plate girders providing additional depth, the bridge's roadway section was also shallow.
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| The decision to use such shallow and narrow girders proved to be the original Tacoma Narrows 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 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 bridge|toll]]-paid traffic started) felt these motions on the day that the bridge opened on July 1, 1940.
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| ==Attempt to control structural vibration==
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| [[File:Systems to save the Tacoma Narrows bridge.jpg|thumb|Cartoon from the Seattle Times, November 8, 1940. PH Coll. 290.159 University of Washington Libraries. Manuscripts, Special Collections, UW23030]]
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| Since the structure experienced considerable vertical oscillations 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: Gale Research, 1994.</ref>
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| * attachment of tie-down cables to the plate girders, which were anchored to 50-ton concrete blocks on the shore. This measure proved ineffective, as the cables snapped shortly after installation.
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| * 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.
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| * 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.
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| The [[Washington Toll Bridge Authority]] hired Professor Frederick Burt Farquharson, an engineering professor at the [[University of Washington]], to make 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:
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| * To drill holes in the lateral girders and along the deck so that the air flow could circulate through them (in this way reducing lift forces).
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| * To give a more aerodynamic shape to the transverse section of the deck by adding fairings or deflector vanes along the deck, attached to the girder fascia.
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| 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"/>
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| ==Collapse==
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| [[File:TacomaNarrowsBridgeCollapse in color.jpg|thumb|The [[Tacoma Narrows Bridge|1940 Tacoma Narrows Bridge]] collapsing, in a frame from a 16mm Kodachrome motion picture film taken by Barney Elliott]]
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| The wind-induced collapse occurred on November 7, 1940, at 11:00 a.m. (Pacific time), because of a physical phenomenon known as [[aeroelasticity#Flutter|aeroelastic flutter]].<ref name="BillahScanlan91"/>
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| Leonard Coatsworth, a ''[[Tacoma News Tribune]]'' editor, was the last person to drive on the bridge:
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| {{bquote|"Just as I drove past the towers, the bridge began to sway violently from side to side. Before I realized it, the tilt became so violent that I lost control of the car...I jammed on the brakes and got out, only to be thrown onto my face against the curb...Around me I could hear concrete cracking...The car itself began to slide from side to side of the roadway.<p>On hands and knees most of the time, I crawled {{convert|500|yd|m}} or more to the towers...My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb...Toward the last, I risked rising to my feet and running a few yards at a time...Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows."<ref>{{Cite web|url= http://www.wsdot.wa.gov/tnbhistory/people/eyewitness.htm|title= Tacoma Narrows Bridge: Eyewitness accounts of November 7, 1940|accessdate= 2008-08-17|author= Washington State Department of Transportation|authorlink= Washington State Department of Transportation|year= 2005}}</ref>}}
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| No human life was lost in the collapse of the bridge. Tubby, a black male [[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|work=Tacoma Narrows Bridge History|publisher=WDOT}}</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 were ever recovered.<ref name="tubby">{{Cite web|url=http://www.wsdot.wa.gov/TNBhistory/tubby.htm|title=Tubby Trivia|work=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 [[United States dollar|US$]]450.00 (US${{formatnum:{{Inflation|US|450|1940|r=-2}}}} with inflation{{Inflation-fn|US}}) for his car and US$364.40 (US${{formatnum:{{Inflation|US|364.40|1940|r=-2}}}} with inflation{{Inflation-fn|US}}) in reimbursement for the contents of his car, including Tubby.<ref name="coatsworth">{{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".}}</ref>
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| ==Inquiry==
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| [[File:Galloping Gertie fragment 01.jpg|thumb|left|A fragment of the collapsed bridge, in the [[Washington State History Museum]], Tacoma, Washington.]]
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| [[Theodore von Kármán]], the director of the [[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 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. The bridge, however, was insured by many other policies that covered 80% of the $5.2 million structure's value. Most of these were collected without incident.<ref>{{Cite web| title = Tacoma Narrows Bridge | publisher = University of Washington Special Collections | url = http://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html | accessdate = 2006-11-13}}</ref>
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| On November 28, 1940, the [[U.S. Navy]]'s Hydrographic Office reported that the remains of the bridge were located at geographical coordinates {{Coord|47|16|00|N|122|33|00|W}}, at a depth of 180 feet (55 meters).
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| ===Film of collapse===
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| [[File:Tacoma Narrows Bridge destruction.ogg|thumb|Footage of the Tacoma Narrows bridge collapsing. (19.1 [[Mebibyte|MiB]] video, 2:30)]]
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| The collapse of the bridge was recorded on film by Barney Elliott, owner of a local camera shop. The film shows Leonard Coatsworth leaving the bridge after exiting his car. 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]], [[architecture]], and [[physics]] students as a [[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 [[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|work=Tacoma Narrows Bridge History|publisher=Washington State Department of Transportation}}</ref> Elliot's original films of the construction and collapse of the bridge were shot on 16 mm [[Kodachrome|Kodachrome film]], but most copies in circulation are in black and white because newsreels of the day copied the film onto 35 mm black-and-white stock.
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| ===Commission of the Federal Works Agency===
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| A commission formed by the [[Federal Works Agency]] studied the collapse of the bridge. It included [[Othmar Ammann]] and [[Theodore von Kármán]]. Without drawing any definitive conclusions, the commission explored three possible failure causes:
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| * Aerodynamic instability by self-induced vibrations in the structure
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| * Eddy formations that might be periodic in nature
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| * Random effects of turbulence, that is the random fluctuations in velocity and direction of the wind.
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| ===Cause of the collapse===
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| The original Tacoma Narrows Bridge was <!-- avoid weasel words --> solidly built, with girders of carbon [[steel]] anchored in huge blocks of [[concrete]]. Preceding designs typically had open lattice beam trusses underneath the roadbed. This bridge was the first of its type to employ plate girders (pairs of deep [[I-beam]]s) to support the roadbed. 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. 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 buckle dangerously in relatively mild windy conditions that are common for the area, and worse during severe winds. This vibration was [[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 to be sufficient to keep it structurally sound.
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| The failure of the bridge occurred when a never-before-seen twisting mode occurred, from winds at a mild {{convert|40|mph}}. This is a so-called torsional [[vibration mode]] (which is different from the [[transverse mode|transversal]] or [[longitudinal mode|longitudinal]] vibration mode), whereby when the left side of the roadway went down, the right side would rise, and vice versa, with the center line of the road remaining still. Specifically, it was the "second" torsional mode, in which the midpoint of the bridge remained motionless while the two halves of the bridge twisted in opposite directions. Two men proved this point by walking along the center line, unaffected by the flapping of the roadway rising and falling to each side. This vibration was caused by [[Aeroelasticity#Flutter|aeroelastic fluttering]].
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| Fluttering is a physical phenomenon in which several [[Degrees of freedom (mechanics)|degrees of freedom]] of a structure become coupled in an unstable oscillation driven by the wind. This movement inserts energy to the bridge during each cycle so that it neutralizes the natural [[damping]] of the structure; the composed system (bridge-fluid) therefore behaves as if it had an effective negative damping (or had [[positive feedback]]), leading to an exponentially growing response. In other words, the oscillations increase in amplitude with each cycle because the wind pumps in more energy than the flexing of the structure can dissipate, and finally drives the bridge toward failure due to excessive deflection and stress. The wind speed that causes the beginning of the fluttering phenomenon (when the effective damping becomes zero) is known as the flutter velocity. Fluttering occurs even in low-velocity winds with steady flow. Hence, bridge design must ensure that flutter velocity will be higher than the maximum mean wind speed present at the site.
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| Eventually, the amplitude of the motion produced by the fluttering increased beyond the strength of a vital part, in this case the suspender cables. Once several cables failed, the weight of the deck transferred to the adjacent cables that broke in turn until almost all of the central deck fell into the water below the span.
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| ===Resonance (due to Von Kármán vortex street) hypothesis===
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| [[Image:Vortex-street-animation.gif|thumb|right|Vortex shedding and [[Von Kármán vortex street]] behind a circular cylinder. The first hypothesis of failure of the Tacoma Narrows Bridge was resonance (due to the Von Kármán vortex street).<ref>{{Cite news|first= |last= |authorlink= |coauthors= |title=Big Tacoma Bridge Crashes 190 Feet into Puget Sound. Narrows Span, Third Longest of Type in World, Collapses in Wind. Four Escape Death. |url= |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. |publisher=New York Times |date=November 8, 1940, Friday |accessdate=2007-07-21 }}</ref>
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| This is because it was thought that the [[Von Kármán vortex street]] frequency (the so-called [[Strouhal number|Strouhal frequency]]) was the same as the torsional natural vibration frequency. This was found to be incorrect. The actual failure was due to [[aeroelasticity#Flutter|aeroelastic flutter]].<ref name="BillahScanlan91">{{Cite journal|last=Billah|first=K.|coauthors=[[Robert H. Scanlan|R. Scanlan]]|year=1991|title=Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks|journal=American Journal of Physics|volume=59|issue=2|pages=118–124|url=http://www.ketchum.org/billah/Billah-Scanlan.pdf|format=PDF|doi=10.1119/1.16590|bibcode = 1991AmJPh..59..118B }}</ref> ]]
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| The bridge's spectacular destruction is often used as an object lesson in the necessity to consider both [[aerodynamics]] and [[resonance]] effects in [[civil engineering|civil]] and [[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| author=Halliday, David; Resnick, Robert; Walker, Jearl | authorlink= | coauthors= | title=Fundamentals of Physics, (Chapters 21-44) | date= | publisher=John Wiley & Sons | location= | isbn=0-470-04474-8 | pages=}}</ref> and Tipler et al.<ref name="TiplerMosca">{{Cite book| author=Tipler, Paul Allen; Mosca, Gene | authorlink= | coauthors= | title=Physics for Scientists and Engineers : Volume 1B: Oscillations and Waves; Thermodynamics (Physics for Scientists and Engineers) | date= | publisher=W. H. Freeman | location= | isbn=0-7167-0903-1 | pages=}})</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.
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| Usually, the approach taken by those physics textbooks is to introduce a first order forced oscillator, defined by the second order differential equation
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| :<math>m\ddot{x}(t) + c\dot{x}(t) + kx(t) = F cos (\omega t)</math> (eq. 1)
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| where <math>m</math>, <math>c</math> and <math>k</math> stand for the [[mass]], [[damping coefficient]] and [[stiffness]] of the linear system and <math>F</math> and <math>\omega</math> represent the amplitude and the angular frequency of the exciting force. The solution of such [[ordinary differential equation]] as a function of time <math>t</math> represents the displacement response of the system (given appropriate initial conditions). In the above system resonance happens when <math>\omega</math> 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—such as an airplane, a building or a bridge—is based on the linearization of the equation of motion for the system, which is 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 ''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.
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| 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 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|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.
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| The frequency of the vortices in the von Kármán vortex street is called the Strouhal frequency <math>f_s</math>, and is given by
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| :<math>\frac{f_s D}{U} = S</math> (eq. 2)
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| Here, <math>U</math> stands for the flow velocity, <math>D</math> is a characteristic length of the bluff body and <math>S</math> is the dimensionless [[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, <math>D</math> was approximately {{convert|8|ft}} and <math>S</math> was 0.20.
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| 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|vortex-induced vibration]].
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| 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 [[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.
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| 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 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 itself linear behaviour, since the exciting force amplitude is a nonlinear force of the structural response.<ref>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. Princeton University. April 1989</ref>
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| ====Origin of the confusion in failure modes====
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| It is not clear what is the original source of the confusion {{Clarify|date=July 2009}}. Billah and Scanlan cite that Lee Edson in his biography of [[Theodore von Kármán]]<ref>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</ref> is a source of misinformation: "The culprit in the Tacoma disaster was the Karman vortex Street."
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| However, the Federal Works Administration report of the investigation (of which von Kármán was part) concluded that
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| {{bquote|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> }}
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| ==Fate of the collapsed superstructure==
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| Efforts to salvage the bridge began almost immediately after its collapse and continued into May 1943.<ref name="WSDOT">[http://www.wsdot.wa.gov/tnbhistory/Connections/connections4.htm#4 Tacoma Narrows Bridge Aftermath – A New Beginning: 1940 – 1950]</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 superstructure built.<ref>[http://www.lib.washington.edu/specialcoll/exhibits/tnb/page5.html University of Washington Special Collections]</ref> 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.<ref name="WSDOT"/>
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| 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.
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| ==="Preservation" of the collapsed roadway===
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| [[Image:GertieRemains 1.jpg|thumb|upright|right|Remains of the collapsed bridge]] | |
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| 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|accessdate=2007-10-23}}</ref>
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| ===A lesson for history===
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| [[Othmar Ammann]], a leading bridge designer and member of the Federal Works Agency Commission investigating the collapse of the Tacoma Narrows Bridge, wrote:
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| {{bquote|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.<ref>
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| Othmar H. Ammann, Theodore von Kármán and Glenn B. Woodruff. The Failure of the Tacoma Narrows Bridge, a report to the administrator. Report to the Federal Works Agency, Washingthon, 1941</ref>}}
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| 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.
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| == Replacement bridge ==
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| {{Main|Tacoma Narrows Bridge}}
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| 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 {{convert|5979|ft}} long—{{convert|40|ft}} 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.
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| 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.
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| ==See also==
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| *[[List of bridge disasters]]
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| *[[List of structural failures and collapses]]
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| *[[Engineering disasters]]
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| ==References==
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| ;Notes
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| {{Reflist|2}}
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| ==External links==
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| {{Commons category|Tacoma Narrows Bridge}}
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| *[http://www.youtube.com/watch?v=3mclp9QmCGs Color video of the original bridge's construction and collapse with narration]
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| *[http://www.lightandmatter.com/html_books/3vw/ch02/ch02.html Physics behind the collapse of the bridge]
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| *[http://failurebydesign.info failurebydesign.info] – physics presentation and resources
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| *[http://www.citynoise.org/article/5410 Photos of the bridge and the new span under construction]
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| *{{Structurae|id=s0000074|title=Tacoma Narrows Bridge (1940)}}
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| *[http://www.sciencedirect.com/science/article/pii/S0022460X13001247 Josef Malík. Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge. Journal of Sound and Vibration. Vol 332, Issue 15, 22 July 2013, p. 3772--3789]
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| ===Historical===
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| *[http://www.wsdot.wa.gov/tnbhistory/machine/machine2.htm 1940 Narrows Bridge] (1940 bridge construction) Washington Sate Department of Transportation
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| *[http://www.lib.washington.edu/specialcoll/exhibits/tnb/ History of the Tacoma Narrows Bridge]
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| *[http://content.lib.washington.edu/farquharsonweb/index.html University of Washington Libraries Digital Collection – Tacoma Narrows Bridge Collection] More than 152 images and text documenting the infamous collapse in 1940 of the Tacoma Narrows Bridge. Also covers Galloping Gertie's creation, subsequent studies involving its aerodynamics, and finally the construction of a second bridge spanning the Narrows.
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| *[http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html The Tacoma Narrows Bridge Disaster, November 1940]
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| *[http://www.ketchum.org/bridgecollapse.html Images of failure]
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| *[http://www.civeng.carleton.ca/Exhibits/Tacoma_Narrows/ Information and images of failure]
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| *[http://www.wsdot.wa.gov/TNBhistory/ Official site of the Tacoma Narrows Bridge]
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| *[http://www.wsdot.wa.gov/TNBhistory/spanning_time.htm Timeline of the bridges]
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| *[http://www.nwrain.net/~newtsuit/recoveries/narrows/narrows.htm Tacoma Narrows Bridge]
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| *[http://failuremag.com/feature/article/suspended_animation/ Suspended Animation ] – ''[[Failure Magazine]]'' (November 2000)
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| *{{Internet Archive film clip|id=Pa2096Tacoma|description=of the Tacoma Narrows bridge wobbling and eventually, collapsing}}
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| {{Washington State bridge disasters}}
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| [[Category:1940 establishments in the United States]]
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| [[Category:1940 disestablishments in the United States]]
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| [[Category:Articles containing video clips]]
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| [[Category:Artificial reefs]]
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| [[Category:Bridge disasters caused by engineering error]]
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| [[Category:Bridge disasters in the United States]]
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| [[Category:Bridges completed in 1940]]
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| [[Category:Bridges in Tacoma, Washington]]
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| [[Category:Bridges on the National Register of Historic Places in Washington (state)]]
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| [[Category:Transportation disasters in Washington (state)]]
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| [[Category:Engineering failures]]
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| [[Category:National Register of Historic Places in Tacoma, Washington]]
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| [[Category:North Tacoma, Washington]]
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| [[Category:Road bridges in Washington (state)]]
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| [[Category:Suspension bridges in the United States]]
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| [[Category:United States National Film Registry films]]
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| [[Category:1940 in the United States]]
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| [[Category:1940 in Washington (state)]]
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