Tacoma Narrows Bridge (1940)

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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.

1004:. Billah and Scanlan (1991) reported that, in fact, many physics textbooks (for example Resnick et al. and Tipler et al.) 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. 42: 490: 522:

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 $ 129.9 million today) for the Tacoma Narrows Bridge. Another $ 1.6 million ($ 34.6 million today) was to be collected from tolls to cover the estimated total $ 8 million cost ($ 173.2 million today).

1615:...bridge engineering is not, as popularly assumed, an exact science. While ordinary structures are closely controlled by ample experience and experiments, every structure which projects into new and unexplored fields of magnitude involves new problems, for the solution of which neither theory nor practical experience can furnish an adequate guide. It is then that we must rely largely on our judgment and if as a result errors or failures occur we must accept them as a price for human progress. 780: 1551:

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.

3188: 639: 224: 435:. Steinman made several Chamber-funded visits and presented a preliminary proposal in 1929, but by 1931 the Chamber had cancelled the agreement because 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 the construction of the bridge, with its designer in the audience, Steinman predicted its failure. 884: 1630:, 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. 767:. 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. 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. 355: 961: 665:

time, I crawled 500 yards 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 ... Towards 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.

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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.

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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

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researchers were able to prove that the original 16 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.

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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.

899:) to support the roadbed. With the earlier designs, any wind would 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 771:

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.

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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 the main cables and road deck, suffered major damage at their bases from

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tests and recommend solutions to reduce the oscillations of the bridge. 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

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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

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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

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measures. The bridge's main span finally collapsed in 40-mile-per-hour (64 km/h) 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 violent swaying and eventual collapse resulted

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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

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into Puget Sound on November 7 of the same year. The bridge's collapse has been described as "spectacular" and in subsequent decades "has attracted the attention of engineers, physicists, and mathematicians". Throughout its short existence, it was the world's third-longest suspension bridge by

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Efforts to salvage the bridge began almost immediately after its collapse and continued into May 1943. 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

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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

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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

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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

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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

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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

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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 5,979 feet (1,822 m) long, forty feet

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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

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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

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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

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Because planners expected fairly light traffic volumes, the bridge was designed with two lanes, and it was just 39 feet (12 m) wide. This was quite narrow, especially in comparison with its length. With only the 8-foot-deep (2.4 m) plate girders providing additional depth, the bridge's

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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; 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

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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

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To some degree the debate is due to the lack of a commonly accepted precise definition of resonance. Billah and Scanlan 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

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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

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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

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laterally. This energy would then be transmitted to the anchorages and towers. Using this theory, Moisseiff argued for stiffening the bridge with a set of eight-foot-deep (2.4 m) plate girders rather than the 25-foot-deep (7.6 m) trusses proposed by the Washington State Toll Bridge

907:, 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. 561:

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

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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 centre line of the road remaining still (motionless). This vibration was caused by

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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

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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 Farquharson, the wind was steady at 42 miles per hour (68 km/h) and the frequency of the destructive mode was 12 cycles/minute (0.2

1569:, 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 $ 5,919,000 in 2022). 1280:

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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On November 7, 1940, at around 9:45 a.m. PST, especially strong winds caused the bridge to sway wildly from side to side. At least two vehicles were on the bridge at the time – a delivery truck driven by Ruby Jacox and Arthur Hagen, employees of

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Washington State engineer Clark Eldridge produced a preliminary tried-and-true conventional suspension bridge design, and the Washington State Toll Bridge Authority requested $ 11 million (equivalent to $ 220 million today) from the federal

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Following Moisseiff's design, bridge construction began on November 23, 1938. Construction took only nineteen months, at a cost of $ 6.4 million ($ 138.5 million today), which was financed by the grant from the PWA and a loan from the RFC.

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behind a circular cylinder. The first hypothesis of the failure of the Tacoma Narrows Bridge was resonance (due to the Kármán vortex street). This is because it was thought that the Kármán vortex street frequency (the so-called

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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

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to $ 17.4 million today). The bridge was insured by many other policies that covered 80% of the $ 5.2 million structure's value (equivalent to $ 113.1 million today). Most of these were collected without incident.

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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. Their footage was subsequently sold to

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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

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The first option was not favoured, 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.

934: 938: 937: 933: 932: 694: 939: 675:, walked onto the bridge to try to save Tubby, but was forced to turn back when the span began to break apart in the center. At approximately 11:00 a.m., the bridge collapsed into the strait. 3109: 2308:

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!'

1301:. 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. 895:

anchored in concrete blocks; 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

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being deflected 12 feet (3.7 m) 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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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

1379: 936: 3296: 3276: 1672:(12 m) 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. 509:, had published a paper that was probably the most important theoretical advance in the bridge engineering field of the decade. Their theory of elastic distribution extended the 1469: 2380: 333:) flutter oscillation would continuously increase, with a negative damping factor, i.e., a reinforcing effect, opposite to damping. The collapse boosted research into bridge 3336: 3316: 3271: 1263: 798:

and a world-renowned aerodynamicist, was a member of the board of inquiry into the collapse. He reported that the State of Washington was unable to collect on one of the

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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.

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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

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The failure of the bridge occurred when a never-before-seen twisting mode occurred, from winds at 40 miles per hour (64 km/h). This is a so-called torsional

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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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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.

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How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States: Addenda et Corrigenda

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The Tacoma Narrows Bridge, with a main span of 2,800 feet (850 m), was the third-longest suspension bridge in the world at that time, following the

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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

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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".

501:, the noted New York bridge engineer who served as designer and consultant engineer for the Golden Gate Bridge — petitioned the PWA and the 2772: 2551:"Big Tacoma Bridge Crashes 190 Feet into Puget Sound. Narrows Span, the Third Longest Type in the World, Collapses in Wind. Four Escape Death". 2447: 3311: 2743: 935: 439: 1611:, a leading bridge designer and member of the Federal Works Agency Commission investigating the collapse of the Tacoma Narrows Bridge, wrote: 2609: 2580: 1526:, 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 3070: 2300:, 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 3356: 3229: 2806: 1590: 2172: 1018: 329:

that was self-exciting and unbounded: for any constant sustained wind speed above about 35 mph (56 km/h), the amplitude of the (

41: 2917: 505:(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 59: 3286: 2980: 1530:, the mathematical distinction ... is quite clear, self-exciting systems differing strongly enough from ordinary linear resonant ones." 3306: 3022: 3006: 2852: 1297:

or von Kármán vortex street. The body will in consequence try to move toward the low-pressure zone, in an oscillating movement called

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represents the displacement response of the system (given appropriate initial conditions). In the above system resonance happens when

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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

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named "Tubby", as well as inflicting injuries on people fleeing the disintegrating bridge or attempting to rescue the stranded dog.

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How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States

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dangerously in relatively mild windy conditions that are common for the area, and worse during severe winds. This vibration was

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Coatsworth received $ 814.40 (equivalent to $ 17,700 today in reimbursement for his car and its contents, including the dog, a

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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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To drill holes in the lateral girders and along the deck so that the airflow could circulate through them (in this way reducing

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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.,

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Usually, the approach taken by those physics textbooks is to introduce a first order forced oscillator, defined by the

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Full-scale, two-way Fluid Structure Interaction (FSI) model of the Tacoma Narrows Bridge exhibiting aeroelastic flutter

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while it was still under construction, several strategies were used to reduce the motion of the bridge. They included:

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Malík, Josef (2013). "Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge".

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Traffic was stopped to prevent additional vehicles from entering the bridge. Howard Clifford, a photographer for the

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Leon S. Moisseiff and Frederick Lienhard. "Suspension Bridges Under the Action of Lateral Forces," with discussion.

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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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On November 28, 1940, the U.S. Navy's Hydrographic Office reported that the remains of the bridge were located at

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

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analysis to reveal all the degrees of freedom of the particular structure and the set of design loads imposed.

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A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge

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The Washington State Toll Bridge Authority hired Frederick Burt Farquharson, an engineering professor at the

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

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greater than 1000, the Strouhal number is approximately equal to 0.21. In the case of the Tacoma Narrows,

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Following the incident, engineers took extra caution to incorporate aerodynamics into their designs, and

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The bridge's spectacular destruction is often used as an object lesson in the necessity to consider both

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was built, it began to move vertically in windy conditions, so construction workers nicknamed the bridge

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analysis and thereafter the natural frequencies of the structure are found, together with the so-called

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began campaigning and funding studies in 1923. Several noted bridge engineers were consulted, including

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blocks on the shore. This measure proved ineffective, as the cables snapped shortly after installation.

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Henry Petroski. Engineers of Dreams: Great Bridge Builders and the Spanning of America. New York:

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Plaut, R.H. (2008). "Snap Loads and Torsional Oscillation of the original Tacoma Narrows Bridge".

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is a source of misinformation: "The culprit in the Tacoma disaster was the Karman vortex street."

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built. With steel being a valuable commodity because of the involvement of the United States in

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as being culturally, historically, or aesthetically significant. This footage is still shown to

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The frequency of the vortices in the von Kármán vortex street is called the Strouhal frequency

779: 325:, but it was more complicated in reality; the bridge collapsed because moderate winds produced 3202: 3082: 2925: 2861: 2722: 2605: 2576: 2001: 1899: 1833: 1642: 1178: 997: 977: 919: 861: 428: 240: 107: 2990: 2221: 566:-paid traffic started) felt these motions on the day that the bridge opened on July 1, 1940. 3158: 2985: 2950: 2712: 2151: 2131: 2090: 1891: 1862: 1718: 1406: 653: 442:

and appropriated $ 5,000 (equivalent to $ 100,000 today) to study the request by Tacoma and

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studied the collapse of the bridge. The board of engineers responsible for the report were

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Random effects of turbulence, that is the random fluctuations in velocity of the wind.

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The wind and Beyond. Theodore von Karman: Pioneer in Aviation and Pathfinder in Space

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Elliott and Monroe's footage of the construction and collapse was shot on 16 mm

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Proposals for a bridge between Tacoma and the Kitsap Peninsula date at least to the

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attachment of tie-down cables to the plate girders, which were anchored to 50-ton

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In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability

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A key consequence was that suspension bridges reverted to a deeper and heavier

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The bridge's collapse had a lasting effect on science and engineering. In many

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Road bridges on the National Register of Historic Places in Washington (state)

3059: 3053: 2954: 2349:"A Tacoma Narrows 'Galloping Gertie' bridge-collapse surprise, 75 years later" 2021: 1581: 756: 534: 290:. The motion continued after the bridge opened to the public, despite several 17: 2865: 2726: 1903: 1693:, suspension bridge that shook violently until weight limits were implemented 1653:, which gave the necessary stiffness together with reduced torsional forces. 891:

The original Tacoma Narrows Bridge was the first to be built with girders of

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The effects of Galloping Gertie's fall lasted long after the catastrophe.

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The weather system that caused the bridge collapse went on to cause the

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However, "Eastern consulting engineers" — by which Eldridge meant

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Simplistic representation of the collapse of the Tacoma Narrows Bridge

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textbooks, the event is presented as an example of elementary forced

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The original Tacoma Narrows Bridge on its opening day on July 1, 1940

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film, but most copies in circulation are in black and white because

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Aerodynamic instability by self-induced vibrations in the structure

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Program for the opening of the Tacoma Narrows Bridge, June 30, 1940

341:, which has influenced the designs of all later long-span bridges. 1715:, a bridge that collapsed in 1967 on the West Virginia–Ohio border 1634: 1580: 1476: 959: 929: 882: 778: 689: 637: 488: 483: 222: 2801: 2689:

Olson, Donald W.; Wolf, Steven F.; Hook, Joseph M. (2015-11-01).

2981:"Suspended Animation: The collapse of the Tacoma Narrows Bridge" 2301: 1154: 980:. This was found to be incorrect. The actual failure was due to 704: 642:

The main bridge span falling into the strait on November 7, 1940

154: 3091: 2971: 2744:"Tacoma Narrows Bridge: Aftermath – A New Beginning: 1940–1950" 2670:

Construction Disasters: Design Failures, Causes, and Prevention

2283:. Washington State Department of Transportation. Archived from 2196:. Washington State Department of Transportation. Archived from 348: 1471:, to cause resonance and therefore vortex-induced vibration. 1496:

Billah and Scanlan state that Lee Edson in his biography of

513:

theory that was originally devised by the Austrian engineer

415:

proposal, but concerted efforts began in the mid-1920s. The

3073:

at the Gig Harbor Peninsula Historical Society & Museum

763:

of the day copied the film onto 35 mm black-and-white

3327:

National Register of Historic Places in Tacoma, Washington

2528:"Opening and Experiments to study 'ripple' — UW Libraries" 266:. It opened to traffic on July 1, 1940, and dramatically 2571:

Halliday, David; Resnick, Robert; Walker, Jearl (2008).

651:, and a vehicle driven by Leonard Coatsworth, editor at 282:

Construction began in September 1938. From the time the

1982:

Transactions of the American Society of Civil Engineers

2850:

Amann, Othmar H. (1945-06-01). "Bridges of New York".

956:

Resonance (due to Von Kármán vortex street) hypothesis

574:

Since the structure experienced considerable vertical

438:

In 1947, the Washington State legislature created the

2247:"Tacoma Narrows Bridge: Art of the Bridges Continues" 1438: 1342: 1310: 1244: 1199: 1021: 947:

Fluttering is a physical phenomenon in which several

2222:"::: Tacoma Narrows Bridge Film Collection :::" 3222: 3195: 3127: 2085:Rita Robison. "Tacoma Narrows Bridge Collapse." In 728:was selected for preservation in the United States 701:

Footage of the old Tacoma Narrows Bridge collapsing

212: 202: 194: 186: 178: 168: 163: 149: 139: 131: 123: 113: 103: 98: 90: 51: 32: 1880:Petroski, Henry (2009). "Tacoma Narrows Bridges". 1623:testing of designs was eventually made mandatory. 1463: 1373: 1323: 1257: 1230: 1117: 1827: 1825: 1823: 1821: 1727:, a suspension bridge that collapsed in Indonesia 2853:Journal of the Boston Society of Civil Engineers 2826:"WSDOT – Tacoma Narrows Bridge: Extreme History" 2089:, edited by Neil Schlager, pp. 18–190. Detroit: 1425: was approximately 8 feet (2.4 m) and 2999:"Exhibit: History of the Tacoma Narrows Bridge" 2422:"Prison Minimum Set For Ex-Insurance Executive" 2408:Father of Supersonic Flight: Theodor von Kármán 2063:"BUILDING BIG: Databank: Tacoma Narrows Bridge" 1917: 1915: 1913: 1613: 1548: 1514: 1505: 662: 3362:Transportation disasters in Washington (state) 2668:Steven Ross, et al. "Tacoma Narrows 1940." In 1785:"Tacoma Narrows Bridge history - Tubby trivia" 302:Efforts to replace the bridge were delayed by 3103: 3035:Washington State Department of Transportation 2830:Washington State Department of Transportation 2634: 2632: 2117:Washington State Department of Transportation 2103: 2101: 2099: 1417:, which depends on the body in question. For 8: 3297:Bridge disasters caused by engineering error 3277:1940 disestablishments in Washington (state) 1771:. Applied Mathematical Modelling, Jan 2015. 1181:of the exciting force. The solution of such 2964:"Two Case Studies of Bridge Design Failure" 2651:Theodore von Karman with Lee Edson (1963). 1585:Remains of the collapsed bridge in May 2008 783:A fragment of the collapsed bridge, in the 660:Coatsworth later described his experience. 486:to sit beneath the roadway and stiffen it. 3151:Tacoma Narrows Bridge ("Galloping Gertie") 3110: 3096: 3088: 2428:. Roseburg, Oregon. 22 May 1941. p. 1 423:, who went on to be chief engineer of the 29: 27:Failed suspension bridge in Washington, US 3337:November 1940 events in the United States 3317:Former toll bridges in Washington (state) 3272:1940 establishments in Washington (state) 2716: 2573:Fundamentals of Physics, (Chapters 21-44) 1702:List of structural failures and collapses 1449: 1437: 1350: 1343: 1341: 1315: 1309: 1249: 1243: 1231:{\displaystyle \omega _{r}={\sqrt {k/m}}} 1218: 1213: 1204: 1198: 1053: 1052: 1026: 1025: 1020: 541:, and the Golden Gate Bridge, connecting 395:Learn how and when to remove this message 3352:Suspension bridges in Washington (state) 3248:List of Washington state bridge failures 2738: 2736: 2596:Tipler, Paul Allen; Mosca, Gene (2004). 1984:, No. 98, 1933, pp. 1080–1095, 1096–1141 1641:, until the development in the 1960s of 1508:frequency depends on the wind velocity. 1492:Resonance vs. non-resonance explanations 2173:"Consumer Price Index (estimate) 1800–" 2000:. American Society of Civil Engineers. 1925:was invoked but never defined (see the 1811:was invoked but never defined (see the 1736: 570:Attempt to control structural vibration 2802:"National Register Information System" 2773:"Subject Guides & Online Exhibits" 1561:dismantled and an entirely new bridge 871:Eddy formations that might be periodic 840:, at a depth of 180 feet (55 meters). 720:also received distribution rights for 440:Washington State Toll Bridge Authority 3302:Bridge disasters in the United States 3061:Collapse of the Tacoma Narrows Bridge 2684: 2682: 2474: 2472: 2320:Pasternack, Alex (14 December 2015). 2171:Federal Reserve Bank of Minneapolis. 1577:Preservation of the collapsed roadway 1374:{\displaystyle {\frac {f_{s}D}{U}}=S} 7: 3322:Historic Civil Engineering Landmarks 2807:National Register of Historic Places 2691:"The Tacoma Narrows Bridge collapse" 2598:Physics for Scientists and Engineers 2194:"Tacoma Narrows Bridge: Weird Facts" 1952: 1950: 1591:National Register of Historic Places 1556:Fate of the collapsed superstructure 1333: 1012: 377:adding citations to reliable sources 3282:1940 disasters in the United States 2454:Special Collections. Archived from 1920: 1806: 1767:Gianni Arioli and Filippo Gazzola. 3347:Steel bridges in the United States 3023:University of Washington Libraries 3015:"Tacoma Narrows Bridge Collection" 3007:University of Washington Libraries 1637:design, including the replacement 1534:Link to the Armistice Day blizzard 1528:externally forced linear resonance 1464:{\displaystyle 2\pi f_{s}=\omega } 1405:is a characteristic length of the 1009:second-order differential equation 796:Guggenheim Aeronautical Laboratory 726:The Tacoma Narrows Bridge Collapse 597:sand-blasted before being painted. 553:roadway section was also shallow. 503:Reconstruction Finance Corporation 227:Map showing location of the bridge 25: 1745:"Tacoma Narrows Bridge collapses" 3186: 1943:. doi: 10.1016/j.jsv.2007.07.057 1921:Cite error: The named reference 1807:Cite error: The named reference 1593:with reference number 92001068. 1177:represent the amplitude and the 353: 40: 3031:"Tacoma Narrows Bridge history" 1994:Scott, Richard (June 1, 2001). 1268: 844:Federal Works Agency Commission 785:Washington State History Museum 589:addition of a pair of inclined 446:for a bridge over the Narrows. 364:needs additional citations for 3175:Fourth Avenue Bridge (Olympia) 3078:Tacoma Narrows Bridge Collapse 2935:Journal of Sound and Vibration 2504:"The Aftermath — UW Libraries" 1941:Journal of Sound and Vibration 1542:that killed 145 people in the 1401:stands for the flow velocity, 1183:ordinary differential equation 1112: 1103: 1088: 1082: 1070: 1064: 1043: 1037: 150: 1: 3312:Bridges in Tacoma, Washington 2480:"Construction — UW Libraries" 2281:Tacoma Narrows Bridge History 2113:Tacoma Narrows Bridge history 914:(which is different from the 3049:Tacoma Narrows Bridge (1940) 2165:American Antiquarian Society 2145:American Antiquarian Society 1663:Tacoma Narrows Bridge (1950) 1639:Tacoma Narrows Bridge (1950) 431:, later the designer of the 207:Tacoma Narrows Bridge (1950) 3357:Transport disasters in 1940 3230:West Spokane Street Bridge 2436:– via Newspapers.com. 1890:(2) (2 ed.): 103–107. 1846:American Journal of Physics 1540:1940 Armistice Day Blizzard 1258:{\displaystyle \omega _{r}} 978:natural vibration frequency 848:A commission formed by the 480:Public Works Administration 127:5,939 feet (1,810.2 m) 3378: 3287:1940 in Washington (state) 2916:Crowell, Benjamin (2006). 2406:Halacy Jr., D. S. (1965). 2354:Seattle Post-Intelligencer 2226:content.lib.washington.edu 1709:, for an engineering error 1660: 456:Puget Sound Naval Shipyard 417:Tacoma Chamber of Commerce 233:1940 Tacoma Narrows Bridge 3307:Bridges completed in 1940 3245: 3184: 2955:10.1016/j.jsv.2013.02.011 2880:"Tacoma Bridge collapses" 2657:Little, Brown and Company 2602:W. H. Freeman and Company 2575:. John Wiley & Sons. 1707:Millennium Bridge, London 1272:). The analysis requires 308:new Tacoma Narrows Bridge 217: 135:2,800 feet (853.4 m) 39: 3332:North Tacoma, Washington 2962:Meador, Granger (2008). 2452:University of Washington 1725:Kutai Kartanegara Bridge 1691:Humen Pearl River Bridge 1544:Midwestern United States 1299:vortex-induced vibration 808:geographical coordinates 603:University of Washington 531:George Washington Bridge 409:Northern Pacific Railway 277:George Washington Bridge 3211:I-5 Skagit River bridge 2448:"Tacoma Narrows Bridge" 1697:List of bridge failures 1628:Bronx–Whitestone Bridge 345:Design and construction 237:bridge at this location 3135:Division Street Bridge 2918:"Vibrations and Waves" 2532:www.lib.washington.edu 2508:www.lib.washington.edu 2484:www.lib.washington.edu 1676:traffic in July 2007. 1617: 1586: 1553: 1519: 1510: 1465: 1375: 1325: 1259: 1232: 1185:as a function of time 1119: 1002:structural engineering 985: 973:) was the same as the 944: 925:aeroelastic fluttering 888: 794:, the director of the 788: 730:National Film Registry 708: 667: 649:Rapid Transfer Company 643: 494: 271:main span, behind the 228: 145:195 feet (59.4 m) 3267:Tacoma Narrows Bridge 3071:Tacoma Narrows Bridge 2812:National Park Service 2109:"Eyewitness accounts" 2087:When Technology Fails 1686:Engineering disasters 1667:Tacoma Narrows Bridge 1598:Harbor History Museum 1584: 1466: 1413:is the dimensionless 1376: 1326: 1324:{\displaystyle f_{s}} 1260: 1233: 1120: 963: 942: 886: 879:Cause of the collapse 782: 700: 641: 557:halves of the centre 492: 454:, which operated the 243:in the U.S. state of 226: 34:Tacoma Narrows Bridge 3167:I-90 Floating Bridge 2676:, 1984, pp. 216–239. 2640:Princeton University 2557:. November 8, 1940. 1604:A lesson for history 1436: 1340: 1308: 1295:Kármán vortex street 1242: 1197: 1019: 966:Kármán vortex street 964:Vortex shedding and 850:Federal Works Agency 373:improve this article 323:mechanical resonance 3143:Allen Street Bridge 3019:Digital Collections 3003:Special Collections 2947:2013JSV...332.3772M 2814:. January 23, 2007. 2709:2015PhT....68k..64O 2410:. pp. 119–122. 2387:. February 28, 2019 1896:10.1511/2009.77.103 1859:1991AmJPh..59..118B 1498:Theodore von Kármán 1159:damping coefficient 982:aeroelastic flutter 858:Theodore von Kármán 824: / 792:Theodore von Kármán 734:Library of Congress 672:Tacoma News Tribune 519:suspended structure 327:aeroelastic flutter 71: / 2993:on 1 January 2014. 2974:on 4 October 2008. 2928:on 3 January 2007. 2922:Lightandmatter.com 2848:Othmar H. Ammann. 2554:The New York Times 1883:American Scientist 1657:Replacement bridge 1649:shape such as the 1643:box girder bridges 1587: 1461: 1371: 1331:, and is given by 1321: 1255: 1228: 1115: 986: 971:Strouhal frequency 949:degrees of freedom 945: 889: 828:47.267°N 122.550°W 789: 714:Paramount Pictures 709: 644: 495: 464:United States Army 452:United States Navy 425:Golden Gate Bridge 295:in the death of a 273:Golden Gate Bridge 229: 179:Construction start 75:47.267°N 122.550°W 3254: 3253: 3238: 3215: 3207: 3203:Hood Canal Bridge 3179: 3171: 3163: 3155: 3147: 3139: 3118:Washington State 2968:Failure By Design 2941:(15): 3772–3789. 2890:. August 21, 2018 2718:10.1063/PT.3.2991 2611:978-0-7167-0903-9 2582:978-0-470-04474-2 2357:. 7 November 2015 1395: 1394: 1363: 1278:fundamental modes 1226: 1193:is approximately 1179:angular frequency 1139: 1138: 1061: 1034: 940: 862:Glenn B. Woodruff 698: 429:David B. Steinman 421:Joseph B. Strauss 405: 404: 397: 306:, but in 1950, a 247:that spanned the 241:suspension bridge 221: 220: 182:November 23, 1938 16:(Redirected from 3369: 3292:Artificial reefs 3236: 3213: 3205: 3190: 3177: 3169: 3161: 3159:Chow Chow Bridge 3153: 3145: 3137: 3120:bridge disasters 3112: 3105: 3098: 3089: 3062: 3038: 3026: 3010: 2994: 2989:. Archived from 2986:Failure Magazine 2975: 2970:. Archived from 2958: 2929: 2924:. Archived from 2904: 2903: 2897: 2895: 2876: 2870: 2869: 2846: 2840: 2839: 2837: 2836: 2822: 2816: 2815: 2798: 2792: 2791: 2789: 2788: 2779:. Archived from 2769: 2763: 2762: 2760: 2759: 2750:. 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J. 2150: 2140: 2132:McCusker, J. J. 2130: 2128: 2124: 2107: 2106: 2097: 2084: 2080: 2071: 2069: 2061: 2060: 2056: 2046:Alfred A. 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Scanlan 1809:Coatsworth 1794:2024-06-04 1754:2020-07-12 1732:References 1429:was 0.20. 1407:bluff body 1274:eigenvalue 905:transverse 757:Kodachrome 605:, to make 535:New Jersey 511:deflection 472:Fort Lewis 385:April 2015 245:Washington 108:Suspension 3223:Incidents 2866:0361-087X 2727:0031-9228 1927:help page 1904:0003-0996 1813:help page 1459:ω 1443:π 1247:ω 1202:ω 1163:stiffness 1107:ω 1101:⁡ 1059:˙ 1032:¨ 994:resonance 975:torsional 800:insurance 787:in Tacoma 761:newsreels 460:Bremerton 331:torsional 268:collapsed 195:Collapsed 2659:. p. 213 2154:(1992). 2134:(1997). 2119:(WSDOT). 1836:(1991). 1680:See also 1645:with an 1238:, i.e., 819:122°33′W 634:Collapse 584:concrete 533:between 411:'s 1889 275:and the 262:and the 258:between 239:, was a 213:Location 169:Designer 114:Material 66:122°33′W 3234:ramming 3066:YouTube 2943:Bibcode 2888:A&E 2884:HISTORY 2705:Bibcode 2385:KING-TV 2093:, 1994. 2052:, 1995. 1855:Bibcode 1773:doi.org 1749:HISTORY 1647:airfoil 1165:of the 897:I-beams 816:47°16′N 775:Inquiry 746:physics 732:by the 484:trusses 413:trestle 319:physics 292:damping 164:History 63:47°16′N 3237:(1978) 3232:Chavez 3214:(2013) 3206:(1979) 3178:(2001) 3170:(1990) 3162:(1988) 3154:(1940) 3146:(1923) 3138:(1915) 2864: 2725: 2608: 2579: 2004: 1902: 1397:Here, 1141:where 901:buckle 860:, and 744:, and 427:, and 260:Tacoma 252:strait 187:Opened 104:Design 2748:WSDoT 2626:1954. 2161:(PDF) 2141:(PDF) 1841:(PDF) 1635:truss 1388:eq. 2 1291:wakes 1269:eq. 1 1132:eq. 1 998:civil 765:stock 545:with 155:lanes 3083:IMDb 2896:2018 2862:ISSN 2723:ISSN 2606:ISBN 2577:ISBN 2434:2017 2393:2019 2363:2015 2335:2020 2327:Vice 2302:Guam 2263:2020 2233:2020 2180:2024 2002:ISBN 1900:ISSN 1665:and 1626:The 1596:The 1409:and 1173:and 1169:and 1161:and 1155:mass 1149:and 1000:and 992:and 564:toll 559:span 537:and 470:and 284:deck 231:The 3081:at 3064:on 3051:at 2951:doi 2939:332 2713:doi 1892:doi 1863:doi 1098:cos 918:or 458:in 375:by 254:of 153:of 151:No. 3263:: 3033:. 3021:. 3017:. 3005:. 3001:. 2983:. 2966:. 2949:. 2937:. 2920:. 2898:. 2886:. 2882:. 2858:32 2856:. 2828:. 2810:. 2804:. 2775:. 2746:. 2735:^ 2721:. 2711:. 2701:68 2699:. 2693:. 2681:^ 2672:. 2631:^ 2604:. 2530:. 2506:. 2482:. 2471:^ 2450:. 2424:. 2383:. 2365:. 2351:. 2324:. 2294:. 2279:. 2249:. 2224:. 2207:. 2163:. 2143:. 2115:. 2111:. 2098:^ 2065:. 2024:. 1960:. 1949:^ 1929:). 1912:^ 1898:. 1888:97 1886:. 1861:. 1851:59 1849:. 1843:. 1820:^ 1815:). 1787:. 1747:. 1546:: 1477:Hz 1157:, 1145:, 984:. 927:. 856:, 752:. 740:, 705:MB 618:). 314:. 279:. 3111:e 3104:t 3097:v 3037:. 3025:. 3009:. 2957:. 2953:: 2945:: 2868:. 2838:. 2790:. 2761:. 2729:. 2715:: 2707:: 2616:) 2614:. 2585:. 2540:. 2516:. 2492:. 2465:. 2395:. 2337:. 2265:. 2235:. 2182:. 2167:. 2147:. 2075:. 2048:/ 2034:. 2010:. 1970:. 1906:. 1894:: 1869:. 1865:: 1857:: 1797:. 1757:. 1456:= 1451:s 1447:f 1440:2 1427:S 1423:D 1411:S 1403:D 1399:U 1390:) 1386:( 1369:S 1366:= 1361:U 1357:D 1352:s 1348:f 1317:s 1313:f 1266:( 1251:r 1224:m 1220:/ 1216:k 1211:= 1206:r 1191:ω 1187:t 1175:ω 1171:F 1151:k 1147:c 1143:m 1134:) 1130:( 1113:) 1110:t 1104:( 1095:F 1092:= 1089:) 1086:t 1083:( 1080:x 1077:k 1074:+ 1071:) 1068:t 1065:( 1056:x 1050:c 1047:+ 1044:) 1041:t 1038:( 1029:x 1023:m 398:) 392:( 387:) 383:( 369:. 337:- 159:2 20:)

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