Tacoma Narrows Bridge (1940)
The 1940 Tacoma Narrows Bridge, the first Tacoma Narrows Bridge, was a suspension bridge in the U.S. state of Washington that spanned the Tacoma Narrows strait of Puget Sound between Tacoma and the Kitsap Peninsula. It opened to traffic on July 1, 1940, and dramatically collapsed into Puget Sound on November7 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 main span, behind the Golden Gate  Bridge and the George Washington Bridge. Construction began in September 1938. From the time the deck was built, it began to move vertically in windy conditions, so construction workers nicknamed the bridge Galloping Gertie. The motion continued after the bridge opened to the public, despite several damping measures. The bridge's main span finally collapsed in winds on the morning of November 7, 1940, as the deck oscillated in an alternating twisting motion that gradually increased in amplitude until the deck tore apart. The portions of the bridge still standing after the collapse, including the towers and cables, were dismantled  and sold as scrap metal. Efforts to replace the bridge were delayed by the United States' entry into World War II, but in 1950, a new Tacoma Narrows Bridge opened in the same location, using the original bridge's tower pedestals and cable anchorages. The portion of the bridge that fell into the water now serves as an artificial reef. The bridge's collapse had a lasting effect on science and engineering. In many physics textbooks, the event is presented as an example of elementary forced mechanical resonance, but it was more complicated in reality; the bridge collapsed because moderate  winds produced aeroelastic flutter that was self-exciting and unbounded: for any constant sustained wind speed above about , the amplitude of the (torsional) flutter oscillation would continuously increase, with a negative damping factor (i.e., a reinforcing effect, opposite to damping). The collapse boosted research into bridge aerodynamics-aeroelastics, which has influenced the designs of all later long-span bridges. Design and construction. Proposals for a bridge between Tacoma and the Kitsap Peninsula date at least to the Northern Pacific Railway's 1889 trestle proposal, but concerted efforts began in the mid-1920s. The Tacoma Chamber of Commerce began campaigning and funding  studies in 1923. Several noted bridge engineers were consulted, including Joseph B. Strauss, who went on to be chief engineer of the Golden Gate Bridge, and David B. Steinman, later the designer of the Mackinac Bridge. Steinman made several Chamber-funded visits and presented a preliminary proposal in 1929, but by 1931 the Chamber had canceled the agreement on the grounds that Steinman was not working hard enough to obtain financing. At the 1938 meeting of the structural division of the American Society of Civil Engineers, during construction of the bridge, with its designer in the audience, Steinman predicted its  failure. In 1937, the Washington State legislature created the Washington State Toll Bridge Authority and appropriated $5,000 (equivalent to $ today) to study the request by Tacoma and Pierce County for a bridge over the Narrows. 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. But there was strong support for the bridge from the U.S. Navy, which operated the Puget Sound Naval Shipyard in  Bremerton, and from the U.S. Army, which ran McChord Field and Fort Lewis near Tacoma. Washington State engineer Clark Eldridge produced a preliminary tried-and-true conventional suspension bridge design, and the Washington Toll Bridge Authority requested $11 million (equivalent to $ million today) from the federal Public Works Administration (PWA). Preliminary construction plans by the Washington Department of Highways had called for a set of trusses to sit beneath the roadway and stiffen it. However, "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 and Frederick Lienhard, the latter an engineer with what was then known as the Port of New York Authority, 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 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. Using this theory, Moisseiff argued for stiffening the bridge with a set of plate girders rather than the trusses proposed by the Washington Toll Bridge Authority. This approach meant a slimmer, more elegant design, and also reduced the construction costs as compared with the Highway Department's design proposed by Eldridge. Moisseiff's design won out, inasmuch as the other proposal was considered to be too expensive. On June 23, 1938, the PWA approved nearly $6  million (equivalent to $ million today) for the Tacoma Narrows Bridge. Another $1.6 million ($ million today) was to be collected from tolls to cover the estimated total $8 million cost ($ million today). Following Moisseiff's design, bridge construction began on September 27, 1938. Construction took only nineteen months, at a cost of $6.4 million ($ million today), which was financed by the grant from the PWA and a loan from the RFC. The Tacoma Narrows Bridge, with a main span of , 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 to its north. Because planners expected fairly light traffic volumes, the bridge was designed with two lanes, and it was just wide. This was quite narrow, especially in comparison with its length. With only the plate girders providing additional depth, the bridge's roadway section was also shallow. The decision to use such shallow and narrow girders proved the bridge's undoing. With such minimal girders, the deck of the bridge was insufficiently rigid and was easily moved about  by winds; from the start, the bridge became infamous for its movement. A mild to moderate wind could cause alternate halves of the center span to visibly rise and fall several feet over four- to five-second intervals. This flexibility was experienced by the builders and workmen during construction, which led some of the workers to christen the bridge "Galloping Gertie". The nickname soon stuck, and even the public (when the toll-paid traffic started) felt these motions on the day that the bridge opened on July 1, 1940. Attempt to control structural vibration. Since the structure experienced considerable vertical  oscillations while it was still under construction, several strategies were used to reduce the motion of the bridge. They included 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 : 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. Collapse. Leonard Coatsworth, a "Tacoma News Tribune" editor, was the last person to drive on the bridge: Tubby, Coatsworth's cocker spaniel, was the only fatality of the Tacoma Narrows Bridge disaster; he was lost along with Coatsworth's car. Professor Farquharson and a news photographer attempted to rescue Tubby during a lull, but the dog was too terrified to leave the car and bit  one of the rescuers. Tubby died when the bridge fell and neither his body nor the car was ever recovered. Coatsworth had been driving Tubby back to his daughter, who owned the dog. Coatsworth received $450.00 for his car (equivalent to $ today) and $364.40 ($ today) in reimbursement for the contents of his car, including Tubby. Inquiry. 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. 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 (equivalent to $ million today). The bridge was insured by many other policies that covered 80% of the $5.2million structure's value (equivalent to $million today). Most of these were collected without incident. On November 28, 1940, the U.S. Navy's Hydrographic Office reported that the remains of the bridge were located  at geographical coordinates , at a depth of 180 feet (55 meters). Film of collapse. At least four people captured the collapse of the bridge. The collapse of the bridge was recorded on film separately by Barney Elliott and by Harbine Monroe, owners of The Camera Shop in Tacoma. The film shows Leonard Coatsworth attempting to rescue his dog—without success—and then leaving the bridge. The film was subsequently sold to Paramount Studios, which duplicated it for newsreels in black-and-white and distributed it worldwide to movie theaters. Castle Films also received distribution rights for 8 mm home video . 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. Elliott and Monroe's films of the construction and collapse were shot on 16 mm 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. 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. A second reel of film emerged in February 2019, taken by Arthur Leach from the Gig Harbor (westward) side of the bridge, and one of the few known images of the collapse from that side. Leach was a civil engineer who served as toll collector  for the bridge, and is believed to have been the last person to cross the bridge to the west before its collapse, trying to prevent further crossings from the west as the bridge started to collapse. Leach's footage (originally on film but then recorded to video cassette by filming the projection) also includes Leach's commentary at the time of the collapse. Federal Works Agency commission. 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: Cause of the collapse. The original Tacoma Narrows Bridge was the first to be built with girders of carbon steel 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 I-beams) 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, one-half of the central span rising while the other lowered. Drivers would see cars approaching from the other direction rise and fall, riding the violent energy wave through the bridge. However, at that time the mass of the bridge was considered sufficient to keep it structurally sound. The failure of the bridge occurred when a never-before-seen twisting mode occurred, from winds at . This is a so -called torsional vibration mode (which is different from the transversal or longitudinal vibration mode), whereby when the left side of the roadway went down, the right side would rise, and vice versa (i.e., the two halves of the bridge twisted in opposite directions), with the center line of the road remaining still (motionless). This vibration was caused by aeroelastic fluttering. Resonance (due to Von Kármán vortex street) hypothesis. The bridge's spectacular destruction is often used as an object lesson in the necessity to consider both aerodynamics and resonance effects in civil and structural engineering. 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. Usually, the approach taken by those physics textbooks is to introduce a first order forced oscillator, defined by the second-order differential equation where , and stand for the mass, damping coefficient and stiffness of the linear system and and represent the amplitude and the angular frequency of the exciting force. The solution of such ordinary differential equation as a function of time represents the displacement response of the system (given appropriate initial conditions ). In the above system resonance happens when is approximately formula_1, i.e. formula_2 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 (). 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. 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 (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. Eventually, if the frequency of vortex shedding matches the natural frequency of the structure, the structure will begin to resonate and the structure's movement can become self-sustaining. The frequency of the vortices in the von Kármán vortex street is called the Strouhal frequency formula_3, and is given by Here, stands for the flow velocity, is a characteristic length of the  bluff body and is the dimensionless Strouhal number, which depends on the body in question. For Reynolds Numbers greater than 1000, the Strouhal number is approximately equal to 0.21. In the case of the Tacoma Narrows,   was approximately and was 0.20. It was thought that the Strouhal frequency was close enough to one of the natural vibration frequencies of the bridge i.e. formula_4, to cause resonance and therefore vortex-induced vibration. In the case of the Tacoma Narrows Bridge, this appears not to have been the cause of the catastrophic damage. According to Professor Frederick Burt Farquharson, an engineering professor  at the University of Washington and one of the main researchers into the cause of the bridge collapse, the wind was steady at and the frequency of the destructive mode was 12 cycles/minute (0.2 Hz). 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. 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. Resonance vs. non-resonance explanations. Billah and Scanlan state that Lee Edson in his biography of Theodore von Kármán is a source of misinformation: "The culprit in the Tacoma disaster was the Karman vortex Street." However, the Federal Works Administration report of the investigation (of which von Kármán was part) concluded that A group of physicists cited "wind-driven amplification of the torsional oscillation" as distinct from resonance: 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 nearly equal to one of the natural frequencies of the oscillation of the system, the system is set into oscillation with a relatively large amplitude." They then state later in their paper "Could this be called a resonant phenomenon? It would appear not to contradict the qualitative definition of resonance quoted  earlier, if we now identify the source of the periodic impulses as "self-induced", the wind supplying the power, and the motion supplying the power-tapping mechanism. If one wishes to argue, however, that it was a case of "externally forced linear resonance", the mathematical distinction ... is quite clear, self-exciting systems differing strongly enough from ordinary linear resonant ones." Link to the Armistice Day blizzard. The weather system that caused the bridge collapse went on to cause the Armistice Day Blizzard that killed 145 people in the Midwest: Fate of the collapsed superstructure. 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 dismantled and an entirely new bridge superstructure built. 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 (equivalent to $ million today). The cable anchorages, tower pedestals and most of the remaining substructure were relatively undamaged in the collapse, and were reused during construction of the replacement span that opened in 1950. The towers, which supported the main cables and road deck, suffered major damage at their bases from being deflected towards shore as a result of the collapse of the mainspan and the sagging of the sidespans. They were dismantled, and the steel sent to recyclers. Preservation of the collapsed roadway. 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. The Harbor History Museum has a display in its main gallery regarding the 1940 bridge, its collapse, and the subsequent two bridges. A lesson for history. Othmar Ammann, a leading bridge designer and member of the Federal Works Agency Commission investigating the collapse of the Tacoma Narrows Bridge, wrote: Following the incident, engineers took extra caution to incorporate aerodynamics into their designs, and wind-tunnel testing of designs was eventually made mandatory . The Bronx Whitestone Bridge, which is of similar design to the 1940 Tacoma Narrows Bridge, was reinforced shortly after the collapse. Fourteen-foot-high (4.3 m) steel trusses were installed on both sides of the deck in 1943 to weigh down and stiffen the bridge in an effort to reduce oscillation. In 2003, the stiffening trusses were removed and aerodynamic fiberglass fairings were installed along both sides of the road deck. A key consequence was that suspension bridges reverted to a deeper and heavier truss design, including the replacement Tacoma Narrows Bridge (1950), until the development in the 1960s of  box girder bridges with an aerofoil shape such as the Severn Bridge, which gave the necessary stiffness together with reduced torsional forces. Replacement bridge. Because of shortages in materials and labor as a result of the involvement of the United States in World War II, it took 10 years before a replacement bridge was opened to traffic. This replacement bridge was opened to traffic on October 14, 1950, and is long, longer than the original bridge. The replacement bridge also has more lanes than the original bridge, which only had two traffic lanes, plus shoulders on both sides. Half a  century later, the replacement bridge exceeded its traffic capacity, and a second, parallel, suspension bridge was constructed to carry eastbound traffic. The suspension bridge that was completed in 1950 was reconfigured to carry only westbound traffic. The new parallel bridge opened to traffic in July 2007.