Tacoma Narrows Bridge (1940)

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.[21] Their footage was subsequently sold to Paramount Pictures, 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.[22] 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.[23]

Elliott and Monroe's footage of the construction and collapse was 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.[24] 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.[25]

Another reel of film emerged in February 2019, taken by Arthur Leach from the Gig Harbor (westward) side of the bridge, and one of the few known images of the collapse from that side. Leach was a civil engineer who served as toll collector for the bridge, and is believed to have been the last person to cross the bridge to the west before its collapse, trying to prevent further crossings from that side as the bridge became unstable. Leach's footage (originally on black-and-white film but then recorded to video cassette by filming the projection) also includes Leach's commentary at the time of the collapse.[26]

A commission formed by the Federal Works Agency studied the collapse of the bridge. It included engineers Othmar Ammann and Theodore von Kármán. Without drawing any definitive conclusions, the commission explored three possible failure causes:

• Aerodynamic instability by self-induced vibrations in the structure
• Eddy formations that might be periodic in nature
• Random effects of turbulence, that is the random fluctuations in velocity of the wind.

Simplistic representation of the collapse of the Tacoma Narrows Bridge

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.[27] This bridge was the first of its type to employ plate girders (pairs of deep I-beams) to support the roadbed.[27] 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.[28] 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.[29] 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 40 miles per hour (64 km/h). 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.

Fluttering is a physical phenomenon in which several degrees of freedom of a structure become coupled in an unstable oscillation driven by the wind. Here, unstable means that the forces and effects that cause the oscillation are not checked by forces and effects that limit the oscillation, so it does not self-limit but grows without bound. Eventually, the amplitude of the motion produced by the fluttering increased beyond the strength of a vital part, in this case the suspender cables. As several cables failed, the weight of the deck transferred to the adjacent cables, which became overloaded and broke in turn until almost all of the central deck fell into the water below the span.

Vortex shedding and Kármán vortex street behind a circular cylinder. The first hypothesis of failure of the Tacoma Narrows Bridge was resonance (due to the Kármán vortex street).[30] This is because it was thought that the Kármán vortex street frequency (the so-called Strouhal frequency) was the same as the torsional natural vibration frequency. This was found to be incorrect. The actual failure was due to aeroelastic flutter.[3]

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)[3] reported that, in fact, many physics textbooks (for example Resnick et al.[31] and Tipler et al.[32]) 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

${\displaystyle m{\ddot {x}}(t)+c{\dot {x}}(t)+kx(t)=F\cos(\omega t)}$

(eq. 1)

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

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 produce (or "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 Kármán vortex street or von Kármán vortex street. The body will in consequence try to move toward the low-pressure zone, in an oscillating movement called 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 ${\displaystyle f_{s}}$, and is given by

${\displaystyle {\frac {f_{s}D}{U}}=S}$

(eq. 2)

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

It was thought that the Strouhal frequency was close enough to one of the natural vibration frequencies of the bridge, i.e., ${\displaystyle 2\pi f_{s}=\omega }$, 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 42 miles per hour (68 km/h) and the frequency of the destructive mode was 12 cycles/minute (0.2 Hz).[33] 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 linear behaviour, since the exciting force amplitude is a nonlinear force of the structural response.[34]

Billah and Scanlan[34] state that Lee Edson in his biography of Theodore von Kármán[35] is a source of misinformation: "The culprit in the Tacoma disaster was the Karman vortex street."

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

It is very improbable that the resonance with alternating vortices plays an important role in the oscillations of suspension bridges. First, it was found that there is no sharp correlation between wind velocity and oscillation frequency such as is required in case of resonance with vortices whose frequency depends on the wind velocity.[36]

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

Subsequent authors have rejected the resonance explanation, and their perspective is gradually spreading to the physics community. The user's guide for the current American Association of Physics Teachers (AAPT) DVD states the bridge collapse "was not a case of resonance." Bernard Feldman likewise concluded in a 2003 article for the Physics Teacher that for the torsional oscillation mode, there was "no resonance behavior in the amplitude as a function of the wind velocity." An important source for both the AAPT user's guide and for Feldman was a 1991 American Journal of Physics article by K. Yusuf Billah and Robert Scanlan. According to the two engineers, the failure of the bridge was related to a wind-driven amplification of the torsional oscillation that, unlike a resonance, increases monotonically with increasing wind speed. The fluid dynamics behind that amplification is complicated, but one key element, as described by physicists Daniel Green and William Unruh, is the creation of large-scale vortices above and below the roadway, or deck, of the bridge. Nowadays, bridges are constructed to be rigid and to have mechanisms that damp oscillations. Sometimes they include a slot in the middle of the deck to alleviate pressure differences above and below the road.[37]

To some degree the debate is due to the lack of a commonly accepted precise definition of resonance. Billah and Scanlan[3] 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."

Remains of the collapsed bridge in May 2008

The underwater remains of the highway deck of the old suspension bridge act as a large artificial reef, and these are listed on the National Register of Historic Places with reference number 92001068.[40][41]

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

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

The Tacoma Narrows bridge failure has given us invaluable information ... It has shown [that] every new structure [that] projects into new fields of magnitude involves new problems for the solution of which neither theory nor practical experience furnish an adequate guide. It is then that we must rely largely on judgment and if, as a result, errors, or failures occur, we must accept them as a price for human progress.[42]

Following the incident, engineers took extra caution to incorporate aerodynamics into their designs, and wind tunnel testing of designs was eventually made mandatory.[43]

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 airfoil shape such as the Severn Bridge, which gave the necessary stiffness together with reduced torsional forces.