in Tacoma
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 $5.2 million 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 .
Federal Works Agency Commission A commission formed by the
Federal Works Agency studied the collapse of the bridge. The board of engineers responsible for the report were
Othmar Ammann,
Theodore von Kármán, and
Glenn B. Woodruff. 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 • Random effects of turbulence, that is the random fluctuations in velocity of the wind.
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. 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
transverse 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.
Resonance (due to Von Kármán vortex street) hypothesis 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
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. 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) 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 {{Numbered block|:|m\ddot{x}(t) + c\dot{x}(t) + kx(t) = F \cos (\omega t) | }} 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 \omega_r = \sqrt{k/m}, i.e., \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 (). 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 f_s, and is given by {{Numbered block|:|\frac{f_s D}{U} = S |}} 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., 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 Farquharson, 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 linear behavior, since the exciting force amplitude is a nonlinear force of the structural response.
Resonance vs. non-resonance explanations Billah and Scanlan 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 ==