Bridges have been built since they time of the Greek and Roman empires. Since bridges have been built, they have collapsed each due to different reasons. The Tacoma Narrows Bridge collapse is an fascinating bridge collapse because it did not collapse for a straight forward reason, like carrying to heavy of a load. Many theories have been presented as to why this phenomena occurred and can be explained using physics. To truly understand the collapse of the Tacoma Narrows bridge we first have to understand it's history.
The bridge in 1940 looking west from the Tacoma side
History
The Tacoma Narrows Bridge was built between November 23, 1938 and July 1, 1940. It was originally supposed to cost 11 million dollars to build but the bridge was redesigned to lower costs and ended up costing 7 million dollars. The design was for a suspension bridge because it can carry a large load across a long span. This is because the weight is distributed through several tension cables which are connected into the ground at several anchorage points. The design featured anchors at each end and support in the middle by two raised towers. During construction, after laying the floor, oscillations started occurring.
The design of the bridge in 1940 highlight key features and forces.
Some important information about the completed bridge:
Total Length: 5000 feet
Span Length: 2800 feet
Width (center to center): 39 feet
Height of side girders: 8 feet
Height of tower: 425 feet
The Collapse
The collapse of the Tacoma Narrows Bridge occurred on then morning of November 7, 1940. Early that morning, the winds picked up blowing at 40-45 miles per hour causing traffic on the bridge to be shut down. Measurements and video began to be recorded so that the phenomena could be examined later. The bridge developed larger than usual vertical vibrations. By 9:30a.m. the bridge was vertically vibrating at a frequency of 36 vibrations per minute with the amplitude of the waves being 1.5 feet. The span began to vibrate torsionally (twisting and turning) around 10a.m. that morning with a frequency of 14 vibrations/ minute. The torsional vibrations occurred int two halves, which were out of phase. This is because of the 'law of minimum energy' which in this case causes the torsional vibratiobs to either twist as a whole or as two halves. However, nature prefers the two half torsional vibrations so that is what the bridge took on. The centre line (yellow line in video below) which sperated the traffic acted as the nodual line and appears not to move. The amplitude of the waves climbed to an amazing 28 feet before the collapse. Finally a 600 foot length broke loose and fell into the water.
In the video below you can see the bridge vibrating and twisting and then finally collapsing.
It was determined after the
collapse that the bridge had been weakened by a storm a few nights earlier. As well, it was discovered that the bridge could not resist the static forces and that it was to flexible and light to absorb dynamic forces. These things ultimately led to it's collapse.
Why the Bridge Collapsed?
There is no clear answer as to what caused the bridge to collapse. Extreme weather conditions that day and several days before play a factor in the weakness of the structure, however, they do not completely explain the collapse. There are three main levels at which the bridge collapse can be examined:
The earliest theory is that the collapse of the bridge was a resonant effect. Resonance is the increase in amplitude of oscillation of a system exposed to periodic force which has a frequency that is equal or very close to the natural undamped frequency of a system. The theory was that the wind provided an external frequency periodically that matched the natural frequency of the bridge. This caused large oscillations, the vertical ones causing the damage. The torsional (twisting) oscillation was destructive to the bridge. The frequency can be calculated by using the equation f=1/T(period).
f=1/T
=1/5
=0.2vibrations/second.
The wavelength of the torsional waves was equal to 2800 feet. The velocity can be calculated at using velocity=frequency x wavelength
v=f x w
=(0.20)(2800)
=580ft/s.
This means that that the waves were moving at a speed of 580 feet per second which is strong enough cause wear on the bridge, enough to eventually make it collapse.
The bridges oscillations are often compared to those of a forced harmonic oscillator.
This is the theory that is most often presented however, it is not the most accurate analysis. Resonance cannot fully explain what happened to the bridge because it assumes that the wind gusts would be regular in phase which is extremely unlikely. As well the wind was blowing side to side rather than the up and down motion the oscillations took on. This leaves us with no explination for the oscillating vertical forces.
f=1/T
=1/5
=0.2vibrations/second.
The wavelength of the torsional waves was equal to 2800 feet. The velocity can be calculated at using velocity=frequency x wavelength
v=f x w
=(0.20)(2800)
=580ft/s.
This means that that the waves were moving at a speed of 580 feet per second which is strong enough cause wear on the bridge, enough to eventually make it collapse.
The bridges oscillations are often compared to those of a forced harmonic oscillator.
Animation of the Tacoma Narrows Bridge while it was oscillating.
2.Vortex Shedding:
3.Self-Excitation:
Vortex shedding can account for these oscillating vertical forces. This is the period shredding of air vortices creating a wake which reinforced the structural oscillations due to air at different speeds combining. Vortices will be formed on the back side of an object when a wind that exceeds a minimum speed blows around any object.
Vortices forming on the back side of objects when the wind blows around them (exceeding a minimum speed).
Vortex shedding allows us to understand the fluctuating vertical forces on the bridge even though the wind was blowing horizontally. Blunt bodies can shed periodic vortices in their wake. As the speed of the wind increases, vortices form on alternating sides of the down-wind side of the object. They then break loose and flow downstream as seen in the picture above. When this happens a transverse force is exerted on the object. The vortices generate both low and high pressure regions on the backside of the bridge. These are called Strouhal vortices and shed at a specific rate. This rate can be modelled by the equation: fs=SU/D. Where fs is the frequency of vortex shedding. S is Strouhal constant, a constant given for a body shape. U represents the velocity of air flow. D is a characteristic dimension of the body. The frequency of vortex shedding of the bridge can be determined by calculating it for the cross section depicted below.
Cross section of the Tacoma Narrows Bridge. The wind is blowing at 42mph (61.1ft/sec).
fs=SU/D
=(0.11)(61.1ft/sec)/(8ft)
=0.84 Hz
Therefore the Strouhal frequency was 0.84 Hz. However, it is generally accepted that the frequency that caused the collapse of the bridge was 0.2 Hz. The value calculated is also 2 octaves higher than the torsional mode frequency. It was found that the as the amplitude built up other changes in airflow would produce compensating, self eliminating forces. When these forces were considered the frequency that cause the collapse was calculated to be the 0.2 Hz that was expected. This shows us that the collapse can't be fully explained by the natural vortex shedding of the bridge structure.
The driving force for the oscillations was not just a function of time as with vortex shedding but also based on a function of bridge angle during the oscillation, known as self-excitation. The bridges motion built up destructive amplitudes because the way the wind and the structure interacted. The Strouhal theory could not predict the vortices shedding because if the way the bridge was twisting. The new pattern excited the torsional mode. Aerodynamic forces cannot affect the natural frequencies of a structure like the bridge but they do affect the damping, therefore, increasing amplitudes. It is not a resonant oscillation since the wind does not have a forcing frequency.
Conclusion
The collapse of the Tacoma Narrows bridge was a devastating phenomena that physicist have tried to explain since its collapse. We can see that something that bridge failures is not simple and many possible explanations must be examined.
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