# Tacoma Narrows Bridge Collapse, the Physics

It seems that nearly everyone has seen the short home movie of the vibration and collapse of the nearly new Tacoma Narrows Bridge in 1940, usually in a high school science class. It is a popular video, as kids really pay attention for a minute! But few teachers are then able to give any description as to what happened or why. It is usually used as an introduction to the subject of vibration in machinery or in Church pipe organs or in a pop bottle with some liquid in it.

This presentation is meant as a relatively non-technical discussion of what happened, and why it hasn't happened again, and other applications of the concepts involved.

This last statement seems to have been altered recently! In May 2010, a videotape of a brand-new bridge built in Russia shows the same amazing flexing of the road surfaces. It is also a very narrow bridge, suggesting that maybe we don't learn very well!

The Tacoma Narrows Bridge happened to be made rather narrow for how long it was. At the time (1940) no one realized there was any disadvantage in that! That disaster caused extensive research to be done on vibrations, resonances and oscillations, and their relationships with physical forces. Specifically, a relationship was found between the speed of a constant wind and various "natural frequencies" of structures or flows. That analysis involved a new parameter called the Strouhal number.

Let's consider the situation of that day. There was a relatively constant strong wind flowing crossways to the bridge, at around 40 mph. That is also around 59 ft/sec. The relationship mentioned above involves a Strouhal number. I'm not sure if anyone has ever discovered why, but the Strouhal number is consistently around 0.2 for many situations, and so that is the normal "design value". Keep in mind that no one knew about the Strouhal number in 1940!

The constant windspeed is multiplied by the Strouhal number (59 * 0.2) to get 11.8 ft/sec, a resonance speed that must be avoided.

From the movie, it appears that the rather narrow, two-lane bridge was around 25 feet wide. Across and back is therefore 50 feet. And it was oscillating at maybe once every 4 seconds, or 0.25/second. We can multiply these two values to get 12.5 ft/second as a transverse speed of the resonance of the bridge.

These values are also actually dependent on the length of the bridge (as to the natural frequency of the structure lengthwise and in torsion). The fact that the bridge was so very narrow allowed it to be very flexible in being able to resonate (twist) at the natural frequencies of the structure of the bridge.

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The fact that these two values are so similar (11.8 and 12.5) means that the bridge structure was very susceptible to developing vibrations which would continuously increase in strength (another subject, somewhat more complex, called "Forced Vibration"). So it was really just a matter of time before that bridge developed large vibrations. It only remained for a constant velocity of crosswind to occur at the specific dangerous velocity.

They had actually known about this earlier! Even as the bridge was being built, it would sometimes be twisting fairly seriously. But then the windspeed would change, or the direction, and the effect would stop. On that fateful day in 1940, the wind stayed at the unfortunate speed for many hours, and the bridge kept increasing its twisting more and more.

The bridge had actually been open to traffic for several months. There were newspaper reports of some drivers saying that it was very unsettling when the bridge was oscillating in a different mode (like a guitar string) where resonant waves were not twisting but traveling lengthwise back and forth along the bridge, because sometimes the "hills" in the bridge were sometimes so great that the car ahead would disappear behind such a hill! The bridge may have soon failed due to that other mode of longitudinal resonance, because the concrete of the roadway was too brittle and rigid to tolerate such flexing for very long, and the cables and towers might have also developed structural damage from repeated flexing like that. However, those failures never had time to happen, because there was that day of constant wind where the twisting mode of resonance overcame the strength of the bridge.

As to the twisting mode of resonance, one interesting aspect is that as the resonant condition continued, the slanted position of the bridge roadway also enabled the constant wind to have enhanced Bernoulli effects regarding increasing the twisting effect. So the bridge design was bad for quite a few different reasons, nearly all closely related to the very narrow width for such a long bridge.

If you are familiar with heavy metal rock and roll music, guitar players sometimes hit a string and then take the guitar over right in front of the speaker of the amplifier. This is a slightly different but closely related effect, where in this case the air motion in front of the moving speaker cones are in exact constant relationship with the motion of the guitar string that is actually creating them. Depending on where you hold the guitar and what angle and distance it is, the result can be the sound getting louder (as the string vibrates more violently) or stays at a constant loudness or fades away. Resonance is an interesting subject, that affects many fields!)

## Quiz!

ALL modern long suspension bridges are at least 6 lanes wide. Say one is 90 feet wide. Is that safe, regarding this problem?

Answer: Yes, it is! Calculations like the above indicate that the 40 mph wind still gives the 11.8 ft/sec resonant speed. But now we have a bridge that is 90 feet wide, or 180 feet across and back. This means the bridge would have to oscillate at a REALLY slow speed, once every 16 seconds, which is far slower than any real bridge structure would vibrate at.

If the bridge deck has a natural twisting frequency similar to the Tacoma Narrows bridge, around once per four seconds, (where the bridge deck natural resonance speed would be 180/4 or 45 ft/sec) the wind would have to stay constant at around 160 mph (severe hurricane strength) (235 ft/sec) for many hours to cause a resonance speed of 235 * 0.2 or 47 ft/sec, to inspire the bridge to build up twisting oscillations over many hours. If ANY man-made structure has to try to withstand many hours of constant 160 mph winds, it figures to have problems!

## Other Related Situations

There have been many really tall smokestacks that have suddenly collapsed, in a constant moderate speed wind, without any apparent reason. The diameter of the smokestack defines a specific resonance situation that needs to be avoided. The height of the smokestack defines another (lower) resonant frequency to avoid. In some cases, the designers did not realize that they had made that smokestack of a height that had a natural resonant frequency that was similar, such that the body of the (concrete or brick) smokestack would develop initially small oscillations, which rapidly increased, essentially shaking the smokestack apart! Once the resonant vibration developed to a certain level, the rigidity of the masonry joints of the smokestack are overcome, and they shatter.

There is actually another, somewhat unrelated, Physics concept involved there! When such smokestacks are destroyed, they virtually always fall over in the same way, breaking apart about in the middle. Part of that is because the mortar joints in that area had taken the most abuse by the vibration, but part of it is for a different reason. The smokestack intially falls as a single unit, where its center of gravity attempts to accelerate downward at a rate due to gravity. However, this causes the very top end to have to have an acceleration about TWICE that of gravity! (just to keep up with the acceleration of the falling!) This effect puts enormous new stresses in the mortar joints of the smokestack, and the top portion (usually about half) of the smokestack appears to break away upward. The result is that the lower half of the smokestack generally falls to the ground simultaneously, as one might expect of a solid object falling over, but that the upper half gets twisted back somewhat upright. Depending on how strong it is, it either then hits and falls over as the lower half just did, or the shock of the impact of the bottom end with the ground can cause this half to break apart (for a different reason than before).

So the next time you see any movie of a smokestack starting to fall over, you now know what is about to happen, and why. (The butler did it!)

This presentation was first placed on the Internet in March 2006.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago