Control Engineering

Did oscillations cause the I-35 bridge collapse?
August 20, 2007

For those familiar with the story, every bridge collapse evokes the infamous film clip showing the Tacoma Narrows bridge, which collapsed in 1940 after only four months in service. That disaster was clearly caused by complex oscillation that involved resonance modes fed by aerodynamic forces. While the cause of the recent collapse of the I-35 bridge in Minneapolis, MN, has not yet been determined, it clearly is quite different.

Let’s start with the Tacoma Narrows incident. The film clip clearly shows a torsional standing wave with one full wavelength fitting between the piers supporting the center span. The piers pinned the center span’s ends, forcing them to be nodes for the torsional vibration and setting up a resonance condition. Resonance is the first element needed for an oscillator.

The second element is gain. That is, a power source capable of adding energy into the oscillation. In this case, it was a cross wind providing lift when the roadway tilted. The lift equation shows the variables of interest in this case:

L = C v² sin α,

where v is the crosswind speed, α is the angle between the bridge and the (presumed horizontal) wind velocity vector, and C is a constant that accounts for the roadway shape, its width, the density of air, and so forth. This equation applies separately to every cross section of the span.

The main thing to notice is that nothing happens until the wind blows, and the bridge tilts. When both occur, aerodynamic forces push the bridge up or push it down depending on whether the particular cross section presents its underside or top to the wind.

The third element needed for oscillation is positive feedback.

Most people incorrectly think that lift itself would be enough to cause the problem, but that is not true. Lift would try to raise or lower the roadway’s centerline in a transverse oscillation mode, which was observed often during the bridge’s life without serious problems. Lift alone would not couple energy into the torsional mode that caused the bridge to fail. If you watch the film clip carefully, you will see that the bridge’s centerline hardly moves at all.

What coupled energy into the torsional mode was the fact that the center of pressure on a flat plate airfoil is about 25% of the chord (length in the wind direction) behind the leading edge. That assymmetry—which applies to all similar airfoils—creates a moment (torque) that twists the airfoil in such a way as to increase the angle of attack.

I like to call this the “falling-leaf effect.” It is the reason that leaves fall flat, as do plastic toy flying disks, pieces of paper, glass windows popping out of Boston’s John Hancock tower, and all similar extended objects. In the case of the Tacoma Narrows Bridge, the falling-leaf effect coupled energy directly into the torsional resonance.

In the end:

Resonance + Gain + Positive Feedback = Oscillation.

The mechanism for the I-35 Minneapolis bridge collapse was quite different. While the Tacoma Narrows Bridge oscillation let everyone around know the bridge was in trouble, there was no reported portentious movement of the I-35 bridge. It was recorded on video only because a security camera happened to be looking at it. Eyewitnesses and survivors described how suddenly it all happened.

This sudden collapse seems to indicate a single-point failure that triggered the structure’s destruction. The facts that the bridge had been in service since 1967 and was cited as “structurally deficient” in 1990 would lead us to believe the cause related to maintenance issues, rather than a systematic problem, such as the Tacoma Narrows Bridge oscillation.
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Posted by on August 20, 2007 | Comments (0)