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October 1, 2001

McKenna Uses Math to Solve Mystery of Bridge Collapse

Joe McKenna has a flair for mysteries. Recently, he cracked a 60-year-old case: the collapse of the Tacoma Narrows Bridge. Now the professor of mathematics is investigating London's undulating Millennium Bridge.

Back in 1940, Washington State's Tacoma Narrows Bridge swayed and twisted. After four months of operation, a windstorm caused it to twist violently and repeatedly and then collapse. Last year, the much-heralded Millennium Bridge was closed after only three days of operation, owing to unexpected swaying.

McKenna's sleuthing tools are known as nonlinear partial differential equations and he fills his chalkboard with mathematical symbols. Yet despite the esoteric nature of these tools, when he explains bridge dynamics he resorts to everyday objects, making analogies to a rubber band or using his tie as a prop.

Although the design and analysis of structures like bridges typically fall within the purview of engineering, McKenna's answer to the Tacoma Narrows conundrum came out of a new mathematical field called nonlinear theory.

A spring is an example of a linear system: among other properties, the spring's resistance to compression is comparable to its resistance to expansion. It turns out, however, that not all springs are linear.

"Until 20 years ago," says McKenna, "almost all science was linear theory." There is no way to write down an exact solution to a nonlinear problem. But then computers came along and changed the rules of the game by allowing approximate solutions to nonlinear problems. "Suddenly," says McKenna, "there was an open front of problems no one had looked at."

To solve problems like the Tacoma Narrows Bridge, McKenna combined nonlinear theory with his established expertise in differential equations. Differential equations, he explains, have to do with rate of change. "If you're interested in the rate at which things change," he says, "then you're interested in differential equations, which describe all of life." With differential equations, McKenna can relate the position of the bridge to how fast it is swaying and how fast the swaying is speeding up, for example.

The Tacoma Narrows Bridge collapsed because it had nonlinear springs. The cables that supported the suspension bridge acted like springs, but their behavior could not be understood with linear theory. Cables offer resistance if you stretch them, but simply go limp if you bring the ends closer together.

Thus, the bridge was nonlinear and responded violently to a small stimulus, in this case the wind. McKenna made a mathematical model which correctly described the observed behavior of the bridge. Case closed.

That is, case closed again. Others before now have been convinced they had cracked the mystery.

High school physics teachers typically explain the Tacoma Narrows Bridge collapse as an incident of a phenomenon called resonance. Says McKenna: "This explanation has enormous appeal in the mathematical and scientific community. It is plausible, remarkably easy to understand, and makes a nice example." But resonance, he says, does not explain the bridge's collapse.

The engineering community has a more sophisticated explanation of the collapse. Proposed soon after the bridge collapsed, the theory, a linear theory, has to do with an aerodynamic phenomenon called vortex shedding.

Erling Murtha-Smith, professor and head of civil and environmental engineering, says engineers generally hold to the vortex shedding theory, but calls McKenna's work "state of the art." Murtha-Smith says the design of bridges was changed to make them more aerodynamic and eliminate vortex shedding. This eliminated the problem but, he says, doesn't mean vortex shedding was the primary cause of the Tacoma Narrows Bridge collapse.

McKenna is not convinced by the vortex shedding theory. He is familiar with the idea because of his work in fluid dynamics with the Air Force during the 1980s, but says it fails to explain the bridge's violent twisting.

He adds that some prominent engineers - including a man considered the guru of structure failure, the late Mario Salvadori - have been pleased with his work, and that some of his many talks on the Tacoma Narrows Bridge have been well attended by engineers. He recalls Salvadori's remarks: "Engineers are by nature forward-lookin g. When they eliminate a problem, they don't worry too much about the cause."

McKenna got his start as an investigator because science was where his aptitude lay and, he says, because "science was the big thing in the '60s." After he earned his Ph.D. from the University of Michigan, he gained experience in a variety of areas. He was an actuary for a short while, worked on claiming oil from Rocky Mountain shale with the Department of Energy, and investigated environmental problems and porous flow with the Army Corps of Engineers.

McKenna says there is a wealth of new problems to explore in mathematics, especially in nonlinear theory. He is currently trying to explain why London's new Millennium Bridge is swaying. This problem, he explains, is more complex than the Tacoma Narrows Bridge. The older bridge was swaying from side to side and twisting. The Millennium Bridge, still standing, is swaying and twisting as well as undulating up and down.

The design and opening of the Millennium Bridge was accompanied by great fanfare. Yet the design of the bridge, according to McKenna, incorporated only linear analysis.

Gumshoe McKenna polishes his nonlinear partial differential equations and opens another case.

Brent C. Evans

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