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Workers check some of the signal amplifiers on the Eiffel Tower's structuralmonitoring-system. |
The monitoring system sends back data in real time about daily, seasonal and annual deformation cycles, and behavior under wind stress. It also checks the effects of fatigue and the risk of frost formation, among other things.
The system, devised by the French company Osmos Group (www.osmos-group.com), uses twelve 32.8-ft strands installed on each of the main rafters above the Tower's second floor and 63 additional optical strands installed on the four pillars and large collar beams on the ground level and first floor. They are anchored into place in the center of rivets and protected by copper tubes.
A monitoring station captures, transmits, and records the data received from sensors installed in the structure. Data can also be exported for further processing and managed directly via an Internet browser.
Similar Osmos systems are monitoring 475 structural sites that include the Manhattan Bridge, the Channel Tunnel between France and the U.K., and the Nagoya power plant in Japan.
Eiffel equations are rediscovered
Debate has simmered among engineers over just why Gustave Eiffel designed his famous tower the way he did. Now it appears the matter has been put to rest, thanks in part to an analysis by Michigan Technological University mathematician Iosif Pinelis.
Pinelis first became intrigued by the problem when a colleague presented two competing theories, each purporting to explain the Eiffel Tower's elegant design.
One, by Christophe Chouard, argued that Eiffel engineered his tower so its weight would counterbalance the force of the wind. According to the other theory, the wind pressure is counterbalanced by tension between the elements of the tower itself.
Chouard had developed a nonlinear integral equation to support his theory, but "Mathematicians could only find one solution, a parabola, of the infinitely many solutions that Chouard's equation must have," Pinelis said. Of course, the Eiffel Tower's profile doesn't look anything like a parabola.
Pinelis soon found an answer confirming a conjecture by Patrick Weidman, an associate professor of mechanical engineering at the University of Colorado at Boulder. Weidman figured that Chouard's theory was wrong. It turns out that all existing solutions to Chouard's equation must either be parabolalike or explode to infinity at the top of the tower.
"The Eiffel Tower does not explode to infinity at the top, and its profile curves inward rather than outward," Pinelis notes. "That pretty much rules out Chouard's equation."
Weidman then found an 1885 memoir delivered by Eiffel to the French Civil Engineering Society affirming he had indeed planned to counterbalance wind pressure with tension between the construction elements. Using that information, Weidman and Pinelis developed a nonlinear integral-differential equation whose solutions yielded the true shape of the Eiffel Tower. That shape is exponential.
The work by Weidman and Pinelis, "Model Equations for the Eiffel Tower Profile: Historical Perspective and New Results," has appeared in the French journal Comptes Rendus Mecanique. An abstract may be viewed at www.elsevier.fr/html/index.cfm?act=abstract&cle=49158.
"The funny thing for me was that you didn't have to go into the historical investigation to disprove a wrong theory," Pinelis says. "The math confirms the logic behind the design."