Jan. 7, 2005 An American engineer has produced a mathematical model explaining the elegant shape of the Eiffel Tower that was derived from French engineer Gustave Eiffel's writings regarding his own fears about the effects of wind on such a structure.
University of Colorado at Boulder Associate Professor Patrick Weidman said Eiffel, one of the premier structural engineers in history, was determined to build the world's first tower reaching 300 meters, the nearest metric equivalent to 1,000 feet, into the sky. The tower was designed to be the centerpiece of the World's Exposition in Paris, marking the centennial of the French Revolution.
But such a tower, never having been successfully erected, raised a chronic concern of Eiffel that he expressed frequently in his communications.
"Eiffel was worried about the wind throughout his building career," said Weidman of the CU-Boulder mechanical engineering department. "Although he was astoundingly bright, he was forced to rely on practical experience rather than mathematical calculations to estimate the effects of wind forces on structures."
Weidman said the Eiffel Tower was not designed according to a single, overarching mathematical formula. Instead, Eiffel's engineers used graphical results to calculate the strength needed to support its tremendous weight, as well as empirical evidence to account for the effects of wind. "He built it section by section, and did not have an equation for its description," said Weidman.
But the spectacular tower, completed in 1889 and which remains one of the most romantic and recognizable structures in the world, has long been believed to be explainable using a mathematical equation, albeit a very complex one.
Weidman began researching the problem when he received a second edition copy of the textbook, "Advanced Engineering Mathematics," in 2001. The book's cover contains photographs of various stages of the Eiffel Tower's construction, and the book's preface contains a non-nonlinear integral equation -- a formula with a number of possible solutions -- for the tower's shape.
The equation was created by French Eiffel Tower aficionado Christophe Chouard, who posted it on his Web site and challenged engineers and mathematicians worldwide to find its solution, said Weidman. In terms of known mathematical functions, Weidman found one solution -- a downward facing parabola, but it has the wrong curvature for the legendary structure.
After giving a talk at Michigan Technological University in 2003, Weidman was introduced to Professor Iosif Pinelis, an expert in mathematical analysis, who offered his help in understanding the underlying features of the integral equation. Calculations by Pinelis showed that all existing solutions to Chouard's equation must be either parabola-like -- which the Eiffel Tower is not -- or "explode to infinity" at the top of the tower.
Weidman, who said he became obsessed with the problem, began to read more about the life of Eiffel and his construction efforts. He contacted Henri Loyette, author of a 1985 book on the life of Eiffel and now the curator of the Louvre in Paris, who suggested Weidman search the historical archives.
Weidman tracked down a copy of a communication from Eiffel to the French Society of Civil Engineers dated March 30, 1885. Written in French, the document affirmed that Eiffel planned to counterbalance wind pressure with the tension between the tower's construction elements.
After translating the 26-page document with the help of professional translator Claudette Roland, Weidman finally deduced the basis for tower construction. A key factor for Eiffel was determining where the tangents to the skyline profile -- which run from given horizontal sections of the tower -- intersect the resulting wind forces acting above those sections.
"Eiffel discovered this form of construction produces no load in the diagonal truss elements commonly used to counteract the bending moment, or torque, of the wind, and hence those truss members could be eliminated," Weidman said. "This allows for a reduction of the tower we