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The shape of the Eiffel Tower
I've got interested, since I am going with about 10 of my students to Paris for the Euromech conference I am coorganizing in July, by the design of the Eiffel Tower. Eiffel was a great engineer, and indeed his shape of the tower apparently comes from his ingenious idea of balancing weigth and wind pressure so that there is little need of foundations. WIKIPEDIA has an account of this design idea, and the resulting integro-differential equation.
I added a preprint with permission of the Solution of Eiffel-Chouard equations by Weidman and Pinelis as attachment here, together with actual Excel file of the Eiffel tower profile measurement by Eiffel himself and others.... Have fun!
More infos on the great Engineer Gustave Eiffel for example at http://en.wikipedia.org/wiki/Gustave_Eiffel
Design of the
The metal structure of the Eiffel Tower weighs 7,300 tonnes while the
entire structure, including non-metal components, is approximately
10,000 tonnes. As demonstration of the economy of design, if the 7,300
tonnes of the metal structure were melted down it would fill the 125
metre square base to a depth of only 6 cm (2.36 in), assuming a density
of the metal to be 7.8 tonnes per cubic metre. Depending on the ambient
temperature, the top of the tower may shift away from the sun by up to
18 cm (7.1 in) because of thermal expansion of the metal on the side
facing the sun.
At the time the tower was built many people were shocked by its
daring shape. Eiffel was criticised for the design and accused of trying
to create something artistic, or inartistic according to the viewer,
without regard to engineering. Eiffel and his engineers, however, as
experienced bridge builders, understood the importance of wind forces
and knew that if they were going to build the tallest structure in the
world they had to be certain it would withstand the wind. In an
interview reported in the newspaper Le Temps, Eiffel said:
Now to what phenomenon did I give primary concern in designing the
Tower? It was wind resistance. Well
then! I hold that the curvature of the monument's four outer edges,
which is as mathematical calculation dictated it should be [...] will
give a great impression of strength and beauty, for it will reveal to
the eyes of the observer the boldness of the design as a whole.
The shape of the tower was therefore determined by mathematical
calculation involving wind resistance. Several theories of this
mathematical calculation have been proposed over the years, the most
recent is a nonlinear integral differential equation based on
counterbalancing the wind pressure on any point on the tower with the
tension between the construction elements at that point. That shape is
exponential. A careful plot of the tower curvature however, reveals two
different exponentials, the lower section having a stronger resistance
to wind forces.
The tower sways 6–7 cm (2–3 in) in the wind.
Do you have the original files or PDF? Can you share them with me? Any interest to discuss about this question on imechanica, and/or with my students? Incidentally, my students are invited to comment in particular!!