“Is it not true that the very conditions which give strength also conform to the hidden rules of harmony ? … Now to what phenomenon did I have to 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.”
-Eiffel in an interview in Le Temps of February 14th 1887
Gustave Eiffel was proud of the good-looking Tower he created whose shape resulted from mathematical calculation: at any height on the Tower, the moment of the weight of the higher part of the Tower, up to the top, is equal to the moment of the strongest wind on this same part. By writing the differential equation of this equilibrium, we can find the harmonious equation that describes the shape of the Tower.
Let A be a point on the edge of the tower, and x be the distance between the top of the Tower and A. Let P(x) be the weight of the part of the Tower above A up to the top of the tower. If f(x) is half the width of the Tower at A, then the moment of the weight of the Tower relative to A = P(x)f(x).
Consider a piece of the Eiffel Tower at a distance t from the top of the Tower, with its thickness equal to dt. Then, viewed from the top, this looks like a square with width 2f(t).
The forces on this piece are:
- Weight dP(t): proportional to its volume – dP(t) = 4kf(t)^2dt
- horizontal wind dV(t): proportional to the part of the surface in direction of the wind – dV(t) = 2Kf(t)dt
The absolute value of the moment of dP(t) relative to A = dP(t)f(x) and the absolute value of the moment of dV(t) relative to A = dV(t)(x-t). These should be equal so
Let a = 2k/K. Then f, the function which give the width of the Eiffel Tower as a function of the distance from the top, is a solution to
Hence giving the equation for the shape of the Eiffel Tower.