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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

tower

Material

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.

Wind

considerations

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.[22]

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.[23][24]

The tower sways 6–7 cm (2–3 in) in the wind.[25]

**23^**

Translated from the French newspaper *Le Temps* of 14 February

1887. Extrait de la réponse d'Eiffel**
**

**24 ^** Elegant Shape Of Eiffel Tower

Solved Mathematically By University Of Colorado Professo**r**

**25 ^** The Virginia Engineer: Correct

Theory Explaining The Eiffel Tower’s Design Revealed

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!!

Attachment | Size |
---|---|

Eiffel_Equation_Weidman_2004.pdf | 9.4 MB |

Eiffel_Tower_measurements.xls | 106 KB |

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## Comments

## some of the reports of the new findings

Correct Theory Explaining The Eiffel Tower’s Design Revealed

January 31, 2005

Iosif Pinelis, a professor of mathematical sciences, first became

intrigued by the problem in 2002, when Patrick Weidman, an associate

professor of mechanical engineering at the University of Colorado at

Boulder, visited Michigan Tech (MTU). Prof. Weidman presented two

competing mathematical theories, each purporting to explain the Eiffel

Tower’s elegant design.

One, by Christophe Chouard, argued that Eiffel engineered his tower

so that 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, Prof. Pinelis said.

Chouard had developed a nonlinear integral equation to support his

theory, but finding its solutions was proving difficult. “Weidman and

the mathematicians whom he had consulted could only find one solution, a

parabola, of the infinitely many solutions that Chouard’s equation must

have,” Prof. Pinelis said. As anyone who has survived high-school

geometry can quickly testify, the Eiffel Tower’s profile doesn’t look

anything like a parabola. Prof. Weidman asked MTU mathematicians if they

could come up with any other solutions.

Prof. Pinelis went back to his office and soon found an answer

confirming Prof. Weidman’s conjecture that Chouard’s theory was wrong.

It turns out that all existing solutions to Chouard’s equation must

either be parabola-like or explode to infinity at the top of tower.

“The Eiffel Tower does not explode to infinity at the top, and its

profile curves inward rather than outward,” Prof. Pinelis notes. “That

pretty much rules out Chouard’s equation.”

Prof. Weidman then went to the historical record, and found an 1885

memoire delivered by Eiffel to the French Civil Engineering Society

affirming that Eiffel had indeed planned to counterbalance wind pressure

with tension between the construction elements.

Using that information, Prof. Weidman and developed an nonlinear

integral-differential equation whose solutions yielded the true shape of

the Eiffel Tower. That shape is exponential.

The work by Prof. Weidman and Prof. Pinelis, “Model Equations for

the Eiffel Tower Profile: Historical Perspective and New Results,” has

appeared in the French journal Comptes Rendus Mecanique, published by

Elsevier and the French Academy of Sciences.

“The funny thing for me was that you didn’t have to go into the

historical investigation to disprove a wrong theory,” Prof. Pinelis

says. “The math confirms the logic behind the design. For me, it was

more fun to go to the math.”

Permanent LinkMichele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## Some videos of interest, but distracting from the problem

Stages of construction of the Eiffel Tower

Eiffel Tower Construction view: girders at the first story

Panoramic view during ascension of the Eiffel Tower by the Lumière brothers, 1898

Franz Reichelt's preparations and fall from the Eiffel

Tower.

Lightning strikes the Eiffel Tower on June 3, 1902, at 9:20 P.M.

Adolf Hitler with the Eiffel Tower in the background

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## Let's return to the problem: the equation for the shape!

Elegant Shape Of Eiffel Tower Solved

Mathematically By University Of Colorado Professor

ScienceDaily (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

weight and reduces the surface area exposed to the wind."

Based on the information, Weidman derived a new equation for the

skyline profile -- one that "embraces Eiffel's deep concern for the

effects of wind-loading on the tower," he said. Weidman found an exact

solution of the equation in the form of an exponential function that

closely matches the shape of the tower's upper half.

The tower is composed of four arched, wrought-iron legs tapering

inward to form a single column that rises to 300 meters, or 986 feet.

The top level was built with a large room that Eiffel used for

meteorological studies, capped by a spiral staircase and a television

antenna that reaches to 1,052 feet today.

Plotting the actual shape of the tower reveals two separate

exponential sections that are hooked together, he said. Since Eiffel did

not seem confident in estimating the wind torque on the tower, he

"overdesigned" the bottom section, beefing it up for safety reasons.

"The structural factor of safety is responsible for the second

exponential equation describing the lower half of the tower," Weidman

said.

Weidman and Pinelis presented their findings in a paper titled "Model

Equations for the Eiffel Tower Profile: Historical Perspective and New

Results."

The paper appeared in the July 2004 issue of the journal, "Comptes

Rendus Mecanique," published by Elsevier and the French Academy of

Sciences. In addition, the English translation of Eiffel's 1885

communication to the Society of French Engineers by Weidman and Roland

recently was accepted for publication in the Architectural Research

Quarterly published in Great Britain.

"While the events of the French Revolution are captured by Charles

Dickens in his poignant novel, "A Tale of Two Cities," the centennial of

the French Revolution is commemorated by Eiffel's graceful tower, the

skyline profile of which is "A Tail of Two Exponentials," Weidman and

Pinelis wrote in their 2004 paper.

The Eiffel tower, notes Weidman, "is a structural form molded by the

wind." This was Eiffel's point more than a century ago, when he wrote

about the four stout legs supporting the legendary tower: "Before they

meet at such an impressive height, the uprights appear to spring out of

the ground, moulded in a way by the action of the wind itself."

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## And here is Chouard Eiffel's equation!

http://www.chouard-eiffel-equation.com/

ntroduction:

Gustave Eiffel was proud of his good-loking Tower whose shape resulted

from

mathematical

calculation, as he

said.

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.

Writing the differential equation of this equilibrium allows us to

find

the "harmonious equation"

that describes the shape of the Tower.

Writing the equation:

Let A be a point on the edge of the Tower. Let 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.

Let f(x) be the half-width of the Tower at A. The moment of

the weight of the Tower relative to point A is equal to P(x)·f(x).

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## And the resulting equation is

The function

, which gives the width of the Eiffel Tower as a function

of the distance from top, is a solution of the following equation:

How can function f(x) be written as a combination of usual

functions

?

If you can solve this equation, please E-mail me the solution: christophe@chouard.com

This has resulted in many replies including those of the double exponential solution explained above

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## Some more impressive images of the Tower and its shape..

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## I received a nice email from Christophe Chouard

I've received a number of files from Christophe

Chouardon the Eiffel Tower Equation, and also an Excel file where all

dimensions and angles of the various sections of the Tower are listed. Anyone interested can write to me.

Christophe

Chouard also said that my students can feel free to make any use you want

of the URL of his site:

http://www.chouard-eiffel-equation.com/

He also said "By

the way, I am just an engineer by training who works in finance (!). I

fell in love with the Tower 15 years ago and I have used my souvenirs

of math in order to write the equation, but then I failed to solve it

since my

knowledge in math is limited. I am just glad that this challenge has

triggered some interest from real mathematicians around the globe,

thank-you internet."

So my students and all Imechanicians can have fun if they want

Thanks to Christophe

Chouard!

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal

## I've added some files to the main post which go into details

I've added the actual PDF and excel files which you need to gather all infos in the main post

http://imechanica.org/node/8117

I've noticed a certain similarity with equation of the classical problem of a gear loaded on its tip (usually solved with beam theory, leading to the classical modulus equation by Lewis). But I have to investigate further possible analogies, also because I fail to remember that gear teeth have exponential shape. I will think about it, but obviously the analogy Eiffel tower profile -- gear tooth profile is limited by the fact that gears are usually obtained by cutting and so usually form enveloping curves...

The module system

Countries which have adopted the metric

system generally use the module system. As a result, the term

module is usually understood to mean the pitch diameter in millimeters

divided by the number of teeth. When the module is based upon inch

measurements, it is known as the

English moduleto avoidconfusion with the metric module. Module is a direct dimension, whereas

diametral pitch is an inverse dimension (like "threads per inch"). Thus,

if the pitch diameter of a gear is 40 mm and the number of teeth 20,

the module is 2, which means that there are 2 mm of pitch diameter for

each tooth.[18]

[edit] Manufacture

Gear Cutting simulation (length 1m35s) faster, high bitrate version.

This section requires expansion.

Gear are most commonly produced via hobbing,

but they are also shaped, broached, cast,

and in the case of plastic gears, injection molded. For metal gears the teeth are usually heat treated to make them hard and more wear resistant while leaving the

core soft and tough.

For large gears that are prone to warp a quench

press is used.

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## The analogy with Lewis formulae however appears

Bending Stresses - The Lewis FormulaAlthough this was published in

1893, it is still very widely used for assessing

bending stresses when designing gears. The method involves moving the

tangential force and

applying it to the tooth tip and assuming the load is uniformly

distributed accross the

tooth width with the tooth acting as a simple cantilever of constant

rectangular cross

section, the beam depth being put equal to the thickness of the tooth

root (t) and

the beam width being put equal to the tooth, or gear, width (bw).

The

section modulus is I/c = bwt2/6 so the bending

stress is given by:

sigmabending

= M/(I/c) = 6WtL/(bwt2) eqn.1.

Assuming that the

maximum bending stress is at point 'a'.

By

similar triangles:

or eqn.2

Rearranging eqn.1 gives:

Substitute the

value for 'x' from eqn.2 and multiply

the numerator and denominator by the circular pitch, 'p' gives:

let y=2x/3p then

This

is the original Lewis equation and 'y' is called the

Lewis form factor which may be determined graphically or by computation.

Engineers often now work with the 'diametral pitch', 'P',

and or the 'module', 'm', which is 1/diametral pitch

= 1/P

Then where

Y = 2xP/3

Written in terms of the module:

sigmabending = Wt/(bwmY)

The Lewis form

factor considers only static loading, it is dimensionless,

independent of tooth size and is a only a function of tooth shape. It

does not take into

account the stress concentration that exists in the tooth fillet.

(from http://www.tech.plym.ac.uk/sme/desnotes/gears/desgear1.htm)

This is all interesting... but needs thinking... any suggestions from imechanicians?

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878