Bridges
Strength and Design
Strength of Materials
Many mathematicians before Galileo had dealt with the problem of statics – how forces are transmitted by structural members. Galileo proposed a new science, the study of the strength of materials, that considered how the size and shape of structural members affects their ability to carry and transmit loads. He discovered that as the length of a beam increases, its strength decreases, unless you increase the thickness and breadth at an even greater rate. You cannot, therefore, simply double or triple the dimensions of a beam, and expect it to carry double or triple the load.
This led Galileo to recognize what we now call the scaling problem – there are limits to how big nature can make a tree, or an animal, for beyond a certain limit, the branches of the tree or the limbs of the animal, will break under their own weight.
The illustration of a cantilever beam demonstrates Galileo’s discovery that the breaking force on a beam increases as the square of its length.
Designing an Arch
It was Robert Hooke who discovered that the line of an arch, for supporting any weight assigned, should be the inversion of the shape of a catenary, or hanging chain, which is bearing that weight. He apparently announced that he had made the discovery to the Royal Society of London around 1671, but he did not provide any details until 1675, and then the details were encrypted. In an appendix to his Description of Helioscopes, he stated that he had found "a true mathematical and mechanical form of all manner of Arches for Building," and the solution was:
"abcccddeeeeeefggiiiiiiii-illmmmmnnnnnooprrsssttttttuuuuuuuux."
Unlike Hooke’s law of the spring, which he announced with a similar anagram, Hooke did not provide a translation in his lifetime, but it was provided by his executor in 1705: "Ut pendet continuum flexile, sic stabit contiguum rigidum inversum--As hangs a flexible cable, so inverted, stand the touching pieces of an arch."
We display the page showing Hooke’s announcement; the "law of the arch" anagram is in paragraph 2. Paragraph 3, which follows immediately, contains the much more succinct anagram for Hookes’ law of springs: "ceiiinossssttuu," or "ut tensio, sic vis as the deflection, so is the force."
Dresden Bridge
In 1731, a bridge heralded as the largest structure of its time was completed over the Elbe at Dresden. To celebrate the event, Schramm wrote this book about it and other great bridges of the world. One arch of the completed Dresden Bridge is shown framing this allegorical engraving on bridge building. Surrounding the shield in the center of the arch are all of the hand tools used in bridge building, including mason's trowels, compasses, drills, and various saws, axes, and hammers. A bridge under construction is seen through the arch, with a floating pile driver in use to drive piles into the firm riverbed. Workmen at various tasks along the riverbank are shaping timbers, cutting stone, and draining water with a water screw.
On the right Minerva, patroness of arts and crafts and a goddess of wisdom and skill, holds a leveling instrument aloft and a drawing board and compass in her lap. Building a bridge successfully, her presence suggests, requires both skill and intelligence, and perhaps even divine assistance.
Pont de Neuilly
Jean-Rodolphe Perronet was the first director of the Ecole des Ponts et Chaussees in Paris, and one of the first great masters of the low-arch bridge. He was the first to recognize that for bridges of equal spans, the intermediate piers carry only vertical loads and can be made quite thin. His bridges were consequently quite light and elegant. The Pont de Neuilly, built over the Seine by Perronet, has 5 arches of 128 feet span each, yet the piers are only 13 feet thick. So that the piers did not thrust sideways during construction, all five arches had to be constructed simultaneously. King Louis XV was impressed by all the centering that had been erected to support the arches and wanted to see the decentering process, so Perronet arranged for all the centering to collapse simultaneously into the Seine before the eyes of the King and the rest of Parisian society. It was quite an occasion.
The engraving shows the Pont de Neuilly just after the centering had been released and fallen into the river.