Basket-handle arch
From the early Middle Ages onwards, the segmental arch, an incomplete half-circumference, was used to build vaults that were less than half the height of their opening.
[6] The pointed arch, which emphasizes height by rising above half the opening, did not see use in bridge construction until the Middle Ages.
[7] By the 18th century, the use of basket-handle arches became prevalent, particularly with three centers, as exemplified by the bridges at Vizille, Lavaur, Gignac,[8] Blois (1716–1724), Orléans (1750–1760), Moulins (1756–1764), and Saumur (1756-1770).
[7] Notable architect Jean-Rodolphe Perronet designed bridges with eleven centers during the latter half of the 18th century, including those at Mantes (1757–1765), Nogent (1766–1769), and Neuilly (1766–1774).
The flexibility inherent in these processes allowed for a wide variety of configurations, leading many architects to favor this type of curve over the ellipse, whose contour is rigidly determined by geometric principles.
[17] In the case of an ellipse, the opening of a vault and the height at the center—corresponding to the major and minor axes—result in fixed points along the intrados curve, leaving no room for architectural modification.
Heron of Alexandria, who authored mathematical treatises more than a century before the Common Era, outlined a straightforward method for tracing this arch.
[19] As the basket-handle arch became more prevalent in bridge construction, numerous procedures for tracing it emerged, leading to an increase in the number of centers used.
Analysis of the figure reveals that the three arcs—Am, mEm', and m'B—comprise the curve and correspond to equal angles at the centers Anm, mPm', and m'n'B, all measuring 60 degrees.
In this method, AB represents the opening and OE denotes the arrow of the vault, serving as the long and short axes of the curve.
[22] When using the same opening and rise, the curve produced by this method exhibits minimal deviation from those generated by previous techniques.
For curves with more than three centers, the methods indicated by Bérard, Jean-Rodolphe Perronet, Émiland Gauthey, and others consisted, as for the Neuilly bridge, in proceeding by trial and error.
Tracing a first approximate curve according to arbitrary data, whose elements were then rectified, using more or less certain formulas, so that they passed exactly through the extremities of the major and minor axes.
He developed tables containing the necessary data to draw curves with 5, 7, and 9 centers, achieving precise results without the need for trial and error.
Additionally, the overhang is defined as the ratio of the arrow (the vertical distance from the highest point of the arch to the line connecting its endpoints) to the total opening.
[23] The table provided allows for the straightforward construction of a basket-handle arch with any specified opening using five, seven, or nine centers, eliminating the need for extensive calculations.
To inscribe the curve within a rectangle labeled ABCD, one would start by describing a semicircle on line segment AB, which serves as the diameter, and divide it into seven equal parts.
This requirement means that the angles formed between the radii do not necessarily need to be equal, allowing for greater flexibility in the design of the curves.
