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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Basket-handle arch | 2/3 | https://en.wikipedia.org/wiki/Basket-handle_arch | reference | science, encyclopedia | 2026-05-05T13:59:06.280548+00:00 | kb-cron |
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. The objective was to create perfectly continuous curves with an aesthetically pleasing contour. Given the indeterminate nature of the problem, certain conditions were often imposed arbitrarily to achieve the desired result. For instance, it was sometimes accepted that the arcs of circles composing the curve must correspond to equal angles at the center, while at other times, these arcs were required to be of equal length. Additionally, either the amplitude of the angles or the lengths of the successive radii were allowed to vary according to specific proportions. A consistent ratio between the lowering of the arch and the number of centers used to trace the intrados curve was also established. This lowering is measured by the ratio of the rise (b) to the width of the arch (2a), expressed as b/2a. Acceptable ratios may include one-third, one-quarter, or one-fifth; however, if the ratio falls below one-fifth, a circular arc is generally preferred over the basket-handle arch or ellipse. For steeper slopes, it is advisable to employ at least five centers, with some designs utilizing up to eleven centers, as seen in the curve of the Neuilly Bridge, or even up to nineteen for the Signac Bridge. As one of the centers must always be positioned on the vertical axis, the remaining centers are symmetrically arranged, resulting in an odd total number of centers.
=== The Huygens method ===
For constructing curves with three centers, Huyghens outlines a method that involves tracing arcs of varying radii corresponding to equal angles, specifically angles of 60 degrees. To begin, let AB represent the opening and OE signify the arrow of the vault. From the center point O, an arc AMF is drawn using radius OA. The arc AM is then taken to be one-sixth of the circumference, meaning its chord equals the radius OA. The chords AM and MF are drawn, followed by a line Em through point E, which is the endpoint of the minor axis, parallel to MF. The intersection of chords AM and Em determines point m, the boundary of the first arc. By drawing the line mP parallel to MO, points n and P are established as the two centers required for the construction. The third center n is positioned at a distance n'O from the axis OE, equal to nO. 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.
=== The Bossut method ===
Charles Bossut proposed a more efficient method for tracing a three-center curve, which simplifies the process. 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. To begin, the line segment AE is drawn. From point E, a segment EF' is taken, equal to the difference between OA and OE. A perpendicular line is then drawn from the midpoint m of AF'. The points n and P, where this perpendicular intersects the major axis and the extension of the minor axis, serve as the two centers required for the construction. When using the same opening and rise, the curve produced by this method exhibits minimal deviation from those generated by previous techniques.
== Curves with more than three centers == 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.
=== The Michal method ===
In a paper published in 1831, mathematician Michal addressed the problem of curve construction with a scientific approach. 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. Michal's calculation method is applicable to curves with any number of centers. He noted that the conditions required to resolve the problem can be somewhat arbitrary. To address this, he proposed that the curves be constructed using either arcs of a circle that subtend equal angles or arcs of equal length. However, to fully determine the radii of these arcs, he also posited that the radii should correspond to the radii of curvature of an ellipse centered at the midpoint of each arc, where the opening serves as the major axis and the ascent functions as the minor axis. As the number of centers increases, the resulting curve approximates the shape of an ellipse with the same opening and slope. The following table illustrates the construction of a basket-handle arch, characterized by equal angles subtended by the various arcs that comprise it. The proportional values for the initial radii are calculated using half the opening as the unit of measurement. 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.