Basket-handle arch



A basket-handle arch is an arch with the profile of its intrados (inner surface) formed by a sequence of circular arcs with neighboring ones being tangent to each other (smoothly transitioning), and the end ones tangent with supports. For example, a three-centered arch contains three arc segments with different centers; the other common type is five-centered. The basket-handle arch is used in architecture, especially bridges. Its shape is similar to that of a semi-ellipse, which has a continuous curvature variation from its origin to its apex, i.e. from the extremities of the long axis to the apex of the short axis. Also known as a depressed arch, basket arch.

History
Since Roman times, bridges vaults have been built with semicircular arches, forming a complete half-circumference. 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.

The pointed arch, which, instead of reducing the excess height of vaults, accentuates it (since the rise is greater than half the opening), was not used in bridge construction until the Middle Ages.

The basket-handle arch appeared at the beginning of the Renaissance, offering an undeniable aesthetic advantage over the segmental vault: the fact that its end arches are vertically tangential to the supports. The Pont-Neuf in Toulouse in the 16th century and the Pont Royal in the following century are the earliest applications in France.

In the 18th century, the use of basket handle arches was common, often with three centers: the bridges of Vizille, Lavaur, Gignac, Blois (1716–1724), Orléans (1750–1760), Moulins (1756–1764), Saumur (1756-1770 ).

Jean-Rodolphe Perronet designed the arches of the bridges of Mantes (1757–1765), Nogent (1766–1769) and Neuilly (1766–1774) with eleven centers in the second half of the 18th century. There were also eleven centers in Tours (1764–1777). The others were reduced to 1/3 or a little more, except for Neuilly, which was reduced to 1/4.

In the 19th century, the first major French railroad bridges were basket-handle arches: Cinq-Mars bridge (1846–1847), Port-de-Piles bridge (1846–1848), Morandière bridges: Montlouis (1843–1845), Plessis-les-Tours (1855–1857).

In England, while the Gloucester Bridge (1826–1827) and the London Bridge (1824–1831) were elliptical, the Waterloo Bridge in London (1816–1818) was still basket-handle arches.

A few basket-handle arches remained in the second half of the 19th century and early 20th century:


 * With three centers: Edmonson Avenue Bridge in Baltimore (1908–1909).
 * With five centers: Annibal Bridge (1868–1870) and Devil's Bridge (1870–1872).
 * With seven centers: Emperor Francis Bridge in Prague (1898–1901).
 * With nineteen centers: Signac Bridge (1871–1872).

One basket-handle arch railway bridge in the United States, the Thomas Viaduct, was constructed in 1833-1835. The viaduct is now owned and operated by CSX Transportation and is still in use today, making it one of the oldest railroad bridges still in service.

Aesthetics
The ancient architects attached a certain importance to the processes that were used to define the outline of the basket-handle arch. It's easy to understand that these processes can vary ad infinitum, but precisely because of this kind of elasticity, architects have often preferred the curve thus traced to the ellipse, whose contour is determined geometrically.

In the case of an ellipse, given the opening of a vault and the height at the center, i.e., the major and minor axes, all the points of the intrados curve are fixed, without the architect being able to change anything at will. On the other hand, the multi-center curve can be more or less rounded at the base and more or less flattened at the top, depending on the arrangement of the centers, leaving a certain amount to the architect's taste.

Advantages and disadvantages
The advantages in terms of layout were undeniable: the layout of full-scale grooves was considered easier and more precise, and the layout of the normals, and thus the segment joints, was immediately on site.

The number of voussoir shapes was limited to the number of different radii, whereas for the ellipse, it was equal to half the number of voussoirs plus one.

However, the discontinuity of the layout led to the appearance of unsightly voussoirs, which could not always be removed during restoration work.

The ancient oval
Although the basket-handle arch was not used for bridge vaults in ancient times, it was sometimes used in the construction of other vaults. And Heron of Alexandria (who wrote his mathematical treatises more than a century before our era) had already defined a simple method for tracing it.

If AB is the width of the vault to be built, its height (or rise, or spire) being indeterminate, we describe a half-circumference on AB, and through the point C of this, taken on the vertical OC, we draw the tangent mn, on which we take the lengths Cm and Cn equal to half the radius. By joining mO and nO, we determine points D and E, through which we trace the isosceles triangle DOE, whose base is equal to the height. Once this has been done, we take the line DA, divide it into four equal parts, and draw parallels to DO through the points of division a, b, c. The points where these parallels intersect the horizontal axis AB and the extended vertical axis CO give the centers we need to trace various curves with 3 centers on AB, as shown in the figure. These curves are what we usually call the ancient oval. Since the basket-handle arch has been widely used in bridge construction, the procedures proposed for tracing it have multiplied, and the number of centers has increased. The following is a brief description of the most widely used of these procedures.

The goal was to create perfectly continuous curves with an elegant contour. Due to the indeterminate nature of the problem, certain conditions were arbitrarily imposed on the assumption that they would lead more reliably to the desired result.

Sometimes, for example, it was accepted that the various arcs of a circle, of which the curve is composed, must correspond to equal angles at the center; sometimes, these partial arcs were assumed to be of equal length, or again, either the amplitude of the angles or the length of the successive rays was allowed to vary according to certain proportions.

Moreover, it has always been accepted that a certain ratio should be maintained between the lowering of the arch and the number of centers used to trace the intrados curve, thus lowering being measured, for the basket-handle arch as for the circular arc, by the ratio of the rise to the opening, i.e. by the ratio b/2a, where b is the rise and 2a is the width of the arch.

This ratio may be one-third, one-quarter, one-fifth, or less, but as soon as it falls below one-fifth, the circular arc should generally be preferred to the basket-handle arch or ellipse. With a larger slope, it's a good idea to have at least five centers, and we've sometimes allowed up to eleven, as in the case of the curve of the Neuilly Bridge or even up to nineteen for the Signac Bridge. Since one of the centers must always be on the vertical axis, and the others symmetrically arranged in equal numbers to the right and left, the total number is always odd.

The Huygens method
For curves with three centers, the following procedure, according to Huyghens, consists in tracing them by making arcs of different radii correspond to equal angles, i.e., angles of 60.

Given AB as the opening and OE as the arrow of the vault, from the center O, with OA as the radius, we describe the arc AMF, from which we take the arc AM, equal to one sixth of the circumference, and whose chord is therefore equal to the radius OA. Draw this chord AM and the chord MF, then draw Em through point E, the end of the minor axis, parallel to MF.

The intersection of AM and Em determines the boundary m of the first arc. By drawing the line mP parallel to MO through this point m, the points n and P are the two centers we're looking for. The third center n is located at a distance n'O from the axis OE equal to nO. It's enough to study the figure to see that the three arcs of the circle Am, mEm', m'B that make up the curve correspond to angles at the centers Anm, mPm,' and m'n'B that are equal to each other and all three of 60°.

The Bossut method
The following method by Charles Bossut for tracing the same 3-center curve is faster.

AB and OE are again the opening and the arrow of the vault, i.e., the long and the short axis of the curve to be traced. We join AE and from point E we take EF' to be equal to OA-OE, then we draw a perpendicular through the middle m of AF' and the points n and P, where this perpendicular meets the major axis and the extension of the minor axis, are the two centers we're looking for.

With the same opening and rise, the curve drawn in this way differs very little from the previous one.

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, Mr. Michal dealt with the question more scientifically and prepared tables containing the data necessary to draw curves with 5, 7, and 9 centers without trial and error and with perfect accuracy.

His method of calculation can also be applied to curves with any number of centers.

Since the conditions that must be met for the problem to cease to be indefinite are partly arbitrary, Mr. Michal proposes that the curves be composed sometimes of arcs of a circle subtending equal angles, sometimes of arcs of equal length. Since this is not sufficient to determine all the radii, he also assumes that the radii of each arc are equal to the radii of curvature of the ellipse described in the center of these arcs, with the opening as the major axis and the ascent as the minor axis.

As the number of centers increases, the curve becomes closer and closer to the ellipse with the same opening and slope.

The following table refers to the drawing of the basket-handle arch with equality of the angles subtended by the parts of the arcs of which it is composed. The proportional values it gives for the first radii are calculated using the half-opening as the unit. The overhang is the ratio of the arrow to the whole opening. It's easy to see how you can use this table to draw a basket-handle arch with any opening at five, seven, or nine centers without doing any research. The only requirement is that the drop is exactly one of those predicted by Mr. Michal.

For example, we need to draw a curve with seven centers, a 12-meter opening, and a 3-meter slope corresponding to a quarter or twenty-five-hundredth drop. The first and second radii are 6 x 0.265 and 6 x 0.419, or 1.594 and 2.514.

If ABCD is the rectangle in which the curve is to be inscribed, we describe a half-circumference on AB as the diameter, dividing it into seven equal parts and tracing the chords Aa, ab, bc, cd, the latter corresponding to a half-division.

On the AB axis, starting from point A, we take a length equal to 1.590 m and have the first center m1. A parallel of radius Oa is drawn through this point, and the point n where it meets the chord Aa is the limit of the first arc. From point n, we take a length nm2 equal to 2.514 m, and point m2 is the second center. From this point m2, we draw a parallel to the radius Ob, from point n a parallel to the chord ab, and the point of intersection n' of these two parallels is the limit of the second arc. Then, through point n', we draw a parallel to the chord bc, and through point E, a parallel to the chord cd.

Finally, at the point of intersection n '' of these two lines, a parallel is drawn to the radius Oc, and the points m3, m4 where it intersects the extension of the radius n' m2 and the extension of the vertical axis give the third and fourth centers. The last three centers m5, m6 and m7 are symmetrical concerning the first three m1, m2, and m3.

As the figure shows, the arcs An, nn', n'n '', etc. correspond to equal center angles and da 51° 34' 17" 14. What's more, if we were to construct a semi-ellipse with AB and OE as the major and minor axes, the arcs of this semi-ellipse, contained within the same angles as the arcs of the circle, would have a radius of curvature in their center equal to the radius of the latter.

This method can construct curves with five, seven, and nine centers with the same ease.

The Lerouge method
After Mr. Michal, the topic was taken up again by Mr. Lerouge, chief engineer of the Ponts et Chaussées, who also drew up tables for tracing curves with three, five, seven, and up to fifteen centers.

However, his calculations are based on the condition that the successive radii increase according to an arithmetic progression, regardless of the equality of the angles they form between them.