The Arch

A beam of stone is a poor thing. Stone is immensely strong when it is squeezed and weak when it is bent, and a lintel laid across two posts is bent: its underside is stretched, and it cracks there long before the stone is crushed. That is why the stone architecture of Egypt and Greece is a forest of closely spaced columns, and why the widest stone lintels of antiquity span only a few metres (see the previous chapter). The arch escapes this limit by a simple change of shape. Cut the span into many small wedges (voussoirs) and arrange them on a curve, and every stone is squeezed and none is bent. The arch turns the weight it carries into compression, the one kind of force stone resists superbly.

The price of this trick is that an arch does not simply press down on its supports; it also pushes them apart. That outward push, the thrust, is the central fact of all arched and vaulted architecture. Everything in this chapter and the next (the shape of arches, the thickness of walls and piers, the buttresses of Romanesque churches and the flying buttresses of Gothic cathedrals) is a way of living with it.

The anatomy of an arch

An arch has its own vocabulary. It begins at the springing, where the curve leaves the vertical of its support; the stone on which it rests is the impost, and the lowest wedge is the springer. The inner curve is the intrados, the outer the extrados; the distance between the springings is the span, and the height of the intrados above them the rise. The central wedge at the crown is the keystone, the zone about halfway up the haunch. The triangular patches of wall between arches, or between an arch and the square frame around it, are spandrels. Whatever carries the arch and resists its thrust (pier, wall or buttress) is its abutment.

℟Wedges that lock by their own weight

Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.

joints ⟂ intrados (radial to the centre)

Vitruvius already knew the essential rule: the joints between the voussoirs must point towards the centre of the curve. Then no stone can drop without forcing its neighbours apart, and the ring locks itself. What a finished arch cannot do is stand before it is finished. Until the keystone is set, the half-rings would simply fall inward, so every arch is built on a temporary timber frame, the centring, which is struck only when the ring is closed. The cost of centring shaped building practice for two thousand years: it rewarded equal spans, which could reuse one frame, and it was one of the great attractions, in the twelfth century, of the rib vault.

℟Every arch is first built on a frame

An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.

Drawing arches

Every traditional arch is drawn with compasses, and the clearest way to name an arch is by the centres from which its curve is struck. The semicircular arch of Rome and of the Romanesque has one centre, in the middle of the springing line; its rise is exactly half its span, and it cannot be otherwise. A segmental arch is a smaller part of a larger circle, its centre below the springing line; it rises less and pushes harder. A stilted arch sets a semicircle on vertical legs. The horseshoe arch continues the circle below its centre, so that the arch is wider inside than at its springing; it is the arch of Visigothic Spain and of the Great Mosque of Córdoba.

The pointed arch has two centres. When each half is struck from the opposite springing, with a radius equal to the span, the crown and the two springings make an equilateral triangle and the arch is called equilateral; with centres outside the span it becomes a sharp lancet, with centres inside it a blunt drop arch. Late medieval builders added the low four-centred arch of Tudor England, with two short arcs at the springings and two long, flat ones meeting at the crown, and the ogee, whose double curve bends outward before it rises to a point. Some arches are defined not by centres at all but by a continuous curve (the ellipse, the parabola, the catenary), and some are not true arches: the corbelled arch is built of horizontal courses, each oversailing the one below, and holds up by cantilever, not by wedging.

Why an arch pushes

Imagine an arch cut in half at its crown. Each half leans on the other; take one away and the other falls inward. What holds each half up is the push of its partner, a horizontal force at the crown. Since the half-arch is not accelerating sideways, its foot must be pushed back by an equal and opposite force from the abutment. That is the thrust. Its size depends on the load and on the shape: for the same span and the same load, the lower the rise, the greater the thrust. A shallow segmental arch or a flat jack arch pushes very hard; a tall pointed arch hardly at all by comparison.

℟The flatter the arch, the harder it pushes

An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.

H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)

Alberti, writing in the 1440s, distinguished the full semicircular arch, the “diminished” segmental arch and the pointed arch of two arcs, and regarded the semicircle as the strongest; the flatter arch, he observed, depends more on the strength of what holds its ends. Vitruvius, fifteen centuries earlier, had recommended the arch’s humblest use, the relieving arch (an arch turned in the wall above a lintel, so that the weight of the masonry is carried round the opening), and Roman bricklayers built it everywhere.

℟Relieve a lintel with an arch

Over a doorway or window spanned by a lintel, turn an arch in the wall above so that the weight of the masonry is carried round the opening onto the jambs, leaving the lintel to carry only the small triangle of wall beneath the arch.

The line of thrust

Inside every arch there runs an invisible line: the path of the resultant force as it passes from stone to stone, from the crown down to the foundations. Draw the weights of the voussoirs and whatever they carry, combine them one by one with the thrust at the crown, and the resultant traces a polygon, the same shape a chain would take if the loads hung from it. This is the line of thrust. If it can be found lying everywhere within the masonry, the arch will stand; if it cannot, the arch will open hinges where the line escapes, and with four hinges it becomes a mechanism and falls.

The model draws the line through the middle of the ring at the crown and at the springings, then continues it through the piers, adding their weight as it goes. Two lessons appear at once. A thin semicircular ring cannot contain its own thrust line: the line swings outside the intrados near the haunches. Real Roman and Romanesque arches are therefore thick, and their haunches are loaded with walling that pulls the line back inside. And the piers matter as much as the arch: make them too slender and the line walks out of their base, and the arch spreads its supports and falls. Pointed arches, whose shape lies closer to the thrust line of their own weight, can be much thinner.

℟An arch stands if its thrust line fits inside it

Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.

The ideal shape for an arch carrying only itself was found by the English scientist Robert Hooke in the 1670s: it is the curve of a hanging chain turned upside down. A chain can only pull, so it hangs in exactly the shape in which every link is in pure tension; invert it, and the same shape is one in which every stone is in pure compression. Hooke published the idea as a Latin anagram. Two centuries later Antoni Gaudí designed churches by hanging weighted strings from his ceiling and photographing them upside down.

Ut pendet continuum flexile, sic stabit contiguum rigidum inversum. (As hangs the flexible line, so but inverted will stand the rigid arch.)

℟As hangs the chain, so stands the arch

A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.

free-hanging chain (catenary) ↔ ideal arch under its own weight

How thick must the pier be?

Before anyone could calculate a thrust, builders needed rules that told them how strong a support had to be, and they found them in geometry. The most famous was printed by the Jesuit François Derand in 1643 and taught by François Blondel at the Royal Academy of Architecture in Paris: divide the intrados into three equal parts, draw a line from the springing through the first point of division, and extend it beyond the springing by its own length; a vertical dropped from the end of that line gives the outer face of the pier. For a semicircle the rule gives a pier a quarter of the span thick; for an equilateral pointed arch a little less; for a lancet less still.

The rule knows nothing of loads, stresses or materials, yet it is sound. It works because stability in masonry is a matter of geometry: the stresses in stone are far below what it can bear, and an arch fails not by crushing but by changing shape. A design that is stable at one size is therefore stable at any size, if all its parts grow together. The medieval masons knew this in their bones. Their rules (Lorenz Lechler’s wall of one tenth the span of a choir, Villard de Honnecourt’s constructions, the lodge procedures argued over at Milan) are all rules of proportion between structure and span (The width of the choir is the measure of all).

℟The Derand–Blondel rule for abutments

To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.

pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet

Arches in a row

Set arches side by side and a new economy appears. Over each intermediate pier the thrust of one arch meets the equal and opposite thrust of its neighbour; the horizontal forces cancel, and the pier has only to carry weight. It can be slender, even a single column. Only at the ends of the row is there nothing to balance the thrust, and there the abutments must be massive. Vitruvius told builders to make the outer piers of an arcade the widest, and every Roman aqueduct shows the result: an airy run of arches ending against the hillsides or in solid masses of masonry.

℟Make the end piers of an arcade the strongest

In a row of arches the thrusts of neighbouring arches cancel over each intermediate pier, which may therefore be slender; at the ends of the row nothing balances them, so the outermost piers must be made wider or buttressed.

The round arch imposes one further discipline on an arcade. Because its rise is always half its span, arches of different widths spring from the same line but rise to different heights. A Roman or Romanesque builder who wanted crowns in line had to keep all spans equal, or stilt the narrow arches on raised legs. The pointed arch removes the constraint. Its rise is not fixed: two centres can be placed wherever the builder chooses, and a wide and a narrow pointed arch can both be drawn to reach exactly the same height. It sounds like a detail. It was the key that opened Gothic architecture, because a vault is a set of arches of different spans that must all meet at one crown.

℟Pointed arches reach any height on any span

A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.

round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2

The next chapter follows the arch into three dimensions. Push an arch along its axis and it becomes a barrel vault; cross two barrels and you have a groin vault; build the groins first as ribs and fill between them, and you arrive at the vaults of the Gothic cathedrals, and at their flying buttresses, which are nothing but half-arches carrying the thrust of the vault through the open air to the ground.