Arches and vaults · Pascal Architecture

A brick arch built ring by ring on timber centering, then struck so it stands on its own between two brick piers. The line of thrust runs through the ring and down the piers, and moves as the arch flattens or takes weight. Extended in depth, the same arch becomes a barrel vault; crossed with a second, a groin vault.

Brick and stone are strong when squeezed and weak when pulled. Laid as a beam across an opening, a row of bricks would have to stretch along its underside, and it would crack and fall. Curve the same bricks into an arch and the load is passed from one brick to the next as pure compression, down both sides to the ground. For thousands of years this was the only way to span a wide opening with masonry, and it still shapes how we build in it.

Compression only

Each wedge-shaped piece of an arch, a voussoir, is pressed between its two neighbours. Follow the force from joint to joint and you trace the line of thrust: where the push crosses each joint. If the arch were shaped exactly to this line, every joint would be squeezed evenly and no part would bend. A hanging chain finds this shape on its own, upside down; turned over, the curve a chain takes under a load is the arch that carries that load in pure compression.

Real arches are not shaped that precisely, and they need not be. Mortar joints cannot pull, but they can let the thrust move off-centre. As long as the line stays inside the masonry, every joint is still in compression somewhere across its depth and the arch stands. This is why an arch of loose blocks, with no glue at all, can stand for centuries.

The parts

Voussoirs: The wedges of the arch. In the figure they are bricks laid on edge with their ends showing, in two rowlock rings. The bricks are square; the joints taper, about ¼ in at the inside of the curve and more at the outside, and that taper is what makes each unit a wedge.
Keystone: The voussoir at the crown, set last. Here it is a stone running through both rings and standing a little proud.
Springers and springing line: The first voussoirs on each side, and the level at which the curve leaves its supports.
Intrados and extrados: The inner and outer curves of the arch face. The underside of the arch is its soffit. The crown is the top; the haunches are the regions between crown and springing.
Span and rise: The clear distance between the supports, and the height of the intrados above the springing line.
Skewback: The sloping bed a segmental arch starts from, cut on the radius so the thrust meets it square. A semicircular arch starts from a level bed.
Abutments: The masses on each side that take the arch’s push and carry it to the ground.

Built on centering

Half an arch cannot stand. Until the last piece is in, each side would simply topple inward. So an arch is built on centering, a temporary timber frame cut to the curve of the intrados. In the figure it is two ribs of curved pieces, each on a tie with a king post and two struts, with boards called lagging across them to bed the bricks on. The bricklayer works up both sides at once, the inner ring first and then the outer, and closes the arch with the keystone.

The centering stands on posts, and between each post and the frame sits a pair of folding wedges. Once the mortar has gained some strength, the wedges are knocked out and the whole frame drops away from the soffit. This is striking the centering. It is done gently, because at that moment the arch takes its own weight for the first time and settles slightly as its joints close up. Drag the build slider in the figure to watch the sequence, or press B to run it again.

Thrust, and what holds it

An arch does not only press down on its supports; it also pushes them apart. This sideways push, the thrust, has to be resisted, or the supports spread and the arch drops through. For a given load and span the thrust grows as the arch gets flatter:

H ≈ w L² / 8 fhorizontal thrust from a uniform load w per foot, span L and rise f

There are four ways to deal with it:

Abutments. Heavy masonry on each side, whose own weight bends the thrust down into the ground. The figure’s piers do this. Follow the dashed line out of the arch and down through each pier: it bends toward the vertical as it gathers the pier’s weight, and it has to reach the floor inside the pier’s base.
Buttresses. Masses or walls set against the arch, or against a wall carrying arches, at the points where the thrust arrives. Flying buttresses carry it across an aisle.
Ties. A steel rod or timber across the springing, which holds the two ends together in tension. The arch then only presses down.
Neighbouring arches. In a row of arches each one pushes against the next, and the thrusts cancel everywhere but at the two ends, which need abutments.

Rise against thrust

The figure keeps the span at 6 ft and the ring at 7⅝ in, two rowlocks, and lets you lower the rise from a semicircle to a shallow segmental arch with 1 in of rise for every foot of span. The numbers come from the same model that draws the line of thrust, with brickwork at 120 lb/ft³.

Rise	Thrust	Thickness to spare	Folds under
3′ 0″ · 1:2	99 lb (0.44 kN)	1.8×	83 kg
2′ 6″ · 1:2.4	114 lb (0.51 kN)	2.8×	130 kg
2′ 0″ · 1:3	135 lb (0.60 kN)	5×	204 kg
1′ 6″ · 1:4	169 lb (0.75 kN)	11×	429 kg
1′ 0″ · 1:6	240 lb (1.07 kN)	35×	never
0′ 6″ · 1:12	455 lb (2.02 kN)	270×	never

Thickness to spare compares the ring with the thinnest ring of the same shape that could still stand; folds under is the weight at the quarter point that would break it. The ring itself weighs 2.25 kN (505 lb) as a semicircle and 1.36 kN (306 lb) at 1:12.

Two lessons pull in opposite directions. Flattening the arch makes it lighter and, against a weight set on it, far harder to break: a shallow ring has room for almost any line of thrust. But its thrust grows more than fourfold, and all of it lands on the abutments. In the figure the line through each pier reaches the floor 15 in from the outer edge for the semicircle, 9 in for a rise of 1 ft, and less than an inch at 1:12. Any flatter and these piers would tip over. A flat arch is only as good as what it pushes against.

The line of thrust

For one arch under one load there are many lines of thrust that would balance it; the arch settles into one of them depending on how its supports move. The figure draws the one that keeps furthest from the faces. The traditional design rule asks that it stay within the middle third of the ring, so that every joint is compressed across its whole depth and no joint opens even a hairline. Hover the line in the figure to see the middle third.

A free-standing semicircular ring this thin cannot meet that rule under its own weight: at the crown and the haunches the line swings into the outer thirds. It still stands, with a good margin. The test used for old masonry is geometric: the thinnest semicircular ring that can carry itself is a little under 11% of its radius thick, and this one is 1.8 times that. In a wall, the brickwork filling the spandrels above the haunches presses down there and pulls the line back toward the centre.

Now put weight on one haunch. The line is pushed out toward the top face under the load, toward the bottom face on either side, and toward the top again at the far springing. When it touches the faces at four points no line fits inside the ring any more: the joints there open like hinges and the arch folds into four pieces. Set four plates on the semicircle in the figure, 105 kg, and watch it happen. Masonry arches almost never fail by crushing; they fail by turning into a mechanism, and it is shape, not strength, that prevents it.

Arch forms

Semicircular: Half a circle, struck from one centre on the springing line. The Roman arch: the tallest round arch for its span, and the one that pushes least.
Segmental: Less than half a circle, struck from a centre below the springing line. It needs less height and pushes harder, so it starts from skewbacks.
Pointed: Two arcs from two centres meeting at a point. Gothic builders used it because its rise can be chosen independently of its span, and because it pushes sideways less than a round arch of the same span.
Jack, or flat: A level soffit with voussoirs radiating from a centre well below. It is all thrust and very little arch, usually built with a slight upward camber so that it never sags below level.

Vaults

A vault is an arch extended in depth, to roof a space rather than span an opening. Choose Barrel vault in the figure and six more rings are laid behind the first. Its thrust now arrives along the whole length of both walls, about 2.3 kN for every metre of wall (156 lb per foot) in the semicircular case. The walls must be thick or buttressed along their length, and every opening cut into them weakens the support.

A groin vault crosses two barrels of the same size over a square bay. Where they meet they form groins, the creases running diagonally across the bay. The curved panels between them, the webs, lean on the groins, and the groins bring the whole load down to the four corners. Over a 6 ft bay the figure’s vault weighs about 41 kN, a little over 10 kN on each corner pier, and nothing on the sides between. That is what freed Romanesque and Gothic builders to open up the walls between their piers. A rib vault goes a step further: masonry ribs are built along the groins and edges first, on light centering, and the thin webs are filled in between them.

Spin an arch about its vertical axis and you get a dome. For the same idea in timber and steel, where a straight member is made to work in a curve or a triangle, see beams and lateral stability.