The weight that makes it safer

A masonry arch stands if some line of compression can be drawn inside the stonework. That is a complete account of the arch and it says nothing at all about what happens at the springing, where the line arrives at the abutment travelling at an angle and carrying a horizontal push it has to give to something.

This essay is about the something. It is the same question one free body further down, and the answer has a property the arch’s does not: weight added at the top makes it better.

Which free body produced the number #

Cut the pier horizontally and take everything above the cut. Three things act on it: the arch thrust arriving at the top, the weight of the masonry above the cut, and whatever is standing on top. Their resultant has to be carried by the cut face, and its position is

$$e = \frac{M}{N}$$

with $M$ the moment of all three about the pier’s centre-line at that level and $N$ their total vertical force. Trace $e$ down the pier and the locus is the thrust line.

For this pier the base cut carries 424 kN a metre of self weight plus 16.9 kN of vertical thrust component, against a moment from 36.3 kN of horizontal thrust acting nine metres up. That puts the resultant 0.503 m from the centre of a base 2.68 m wide.

Two thresholds matter, and they are geometric:

Inside the stone, $|e| < w/2$. Beyond this there is no set of compressive forces in equilibrium with the load and the pier is a mechanism. Here $w/2 = 1.34$ m, so there is a factor of nearly three in hand.

Inside the middle third, $|e| < w/6$. Beyond this the joint opens on one side: a material that cannot be pulled cannot hold a resultant outside its kern, so part of the bed joint stops bearing. Here $w/6 = 0.447$ m, and 0.503 is outside it.

The stone’s strength appears in neither. The toe stress at the base is 351 kPa, against a masonry compressive strength of several megapascals — a factor of twenty unused. What the pier is short of is not strength.

Why the joint opening is worth caring about #

A resultant outside the middle third does not mean the pier falls down. It means the bed joint has opened, and three things follow.

The bearing area shrinks, so the stress at the toe rises — faster than linearly, because the resultant is moving towards the toe while the area is shrinking beneath it. That is the same non-linearity every eccentrically loaded no-tension section has, and it is why the stress runs away well before the resultant reaches the edge.

The pier becomes softer, because a section bearing on a third of its width has a fraction of the second moment of the full one. A softer pier deflects more under the thrust, which moves the resultant further out, which opens the joint more — a mild version of the load that makes itself worse, and the reason a slender masonry pier has a genuine second-order problem while a squat one does not.

And an open joint admits water, which is a durability problem that becomes a structural one over a few centuries.

The pinnacle, and the shape of what it buys #

Now add weight on top. A pinnacle sits on the pier’s own centre-line, so its weight contributes to $N$ and contributes nothing to $M$ — it has no lever arm about the axis the eccentricity is measured from.

$$e(P) = \frac{M_0}{N_0 + P}$$

which is a hyperbola. Two things follow from the shape rather than from the numbers.

The eccentricity falls as $1/N$, so doubling the weight on the joint exactly halves it. That is a strong return at first and a weak one later: the first tonne is worth ten of the tenth.

There is no threshold. Any pinnacle helps, and none of it is wasted; the improvement is smooth from zero. That is unusual — most structural interventions have a size below which they do nothing at all, as a brace does.

For this pier the answer is 55 kN per metre of run, which is a stone pinnacle of ordinary Gothic proportions. That is the whole of the mediaeval device: the pinnacle is not ornament with a structural excuse, and it is not ballast in the sense of dead weight resisting an uplift. It is a term in a denominator.

The batter, which is worth twelve times as much #

There is a second way to move the same eccentricity, and on this pier it is far more powerful.

Take the batter away — make the pier a uniform 1.6 m rather than widening to 2.68 m at the base — and the pier does not stand at all. The base eccentricity goes from 0.503 m to 0.978 m against a half-width of 0.80 m: the resultant is outside the stone.

To recover it takes 74 kN per metre of pinnacle merely to get the line back inside the section, and 890 kN — twelve times the battered pier’s requirement — to get it back inside the middle third.

The batter does two things where the pinnacle does one. It adds self weight, which is the pinnacle’s effect. And it widens the section, which raises both thresholds at the level where the resultant is furthest out. The two multiply rather than adding, so widening the base is worth much more per tonne of stone than piling the same tonne on the top.

Push the batter to 0.25 and the pier reaches a 3.85 m base, carries 540 kN of its own weight, and lands at an eccentricity of 0.116 m — comfortably inside the middle third with no pinnacle whatever.

Which is what a Gothic buttress actually looks like, and it is worth noticing what it does not look like. It is not a uniform pier. It is not a pier with a lump on top. It is a wedge that steps outwards as it descends and carries a weight at its head, and both features are doing the same arithmetic on the same equation.

The section that governs, which is not the base #

The base carries the most vertical force and has the widest section, and it is easy to assume it is therefore the critical one. It is not, and the counterexample is sharp.

Take the same pier and make the thrust shallower — five degrees rather than twenty-five, which is what a flatter arch delivers. The base eccentricity falls to 0.594 m, well inside a 2.68 m base. And the pier does not stand.

The reason is that the line leaves the stone near the top, where the pier is only 1.6 m wide and almost none of its own weight has accumulated. Up there the moment from the thrust is small but the vertical force is smaller still, and the ratio is what matters.

So the check has to be made on every cut, and the governing one is wherever $|e|/(w/2)$ peaks. On a battered pier under a steep thrust that is usually near the base; on a uniform pier or under a shallow thrust it is near the top. The critical section moves with the angle of the load, which is not a property any single cut can report — the same trap an envelope sets in a different problem.

That is also the mechanical reason a flying buttress is placed where it is. A flatter arch delivers a flatter thrust; a flatter thrust is harder to turn downwards; and the fix is to deliver it high on a pier that has room to widen below — which is a flyer arriving at the top of a pier, rather than a thrust arriving at the bottom of a wall.

What a buttress is competing against, which is a tie #

There is a completely different way to deal with an arch’s thrust, and comparing the two is the clearest statement of what a buttress is for.

Put a tie across the springings and the thrust never reaches the ground at all: the arch and its tie are a self-contained assembly delivering nothing but a vertical reaction, and the piers below can be as slender as their axial load allows. That is the tied arch, and it is structurally superior in every measurable way — less material, no eccentricity problem, no dependence on the ground.

It is also unusable in the building the buttress was invented for. A tie across the springing of a nave vault runs across the nave at the height of the triforium, and the whole point of the building is that the space is open. So the thrust has to be taken to the ground on the outside, through masonry, by a structure that cannot be pulled.

Seen that way the flying buttress is not a clever solution to carrying a thrust. It is what is left when the good solution has been ruled out by the brief, and every feature of it — the flyer’s angle, the pier’s batter, the pinnacle — is compensation for the absence of a tie. The comparison is worth making because it is the honest reason the device is elaborate.

The other way a pier goes #

Everything above is about the resultant leaving the section. There is a second failure on the same free body, and it is checked on the same cut: sliding.

The horizontal thrust is 36.3 kN a metre. The friction available on the base joint at a coefficient of 0.6 is 264 kN. A factor of seven, and it is comfortable here — but it is not comfortable everywhere, and its behaviour is the opposite of the overturning check’s in one important respect.

Adding a pinnacle helps sliding too, because friction is proportional to the normal force. Widening the base does not help sliding at all beyond the weight it adds, because friction does not care how wide the joint is. So the two interventions that are nearly equivalent for the thrust line are not equivalent for sliding — and on a low pier under a very flat thrust, where sliding is the governing check, the pinnacle is the better purchase of the two.

The other place it bites is a joint that is not level. A bed joint sloping outwards reduces the friction available and adds a component of the weight to the demand at the same time, which is why a buttress’s joints are cut square to the load path rather than to the horizon on any pier of consequence.

What a mediaeval builder was actually optimising #

None of the above was available to the people who built these. There was no thrust line, no eccentricity, no kern; Coulomb’s analysis of the arch is 1773 and the buildings are three hundred years older.

What there was, is a rule of proportion — the pier’s width as a fraction of the vault’s span, the pinnacle’s height as a multiple of the pier’s width — passed on in lodges and refined by what fell down. Those rules encode the arithmetic above surprisingly well, and it is worth seeing why they could.

The thrust from a vault of given shape scales with its span times its load. The pier’s self weight scales with its width times its height. The eccentricity is a ratio of the two, so a proportional rule is dimensionally the right kind of rule: scale the whole building and the ratio holds. That is not true of a strength check, where the demand grows as a volume and the capacity as an area, and it is why proportional rules worked for masonry for centuries and failed for iron almost immediately.

The failures, when they came, were at the edges of the proportion set — the tallest naves, the widest vaults, the thinnest piers. Beauvais fell in 1284, and its choir was the tallest ever attempted.

Where this model stops #

Three limits, and the first is the one that makes the whole approach honest rather than approximate.

The thrust is not a number. An arch with two pinned feet is indeterminate, and the thrust it delivers is anywhere in a range. The pier has to be safe for the whole range, and — this is the useful half — it only has to be safe for one line at each thrust. That is limit analysis: finding any admissible line inside the stone is a proof of safety, and no line has to be the real one. What the drawing above shows is a lower bound and is not a prediction.

The masonry is assumed to have no tension and infinite compressive strength. The first is nearly true and conservative. The second is not true and is why the toe stress was computed above: it is a check that has to be made and that almost never governs, which is a different thing from a check that can be skipped.

The pier is assumed rigid. Real ones deflect, the deflection moves the resultant, and on a slender pier that feedback matters. Squat piers — which is most of them — are unaffected.

What the picture cannot show #

The drawing is a plane. A real buttress is a three-dimensional block, often stepped in two directions, and the thrust it takes arrives at an angle to its own plane wherever the vault is not a simple barrel. The thrust line is then a curve in space and the condition is that it stays inside a solid, which is the same statement and a much harder drawing.

Nor does the drawing show time. A masonry pier settles, its mortar creeps, and the arch above it relaxes; the thrust five hundred years after construction is not the thrust the day the centring came out. Settlement of one support would rewrite the internal forces of a redundant frame; here it rewrites nothing, because the analysis never claimed to know which line the structure had taken. What makes the structure survivable is precisely that it is a lower bound: a pier that can accommodate any thrust in a wide range does not need the range to stay put.

The generalisation #

The habit worth carrying out of this is a way of reading a stability problem.

Every check in this essay has the same form: is a line inside a region? The thrust line inside the stone. The resultant inside the kern. The reaction inside the friction cone. None of them contains a material strength, all of them contain a geometry, and each is improved by moving either the line or the region.

That is a different kind of design variable from the ones most of this site uses. A beam is made safer by raising its capacity; a pier is made safer by moving a line, and the moves available — weight on top, width at the bottom, the angle the load arrives at — are all geometric. It is the reason the mediaeval builders got as far as they did with no theory of stress at all: the problem they were solving does not have stress in it.