The line that must stay inside

There is a widespread and comforting belief that a masonry arch stands because its shape matches the line the load wants to take, and that a badly shaped arch is therefore in trouble. It is a natural belief, because the funicular shape is such a satisfying idea and because the catenary keeps turning up in the history of the subject with an air of inevitability. It is the same instinct that makes depth look like the free variable in a truss: a true and useful idea, promoted to a law it was never entitled to.

It is also wrong, and the correction is more interesting than the belief. An arch stands if there exists any line of compression, in equilibrium with the loads, lying entirely within the masonry. Not the right line; a line. There are infinitely many candidates, one for every value of the horizontal thrust, and the arch needs only one of them to fit.

That existence claim is Heyman’s safe theorem, and it converts a question about what the arch is doing, which is unanswerable, into a question about what the arch could be doing, which is a matter of geometry.

What the line of thrust is #

Take any arch under any set of loads, and consider the resultant of all the forces acting on the part of it to one side of a cut. That resultant passes through some point in the cross-section at the cut. Do this at every cut along the arch, and the locus of those points is the line of thrust: the path the compression takes through the stonework.

Its equation is simple enough to be surprising. For an arch of span $L$ under vertical loads, with a horizontal thrust $H$ at the springings, the line of thrust has ordinate

$$y(x) = \frac{M_{\text{ss}}(x)}{H}$$

where $M_{\text{ss}}$ is the bending moment the same loads would produce on a simply supported beam of the same span. The whole family of possible thrust lines is one beam analysis divided by one number.

This is why the arch and the beam are the same problem seen twice, and why the funicular polygon works: the polygon’s pole distance is the thrust, and moving the pole further from the load line flattens the polygon in exactly the way that dividing by a larger $H$ flattens the thrust line.

Why masonry is a special case #

For a steel arch the interesting question is stress, and the thrust line is a convenience. For masonry it is the whole analysis, and the reason is three assumptions that Heyman set out in 1966 and that are unusually well matched to the material.

Masonry has no tensile strength. Not “little”; for the purpose of the analysis, none. Mortar joints open under the smallest tension, and old arches frequently have no mortar worth the name.

Compressive stresses are low. A masonry arch of ordinary proportions works at a small fraction of the crushing strength of its stone — often below a twentieth. Crushing is therefore not the failure mode, and the material can be treated as infinitely strong in compression.

Sliding does not occur. Friction between the voussoirs is enough to prevent joints slipping, which is generally true for joints not too far from perpendicular to the thrust.

Grant those three and the arch becomes a purely geometrical object. It cannot fail by stress, because stress is irrelevant; it can only fail by the thrust line leaving the masonry, because at the moment it does, a joint opens on the far side and a hinge forms.

The safe theorem, and what it costs to state #

The safe theorem for masonry is the lower-bound theorem of plasticity in a particular disguise, and it is the same theorem that gives the lower bound on a collapse load.

If a set of internal forces can be found which is in equilibrium with the applied loads and which nowhere violates the material’s yield condition, the structure will not collapse.

For masonry the yield condition is “no tension”, which for a line of thrust means “inside the ring”. So: find one thrust line inside the masonry and the arch is safe. The theorem says nothing whatever about which line the arch has, and this is the part that takes getting used to, because engineers are trained to want the actual answer.

The actual answer is genuinely unavailable. A masonry arch is highly redundant, and its true state depends on its construction history, on the settlement of its abutments, on the temperature, on the mortar’s shrinkage — none of which is knowable for a bridge built in 1780. The state is unknowable and the safety is computable, and the theorem is what separates the two.

The thickness is the variable, not the shape #

Once the criterion is “does a line fit”, the arch’s thickness stops being about stress and becomes about the width of the corridor available to the thrust line. Two consequences follow, and both are counter-intuitive if the funicular story is the one in mind.

The first is that shape matters much less than expected. A circular arch is not funicular for its own weight, and a great many perfectly sound arches are circular; the thrust line under self-weight simply wanders a little away from the centreline and the masonry accommodates it. What the shape decides is how much of the ring’s width the wandering consumes, and a good shape is one that leaves some over.

That is a claim with a number attached, and the number is available by drawing the same ring on the right shape. Under a uniform load the funicular is a parabola, so a parabolic ring is the arch the funicular story says the mason should have built.

Six per cent is the whole prize for building the correct shape instead of the convenient one, and it is worth being clear about what it is not. It is not that the shape is irrelevant. Put the same two rings on a diet and they part company completely: the circular one runs out of masonry at a thickness of 4.4% of the span, and the parabolic one never does. A thrust of $wL^2/8r$ puts the line exactly on a parabolic centreline at every station at once, so a parabolic ring of any thickness whatever — a line of masonry with no width at all — admits that one line and stands.

Shape decides the limit and thickness decides the margin, and an arch with a generous ring is nowhere near its limit. The circular ring drawn here has twice the masonry it needs, which is why the six per cent the shape could add is not worth crossing a site for. The mason who built circles was not making a concession; he was spending thickness, which is cheap and available in courses, instead of geometry, which is expensive to set out and impossible to correct.

The second is Heyman’s geometrical factor of safety: the ratio of the actual ring thickness to the smallest thickness for which any line would still fit. It is a factor on geometry rather than on load, and it is the natural measure precisely because load is not what threatens the arch.

For a semicircular arch under its own weight, the limiting thickness works out at about 1/17 of the radius — Couplet found essentially this in 1730, and Heyman rederived it as a limit-state calculation. Below that the arch cannot stand at any thrust; above it, the surplus is the safety.

Which free body produced the number #

The figures take the arch ring as a whole, cut it free at both springings, and ask what internal forces at those cuts are in equilibrium with the self-weight.

Self-weight is applied as a uniform load per unit span, so the moment it produces on a simply supported span of 10 is the parabola $M_{\text{ss}}(x) = wx(L-x)/2$, peaking at $wL^2/8 = 12.5$ per unit weight. The thrust line ordinate at the crown is that peak divided by $H$, and setting the ordinate equal to the centreline rise of 2.8 gives $H = 12.5/2.8 = 4.46$ — which is the thrust that puts the line exactly on the centreline at the crown.

The two extremes come from a search rather than a formula: the generator sweeps the thrust downward until the line touches the extrados somewhere, and upward until it touches the intrados, and reports both. The independent check available on that calculation is the parabolic case, where the closed form $H = wL^2/8r$ is exact and the search has to reproduce it, and it does.

The assumption the figure rests on is that the load is the arch’s own weight distributed uniformly along the span, not along the curve. A real arch is heavier at the haunches, where the ring is longer per unit of span and where spandrel fill sits above it, and that redistribution moves the thrust line down at the haunches — generally a help. The figure’s uniform load is the conventional simplification and it is on the conservative side for the crown, which is not a claim the picture can make about itself.

The low-stress assumption, checked rather than asserted #

The second of Heyman’s three assumptions is the one doing the most work — it is what makes the material infinitely strong and the analysis purely geometrical — and it is stated on this page as a fraction without a calculation behind it. It deserves one, because the answer is comfortable in three places and marginal in the fourth.

Give the arch real dimensions: 10 m span, 2.8 m rise, a 500 mm ring one metre wide, masonry at 22 kN/m³, with some spandrel fill above. Its own ring is 12.1 m along the curve and weighs 133 kN; with the fill, call the total 233 kN, so $w = 23.3$ kN/m and

$$H = \frac{wL^2}{8r} = \frac{23.3 \times 100}{22.4} = 104\ \text{kN}$$

The vertical reaction is 116 kN, so the springing carries a resultant of 156 kN across a joint of 0.5 m².

So the assumption is excellent everywhere the line is comfortably inside and only adequate at exactly the place the analysis sends it — the hinge, where Heyman’s idealisation puts the whole force on a line of zero width and the stress on paper is infinite. What saves it is that the contact spreads: the stone crushes a little, the joint beds down over a few tens of millimetres, and the stress lands at a fifth of the crushing strength rather than beyond it. The theorem’s own worst point is the one place the material has to behave like a material.

And the calculation says exactly when the assumption fails, which the assertion does not. $H$ carries $1/r$, so flattening this arch to a rise of one twentieth of the span multiplies the thrust by 5.6 and takes the hinge stress to 12 N/mm² — at the crushing strength. A flat arch is a crushing problem wearing a geometry problem’s clothes, which is why the theorem’s clean results belong to the deep arches of the eighteenth century and not to the shallow ones of the nineteenth.

What a factor of two costs in bricks #

The geometrical factor of safety was named above and not spent. Spending it is the shortest route from this theorem to a bridge.

Heyman’s recommendation is a factor of 2 on thickness, and the limiting thickness for a semicircular arch under its own weight is about $R/17$. So a safe semicircular arch wants

$$t \ge \frac{2R}{17} = 0.118\,R$$

For a 5 m radius — a 10 m span — that is a ring 590 mm thick, which is two and a half rings of brick on edge, or a course of dressed stone of ordinary depth.

That number is worth holding beside the arches that exist. Old brick arch bridges of this span are built with two or three rings, and the reason has always been given as tradition or as workmanship. It is neither: it is the smallest ring that carries a factor of two on the only quantity the structure can run out of, and the builders arrived at it by two centuries of watching thinner ones fail.

The arch on this page can be asked the same question directly, because the generator finds the admissible range by search and the search can be run against the thickness as well as against the thrust. Thin the circular ring a hundredth of a span at a time and the corridor of admissible thrusts closes; it shuts entirely at 4.39% of the span, which on a 10 m span is a ring 439 mm thick.

Set that against the ring the other figures draw, which is 9% of the span. The geometrical factor of safety is 0.09/0.0439, which is 2.05 — and Heyman’s recommendation is 2. The arch drawn throughout this essay was not chosen to land there; it was chosen as an ordinary-looking ring, and it lands there because ordinary-looking rings are the ones that survived.

It also says what the factor is a factor on, which is the part that stays counter-intuitive. It is not a factor on the load — doubling the traffic does not halve it, because the thrust line’s position under a distributed load barely moves. It is a factor on the geometry, and what it protects against is everything that changes the geometry: an abutment that spreads, a ring that delaminates, a voussoir that weathers, a repair that adds weight in the wrong place. Those are the events that kill masonry arches, and a factor on strength would have covered none of them.

The cracked arch is the working arch #

The most useful practical consequence is that a masonry arch with open joints is not necessarily damaged.

When the thrust line touches the edge of the ring, the joint there opens on the opposite face — the stone is not in tension, it has simply stopped touching. That is a hinge. An arch can carry three hinges and remain a structure, because three hinges plus two abutments is a determinate arch, and a determinate arch is perfectly capable of standing. Only the fourth hinge turns it into a mechanism.

So the cracks that alarm an inspector are, in the majority of cases, an arch that has settled into a particular member of its family of thrust lines and told the world which one it chose. The relevant question is not whether there are cracks but how many hinges they represent and where they are.

Where the hinges are is a question the family of thrust lines answers before any crack has appeared. An arch that has drifted to the high-thrust end of its family has its line pressed against the intrados at the haunches; one that has drifted to the low-thrust end has it pressed against the extrados at the crown. Those are the joints that open, and which pair opens says which way the arch has moved.

A three-pinned arch is the same insight built deliberately rather than discovered. Putting three hinges in on purpose makes the thrust line pass through three known points, which is exactly enough to determine it, and the structure gives up its redundancy in exchange for an answer that does not depend on the abutments having stayed where they were built.

The load that moves, and why arches tolerate it #

A uniform load is the easy case, and the reason arches are interesting under traffic is that a point load somewhere along the span is not.

Adding a concentrated load changes $M_{\text{ss}}(x)$ from a parabola into a shape with a kink under the load, and dividing by the thrust carries that kink into the line. The line therefore develops a local excursion toward the intrados under the load and toward the extrados elsewhere, and the corridor has to accommodate both at once. This is why the worst position for a single wheel on an arch is not the crown, where instinct puts it, but somewhere around the quarter point — the crown load is resisted by a symmetric response the arch is shaped for, and the quarter-point load is not.

The question of where to stand a load to do the most damage is an influence-line question, and the answer for an arch is unusual in that the quantity of interest is not a stress but a geometrical clearance.

Where the model stops #

Sliding. The no-sliding assumption fails where the thrust line crosses a joint at a shallow angle, which happens near the springings of a very flat arch and at the extrados of a steep one. A sliding failure is not covered by any of this and is sudden.

Crushing. The low-stress assumption fails for a high arch carrying a great deal of fill, and for the very thin arches that modern brickwork makes possible. Where stress matters, the thrust line’s position stops being sufficient and its distribution across the joint has to be considered as well.

Spreading abutments. The theorem assumes the supports stay put. Masonry arches are extremely sensitive to abutment movement, because a spread of a few millimetres changes which thrust lines are geometrically available — the span lengthens, the rise falls, and the corridor narrows. Most masonry arch failures are abutment failures wearing an arch’s clothes.

One ring, acting alone. The analysis treats the arch ring as the structure. In a real bridge the spandrel walls, the fill and the parapet all carry load and all stiffen the ring, and the load path through them is a choice the analysis has made by ignoring them.

Three dimensions. A real arch bridge is a barrel vault with spandrel walls and fill, all of which stiffen it and none of which the two-dimensional analysis credits. The two-dimensional answer is conservative for that reason, and quite substantially so.

The figures cannot show the thing that decides the real behaviour, which is time. Every line drawn here is an instant; the arch has had two centuries of temperature cycles, traffic, and slow abutment creep, and it has been moving among the members of its family the whole while. A drawing of the admissible range is honest about what is possible and silent about what has happened.

The generalisation #

The move that makes this work — replace “what is the structure doing” with “is there any admissible state at all” — is the lower-bound theorem, and it is available wherever a material has a yield condition and enough ductility to redistribute.

That is why it turns up in plastic collapse of steel frames, where any bending-moment distribution in equilibrium with the load and nowhere exceeding the plastic moment guarantees the frame will not collapse. It is the licence behind the plastic hinge as a design tool rather than a curiosity, and behind the moment redistribution that continuous beams are permitted. It is why strut-and-tie modelling works for reinforced concrete: invent a plausible truss inside the member, check the struts and ties, and the theorem covers the rest. In all three cases the engineer is permitted to make something up, and the theorem converts the invention into a guarantee.

The intellectual price is precision. A lower-bound answer is safe and it is not the answer; the true collapse load may be considerably higher, and the better the invented state, the closer the bound. What is bought is enormous: a redundant structure whose real internal state depends on unknowable history becomes analysable by anyone with a pencil.

Robert Hooke stated the arch problem in 1675 as an anagram — ut pendet continuum flexile, sic stabit contiguum rigidum inversum, as hangs a flexible cable so, inverted, stand the touching pieces of an arch — and it took until Heyman’s 1966 paper for the existence version to be stated cleanly and the shape version to be demoted. Poleni had used a hanging chain to assess the dome of St Peter’s in 1748 and found the thrust line lay within the masonry, which is the safe theorem applied a little over two centuries before it was proved.

The ladder from here #

Later rungs on this anchor: the collapse mechanism of a masonry arch and the four hinges that make it. Abutment spread as the governing action rather than load. The barrel vault and the spandrel fill, which is where the two-dimensional model’s conservatism lives. Fill as a load and fill as a structure. The pointed arch, and what the shape does to the corridor. And the modern tied arch, where the horizontal thrust is taken by a tie rather than by the ground, and the whole question of abutment movement disappears along with the arch’s chief vulnerability.