Arches, Vaults, Domes & Shells

Hang a chain loosely between two points and it settles into a graceful sagging curve - the catenary. That curve is not arbitrary: it is the exact shape in which the chain can support its own weight using nothing but tension, because a chain has no stiffness and cannot resist bending, so it is forced to find a shape where only pull is needed. Change the loading - hang weights along it - and the chain instantly adjusts to a new curve that keeps every link in pure tension. This load-following shape is called the funicular shape, from the Latin for rope.

Now perform the single most beautiful move in structural thinking: take that hanging tension curve and flip it upside down. The mirror-image curve carries the same loads in exactly the same way, but now every particle is in pure compression instead of pure tension. That flipped catenary is the ideal arch. A masonry arch built to the funicular shape of its load carries that load with no bending at all, purely by the stones pressing on one another - which is why it can be so thin and why it can be built from a material like stone or brick that has almost no tensile strength. Antoni Gaudi famously designed the funicular forms of the Sagrada Familia with upside-down hanging models of strings and weighted bags, letting gravity compute the perfect compression shapes for him.

The lesson generalises. For any given set of loads there is a particular funicular curve, and a form-active structure works best when its shape matches that curve. A pointed load wants a pointed arch; a uniform load wants a parabola; self-weight wants a catenary. Real arches, being stiff, tolerate some departure from the ideal - the internal thrust line simply shifts within the thickness of the arch - but the closer the form follows the funicular, the thinner and more efficient the structure can be, and the further any departure, the more bending creeps in and the thicker it must become.

An arch spans by turning downward load into a chain of compression that flows around its curve and down into its supports. The wedge-shaped stones, the voussoirs, press against each other so tightly that the whole ring behaves as one; the stone at the very top, the keystone, locks the ring closed. Because the forces stay in compression - stone's strong direction - an arch can span far more than any stone lintel, which would crack the instant its underside went into tension. This is why the arch was such a liberating invention: it let masonry, a compression-only material, leap across openings and rivers and naves.

But compression flowing around a curve cannot simply arrive vertically at the base. It arrives at an angle, which means the arch pushes both down and outward on its supports. This outward push is the thrust, and it is the defining challenge of every arch, vault and dome. Left unresisted, the thrust spreads the supports apart, the arch flattens, and it collapses - many a ruined arch failed not because the stone was overloaded but because its abutments moved. An architect who draws an arch has, in the same stroke, committed to resisting a horizontal force at its base, and where that force goes is as much a part of the design as the curve itself.

There are three classic ways to catch the thrust. The first is mass: thick abutments or piers heavy enough that the combined line of their weight and the thrust stays safely within them - the strategy of Roman aqueducts and thick-walled vaulted halls. The second is the buttress, a mass of masonry placed specifically to receive the thrust and carry it to the ground; the Gothic flying buttress is the spectacular refinement, reaching over the aisles to prop the high nave vaults and letting the walls between dissolve into stained glass. The third, and the neatest for a single arch, is the tie: a tension member (a steel rod or bar) strung across the span from base to base, so the outward thrust of the two springings simply pulls against the tie and cancels, needing no heavy abutment at all. The tied arch is everywhere in modern building precisely because it internalises the thrust.

Extend an arch and you get the great roofing forms of masonry. Stretch an arch sideways into a continuous tunnel and you have the barrel vault (or tunnel vault): a half-cylinder that roofs a long space in compression, but one that thrusts outward along its entire length, so it needs continuous support - thick side walls or a row of buttresses down each flank. Cross two barrel vaults at right angles and you get the groin vault, which concentrates the whole roof's thrust neatly onto four corner points, freeing the walls between - the discovery that let Roman and later Gothic builders open up their side walls. Add ribs along the groins and you have the ribbed vault that defines Gothic architecture, where slim stone ribs carry the loads to the piers and thin webs merely fill between them.

Rotate an arch around a vertical axis instead of stretching it, and you get the dome - and a dome behaves in a fascinatingly different way from an arch. It carries load in two directions at once: meridional forces run down the curve from crown to base like the lines of longitude on a globe, always in compression, while hoop forces run around the dome like the lines of latitude. Near the top the hoops are in compression, squeezing the shell together; but lower down, towards the base, the hoops go into tension as the dome tries to splay outward. This is why old masonry domes so often crack in vertical lines near their base (the tension exceeding what mortar can hold) and why they were ringed with chains or heavy drums to contain the hoop tension - the iron chain around the base of St Peter's and the great stone drum of the Pantheon are both devices to catch exactly this outward hoop thrust.

The dome's two-way action makes it more stable and more efficient than a single arch, because load can find many paths around it, and it is why a dome can be relatively thinner than a barrel vault of the same span. But the same outward thrust rule applies, gathered now into a ring: a dome must either sit on a massive drum and walls, or be tied around its base by a tension ring that swallows the hoop thrust. From the Pantheon to the Mughal tombs to the geodesic domes of the twentieth century, every dome is a negotiation between compression flowing to the ground and the tension needed to stop the base from bursting.

The twentieth century took the funicular idea and gave it a new material - reinforced concrete - and a new form: the thin shell. A shell is a curved surface, usually of concrete only a few centimetres thick, that carries load almost entirely as in-plane forces (compression and tension spread through the surface, called membrane forces) rather than by bending, exactly as a curved eggshell does. Because it works by its shape rather than its bulk, a shell can be breathtakingly thin relative to its span - proportionally thinner than an eggshell - and roof huge column-free spaces with a sculptural, continuous surface that is both structure and roof at once.

The masters of the shell made it an art. Pier Luigi Nervi ribbed his concrete domes and roofs with elegant patterns that traced the flow of force; the Spanish-Mexican engineer Felix Candela built swooping hyperbolic paraboloid (hypar) shells - saddle-shaped surfaces that are doubly curved yet made of straight lines, so they could be formed with straight timber shuttering - roofing markets and restaurants with concrete barely four centimetres thick. Heinz Isler in Switzerland used hanging cloth models, frozen and inverted (the funicular idea again), to find pure-compression free-form shells of astonishing thinness and beauty. And concrete lets a shell do what masonry cannot: because reinforced concrete can take tension, a shell can resolve its own edge and hoop tensions internally, with steel, rather than needing massive external buttressing.

Shells are among the most efficient structures ever devised, but they are honest about their difficulties. Their power depends on double curvature - a surface curved in two directions is vastly stiffer than one curved in a single direction, which is why a flat sheet flops but a curved one is rigid, and why singly curved barrel shells are weaker than doubly curved domes and hypars. Thin shells are also prone to buckling (a compression surface can suddenly snap inward if too thin or too flat), sensitive to construction accuracy, and expensive in the elaborate formwork the curves demand - which is one reason the great age of concrete shells faded as labour costs rose. Yet the form-active idea they embody, computing structure from the flow of force, is timeless, and digital design and new fabrication are quietly bringing shells back.

Three principles let an architect design honestly with arches, vaults, domes and shells. The first is match the form to the load. These structures reward you with extreme thinness only when their shape is close to the funicular curve of the loads they carry, so the elegant curve is not a stylistic flourish - it is a structural calculation made visible. A shape chosen purely for looks, far from the funicular, drags bending into the structure and forces it to thicken, losing everything that made the form worthwhile. The best form-active buildings are ones where the eye is really seeing the flow of force frozen into shape.

The second principle is always resolve the thrust, and resolve it visibly and deliberately. Every arch, vault and dome pushes outward, and that push must be caught by mass, by a buttress, or by a tie - there is no exception and no way to wish it away. Historically the thrust generated some of the most expressive architecture ever built (the flying buttresses of Chartres are the thrust made into poetry); today it is most often caught quietly by a steel tie. Either way, the mature designer knows that drawing the curve is only half the job and locating the thrust resistance is the other half, decided with the engineer from the first sketch.

The third principle is honesty about the trade-offs. Form-active structures are peerlessly efficient in material and can be sublime to occupy, but they demand curved, often bespoke construction - expensive formwork for concrete, skilled masonry for stone, careful geometry throughout - and they resist the easy repetition of a rectangular frame. They are therefore best chosen where the span is large, the form is celebrated, and the building can justify the craft: a place of worship, a terminal, a pavilion, a market hall. Used there, with the form matched to the load and the thrust firmly caught, they remain the most efficient and moving way to enclose great space that architecture has ever found - and they lead naturally to the last idea of this module, structures that keep only the tension half of the funicular curve: cables and membranes.