Structural Engineering & Statics
where the load goes
Structural form

Held up by the air inside

A membrane has no bending stiffness at all, so the only thing that can hold it in tension is a pressure difference. The pressure needed to hold up a roof is smaller than the pressure a closed door makes — and the same pressure arrives at the foundation as hundreds of tonnes of uplift.

Assumes The surface that carries by being curved, The shape that carries itself, and the arch that is its reflection and The force nobody put in the model.

19 min read 8 figures The load must go somewhereGeometry beats materialScale changes everything

Every structure in this collection so far has had something in it that resists bending. A beam has a second moment of area, a shell has a thickness, a cable has none but is at least in tension by virtue of the load hanging on it.

A fabric roof has nothing. It is a sheet a millimetre thick, it will not carry the smallest bending moment, and it cannot even be relied on to be in tension — a slack membrane is a rag. The only thing available to keep it taut is a pressure difference across it, and once that exists the roof carries load by the same membrane equation everything else curved does:

N1R1+N2R2=q\frac{N_1}{R_1} + \frac{N_2}{R_2} = q

For a spherical cap of radius RR that gives N=qR/2N = qR/2 in every direction. For a cylinder it gives Nhoop=qRN_{hoop} = qR and nothing along the length, which is the factor of two that decides why air-supported cylinders are always cabled and air-supported domes are not.

Held up by a pressure nobody can feelAn air-supported roof of 60 m span and 9 m rise. The membrane has no bending stiffness whatever, so the only thing that can hold it in tension is a pressure difference, and the pressure has to exceed the load per unit **plan** area and nothing else: 0.25 kN/m² of fabric plus 0.6 of snow is 0.85 kN/m², so 1.19 kN/m² does it — 1190 pascals, which is 1.17 per cent of an atmosphere and 121 millimetres of water. Ears do not notice it. A door does: at 2.1 kN on an ordinary leaf, the building needs an airlock rather than a handle. And the whole of it arrives at the foundation as 3365 kN of uplift — 17.9 kN on every metre of perimeter — which is the bill the pressure's smallness conceals.18 kN/m18 kN/mN = 9.3 kN/m1190 Pa121 mm of water60 mtotal uplift 3365 kNworst membrane force when the snow melts
Fig. 1 A 60-metre air-supported dome, with the pressure that holds it up, the membrane force that follows and the uplift the same pressure delivers to the foundation.

Which free body produced the number

The pressure is set by one free body and it is the simplest on the site: the whole roof.

Cut round the perimeter and take the cap. Acting on it are its own weight, whatever is lying on it, and the internal pressure over the whole of its underside. The vertical resultant of a uniform pressure on any surface is the pressure times the plan area it covers, whatever the shape of the surface — which is the same statement that makes a basement’s uplift depend on its footprint rather than on its shape. So

p⋅πa2≥(g+s)⋅πa2⟹p≥g+sp \cdot \pi a^2 \ge (g + s)\cdot \pi a^2 \qquad\Longrightarrow\qquad p \ge g + s

and the area has cancelled. The pressure needed to hold up an air-supported roof is the load per unit plan area and nothing else — not the span, not the rise, not the fabric, not how big the building is. A fabric at 0.25 kN/m² under 0.6 kN/m² of snow wants 0.85 kN/m², whether the dome is thirty metres across or three hundred.

That is 850 pascals. It is 0.85 per cent of an atmosphere and 87 millimetres of water gauge. Nobody’s ears register it, an ordinary fan maintains it, and the reason such buildings run at more than the minimum — typically 250 to 500 pascals in fair weather, raised under snow — is to keep the membrane taut with a margin rather than to hold it up.

The number the smallness conceals

The same pressure acts on the whole plan area and every newton of it has to be held down at the perimeter.

For a sixty-metre dome at the design pressure that is about 3,400 kilonewtons of uplift, arriving as 18 kN on every metre of perimeter. It goes into a ring beam, and out of the ring beam into ground anchors, a mass foundation or a tension pile — and that anchorage, not the fabric, is where an air-supported building’s structural cost is.

The pressure is negligible and the anchorage is not, and the reversal is exactly the one a basement slab under groundwater presents: a pressure too small to feel, multiplied by an area large enough to lift the building. The arithmetic is the same and so is the failure mode — a structure that is perfectly adequate against every load it was designed for and floats.

A beam, its loads and its reactionsA free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other.1289.510.5ΣM about one support gives the other reaction; ΣF then gives the first
Fig. 2 The free body that settles it. Choosing the whole roof rather than a piece of it is what makes the pressure requirement one line, and it is the same choice that makes an uplift check one line.

The rise, which is the only variable there is

There is no depth to a membrane, so the usual structural lever — make it deeper — is not available. What is available is the radius, and it is set by the rise.

For a cap of half-span aa and rise ff,

R=a2+f22fR = \frac{a^2 + f^2}{2f}

which for a shallow cap is close to a2/2fa^2/2f: halve the rise and double the radius, and with it the membrane force. A hemisphere has R=aR = a and the smallest membrane force any cap can have.

The flatter the roof, the harder the fabric worksThe membrane force in an air-supported cap of 60 m span against its rise, at a net pressure of 340 pascals. A membrane carries load only by curvature — N/R is the pressure it can take — so flattening it lengthens the radius and the force rises in proportion. At a rise of 0.30 of the half-span the fabric carries 9.3 kN/m against a strength of 100; at half that rise it carries 39.7. The curve is why air-supported roofs look the way they do: the shape is not an aesthetic decision and it is not a structural depth either, because there is no depth. It is the radius, and the radius is the only variable there is.00.20.40.60.80102030rise ÷ half-spanmembrane force (kN/m)fabric strength 100 kN/m,off the top of this axisas drawnN = qR/2 with R = (a² + f²)/2f
Fig. 3 Membrane force against rise, at a fixed net pressure. There is no depth to trade, so the curve is the whole of the structural design.

That is why air-supported roofs are shaped the way they are — not for appearance and not for any spanning efficiency, but because the rise is the only structural variable and a flat one is unbuildable. The practical rises are between a fifth and a third of the half-span, and the lower end of that is set by the fabric strength while the upper end is set by wind: a taller dome catches more of it.

The load case nobody would write down

Here is the reversal that makes this subject worth an essay. The membrane carries the net outward pressure, q=p−(g+s)q = p - (g + s), so raising the pressure to cope with snow is raising it against a load that will not always be there.

At the design pressure with the snow on, the net pressure is small and the fabric force is small. Take the snow off and leave the pressure where it is, and the net pressure trebles. The worst membrane force in the life of the structure occurs during a thaw, at a moment when nothing at all is happening to the building.

The hoops change their mind at an angle no proportion choseThe two membrane forces of a spherical dome of radius 30 m under 3 kN/m² of surface, taken from the crown to a base at 60°. The free body for the meridional force is the cap above a cone of half-angle φ, and vertical equilibrium of it gives N_φ = −wR/(1 + cos φ) directly: -45.0 kN/m at the crown falling to -60.0 at the base, compression everywhere. Equilibrium normal to the surface then gives the hoop force, which starts at -45.0 kN/m and reaches 15.0 — it changes sign, and the angle at which it does was found here by bisecting N_θ rather than quoted: 51.827292°. Setting N_θ = 0 gives cos²φ + cos φ − 1 = 0, so cos φ is (√5 − 1)/2, the reciprocal of the golden ratio — an identity this site's solver gate checks against the bisection to nine decimals rather than asserting, because it is too pretty to be believed on sight. Below that parallel the hoops are in tension, which masonry has none of, and that is where every old dome is cracked.0102030405060-60-40-20020angle from the crown (degrees)membrane force (kN/m)N_θ = 0 at 51.8273°meridional N_φ-60.0 kN/mhoop N_θ15.0 kN/mcompression abovetension below
Fig. 4 The membrane forces in a dome under a load that acts on it. A pneumatic roof’s load acts in the opposite direction, which is why the case that governs is the one where the load goes away.

No structure in this collection has that property except by accident. It exists because the pressure is a control variable rather than a load — something the building operates rather than something that happens to it — and the consequence is that an air-supported building is a machine with a control system, not merely a structure. The pressure has to come down as the snow melts, which means somebody or something has to know that it is melting.

And a power failure is a collapse. Not a serviceability problem: the roof deflates, comes down on whatever is under it, and has to be re-inflated with the snow removed. That is why every air-supported building of any size has redundant blowers on standby power, and why the structural engineering of the form has a mechanical engineering half that no other structure here does.

The cables that make big ones possible

The membrane force qR/2qR/2 grows with the radius, so a very large air-supported roof would need a very strong fabric. It does not, because the fabric’s radius is not the roof’s.

Lay a net of cables over the membrane on a grid, anchored to the same ring beam. The fabric between the cables bulges to a small radius of its own — set by the cable spacing and how much it is allowed to billow — and its membrane force falls with that radius rather than with the roof’s. The cables then carry what the fabric hands them, and they are far better at it, because a cable at high tension is a cheap way to carry a line load.

The net is nearly linear right up to the moment half of it lets goLoad against centre deflection for a 36 m square net of cables at 2 m centres, a sagging family 1.8 m deep and a hogging family 1.8 m high, pretensioned to 400 kN. The tangent stiffness at the origin is 13.00 kN/m³ and the curve barely bends: at the design load of 1 kN/m² the centre has moved 76.8 mm. What ends the story is not a stress. At 488 mm the hogging family's tension has fallen to zero and it goes slack, which happens at 6.54 kN/m² — 6.5 times the design load. Past that point half the net has stopped working and the rest has to find the whole load by sagging, so the real limit on a cable roof is a loss of geometry rather than a want of strength.010020030040050001234567deflection at the centre of the net (mm)load on the roof (kN/m²)the hogging family goes slack: 6.54 kN/m²6.5× the design loaddesign: 1 kN/m² at 76.8 mmk₀ = 13.00 kN/m³ at the origin
Fig. 5 A prestressed cable net, whose stiffness comes from the geometry rather than from the material. A cable-restrained pneumatic roof is the same idea with the pressure supplying the prestress.

At a three-metre cable spacing the fabric’s radius is an order of magnitude below the roof’s, and the fabric stress falls by the same factor. That is what made the large air-supported roofs of the 1970s and 1980s possible — the Pontiac Silverdome, the Metrodome, the Tokyo Dome — all of them fabric over a cable net, all of them at pressures of a few hundred pascals, and all of them spanning distances no unrestrained membrane could have.

It also introduces the failure mode those buildings are remembered for. A cable net over a membrane creates pockets, and snow drifts into pockets. A local accumulation raises the load in one place, the net pressure there goes negative, the membrane goes slack, and the pocket deepens — which collects more snow. That is a snap-through in a form that has no bending stiffness to resist it, and it is what tore the Silverdome’s roof in 1985 and the Metrodome’s in 2010.

A load with a maximum in it, and nothing bifurcatesLoad against apex movement for a two-bar frame of half-span 1000 mm and rise 150 mm. The load rises to 133.4 kN at a movement of 64 mm — well short of the 150 mm that would bring the apex level — and then falls. Past that point the frame can only be held by taking load away, so under a dead weight it goes: 260 mm of movement at constant load, arriving inverted and in tension. The minimum on the path is -133.4 kN, the exact negative of the maximum, because the geometry is symmetric about the flat position and the arithmetic knows it.050100150200250300350-150-100-50050100150movement of the apex (mm)load (kN)limit point: 133.4 kNit jumps 260 mmas builtat the limitafter it goes
Fig. 6 An equilibrium path that turns over. A membrane pocket that deepens as it fills is on the falling branch of one of these, and there is nothing in the structure to catch it.

Where the form came from, and where it went

The mechanism is older than the buildings. A pneumatic tyre is an air-supported structure, and so is a balloon; what took a century was fabric that would hold pressure without leaking faster than a fan could replace it.

Frederick Lanchester patented an air-supported field hospital in 1917 and built nothing, for exactly that reason. The form arrived in quantity with Walter Bird’s radomes in the late 1940s — spherical fabric enclosures over radar dishes, where the requirement was a large volume of clear space with nothing structural in the signal path, and where nothing but a membrane would do. Radomes are the perfect case: small, uniformly loaded, and with a customer for whom an internal column was not a compromise but a failure.

The stadium roofs of the 1970s took the same idea two orders of magnitude larger and needed the cable net to do it. They were built because they were cheap: a fabric roof over a cable net cost a fraction of a long-span steel roof and went up in months. Most of them have since been replaced, and the reason is not the mechanism but the operating cost and the snow. A structure that has to be run rather than merely maintained turns out to have a whole-life cost that the erection saving does not cover.

Where the form has stayed is where the alternative is worse: temporary and semi-permanent enclosures, sports halls, warehouses, swimming-pool covers, and anywhere a very large clear volume is wanted for a limited time. The engineering has not changed; the economics decided the application.

The other pneumatic form, which is not air-supported at all

There is a second family that shares the mechanism and not the arithmetic, and it is worth distinguishing because the two get called by the same name.

An air-supported structure pressurises the space people occupy: one membrane, the building’s whole volume behind it, and an airlock at every door. An air-inflated one pressurises only the structure — tubes, cushions, an arch made of a fabric torus — and leaves the occupied space at atmospheric pressure. The second needs a far higher pressure, because the member is small: the same membrane equation with a radius of half a metre instead of fifty needs a hundred times the pressure to reach the same force.

That trade is the whole of the difference. An inflated arch at 30 kilopascals is a genuine structural member with a bending stiffness of its own — it resists by having part of its section go slack under bending, which is a moment–curvature relation like any other and reaches a limit when the compression side reaches zero pressure. The building under it needs no airlock and no pressurisation, and a leak deflates one member rather than the building.

Cushion roofs — three-layer transparent pillows on a steel grid — are the same family, and are now far more common than air-supported roofs. The pressure is a few hundred pascals, the panels are small, and the structure carrying them is ordinary steelwork. The membrane has stopped being the structure and become the cladding, which is where most of this idea has ended up.

Where the model stops

Wind is the load this essay has not mentioned and the one that decides most designs. A dome in wind has pressure on the windward face and suction over most of the rest, and the suction helps — it adds to the net outward pressure and tightens the membrane. The windward face is the problem: a positive external pressure there can exceed the internal one, the membrane goes slack, and a slack membrane in a moving airstream flutters. The internal pressure therefore has to be set against the worst local external pressure rather than against an average, and that is a wind-tunnel question rather than an arithmetic one.

The membrane theory assumes the shape is the shape. Everything above takes the cap as a sphere of known radius, and a real membrane under an unsymmetrical load is not: it changes shape until it can carry what is on it, exactly as a cable does. The analysis of a real fabric roof is therefore geometrically non-linear from the start, with the shape as an unknown, and the closed forms here are the answer for the symmetric case only.

And a fabric is not a material in the sense the rest of this site means. It is a woven fibre with a coating, its stiffness differs by a factor of several between warp and fill, it creeps, and its strength falls with age and ultraviolet exposure. Nothing in this collection’s material section is quite as anisotropic or as time-dependent, and the design stresses used are a small fraction of the tested strength for exactly that reason.

The bending the membrane theory denied, and how far in it reachesBending stress in the wall of a 30 m dome 100 mm thick, along the meridian inward from a fully restrained edge. A membrane solution has two force resultants and no bending, so it cannot satisfy a real boundary condition: the free edge here wants to move out by 0.29 mm and a ring beam does not let it. Closing that gap costs 0.52 MPa of bending at the ring, against 0.60 MPa of membrane stress in the same wall. Near the edge the meridian is a beam on an elastic foundation — flexural rigidity Et³/12(1 − ν²), foundation modulus Et/R² — so it obeys the same fourth-order equation, and the disturbance is down to four per cent of itself at π/β = 2.4440·√(Rt) = 4.23 m. The textbook's “about 2.45√(Rt)” is a rounding of exactly that. Past three of those lengths the shell has forgotten the edge entirely. The wall is drawn 12 times its true thickness.4.23 m = 2.4440·√(Rt)edge bending 0.52 MPamembrane 0.60 MPanothing left of it herethe same fourth-root length a beam on an elastic foundation uses, from an entirely different structure
Fig. 7 Where the bill is presented in a curved surface: at the edge, where the membrane state has to be handed to something that can take bending. A pneumatic roof’s version of that edge is its ring beam and its anchors.

The generalisation

There is one idea in this essay that is not about air at all, and it is worth separating from the rest.

A structure with no bending stiffness has to be prestressed, and the prestress is what makes it a structure. Take the pressure away from a pneumatic roof and it is a sheet. Take the pretension out of a cable net and it is a bundle of wires. Take the pretension out of a spoked wheel and it is a hoop and some sticks. In each case the prestress is not an improvement to a structure that already works — it is the thing that makes the assembly capable of resisting anything at all, and the design quantity is how much of it there is rather than how strong anything is. It is the same relationship a prestressed concrete section has with its tendons, one field further along.

That is a different relationship with force from the one the rest of this site describes. Everywhere else, a load arrives and a structure resists it; here, the structure has to be holding itself in a particular state before a load can be resisted, and the state has to be maintained. The state can be lost — to creep, to relaxation, to a leak — and losing it is not a reduction in capacity but the disappearance of the structure.

A line of thrust, and the masonry it has to stay insideAn arch ring of 9% of the span in thickness, rising 28% of the span, under its own weight as a uniform load. Any horizontal thrust between 3.85 and 5.23 puts a line of compression entirely inside the masonry, so the arch stands — and which of them it actually takes is not decided by statics. The two extremes are drawn: the minimum-thrust line, which rides high at the crown and low at the haunches, and the maximum-thrust line, which does the opposite.thrust anywhere from 3.85 to 5.23 fitsH = 3.85, leastH = 5.23, most
Fig. 8 The compression form of the same idea. An arch’s thrust line has to stay inside the material; a membrane’s stress has to stay in tension. Both are conditions on a state rather than on a strength.

Which is why the honest description of an air-supported building is not that it is held up by air. It is that the air holds it in tension, and the tension holds it up.

Where this goes next

0102030405060-60-40-20020angle from the crown (degrees)membrane force (kN/m)N_θ = 0 at 51.8273°meridional N_φ-60.0 kN/mhoop N_θ15.0 kN/mcompression abovetension below The surface that carries by being curved A flat plate spends its thickness on a lever arm of a few millimetres. Curve the same sheet and the load is carried in the surface itself, at a thirtieth of the thickness — and the entire bill is presented at the edges. 00.20.40.60.811.2020406080100120140160midspan sag (m)total load on the cable (kN)the design load, 150 kNsolved 0.740 m1.125 mtangent here 341.2 kN/mk₀ = 8T₀/L = 133.3 kN/mthe flat-cable law The stiffness that comes from the shape A cable has no bending stiffness whatever, and it still holds up a roof. What resists the load is the change of its own geometry, so its stiffness is a function of the tension already in it — and prestress buys stiffness that no change of material could. water table39.2 kN/m² upward, everywhereweight 18.7 MN6 muplift 23.54 MN · weight 18.72 MNfactor against flotation 0.80no strength appears anywhere in that ratio A basement is a boat Every load in this collection presses down and is resisted by strength. Hydrostatic uplift presses up, is resisted by weight, and does not care what is built on it — so the check contains no material property at all. It is a ratio of two weights, and one of them is water. 00.511.522.5020406080100120140deflection ÷ thicknessload the strip carriesarchingcatenaryyield line× 3.89 at 0.24tnothing to arch against The force nobody put in the model A slab strip whose ends cannot move apart is not the strip in the yield-line calculation. Deflecting shortens the chord between its ends, the ends do not come in, and the strip is forced into an arch — worth four times the load it was designed for, at a movement nobody would see.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

AnchorageCable netCurvatureEquilibriumFree bodyFunicularHoop forceLoad pathMembrane actionPneumatic structurePrestressServiceabilityShell actionSnap throughUplift