Two curvatures of opposite sign

Hang a chain between two points and it takes a shape. Load the middle of it and it takes a different shape. Load a quarter point and it takes a third shape, with a kink in it, and none of those changes required any force at all beyond what the load itself supplied.

That is not a structure. It is a mechanism — an arrangement that changes geometry freely — and the reason it appears to work is that the load it was drawn for happens to be the load it is carrying.

Cross the chain with a second family of cables curved the other way and something changes completely. Not because the second family carries load — under a downward load it is being unloaded — but because it refuses to let the first family change shape without being stretched.

Which free body produced the number #

Take a node where a sagging cable crosses a hogging one and cut both. Four forces act on the node from the cables plus whatever load is applied there.

Vertical equilibrium of that node is the whole of the argument. The sagging cable’s two ends leave the node at slightly different angles, so their resultant is downward if the cable is sagging — it can push the node up. The hogging cable’s resultant is upward if it is rising — it can push the node down. Both resultants are the deviation force $H\kappa$ per unit length that a curved tension path applies, and the two curvatures have opposite signs.

So the node’s vertical equilibrium is

$$H_s\kappa_s + H_h\kappa_h = p$$

with the two terms opposing each other. A downward load is carried by the sagging family increasing its tension and the hogging family decreasing its own — and the second half of that is what makes the net stiff, because it is a resistance that exists before the load arrives.

Why the pretension is the structure #

Differentiate the equilibrium above with respect to the deflection and the tangent stiffness of a net of spacing $s$ and chord $L$ comes out as two terms:

$$k_0 = \underbrace{\frac{8(H_s + H_h)}{sL^2}}_{\text{geometric}} + \underbrace{\frac{16EA(f_s^2 + f_h^2)}{sL^4}}_{\text{elastic}}$$

The first term contains the pretension and no material property whatever. The second contains the cables’ axial stiffness and the sags, and for a flat net it is small.

That is the fact worth carrying. A cable net’s stiffness is mostly bought with force, not with steel. Doubling the cable area raises the second term and leaves the first alone, which on a typical net is a few per cent. Doubling the pretension nearly doubles the whole thing, using exactly the same cables.

There is a limiting case that makes the point sharply. Set the pretension to zero and the first term vanishes; the net still has cables, still has area and still has a modulus, and its stiffness at the origin is the elastic term alone — which for a flat net is nearly nothing. A slack net is a bag. Strong enough and still falls over is the same distinction on the other side of the ledger: strength and stiffness are separate properties, and here the strength is entirely in the steel and the stiffness is entirely in the force.

The stiffness that comes from the shape is the general version of this, and a net is where it is least disguised. The stiffness is geometric: it exists because a taut curved line resists being moved off its curve, and the resistance is proportional to how taut it is.

The same statement in different words is that a net is a structure carrying a self-stress — a set of internal forces in equilibrium with no applied load at all. The forces that are there with nothing applied is that idea in general; here the self-stress is not an accident of redundancy but the entire design intent, and the pair of pretensions is not free: choosing the sagging one fixes the hogging one, because the two have to balance at every node with nothing applied.

The surface, and why it has to be a saddle #

A net’s geometry is not chosen; it is found, and the condition it has to satisfy is the equilibrium above with $p = 0$.

Write it out at every node and the result is a statement about curvature: at every point of the surface, the two families’ curvatures must have opposite signs, in the ratio of their tensions. A surface whose two principal curvatures have opposite signs is anticlastic — negative Gaussian curvature, a saddle — and it is the only kind of surface a prestressed net can take.

That is a strong restriction and it is the reason cable-net roofs all look alike. A dome cannot be a net: both its curvatures have the same sign, so both families would have to be in tension in the same direction and there is nothing to balance them. A flat surface cannot be a net either, because zero curvature means zero deviation force and the pretension has nothing to react against.

The surface that carries by being curved covers the compression case. The two are mirror images: a shell is a synclastic surface in compression and a net is an anticlastic surface in tension, and both are carrying load with no bending because their geometry is doing the work.

There is a third member of the family, and it takes the other way out of the restriction. Held up by the air inside is a synclastic tension surface, which is impossible without something pushing outward from within — so an air-supported roof replaces the second cable family with a pressure, and the design problem becomes keeping the pressure on.

One more consequence of the restriction is worth drawing out, because it explains why these roofs are so often described by their masts. An anticlastic surface needs its two curvatures held apart, and the simplest way to hold them apart over a large span is to push one region up and let the neighbouring region hang — a mast under a high point, cables to low anchorages either side. The mast is not carrying the roof in the way a column carries a floor; it is establishing the geometry that lets the roof carry itself, which is the same job the polygon that finds the shape does for an arch. Take the mast away and the surface is not overloaded, it is impossible.

Form finding, which is a computation and not a drawing #

Because the shape is fixed by the self-stress condition, it cannot be drawn. It has to be computed, and the computation is unusual: the unknowns are the coordinates, the equations are equilibrium, and the material properties do not appear.

The classical method is force density: assign each cable segment a ratio of force to length, and the equilibrium equations become linear in the coordinates. Solve them and the surface falls out. Change the force densities and a different surface falls out, so the designer’s variable is a set of ratios rather than a set of points — which is why designing a net feels less like drawing and more like tuning.

This is where a net differs from every other structure in this collection in a way worth stating plainly. Its geometry is a result rather than an input. A beam’s span is chosen; a truss’s node positions are chosen; a net’s surface is the answer to an equilibrium problem, and a designer who insists on a particular shape is specifying a self-stress state that may not exist.

The load that ends it #

A net’s failure is not a strength failure, and the point at which it stops being a structure arrives well before anything breaks.

Load it downward and the sagging family’s tension rises while the hogging family’s falls. Keep going and the hogging tension reaches zero. At that deflection the second family goes slack, and everything the second family was providing disappears at once: the geometric stiffness halves or worse, the restraint against pattern loading vanishes, and the net becomes what a single family always was — a mechanism, finding whatever shape the load asks for.

So the design case is not the largest load. It is the load that unloads the hogging family, and it can be a long way below the strength of anything. A net designed on stress alone can be perfectly safe and completely unserviceable, flogging in the wind at a fraction of its capacity.

The remedy is the pretension again: the higher the hogging family’s initial force, the more downward load it takes to reach zero. That puts pretension in the position of doing three separate jobs — supplying stiffness, holding the shape, and providing the reserve before slack — and it is why a net’s pretension is usually governed by the third rather than by the first.

It is worth being precise about what “slack” means, because it is not the same as “broken” and the difference decides how a net is checked. A slack cable has lost its tension and is still there: it hangs, it has no stiffness, and it takes force again the moment the deflection reverses. So a net past its knee has not failed — it has changed into a different structure, reversibly, and it will change back. What it cannot do while it is in that state is resist the next thing that happens, and the next thing that happens to a light roof in a storm is usually an uplift, at which point the two families swap roles and the sagging one goes slack instead.

That symmetry is the reason a net’s design cases come in pairs. Every check has a downward version and an upward version, the two govern different families, and a pretension chosen to keep the hogging family taut under snow may be the wrong pretension for keeping the sagging family taut under wind. There is usually a window and it is not always wide.

What holds the edges #

Every force in a net is a tension, and tensions have to be anchored. A net’s boundary carries the sum of everything inside it, and the boundary is where a tension structure becomes an ordinary one.

Three arrangements are used and each has a different failure. A rigid ring in compression closes the force system on itself — the ring carries the net’s pull as a compression round its own circumference, and the whole roof is then self-anchoring, needing nothing from the ground but its own weight. A cable edge between masts spreads the pull along a curve, so the boundary is itself a funicular and its shape is another form-finding problem. And direct anchorage takes the tension into the ground, which is a foundation problem of a kind buildings rarely have: a permanent uplift, resisted by mass or by ground anchors.

The ring case is worth noticing because it is the same idea as a self-stressed net, one level up: the compression ring and the tension net are a single self-equilibrating system, and the whole roof exerts no net force on its supports at all beyond gravity. That is an unusual property for a large structure, and it is why several of the best-known cable roofs sit on very light perimeter columns.

Where the model stops #

The net was treated as a grid of straight segments between nodes. A real net’s cables are continuous through their clamps and can slip, which redistributes force along a cable and changes the stiffness — usually favourably, and unpredictably.

Only two families were considered. A triangulated net has three, which makes it stiffer in shear and much harder to form-find, and a fabric membrane has a continuum of directions and a shear stiffness of its own.

Nothing here is dynamic. A light roof with almost no mass and a low stiffness has a natural frequency in the range the wind excites, and the wind on a flexible curved surface is not a static pressure. That is a coupled problem and it has decided the design of several of these roofs.

Nothing here is about how the net is built. A prestressed structure has to be got into its prestressed state, and the sequence of tensioning is a design problem in its own right: the shape at every intermediate stage is a different equilibrium, the forces in the partly tensioned net are not fractions of the final ones, and the geometry only closes at the end. The structure that was never complete is the general case; here the erection states are not merely different, they are mechanisms.

And the pretension does not stay where it was put. Cables relax, clamps settle, anchorages creep, and temperature changes the force in a member whose stiffness is its force. A net that has lost a fifth of its pretension has lost a fifth of its stiffness, and there is no visible sign of it.

The generalisation #

The idea to carry away is that stiffness does not have to come from a material.

Every other stiffness in this collection is $EA$ or $EI$ divided by a length: a property of the stuff, times a property of the shape. A net’s is a force divided by a length squared, and the force is one the designer chose. That makes it adjustable after the structure exists, which nothing else here is, and it makes it disappear if the force is lost, which nothing else here does either.

The third is a warning about intuition. Nearly everything in this collection is a structure whose stiffness a designer can estimate by looking at it — a deep beam is stiffer than a shallow one, a thick wall than a thin one, a short column than a long one. A net’s stiffness is invisible: two nets of identical geometry and identical cables can differ by a factor of five in stiffness and look exactly the same, because the difference is a force. The only way to know what a cable structure will do is to know what was put into it, and the only record of that is a tensioning schedule.

The second idea is about mechanisms. A count of members and restraints says a net is wildly deficient — it has far more freedoms than equations — and it stands up anyway. The count that does not see it is the essay about that failure of counting, and a prestressed net is the cleanest example of its opposite: a mechanism that is stiff, because a self-stress state exists that its motions would have to work against.