The stiffness that comes from the shape

Cut a cable anywhere and look at what can cross the cut. A force along its own length, and nothing else. No moment, because a moment needs a section with depth and this one folds. No shear, for the same reason. Whatever holds the roof up has to be assembled out of that single component, and the material has no say in the matter.

So a cable pushed sideways cannot resist by getting stiffer in the way a beam does. It resists by moving — by taking up a shape in which the force along its length has a component pointing back against the load. The resistance is a property of the geometry it has adopted, not of the steel it is made of, and the geometry is different at every load.

That curve is the whole essay. It rises, it steepens, and everything on this site that bends gives a straight line instead.

The stiffness is in the shape, and the shape is in the load #

The claim, stated once and plainly: a cable’s transverse stiffness is geometric, and it is proportional to the tension already in it. Put no tension in and there is no stiffness at all. Put tension in and the stiffness arrives, without a single kilogram of extra steel and without changing anything about the steel that is there.

This inverts the rule the rest of the collection runs on. Material far from the middle is how a beam gets stiff, and the rule is so reliable that it produces a factor of forty between two shapes cut from the same plate. A cable has no middle to be far from. What it has is a sag, and a sag is what turns an axial force into a transverse one — which is exactly the argument the funicular shape makes about strength, running here instead into stiffness.

The shape and the force are one statement here, which is the whole of the difference. Reflect the loaded cable and the identical curve carries the identical load in compression, which is why the arch and the cable are one family; but neither can be asked to resist more load without changing shape, because the shape is the resistance. A beam under twice the load is the same beam. A cable under twice the load is a different curve.

The free body, and the constant that comes out of it #

Every number on this page comes from a cut, so here is the cut.

Take the cable between its lowest point and some station along it and keep that piece. Three things act on it — the horizontal pull $H$ at the low point, the tension along the cable at the far cut, and the weight hanging between. Nothing horizontal is applied anywhere in between, so horizontal equilibrium of that piece says the horizontal component of the tension is $H$ at every station. The horizontal force in a cable is constant along it, and that one result from one force sum is what makes the rest tractable.

Under point loads rather than a distributed one the same statement draws a polygon with a vertex at every load, and the segments differ only in their slope: the steep ones near the supports carry more tension than the flat ones at the middle, and every one of them has the same horizontal component. The tension varies along a cable; the horizontal part of it does not.

Now take half the cable and sum moments about the support. That half carries $wL/2$, acting through the quarter point of the span, and $H$ arrives at midspan a distance $\delta$ below the support, so

$$H\delta = \frac{wL}{2}\cdot\frac{L}{4} \quad\Longrightarrow\quad \delta = \frac{WL}{8H}.$$

That is the only equilibrium statement the problem contains, and it has two unknowns in it. The second equation is not an equilibrium statement at all — and that is the interesting part.

The equation that is not equilibrium #

A parabola of sag $\delta$ over a chord $L$ is longer than the chord by approximately $8\delta^2/3L$. If the cable is to sag further, it has to become longer, and the only thing that can lengthen it is elastic extension under the increase in tension, $(H - T_0)L/EA$.

Setting the length the geometry needs equal to the length the steel supplies, and substituting $\delta = wL^2/8H$, gives

$$H^3 - T_0 H^2 - \frac{w^2L^2 EA}{24} = 0.$$

A cubic, for what in any other member would be a division. It is bisected here rather than linearised, on a bracket widened by doubling — Newton from $H = T_0$ has a stationary starting point for a slack cable and a derivative of only $T_0^2$ otherwise, so it lands wherever the load happens to throw it.

Eliminating $H$ between the two equations gives the load–deflection law in closed form, which is the quotable version:

$$W(\delta) = \frac{8T_0\delta}{L} + \frac{64\,EA\,\delta^3}{3L^3}.$$

Two terms, and they are two different mechanisms. The first is the prestress being turned toward the load by the sag. The second is the steel stretching. Differentiate and the tangent stiffness is $8T_0/L + 64EA\delta^2/L^3$, so at zero deflection

$$k_0 = \frac{8T_0}{L}$$

and there is no material property in it whatsoever. A hemp rope and a steel strand of the same length, pulled to the same tension, have exactly the same initial transverse stiffness. They part company only in the cubic term, which is to say only after they have moved.

Prestress is the stiffness #

The consequence is the reason cable structures are tensioned at all, and it is worth seeing at four levels of prestress rather than argued.

The slack cable makes the point. It has no stiffness at all at the origin, so an arbitrarily small load produces a disproportionate movement — with $T_0 = 0$ the law reduces to $W \propto \delta^3$, and the sag grows as the cube root of the load. A washing line has this property and everybody has watched it happen. Adding steel does not fix it; adding tension does.

This is prestress used for a different purpose than a prestressed beam uses it. There, the tendon is a load put on backwards so that the material never has to work in tension. Here, nothing about strength is being bought at all. The tension is there to supply a stiffness the structure does not otherwise possess, and it is spent by relaxation and creep over the life of the building in exactly the way a prestressed beam’s is, with a different symptom — not cracking, but a roof that has gone soft.

The curve that bends the other way #

Every load–deflection curve in this collection that departs from a straight line has so far departed downward.

A structure that has leaned carries its weight off the line it was acting on, so the extra moment bends it further, so it leans more. Its stiffness falls as the load rises, and it runs out entirely at the buckling load. A shallow frame that snaps through does the same thing more violently. Both are geometric nonlinearity, and both are the geometry working against the structure.

A cable is the same phenomenon with the sign reversed. Its deflection improves the geometry — more sag means a steeper cable at the ends, which means more of the tension is pointing the right way — so the structure gets better at its job the harder it is pushed. Two structures, one mechanism, opposite signs — and the sign is decided by whether the deflection lengthens or shortens the lever arm the internal force works on.

The relation $H = wL^2/8f$ that all of this rests on is a reciprocal in the sag, and a reciprocal has no flat part anywhere on it. Drawn at the sag ratios a real structure is built at, it is the least forgiving curve in the subject.

The reciprocal is the argument depth makes everywhere, and the cable’s version of it is unusual only because the depth is a variable rather than a decision. A truss designed with a lever arm of 1 m has a lever arm of 1 m at every load it will ever see; a cable settles on its own sag under every load, and the price of a shallow one is not paid in the ribbon — which is in tension and cannot buckle — but at each end, where the ground has to take 6.25 deck-weights of pull.

Two families, pulling against each other #

A single cable is stiff only against load in the direction that increases its sag. Push it the other way and it goes limp, which makes it useless as a roof surface on its own.

The fix is to cross it with a second family curving the opposite way and pull the two against each other. Both are then in tension with no load on the roof at all, and any deflection adds curvature to both — the sagging family gains sag, the hogging family loses rise, and both resist, because transverse resistance is tension times curvature and the curvature the deflection adds has the same sign for both. That is a surface with stiffness in two directions built entirely out of things with none.

The free body is a unit square of the surface at the centre, cut on all four sides. Vertical equilibrium of the unloaded square requires $n_s f_s = n_h f_h$, and that is not a design decision — the two pretensions are fixed relative to each other by equilibrium, and only their common scale is free. A net is a structure whose internal forces exist before anything is applied and are not arbitrary, which is what a self-stress state is.

The pretension is therefore not a stiffening measure applied to a structure that was already one. It is what makes the two families into a structure at all, and the load–deflection curve of the result is nearly a straight line.

The net’s near-linearity is worth a moment. A single cable stiffens by a factor of two and a half over its range; the net hardly stiffens at all, because most of its stiffness was there before the load arrived.

Most of it was not bought with pretension #

Splitting $k_0$ into its two terms produces the number that reverses the usual account of what a cable net is.

Eighty-one per cent of that net’s stiffness comes from its curvature rather than from its jacks. The pretension term is $8(H_s + H_h)/sL^2$ and contains no curvature at all; the elastic term goes as the square of the sag, so it overtakes the pretension term at a sag of 0.726 m — about a fortieth of the span — and everything deeper than that is mostly shape.

Which recovers the practical rule from the arithmetic rather than from experience: a shallow net has to be tensioned very hard and a deep one hardly at all, because a shallow net has almost none of the second term and has to buy the whole stiffness with the first. It also explains why the anchorages of a flat cable roof are so much more brutal than the surface above them suggests.

The crossover sag has a closed form, and it is worth having because it is a rule rather than a number. Setting the two terms equal, the spacing and the span both cancel and what is left is

$$\frac{f^*}{L} = \sqrt{\frac{\sigma_0}{2E}}$$

with $\sigma_0 = H/A$ the stress the pretension puts in the cable. The crossover sag ratio is the square root of half the prestress strain — no length, no load, no area. At the 240 N/mm² this net is stressed to, that is 1 in 41, which is the 0.726 m the figure reports.

So the rule generalises without recomputation: a net sagging deeper than about a fortieth of its span gets most of its stiffness from its shape, and one shallower than that gets most of it from its jacks — and stressing the cables harder moves the boundary deeper, which is the opposite of what raising a stiffness usually does.

What actually ends it #

The interesting failure of a cable net is not a failure of a cable.

At 407 mm the hogging family has nothing left in it. It goes slack, half the surface stops contributing in one step, and the remainder has to find the whole load by sagging — from a state in which it is already at 925 kN and its geometry has moved. The sagging cables are nowhere near any strength that matters; the structure has run out anyway.

So the design case for a cable roof is the loss of a self-stress state, and the quantity to compute is a displacement rather than a stress. This is unusual enough to be worth naming as a category: the limit is set by a member that stops working rather than by a member that breaks, in the same way that a brace’s job is a stiffness rather than a strength and serviceability rather than strength usually governs a floor. Ponding on a cable roof is the vicious version — water that will not run off collects in the sag it has just deepened, on a surface whose stiffness against that particular deflection is what the ponding is destroying.

Where the same argument turns up #

Geometric stiffness is not a cable speciality. It is what a membrane roof does, what a spoked bicycle wheel does, and what a curved shell does from the other side of the sign — a shell is a surface that carries in its own plane because it is curved, and needs almost no thickness to manage it.

The shell and the cable are the same trade made twice, once in compression and once in tension, and in both cases what is bought is an exponent rather than a factor: a curved wall’s thickness is linear in the pressure it carries and a flat one’s goes as the square root of it, so the advantage of the curve grows as the load falls rather than staying put.

The connection runs in the other direction too. The stiffest path takes the load whenever members share a displacement, and a cable net is the extreme case of that rule, because the stiffness deciding the split is itself a function of the deflection being shared. And a structure whose stiffness depends on its prestress is a structure whose natural frequency does too, which is why the period nobody chose is, for a cable roof, a number set by the tensioning sequence rather than by the mass and the section.

Where the model stops #

The profile is a parabola, at every stage. The solver imposes that shape and computes only its amplitude, which reduces a continuum to one degree of freedom. It is a good approximation for shallow cables under distributed load and a poor one for a deeply sagging chain or a cable under a concentrated load, where the true shape has a kink the parabola cannot represent.

The load is uniform and stays uniform. A cable’s stiffness is directional in the deepest possible sense — it resists what deepens its sag and cannot resist anything else. An asymmetric load on a suspended structure produces a shape change the funicular argument never contemplated, which is what stiffening trusses exist to absorb.

The supports do not move. Every horizontal force here is delivered to something assumed rigid. Real anchorages give, and a support that draws inward reduces the tension, which reduces the stiffness, which increases the deflection — the same runaway loop as a leaning frame’s, this time with the geometry losing.

The prestress is the prestress that was intended. It is applied by jacking, measured indirectly, and then leaks away through relaxation of the strand and creep of whatever it is anchored to. Since $k_0$ is exactly proportional to $T_0$, a net that has lost a fifth of its pretension has lost a fifth of the first term of its stiffness, and there is nothing about the loss visible from underneath.

The state depends on the sequence. A net has no shape until it has been tensioned, and the order in which the cables are pulled decides the self-stress state it lands in — which makes it, like every structure whose analysis assumes it arrived complete, a case where the construction sequence is part of the answer.

What the figures cannot show #

Every picture on this page is a plot. Not one of them is a cable, and that is deliberate, because the argument is entirely about a relationship between a load and a movement and a drawing of a cable would show a curve that looks the same at every load.

Which conceals two things. The first is that the sags being plotted are real and large — 740 mm on a 30 m cable is a sag ratio of one in forty, and 407 mm of net deflection is span over seventy-four. Nothing here is drawn exaggerated, because nothing here is drawn at all; these are movements a person standing underneath would see.

The second is that the axes hide how little of the structure is doing anything different. The steel is the same steel at every point of every curve, at a stress that never approaches anything interesting. The whole of the nonlinearity — the factor of 2.56 in stiffness, the cube-root behaviour of the slack cable, the collapse of one family of a net — is a change of shape, and a plot of stress against load would be a straight line with nothing to say.

The ladder from here #

Later rungs on this anchor: the catenary’s own stiffness, which differs from the parabola’s by the term that made Galileo wrong. The cable-stayed system, where the stay’s apparent modulus falls with its own sag and the Ernst correction repairs it. Form-finding by force density, which makes the nonlinear problem linear by choosing the wrong unknown deliberately. The dynamic relaxation method, which solves a static net by giving it a fictitious mass and damping it. Cable-stayed against suspended, and why the two carry live load so differently. The saddle roof and the ring, where the two families’ thrusts close on themselves rather than on the ground. Wind uplift, which reverses the sign and asks the hogging family to do the sagging family’s job. And the prestress that has to be locked in during erection, where a sequence of jacking operations has to arrive at a state that equilibrium fixed in advance.