Folded until it spans

Hold a sheet of paper out horizontally by one edge and it flops. Put one fold along it and it stands out straight. The demonstration is old enough to be a cliché and the arithmetic behind it is worth doing, because the number that comes out is one of the largest in this collection and it contains nothing but geometry.

Ten thousand. The material is identical, the plan cover has fallen by 20%, and the only thing that changed is where the material sits.

Which free body produced the number #

Cut the profile across and take the section. It is a zigzag of straight legs, each of length $c$ and inclined at $\theta$, so each rises through $h = c\sin\theta$.

A line element of length $c$, thickness $t$, inclined so that it spans a vertical height $h$, has a second moment about the profile’s own mid-height of

$$I_{leg} = \frac{t\,c\,h^2}{12}$$

which is $\int y^2 \,dA$ done along the leg — the same integral the sections field assembles for any shape, with the mass distributed along a line rather than over a rectangle. There are $B/c$ legs across a developed width $B$, and they alternate up and down about the same mid-height, so

$$I_{fold} = \frac{B}{c}\cdot\frac{t\,c\,h^2}{12} = \frac{B\,t\,h^2}{12}$$

The leg length has cancelled. The flat sheet is the same expression with the thickness in place of the fold depth, $I_{flat} = Bt^3/12$, so

$$\frac{I_{fold}}{I_{flat}} = \left(\frac{h}{t}\right)^2$$

No material property appears, no width appears, and the answer is a pure square. That is the cleanest statement of what shape is worth that this collection has, and it is worth setting beside the sections field’s own version: four sections of equal area spanning a factor of forty. The folded plate is that comparison taken to its limit, because a flat sheet is the worst section there is and a fold is the cheapest possible improvement on it.

What it costs #

Nothing in engineering is free and the price here is plan cover. A developed width $B$ folded at $\theta$ covers only $B\cos\theta$, so the sheet that covered 3,000 mm flat covers 2,400 folded — and deepening the fold spends more of it.

The trade is favourable a long way past where it feels favourable. Giving up a further 17% of cover — 2,400 down to 2,000 — buys 54% more second moment, and the diminishing returns arrive slowly because the gain is a square while the cover is a cosine.

A profiled steel deck is that trade made at about 25°, a corrugated sheet at about 45°, and a folded-plate concrete roof at 30° to 60° — and all three are the same calculation with different fabrication constraints deciding where on the table to sit.

Fewer and deeper, which is the wrong instinct #

With the plan cover fixed, $\theta$ is fixed, and the fold depth is the leg length times its sine. So $h \propto c$, and

$$I \propto h^2 \propto c^2 \propto \frac{1}{n^2}$$

Halving the number of folds quadruples the stiffness of the same sheet. That is the opposite of the instinct, which says a sheet with many small folds is better supported and therefore better — and it is worth being clear about why the instinct is wrong. Many small folds put the material near the profile’s own mid-height, where a second moment does not want it. Two deep folds put it as far away as the plan cover allows.

Something must stop it, or the optimum would be one fold and a sheet 1,500 mm deep.

The leg is a plate of width $c$ and thickness $t$, and its critical stress is

$$\sigma_{cr} = \frac{k\pi^2 E}{12(1-\nu^2)}\left(\frac{t}{c}\right)^2$$

which for the six-leg profile drawn — legs 500 mm long in 3 mm sheet — is 27.3 N/mm². The steel’s yield is 275. So that profile is limited by the buckling of its own flat panels at a tenth of what the material could do, and every millimetre of extra depth it has is depth it cannot use.

The optimum is exactly where the two limits meet #

The two effects are both powers of the leg length and they run opposite ways.

$$w \propto \frac{\sigma\,I}{h} \propto \sigma\, h \propto \frac{1}{c^2}\cdot c = \frac{1}{c}$$

while the leg buckles. So the load rises as the folds get shorter — because a shorter leg can take proportionally more stress than the depth it loses. Once $\sigma_{cr}$ reaches $f_y$ the stress stops rising, and

$$w \propto f_y\,h \propto c$$

so the load falls again. The maximum is exactly at the leg length whose buckling stress equals yield, and there is nothing approximate about that: it is the meeting of two straight lines on a log plot.

For this sheet that length is $t\sqrt{k\pi^2 E/12(1-\nu^2)f_y} = 158$ mm, so 19 legs across 3,000 mm — carrying 8.65 N per millimetre of span against the 2.73 of the six-leg profile drawn, a factor of 3.2 from choosing the fold count properly and changing nothing else.

The result generalises past folded plates. Every thin-walled section has this structure: stiffness rewards putting material far apart, local buckling punishes leaving it unsupported over a length, and the optimum is where the two limits coincide. It is the same statement as a section’s classification — a section whose plate elements cannot reach yield is a section that has been made too efficient by half.

The optimum profile fails on deflection by a factor of four #

The peak found above is a peak in strength, and a roof is not usually sold on strength. Run the same two profiles through a deflection check and the answer inverts.

At 6 m span, $\delta = 5wL^4/384EI$:

The optimised profile carries 3.2 times the load and sags four times more than any deflection limit permits. Nineteen folds bought strength by giving away exactly the quantity a roof is checked on, because $I$ went down by ten while $w$ went up by three.

The reason is that the two limits want opposite things. A deflection limit gives an allowable load

$$w_\delta = \frac{384\,E\,I}{1250\,L^3} \;\propto\; I \;\propto\; h^2$$

at $\delta \le L/250$, which rises with fold depth — while the strength-limited load $w_{str} \propto \sigma_{cr} h \propto 1/h$ falls with it. So the strength optimum sits at as many folds as buckling allows and the stiffness optimum sits at as few as the material allows, and the design is at neither.

It is at the crossing. For this sheet and this span, $w_\delta = 2.24 \times 10^{-4}h^2$ and $w_{str} = 819/h$, so

$$h^3 = \frac{819}{2.24\times10^{-4}} = 3.66\times10^6 \quad\Rightarrow\quad h = 154\ \text{mm}$$

— a leg of 257 mm, twelve folds, carrying 5.3 N/mm at exactly $L/250$. That is 39 per cent less than the strength peak and 95 per cent more than the profile drawn, and it is the only one of the three that is a design.

Two things about that result generalise past folded plates. A profile whose fold count was chosen on strength alone will be a serviceability failure, because thin-walled optimisation always trades depth for stress and only one of those appears in a deflection. And the crossing moves with the span, since $w_\delta$ carries $L^{-3}$ and $w_{str}$ does not — so the same sheet wants deeper folds over a longer span, which is why a deck profile is specified against a span table rather than against a capacity. It is the same reasoning that decides a beam’s depth long before its strength does.

At the optimum, a stronger material buys the square root of itself #

The optimum leg length is the one whose buckling stress reaches yield,

$$c^* = t\sqrt{\frac{k\pi^2 E}{12(1-\nu^2) f_y}}$$

and substituting it back into $w \propto f_y h \propto f_y c^*$ gives something worth looking at twice:

$$w_{max} \;\propto\; t\,f_y\sqrt{E/f_y} \;=\; t\sqrt{E f_y}$$

The geometric mean of the two material properties. A folded plate at its own optimum is not a strength structure and not a stiffness structure; it is exactly half of each, because raising $f_y$ raises the stress the legs may carry and shortens the legs that can carry it, and the two effects share the gain equally.

The practical readings are both slightly surprising. Going from S275 to S460 raises $w_{max}$ by $\sqrt{460/275} = 1.29$ rather than 1.67 — a stronger steel again buying less than it charges for, this time by a square root rather than not at all. And aluminium does better here than its yield stress suggests: at $E = 70$ GPa and $f_y = 160$ N/mm² its $\sqrt{Ef_y}$ is 44 per cent of steel’s while its density is 34 per cent, so a folded aluminium sheet at its optimum is 28 per cent better per unit weight than a folded steel one. Corrugated aluminium roofing is not a compromise; it is the material the arithmetic prefers.

The same substitution says what thickness is worth, and it is the one variable that is not squared anywhere. $w_{max} \propto t$, exactly once, because a thicker sheet both carries more and permits longer legs in the same proportion. So doubling the sheet doubles the capacity and doubles the cost of the material, which makes thickness the one dimension in this whole calculation that buys nothing for free — and the fold, which buys a factor of ten thousand for nothing but plan cover, the one that does.

Where the folds have to be held #

Two conditions have been assumed silently and both fail if they are not detailed for.

The profile must keep its shape. A folded plate loaded transversely wants to flatten out: the legs rotate about the folds, the depth falls, and the second moment goes with it as a square. Every real folded-plate structure therefore has diaphragms at its ends and often at intervals along it — a stiff plate across the section that holds the fold angles. Without them the member’s stiffness is not $Bth^2/12$ but something much smaller and much harder to compute.

And the folds themselves must not spread. At the supports, the reaction has to be delivered into the plate, and a folded plate delivered into at one fold and supported at another has to carry the difference transversely — which is a slab spanning between the folds, at right angles to everything computed above.

Where the folds become a curve #

A folded plate and a shell are usually taught as separate subjects, and the relationship between them is the useful thing to notice. A folded plate is a shell that has been made from developable pieces, which is the whole of its practical advantage: flat sheet can be pressed, cut, welded and transported, and a doubly curved surface cannot.

What it gives up is the shell’s membrane action. A curved surface carries load by being curved, with in-plane forces and no bending; a folded plate carries load by having a large second moment about the section, which is bending in the ordinary sense with the material a long way from the axis. The first is more efficient and the second is buildable, and the whole history of thin concrete roofs is the trade between them. A shell’s own weakness is the other half of the comparison: a shell loses most of its theoretical buckling load to imperfections nobody can see, and a folded plate, whose stability question is the ordinary plate buckling of its legs, does not.

What it is worth against the alternatives #

The fold is one of three ways to get depth out of a flat material, and comparing them says what it is for.

The competition is the field’s founding scaling result stated for a plan dimension: a chord force is M/d, so depth is the cheapest strength there is, and a fold buys depth out of a sheet that was going to be there anyway.

A truss buys depth with separate members and the joints to connect them, and reaches efficiencies a folded plate cannot — but every joint is a fabricated detail, and depth costs it nothing at all.

A beam buys depth with a rolled or welded section, and pays in material because the web between the flanges is doing very little.

A fold buys depth with a bend, which costs nothing but plan cover — and gets its depth and its covering surface out of the same piece of material. That is its whole proposition: a folded plate is a structure and a cladding at the same time, and comparing it with a truss on structural efficiency alone misses what it is being bought for.

The same argument in one dimension fewer explains the corrugated iron sheet, which is the most-manufactured structural product in history and is this page’s arithmetic applied to a 0.7 mm sheet with 15 mm of depth: a gain of 460, from a material whose flat form would not hold its own weight over a metre.

One correction the plane-section assumption needs across a wide profile: shear lag means the fold furthest from the support is stressed less than the calculation says, so a very wide folded plate does not develop its full width and the gain flattens off before the arithmetic does.

Where the model stops #

The profile is treated as a plane section. For a member spanning much further than it is wide that is good; for a short wide one the fold nearest the load carries far more than its share, and shear lag applies across the profile exactly as it does across a wide flange.

The legs are taken as flat and the folds as sharp. A pressed fold has a radius, and the radius takes material out of the extreme fibre where it was worth most — a small correction for a sharp fold and a substantial one for a cold-formed section whose radii are several times its thickness.

Only the longitudinal direction is computed. The transverse slab action described above is real, is often what sets the thickness, and appears nowhere in $Bth^2/12$.

The buckling coefficient is taken as 4 throughout. A leg is supported along both its edges by folds, which are rotational restraints rather than simple supports, so $k$ is between 4 and 6.97 and the argument is conservative by up to 74% in the critical stress and 32% in the optimum leg length.

Nothing here is a shell. The moment the fold angle becomes shallow enough that the legs act together as a curved surface, membrane action begins and none of this applies.

And self-weight is ignored. A folded plate is remarkably light — that is its point — but at long spans the profile is carrying mostly itself, and the load capacity quoted is a gross figure.

What the pictures cannot show #

The profile figure draws the folded and flat sections at the same horizontal scale and the same line weight, and the line weight is the sheet thickness. At the drawing’s scale a 3 mm sheet is a fraction of a pixel, so both members are drawn thicker than they are — which flatters the flat sheet enormously, since the whole of its second moment is that thickness cubed.

The gain figure plots a pure power law on log axes and is therefore a straight line with nothing to see. That is deliberate: the content of the figure is that the line is straight and its slope is two, and a figure whose whole message is the absence of a feature is a figure a reader can be forgiven for finding empty.

And the optimum figure draws a peak at 19 legs, which is a fold every 158 mm across a 2,400 mm cover. Nothing in the drawing indicates whether a press brake can make that, whether the folds foul the fixings, or whether anybody wants a roof that looks like that — and all three have decided more real profiles than the arithmetic has.

The ladder from here #

Later rungs on this anchor: the transverse slab action set out properly, with the folds as supports and the legs as continuous spans. Diaphragms and end stiffening, and the loss of stiffness when they are omitted. Folded plates of unequal legs and trapezoidal profiles, where the section is no longer symmetric and the neutral axis moves. Stressed-skin action in a whole building, where the roof deck is the shear diaphragm and the same sheet is doing two structural jobs at once. Cold-formed section design, which is this optimisation applied to a member rather than a surface and which turns on effective widths rather than on full ones. The origami-inspired folded structures whose fold pattern is chosen so that the surface can be deployed, where the geometry is a mechanism by design. And the concrete folded-plate roofs of the 1950s and 60s, which were designed with this arithmetic, built in enormous numbers, and abandoned when formwork became more expensive than steel.

The habit worth carrying is the one the arithmetic makes vivid: a structure’s efficiency is decided by a length and a thickness, and the ratio between them is squared. Everything else — the material, the width, the span — enters somewhere else in the calculation, and none of it enters that ratio at all.