Structural Engineering & Statics
where the load goes
Concept

Form-finding — where it appears

Choosing a structure's geometry so that the load it will carry produces the internal forces it is good at, rather than analysing a given shape. The shape found is funicular for one load case only, so every other case puts bending into a structure whose whole justification was that it had none.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The cable and the arch are the same curve. The shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is.

The shape that carries itself, and the arch that is its reflection

Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is.

structures · Funicular The funicular polygon for five loads. The shape a string takes under 5 point loads, with a vertex at every load and a constant horizontal component of 27.9 throughout. The end segments carry the most — 38.6 against 28.0 in the flattest one — because they are steepest.

The polygon that finds the shape

A hanging string under five loads has no smooth curve in it — it has five vertices and six straight segments, and every slope in it is a running sum divided by one number.

structures · Funicular A cable alone goes to a kink, and a kink is not a road. A point load of 1000 at mid-span of a 900 m suspended deck. The upper shape is the cable with no girder at all: two straight lines meeting under the load, because a cable takes the funicular shape of whatever is on it and the funicular of a point load is a kink — 0.0083 radians of it here. The lower shape is the same cable with the girder present, peaking at 1.125 against the bare cable's 1.873. The girder is not carrying the load — it takes only 17% of it — it is spreading it, over a characteristic length of √(EI/H) = 183 m, and what reaches the cable is spread over that length rather than arriving at a point.

The deck is not there to carry the load

A cable takes the shape of whatever is on it, which is exactly the problem — under a point load its shape is a kink, and a kink is not a road. The stiffening girder exists to spread the load until what reaches the cable is something the cable's own shape is right for.

structures · Stiffening girder The same sheet, twice, and a factor of ten thousand. A 3000 mm developed width of 3 mm sheet, covering 2400 mm in plan — so the legs sit at 36.9° and the fold is 300 mm deep. Flat, its second moment about its own mid-plane is 6750 mm⁴, which spans nothing. Folded, it is 67.50×10⁶ — 10000 times as much, which is exactly the depth in thicknesses squared. The material is identical, the plan cover has fallen by 20%, and the only thing that changed is where the material sits. What limits it is buckling of the leg: at this leg length the flat between the folds goes at 27 N/mm², well below the steel's 275.

Folded until it spans

A flat sheet has a second moment of area of B·t³/12 and will not span anything. Folded, the same material has B·t·h²/12, and the gain is exactly the fold depth over the thickness, squared — a ratio with no material in it and no width in it.

structures · Folded plate The split is where buckling puts it. A branching column carrying 400 kN to two points 3.0 m apart, over 9.0 m. Every member is drawn at the thickness it needs: the area is the larger of N/σ and what Euler asks of a strut of that length, and 3 of 3 members here are sized by buckling rather than by strength. The split sits at 57% of the height, which is where the total volume is least — 51% less than the fan of straight struts that carries the same load with the same stresses.

The tree that strength does not ask for

A branching column carries a roof on many points and reaches the ground on one. Size every member by its stress and the optimum tree turns out to have no trunk at all — the best answer is a fan of straight struts from the base. Put buckling in and the trunk appears, at 57 per cent of the height.

structures · Branching structure Two curvatures of opposite sign, which is what makes it a structure. A cable net over a 36 m square, drawn as the two families of cables that are also the two rulings of the surface. One family sags and carries downward load by hanging; the other rises and carries upward load — wind uplift, and a load reversal anywhere — by the same mechanism upside down. Neither can do anything alone. A single family of cables is a mechanism: it changes shape freely under any load pattern it was not tensioned for, and the shape it moves to is decided by the load rather than by the designer. Put the two together and each is the other's restraint, but only if they are pulled against one another first — the pretension of 520 kN in the sagging family and 715 in the hogging one is a self-equilibrating state that exists with no load on the roof at all, and it is what turns two mechanisms into one structure. The curvatures are drawn four times their true value: a real net of this span sags 2.2 m over 36, which is flatter than it looks anywhere.

Two curvatures of opposite sign

A single family of cables is not a structure. It is a mechanism that takes whatever shape the load asks for, and it will do that under any load pattern it was not tensioned for. Cross it with a second family curved the other way, pull the two against each other, and the pair becomes stiff — with no bending anywhere and no material property involved in the stiffness at all.

structures · Cable net

Named alongside it

The objects these essays reach for when they reach for this one.

FunicularHorizontal thrustSag ratioCatenaryLoad pathOptimisationSelf-weightStiffnessThrust lineAnticlasticBranching structureBuckling

All concepts