The tree that strength does not ask for

A roof needs supporting at many points. The ground is available at few. Between the two there is a family of structures that start as one member and end as many, and the tree is the obvious member of it — a trunk that divides, and divides again, until there are as many tips as the roof has points.

They are built because they look like trees, which is a perfectly good reason and not the one this essay is about — though it returns at the end, with better credentials than it started with. The question here is narrower and has a clean answer: given a load to deliver from a set of points down to one, how much material does a branching structure need, and is it less than the alternatives?

The answer arrives in two halves that point in opposite directions, and the disagreement between them is the whole finding.

Which free body produced the number #

Cut just below a joint and take everything above it.

The loads at the tips are vertical and equal. The branches meeting at the joint carry axial forces along their own axes. For the sum of forces to vanish in both directions with no moment left over, three things have to be true at once: the branch axes have to pass through a common point, the resultant of the branch forces has to lie along the trunk, and the loads have to be symmetric about it.

When they are, the arithmetic is trivial. Each branch carries $N = (P/n)\,\ell/g$, where $\ell$ is its length and $g$ its rise — the vertical load divided by the cosine of its inclination — and the trunk carries the whole of $P$. No member carries any bending at all, which is the property the form exists for and is the reason the same drawing appears as a funicular shape in another field.

Strength alone wants no tree #

Now ask for the volume of material, with every member sized so that its stress is exactly the working stress. Hold the height $H$ and the reach $s$ fixed and let the split happen at any level; call the branch rise $g$, so the trunk is $H - g$ long.

The trunk contributes $P(H-g)/\sigma$. Each branch is $\ell = \sqrt{s^2+g^2}$ long and carries $(P/n)\ell/g$, so the branches together contribute $P\ell^2/(g\sigma) = P(s^2+g^2)/(g\sigma)$. Add them:

$$V = \frac{P}{\sigma}\left(H - g + \frac{s^2 + g^2}{g}\right) = \frac{P}{\sigma}\left(H + \frac{s^2}{g}\right)$$

The $g$ terms cancel except for one, and what is left falls monotonically as $g$ grows. The volume is least when the branch rise is the whole height — that is, when the split is at the ground and the “tree” is a fan of straight struts from a single point to the tips.

The trunk is pure waste. Every millimetre of it carries the full load over a length that could have been part of a member also covering some of the horizontal distance.

That result is a small instance of a much older one. Michell’s theorem, from 1904, says that the minimum-volume truss transferring given loads to given supports is made of members lying along the principal directions of a particular strain field, and it produces fans of straight members radiating from load points rather than hierarchies. Strength-optimal structures are not tree-shaped, and the fact has been known for a century.

Buckling is why trees exist #

Nothing above mentions the length of a member except as something to multiply a force by. Buckling does.

A strut of length $\ell$ carrying $N$ needs a second moment of at least $N\ell^2/\pi^2E$, and for a section of a given shape the second moment goes as the square of the area. So the area buckling demands goes as $\sqrt{N}\,\ell$ rather than as $N$, and — this is the part that matters — it grows with the length while the strength requirement does not.

Put that requirement in and the fan of long struts becomes the expensive option. Each member of the fan is $\sqrt{H^2+s^2}$ long, carrying a share of the load over the full height; each branch of a tree is shorter, and the trunk carries the whole load over a length where it is stubby. The optimum moves up off the ground and settles, for the column drawn here, at 57 per cent of the height — with a volume 51 per cent below the fan’s.

The tree is a stability structure. It is not a way of carrying force efficiently; it is a way of carrying force over a distance without any single member being long.

That asymmetry is what length costs a compression member and it has no counterpart in tension. A tie’s area depends only on its force; a strut’s depends on its length as well, and past a modest slenderness the length is the whole of it. Every branching structure, every space frame and every lattice in this collection is an answer to it.

Every generation is worth less than the last #

If one split helps, more should help further, and they do — with a diminishing return steep enough to explain why real branching structures have two or three levels and not eight.

Measured on the same column: one level of branching gives a volume of 0.100 units, two gives 0.074, three gives 0.064. The first split buys 26 per cent, the second 14, the third would buy less than eight — against a joint count that doubles each time and a fabrication cost that does not.

The reason for the diminishing return is that each generation halves the load and roughly halves the length, and the buckling requirement goes as $\sqrt{N}\ell$ — so each split reduces a member’s demanded area by a factor of about $\sqrt{2}\times 2$, while multiplying the number of members by two. The product improves, and less each time.

The optimisation the second generation is being fitted into is the same optimisation, and it is worth drawing it that way rather than quoting a number out of it.

Branching is not the only way to make every member short. The competing answer arrives at the same objective from a different direction, and it is worth having beside the tree rather than after it.

What the section does to the answer #

The buckling requirement contains a shape factor, and it moves the answer more than anything else does.

A solid circular member has $I = A^2/4\pi$. A circular hollow section of the same area, with a diameter-to-thickness ratio of twenty, has a second moment perhaps twenty times larger — so the area buckling demands falls by a factor of $\sqrt{20}$, and the volume of the whole tree falls with it. The column drawn here needs 0.100 units of material as a solid section and 0.022 as a tube: a factor of 4.5, from a decision about the section rather than about the form.

That is a general and slightly deflating point about form optimisation. The choice between a fan and a tree is worth 51 per cent; the choice between a solid bar and a tube is worth 350. Where the material sits within the section is a bigger lever than where the members sit within the structure, and it is available without any joints at all.

The joint that is only free of moment on one load case #

Everything above assumes the tips are equally loaded. They are not, and the case that is not symmetric is the one that sizes the joint.

Take the load off one branch of a two-branch joint — a partially loaded roof, a snow drift on one side, a maintenance load — and the two branch forces no longer resolve into something along the trunk. The joint carries the difference, and the difference has a lever arm: the horizontal offset from the joint to the tip, which is metres rather than millimetres. On this column that is 300 kNm at a joint whose members were sized for axial force alone.

This is the standing hazard of every form-found structure, and it deserves stating in general terms. A shape found for one load case is only funicular for that load case. An arch is funicular for its own weight and carries bending under anything else; a cable takes the shape of the load it carries and changes shape when the load changes; a branching column is momentless under symmetric load and is a frame under any other.

Masonry states the same lesson with the material rather than the joint. A thrust line that has to stay inside the stonework is a shape found for one loading, and under another it leaves — and what happens next is decided by whether there is enough material around the line to contain it. A branching column’s equivalent of “enough material” is a joint stiff enough to carry the moment the asymmetry produces, and a joint is a thing somebody has to detail rather than a thickness somebody can add.

The load path is chosen, and it is chosen visibly #

There is one property of a branching column that no volume calculation captures and that is probably the real reason they are built.

Most structures hide their load path. A flat slab on columns gives no indication of how the load reaches the ground; a steel frame behind cladding gives none either; even a transfer structure, which is the most dramatic redirection of load in the subject, is usually a beam in a ceiling void that nobody sees. The route the force takes is a matter for the drawings.

A tree is the load path, at full size, in the room. Each split is a place where a force divides in a ratio that is visible from the geometry; each branch’s inclination is the direction the force is travelling; the trunk’s thickness against the branches’ is the accumulation. Somebody standing under one can read the whole structure without being told anything.

That is worth something, and it is worth saying plainly that it is an architectural value rather than a structural one — the volume argument above says the fan is nearly as good and the tube is four times better. But it is not nothing, and it is consistent with the rest of this collection’s habit: the drawing is the calculation, and a structure whose geometry is its own force diagram is a structure that can be understood by looking.

The tips push apart, and something has to hold them #

The volume calculation counts three members and stops. There is a fourth, it is not drawn, and it is not small.

At the optimum split the branch runs from the joint at 3.87 m up to a tip at 9 m, three metres out. Its length is 5.94 m and it carries

$$N = \frac{P}{2}\cdot\frac{\ell}{g} = 200 \times \frac{5.94}{5.13} = 232\ \text{kN}$$

of which 200 kN is vertical and 117 kN is horizontal, pointing outward. The tree pushes its two tips apart with 117 kN, exactly as an arch pushes its springings apart, and for exactly the same reason: the load is being carried by an inclined member and an inclined member has a horizontal component.

Something has to close that. Usually it is the roof — the plate, the purlins, the perimeter beam — which is then carrying a tie force nobody put on a drawing. Count it: at working stress a tie carrying 117 kN over the 6 m between tips is $117 \times 6 = 702$ kN·m of strength-volume, against the tree’s own $P(H + s^2/g) = 4{,}302$. Sixteen per cent, in a member the optimisation never saw.

And it changes the comparison the essay was making. The fan from the ground has the same tips and a steeper branch, so its horizontal component is 66.7 kN rather than 117, and its tie is 400 kN·m against the tree’s 702. The tie is a tension member, so buckling does not touch it — which means it adds to both structures at the same rate and adds more to the tree. The 51 per cent by which the tree beat the fan under the buckling constraint is real and it is smaller than the figure says.

The general form is the one an arch has taught this collection twice: a structure that carries load by inclining its members has a horizontal force at its boundary, and whether that force is expensive depends entirely on what is there to take it. A tree in a roof plane has a diaphragm and it is cheap. A tree standing on its own — a canopy, a single-column shelter — has nothing, and the thrust has to go into the base as a moment, which is the one thing the form was chosen to avoid.

And the roof braces it, or the whole optimisation is void #

There is a second thing the roof is doing, and it is worth checking because the arithmetic collapses without it.

Every buckling length in this essay is a member’s own length. For the branches that is right — each is held at the tip by the roof and at the joint by its sibling. For the trunk it is right only if the joint at the top of it cannot move sideways, and nothing in the tree itself stops it: the two branches are symmetric, so a sway of the joint is resisted only by their bending.

If the roof holds the tips, the joint is held, and the trunk is a 3.87 m strut. If it does not, the tree is a flagpole: a cantilever of the full 9 m, with an effective length of 18 m and a critical load

$$\left(\frac{18}{3.87}\right)^2 = 21.6$$

times smaller — on the member carrying the entire 400 kN. That is not a correction to the optimum; it is a different structure, sized by a sway mode the essay’s whole method has no term for.

Which puts the two findings of this page together in a way neither has on its own. The branching column is a form that only works inside something else. Its efficiency is bought by making every member short, and members are only short if their ends are held; its momentless joints are bought by symmetry, and symmetry is only maintained if the load is shared; its thrust is only cheap if there is a plate to close it. A tree in a roof is an excellent structure. A tree on its own is a cantilever with a decorative top, and its volume is decided by the 18 m rather than by any of the arithmetic above.

Where the model stops #

The tree is planar and regular. Real branching columns are three-dimensional, branch into three or four rather than two, and are rarely symmetric — the roof grid they support decides where the tips must be, and the tips are not conveniently spaced. The branching factor alone is worth drawing, because the sweep over generations above held it at two throughout.

The members are pin-ended. Every buckling length above is the member’s own length, which assumes the joints offer no rotational restraint. A welded tree joint offers a great deal, and taking credit for it moves the optimum down and reduces the volume — at the cost of the joint having to carry the moment that credit implies.

And the objective is material volume. That is the objective for which the mathematics is clean and it is almost never the objective a project has. Fabrication cost scales with joints, not with tonnage; erection cost scales with the number of pieces; and a tree with fourteen members and seven bespoke nodes can be more expensive than a fan with four straight ones twice the weight.

Two structures that are the same argument #

The comparison the volume calculation sets up — a hierarchy against a fan — turns up twice more in this collection with the same arithmetic and different names.

A cable-stayed bridge against a suspension bridge. The stays are a fan from a tower to the deck; the suspension cable is a hierarchy, with the main cable carrying everything and hangers dividing off it. In tension there is no buckling constraint, so the fan is the efficient answer — and the cable-stayed form has displaced the suspension form for every span where the stays are short enough to be practical.

A column tree against a beam. A roof delivered to one point by a branching column is a roof whose spans are short; the same roof on a single column with a long beam is one long member in bending. Bending is the expensive way to carry anything, so the tree wins easily — and the comparison is not between two columns but between a column and a member whose depth would have to grow with its span.

What the three share is a question about whether load should be gathered before it is carried. Gathering costs joints and buys shortness; carrying costs length and buys simplicity. Every structure in this collection that looks like a hierarchy — a tree, a suspension cable, a secondary-and-primary floor framing grid — has answered that question one way, and every fan has answered it the other.

All three are versions of a form found for its own load case and inverted into compression. A branching column is a funicular structure for a symmetric set of tip loads, exactly as a hanging polygon is a funicular for the loads it was drawn from — and it stops being one for the same reason, which is that the loads move and the shape cannot.

What the pictures cannot show #

The members are drawn at a thickness proportional to the area they need, which makes the branching visible and makes the drawing lie about the scale: a 9 m column carrying 400 kN has members of the order of 100 mm across, which at this scale would be hairlines.

Nor can the figures show the foundation. All of this is a comparison of superstructure volume, and the single most persuasive practical argument for a branching column has nothing to do with any of it: one foundation instead of eight. On a poor site that comparison can dominate everything computed above.

The assumption the figure rests on #

The working stress is a single number applied to every member, in tension or compression, and the buckling requirement uses a fixed section shape. Both hide a real choice. The branches of a tree are shorter and less heavily loaded than the trunk, so the section that suits one does not suit the other, and a real tree is fabricated from a small number of sizes rather than from a continuum — which puts its volume above the optimum computed here by whatever the rounding costs.

The ladder from here #

Later rungs on this anchor: Michell’s theorem itself, and why the minimum-volume structure for two loads and two supports is a fan rather than a hierarchy. The three-dimensional tree, where the branch directions have two angles and the joint has three equations. Joint stiffness and the buckling length credit it buys. The tapered branch, which is the fully stressed member when the axial force is constant and the buckling length is not. And the same argument run for tension structures, where the buckling constraint disappears entirely and the fan wins outright — which is why a cable-stayed mast has no branches and a tree does.