A crown sized branch by branch buckles as one at a fifth of its load

A parent sized for its daughters finished a long argument about how thick a crown’s branches have to be not to buckle. A crown that would rather not buckle sized each branch as a column clamped to a parent that does not move. A base that gives let the parent bend, standing each branch on a rotational spring as stiff as its parent’s tip, and found branches no radius could hold. The last essay put the requirement on the parent: size every parent for the base its daughters need, at a margin μ over the load at which a rigid rod would tip on that base. Nothing was left without a radius, and the cheapest crown, in wood, sat at a margin of about 1.7.

Every check in that argument was a branch’s own, on a spring standing in for everything below it. That essay ended by naming what it had not done. A crown is one structure, and the load at which it buckles is the lowest eigenvalue of the whole frame — every branch’s stiffness, every joint’s rotation, every branch’s load at once. It asked whether the whole crown buckles at the load its branches were sized for, above it or below it; which branch the lowest mode concentrates in, and whether that is the branch the spring model marks weakest; and whether the cheapest margin survives the whole-crown check.

The crown as a frame #

The crowns are the earlier ones: symmetric, planar, each branch $2^{-1/2}$ of its parent’s length and turned 20° at every fork, with every load a compression along its branch equal to the tips the branch carries — the axial loading under which the per-branch sizing was cleanest. They are five, six, seven and eight generations deep, sized for their daughters at margins from none (the crown resized without a base term) through 1.25, 1.5, 1.7, 2, 2.5, 3, 4, 5 and 6.

Each crown is assembled as a planar frame. Every branch is a straight member cut into beam-column elements, rigidly joined to its parent and its sibling at the fork, the trunk clamped at the ground. A member’s bending stiffness is its sized $r^4$, as in the sizing, and it carries the compression the sizing gave it. The frame’s buckling load factor λ is the multiple of those loads at which it first buckles: λ = 1 means the crown buckles at exactly the loads it was sized for.

The instrument is checked twice before it is used. A clamped cantilever buckles at the Euler load to three parts in a hundred thousand. A loaded daughter on an unloaded parent — the one case where the spring is exact, since an unloaded parent’s tip resists a moment with exactly its stiffness over its length — buckles where the spring model says, to under two parts in a million. Each branch is cut into four elements on crowns of up to seven generations and two on the eight-generation crown; on the six-generation crown with no base term the two cuts give the same load factor, 0.233677, to six figures, so the cut is not what any result below depends on.

A fifth of the load #

The seven-generation crown sized at the cheapest margin, 1.7, buckles at 0.205 of the loads it was sized for. Every one of its 127 branches was sized to carry its own load on its own parent’s spring, and the frame they make buckles under a fifth of their combined design.

The mode is not local. Drawn over the crown, it leans the trunk, tips the first fork, and carries the whole crown above them over to one side, every branch bending a little. Turned by the dial to no base term, the crown buckles at 0.196; at a margin of 2.5, at 0.255; at 4, at 0.403. A stiffer set of parents helps, and none of them brings the frame anywhere near the loads every branch was sized to carry.

Every crown, every depth #

The finding does not depend on the crown chosen. At the cheapest per-branch margin, crowns of five, six, seven and eight generations buckle at 0.315, 0.248, 0.205 and 0.176 of their design loads: lower with every generation added. Multiplied by the number of generations they are 1.58, 1.49, 1.44 and 1.40, so the load factor falls a little faster than one over the depth — each generation adds a storey of posts to lean. Across every depth and margin read, the highest load factor is 0.88, for the five-generation crown at a margin of 6 — stiffer parents buy a great deal at the bottom of a shallow crown, and still not enough.

Below a margin of 2, where the base term sizes almost nothing, the load factor barely moves with the margin: 0.196 to 0.209 at seven generations. Above it, where the base term takes over the parents, the load factor climbs in proportion to the margin, at about a tenth of it: 0.255 at 2.5, 0.403 at 4, 0.594 at 6. The margin is the one handle the sizing has on the frame, and it works only once it is large enough to be what sizes the parents.

Why the margin works only above two #

The load factor’s two regimes match the two regimes of the sizing. The parent essay found a threshold at a margin of 2.043, above which the base term sizes every interior branch of an axial crown and the junctions stop alternating; below it, the base term only caps the zigzag’s low side. On the seven-generation crown, at margins of 1.25 to 2, the base term sizes exactly one branch — the trunk — and the column term the other 126; at 2.5 it sizes 63, every parent with daughters to support, and the column term the 64 tips.

So below the threshold a larger margin thickens only the trunk, and the frame barely notices, because the trunk is not where the frame is soft: its load factor moves from 0.198 at 1.25 to 0.209 at 2. Above the threshold every parent is sized by the base term in proportion to the margin, every parent’s $r^4$ rises with it, and the frame’s load factor follows — at about a tenth of the margin, which is what a uniform thickening of the parents would give. The alternation three rules and the parent essay traced is not what makes the crown soft. What makes it soft is that the parents, however they alternate, were sized for springs that leave out the crown above.

What the spring left out #

The spring stands in for the parent’s tip: when a daughter’s base turns, its parent resists with its own stiffness over its length. That is right as far as it goes, and it is all a spring can say about the structure below. What it cannot say anything about is the structure above. In the per-branch model each branch carries its load as a point at its own tip. In the frame, the load a branch carries is the load of every tip above it, standing on the branches above it, and when the branch leans the whole crown above it leans with it — so each of those loads swings out by the lean times its height above the branch, and pushes the lean further.

That is the old reason a tall column carrying its load on a rigid post above its tip is weaker than the same column carrying it at its tip, and a crown is nothing but posts on posts. The bending rule met the same arithmetic from the other side: a cube law with a lever arm sized each branch against every tip load’s arm about its base, and sized for how far it bends, against each arm squared. Buckling sized branch by branch had kept only the branch’s own length, and the frame puts the arms back. The seven-generation crown stands about three of its trunk’s lengths tall. Its trunk, sized for a point load at its own tip, would hold 3.66 times its share as an isolated cantilever; made to carry its crown as a rigid body above it, the frame buckles at 0.603.

Where the softness lives #

The frame can be taken apart by making parts of it rigid. At the cheapest per-branch margin, a rigid trunk raises the seven-generation crown’s load factor from 0.205 to only 0.260. Making the first three generations rigid raises it to 0.461; everything above the first fork, to 0.388; everything above the trunk, to 0.603. Without a base term the same five readings are 0.196, 0.280, 0.505, 0.365 and 0.407; at a margin of 4, 0.403, 0.486, 0.820, 0.776 and 1.419.

Across the crowns of six and seven generations at those three margins, a rigid trunk raises the load factor by at most 1.6 times, and a rigid crown above the trunk by 1.8 to 3.5 times. The softness is not concentrated in one member that a thicker trunk would fix. It is spread through the crown — every generation bends a little and carries its crown over — and only at the stiffest margin does a rigid crown let the trunk alone hold its loads.

Where the mode lives #

Where the mode lives can be read from its strain energy. In every crown at every margin and depth read, the trunk carries more of the mode’s strain energy than any other single member, and the trunk with the two generations above it carries at least half: at seven generations, 68 per cent at the cheapest margin. On the seven-generation crown the trunk’s own share is 37 per cent with no base term, 24 at 1.7, 20 at 2.5 and 19 at 4 — the stiffer the parents, the more evenly the mode spreads through the first four generations. As the crown deepens, the share in its bottom three generations falls — 89, 78, 68 and 59 per cent at five, six, seven and eight generations at the cheapest per-branch margin — because there are more storeys above to bend, but at every depth no single member carries as much of the mode as the trunk, and the tips, which carry nothing above them, carry almost none of it.

The branch the spring model marks weakest is somewhere else. At margins up to 2 the branch the per-branch sizing holds at the smallest margin over its own spring is one of the trunk’s two daughters, and it carries 3 to 16 per cent of the mode — a part, not the centre. From a margin of 2.5 the weakest branch by the spring model sits deep in the crown, three to five generations up, and carries under two per cent. The spring model’s ranking of branches says which branch is closest to buckling on its own spring. It does not say where the crown buckles.

The cheapest margin, checked as a frame #

A crown that buckles at a fifth of its load has to be thickened, and the simplest repair is to thicken every branch alike. Scaling every branch’s $r^4$ by 1/λ scales the frame’s buckling load by 1/λ, so the crown then buckles at exactly its design loads, and every branch’s $r^2$ — the wood — grows by $\lambda^{-1/2}$. That gives each margin a second price: the wood it needs to hold as a frame.

Branch by branch the cheapest margin is 1.7 at every depth, as the earlier essay found. As a frame it is 4, at every depth read. But the floor is flat: on the seven-generation crown the wood to hold as a frame is 72.65 at a margin of 1.7 against 72.10 at 4, a difference of 0.8 per cent; on the eight-generation crown 0.5 per cent; on five generations, 2.0. Any margin from 2.5 to 6 costs within half a per cent of the cheapest on the two deeper crowns. What does matter is having a base term at all: with none, the seven-generation crown needs 86.13 to hold, 19 per cent more than the cheapest.

And the price of the whole-crown check itself is large. Held as one frame, the seven-generation crown at its cheapest carries 2.19 times the wood the per-branch sizing asked for; the eight-generation crown 2.38 times.

What it does to the earlier argument #

The argument that ran from the clamped column to the parent sized for its daughters was a sequence of better springs. Each step took one more thing the branch stands on into account, and each found the previous crown short. This is the step the springs could not take, and it finds the crown short by a factor of three to six in load. The exponents and the closed forms the earlier essays read off sized crowns describe crowns sized branch by branch; none of them describes a crown that holds its loads as a structure, and a crown thickened uniformly to do so keeps every ratio between its branches and so every exponent — which means the exponents were never the place to look for whether a crown stands. Fitting the exponent on a crown that holds as a frame would return exactly what it returns on one that buckles at a fifth of its load.

What carries over is the base term. Without it the crown is 19 per cent more expensive to make stand; with it, at any margin from 2 up, the frame-safe crown is within a per cent of its cheapest on the deeper crowns. The parent sized for its daughters was the right idea and the wrong check.

An idealised frame #

The frame is planar and symmetric and its loads run along its branches, the per-branch model’s own idealisation, kept so that the frame is that model assembled rather than a different one; a crown under gravity bends as well as compresses, and a real crown is neither planar nor symmetric. The ground is rigid. The repair is uniform, the simplest and not the cheapest: a crown free to thicken its trunk more than its twigs would hold its loads with less wood, and the margin at which that crown is cheapest is a different optimisation. Nothing here says that any tree is sized against buckling at all.

Readings that would undo it #

A clamped cantilever or a daughter on an unloaded parent buckling anywhere but where the closed forms say. A sized crown that holds its design loads as a frame. A crown whose lowest mode puts more strain energy in some member than in its trunk, or less than half in its bottom three generations. A rigid trunk raising the load factor by more than 1.6 times, or a rigid crown above it by less than 1.8. A frame-holding cost least at a margin other than 4. Each is checked against the assembled frames whenever they are read.

Branches that hold and a crown that does not #

A crown sized branch by branch — each on its parent’s spring, each parent for its daughters’ bases — buckles as one frame at 0.18 to 0.32 of its loads at the cheapest per-branch margin, and lower the deeper it is. The spring stood in for the parent and left out the crown above, which leans with every branch and hangs its loads further out. The mode lives in the trunk and the first forks, not in the branch the spring model marks weakest. Held as a frame, the cheapest margin is 4 rather than 1.7, on a floor flat enough that any margin from 2.5 up will do, and the crown needs more than twice the wood its branches asked for.

Still open: a crown thickened where the mode lives #

The repair here thickens every branch alike. The mode lives in the trunk and the first forks, and thickening a twig costs wood the mode hardly uses. The next measurement thickens each branch in proportion to its share of the mode’s strain energy, repeatedly, until the frame holds its design loads, and asks how much wood that saves over the uniform repair, whether the result still conserves the exponents the per-branch crown did or drifts towards a thicker base and thinner tips, and whether the margin at which it is cheapest moves again.