The chimney that breaks a third of the way up

The ladder that lets go of the wall noted in passing that a rod pivoting about its foot falls in a way no part of it would fall alone: its top accelerates faster than free fall, its lower part more slowly, so that the top is dragged along by the material below it and the lower part holds the rest back. A chimney, it said, is not strong enough to transmit those forces, and breaks partway up. That essay followed the falling rod until it parted from its support. This one is about the chimney: where exactly it breaks, why there, and how far it has to lean first.

The answer is a free-body diagram of a kind a structural engineer draws every day, applied to a structure that is moving. The load nobody applied found forces inside a still structure that no external load accounted for; here the forces inside a falling one come from its motion, and they are larger in some places than any wind ever made them.

What the photographs show #

Photographs of factory chimneys being felled, from the early twentieth century onwards, show the same sequence again and again. The demolition crew cuts a notch at the base on the side towards which the chimney is to fall, and removes the props. The chimney begins to tip, slowly at first, rotating about the uncut part of its base as a hinge. Then, well before it is anywhere near the ground — often when it has leaned only a few degrees — it snaps somewhere in its lower half. The part above the break continues to fall but lags behind, its top trailing, and the two pieces reach the ground separately.

Nothing has struck the chimney. Its only loads are its own weight and the force at the hinge, and it was strong enough to stand for decades under the first. What has changed is that it is moving, and a moving column needs forces inside it that a standing one does not.

The free body above the break #

A rigid column of length L tipping about its base turns with an angular acceleration that follows from its weight and its moment of inertia about the pivot. Its weight acts at its middle and has a lever arm $\tfrac12 L\sin\theta$ at a tilt θ from upright; its moment of inertia about the end is $\tfrac13 mL^2$, the quantity the mass and where it sits computed for a rod turning about its end. So

$$\alpha = \frac{mg\,\tfrac12 L\sin\theta}{\tfrac13 mL^2} = \frac{3g\sin\theta}{2L}.$$

Every point of the column shares that angular acceleration, so a point at distance r from the pivot has a sideways acceleration $\alpha r$, growing linearly from nothing at the base to $\tfrac32 g\sin\theta$ at the top. Gravity, meanwhile, supplies every metre of the column with the same sideways push per unit mass, $g\sin\theta$, its component across the column. The two are equal at one height, where $\alpha r = g\sin\theta$: at $r = \tfrac23 L$. Above that point the material is required to accelerate faster than its weight alone would make it; something must pull it along, and the only thing that can is the material below. Below that point the weight supplies more sideways push than the turning needs; the material there pushes forward on the rest and is itself held back.

The second figure draws exactly this on the part of the column above a cut a third of the way up. The vertical arrows are the weight of each metre; the slanting ones are the force each metre needs to follow the turning, drawn on the same scale. They do not balance metre by metre, and the cut must supply what they lack. Summed over the segment, the two sets of forces happen to cancel at the cut a third of the way up, so no sideways shear acts there; but they do not cancel in their moment. The weight acts mostly on the lower part of the segment, the turning’s demand mostly on the upper part, and the cut must supply the couple that reconciles them. That couple is the bending moment.

The height two-thirds of the way up is familiar from another problem. The point that takes the blow without a jolt found that a rod pivoted at one end can be struck at two-thirds of its length without any jolt at the pivot: its centre of percussion. It is the same point for the same reason. The centre of percussion is where a sideways force produces exactly the acceleration that the rod’s rotation about the pivot requires there, and gravity on a toppling column is a sideways force on every metre, matched only at that one height.

Where the bending peaks #

Carrying the free body through for a cut at any height s gives the moment there:

$$M(s) = -\frac{mg\sin\theta}{4L^3}\, s\,(L - s)^2.$$

It vanishes at the base, where there is nothing below the cut to hold anything back, and at the top, where there is nothing above it to drag along. It is largest where $s(L-s)^2$ is largest, at $s = L/3$, and there it is $mgL\sin\theta/27$. The first figure plots it for three tilts; the shape is the same at every tilt and grows with sinθ.

The comparison with a column standing still at the same tilt is the surprise. Propped at an angle, a column’s weight is carried by bending at its base, where the moment is $mgL\sin\theta/2$, and that is where a leaning chimney would fail. The same column, released and falling, carries none at the base and at most a twenty-seventh at a third of the way up: thirteen and a half times less, in a different place. Falling relieves almost all the bending a lean would cause. The chimney does not break because the fall overloads it; it breaks because its weakest point in falling is somewhere it was never designed to be weak.

Why a pivot is the cause #

The bending comes from the pivot, not from the falling. A column dropped from a height with nothing touching it falls with every part accelerating at $g$ together, and no part needs to push or pull on any other: its internal forces vanish, the state of weightlessness a falling lift shares with everything in it. It does not bend, however long and slender it is. A column falling about its foot is different only because the foot is held. The hinge forces the lower part to accelerate less than it would freely and, through the rigid body, forces the upper part to accelerate more, and the bending is the column’s way of passing those demands along its length.

The hinge also reshapes the motion. Once a toppling column has tipped past about fifty-five degrees from upright, its top accelerates downward faster than a stone would fall — the classic demonstration in which a ball resting on the end of a hinged board is left behind as the board drops — because the column’s lower part, held back by the hinge, passes some of its weight’s effect up to the top. The pile that lands heavier than it weighs met the same effect in a falling chain. A chimney that broke and lost its hinge would fall more gently, and its top, freed from the column below, lags behind exactly because it is no longer being dragged.

What the chimney was built for #

A chimney is designed to stand, and its builders made it strong where a standing chimney needs strength. Its own weight presses on its base; the wind, the largest sideways load it meets, pushes on its whole height and bends it with a moment that is greatest at the base and falls to nothing at the top, growing roughly as the square of the height above each section. So chimneys are built with their thickest walls at the bottom and their thinnest at the top, and a chimney that survives a century of gales has its strength concentrated precisely where the wind’s bending is.

The falling chimney asks for strength somewhere else. Its bending vanishes at the base, where the builders put their bricks, and peaks a third or more of the way up, where the walls have already begun to thin. The chimney is weak exactly where its fall loads it, because it was strong exactly where its standing loaded it. That is the general shape of a surprising failure: not a structure that was too weak, but one whose strength was distributed for a load it no longer carries.

Squeezed along its length, then stretched #

The free body also gives the force along the column. A standing column is compressed by the weight above each section, and early in the fall the toppling column is too. But as it speeds up, every part of it is moving in a circle about the pivot and needs a pull towards the pivot to stay on that circle, growing with its distance and with the square of the rate of turning. Near the top that pull eventually outweighs the component of the weight pressing down along the column, and the top goes into tension — from a tilt of about forty-one degrees, where $\cos\theta = \tfrac34$. By seventy degrees the whole column is being stretched, and the hinge at its base is pulling it round rather than holding it up.

For a chimney this hardly matters, because a chimney has broken long before. But it is a second way the falling body differs from the standing one: the forces inside it change sign as it goes, and a material that carries compression and not tension, like stacked brick, is at risk from every one of those reversals.

Brick, and the first crack #

Brickwork is strong in compression and has almost no strength in tension; the mortar between bricks adheres weakly and cracks open under very small tensile stresses. A brick chimney therefore stays intact only as long as every part of every cross-section remains in compression. The weight of the chimney above any section presses it together uniformly; the bending tries to stretch one face and squeeze the other. The section first cracks when the bending stress on the stretched face exceeds the uniform compression there.

For a hollow chimney of radius R and height L, the bending stress a third of the way up grows as $L^2\sin\theta/R$, with the square of the height because the moment grows with the length times the weight. The compression there grows only with the height. So the taller and slenderer a chimney, the earlier in its fall the bending wins. The last figure computes the tilt at which the first crack opens. A squat chimney, ten times as tall as its radius, must lean thirty-one degrees. A typical factory chimney, forty times as tall as its radius, cracks at twelve. A very slender one, eighty times, cracks at six — in the first second or so of a fall that will take several more. This is why so many photographs show the break early: a tall brick chimney cannot lean far before the bending a third of the way up opens it.

The crack opens on the stretched face, which is the face towards which the chimney is falling: the column bows forward in its lower middle, its leading face convex, while the top, no longer dragged by the material below the crack, falls behind. Steel chimneys and reinforced concrete ones carry tension and do not fail this way at small tilts; they bend, and if they fail at all it is later and lower in the fall.

A tapered chimney breaks higher #

Real chimneys are not uniform. They taper towards the top, with walls that thin as they rise, because the lower courses carry more weight. Tapering changes both sides of the argument. It takes mass away from the top, so the turning drags less material and the bending moment is smaller there. But it takes away strength faster: the resistance of a hollow section to bending grows as the cube of its diameter when its walls are proportional, while its mass per metre grows only as the square. The most-stressed section moves up. In the fourth figure, a chimney whose diameter at the top is under a third of that at the base is most stressed at nearly half its height.

So the third is the uniform column’s answer, and real chimneys break anywhere from a third to more than half of the way up, depending on their taper and on how the thickness of their walls was graded by the builders. Where a particular chimney breaks is a record of how its strength was distributed, written at the moment it fails.

Where the argument holds #

The calculation treats the column as rigid until it breaks, which is accurate for brickwork, whose elastic bending under these loads is a few millimetres over tens of metres. It treats the base as a perfect hinge that holds the chimney’s foot in place. A real demolition hinge is a strip of uncut brickwork that crushes and slides; if the foot slips forward, the motion is no longer a pure rotation and the forces change. The shear that the base must supply is greatest early in the fall, a quarter of the chimney’s weight times sinθ, and a chimney whose foot cannot provide it slides as well as tips.

The calculation also ignores the air. A chimney of tens of metres falling through the air at several metres a second meets a drag that adds to the bending in its upper part; for a slender chimney it is small but not negligible late in the fall, after the first crack has already opened. And it treats the chimney as a single continuous body, where brickwork is many bodies, held together by friction and weak mortar, whose response to bending includes rocking at joints before any crack forms.

What the figures cannot show #

The figures show the moment that opens the first crack, not the fall that follows. Once the chimney breaks, the upper piece is a new body pivoting on its broken end, which is itself moving, and it may break again — some chimneys come apart in three or four pieces, each new break forming in the falling remains by the same argument applied to a body whose pivot is moving. The figures cannot show which of those pieces breaks next or where. Nor can they show the brickwork’s real strength. Old brick and mortar vary widely, and a chimney whose mortar has weathered for a century may have no tensile strength at all, while one with good lime mortar can resist a little. The figures assume none, which gives the earliest crack possible; a real chimney leans somewhat further.

Still open: what a pile of blocks does #

A column of loose blocks with no mortar at all — a stack of children’s bricks, a pile of coins, a tower of dominoes stood on end — cannot bend at all without opening a joint, and cannot carry any shear at its base beyond what friction allows. When such a tower topples it does not behave as one body for long: joints open where the bending would be largest, blocks slide where the shear would be, and the tower comes apart in a sequence that depends on the blocks’ friction, their proportions and how the toppling started. Films of toppling block towers show both kinds of failure, bending openings in the lower half and slipping near the base, and which happens first has been studied as a problem in the mechanics of granular and masonry structures. A general rule for which mode wins, for towers of different slenderness and friction, has not been settled, and the same question decides how ancient masonry columns fall in earthquakes.

The habit worth carrying away is to draw the free body of a part even when the whole is in motion. A uniform column toppling about its base turns at α = 1.5·g·sinθ/L, so only its point at two-thirds of the height moves as its own weight would move it; above, the material is dragged along and below, held back, and the bending this needs is s(L − s)² in shape, greatest at a third of the height at mgL·sinθ/27 — a thirteenth and a half of what the same column carries at its base standing still at that tilt — which a brick chimney forty times as tall as its radius cannot carry past a lean of 12°. A falling chimney is not overloaded. It is loaded somewhere new.