Weight is the only thing resisting it

An advertising hoarding beside a motorway is a simple thing to design and an easy thing to lose. It is eight metres wide, six metres tall, and stands on a base two and a half metres across. The frame behind the panel is a trivial calculation: the wind delivers about 48 kN over the whole face, which two modest columns and a few rails carry without noticing — a strength question with an obvious answer.

Then the storm arrives and the hoarding is found lying in the field behind it, entirely undamaged.

Every number in that figure is a length or a weight. There is no yield stress in it, no second moment of area, no modulus. A body is held down by what it weighs and how wide it stands, and a structure can therefore be comfortably strong and marginally stable at the same time, with no calculation of the first giving any warning about the second.

The free body is the whole object #

The habit this site has spent ninety essays building is to cut something and insist the sums cancel, and the cut here is the one nobody draws: a plane through the ground, with the entire structure above it. Nothing internal appears. The members, the joints and the connections are all inside the body, so their forces cancel in pairs and drop out of every equation.

What is left is four quantities: the weight, the wind, the length between them, and the width of the base. Taking moments about the leeward edge of the base,

$$\sum M = W\frac{B}{2} - F\frac{H}{2} = 0$$

and the factor of safety is the ratio of those two terms. For the hoarding it is 150 against 144.

That the internal forces vanish is not a convenience. It is the reason the answer is independent of the design: two hoardings with the same weight and the same base and completely different frames have identical stability, and a stronger frame that happens to be lighter is worse. Stiffness is not strength is the neighbouring statement; this one is sharper, because stability is not either of them. The engineer’s instinct that more material is more safety has its sign reversed here, and this is the only place in the collection where that is true.

Height is squared and width is not #

The wind on a taller structure is larger and has a longer lever arm, so the overturning moment grows as the square of the height while the restoring moment grows only with the base.

Two metres of extra height costs more than half the safety. That is the arithmetic of a square, and it is the reason scale is a subject rather than a footnote in this collection: a rule that works at one size fails at another for reasons that have nothing to do with the material and everything to do with which power of the length each term carries.

The same reading works the other way. To double the factor of safety, the choices are to double the weight, double the base width, or reduce the height by 30% — and the last of those is the cheapest thing on the list, which is why the hoardings that survive are the low wide ones and not the well-built tall ones.

The failure starts long before it tips #

The factor of 1.04 suggests a structure four per cent away from disaster and otherwise fine. It is not fine, and the reason is that something has already happened at the base which the overturning calculation cannot see.

Take the resultant of the weight and the wind and find where it crosses the base. For the hoarding the eccentricity is

$$e = \frac{M_o}{W} = \frac{144}{120} = 1.20\ \text{m}$$

from the centre of a base whose half-width is 1.25 m. The resultant is not merely outside the middle of the base; it is 5 cm inside the edge.

The threshold in that figure is exact and it is a piece of geometry. A base of width $B$ has a section modulus $B^2D/6$ over an area $BD$, so the eccentricity at which the pressure at the far edge reaches zero is

$$\frac{Z}{A} = \frac{B^2D/6}{BD} = \frac{B}{6}$$

which is the middle third, arriving here from the direction of a whole body rather than a cross-section. It is the same statement as the kern of a section, and this site argues it twice deliberately, because most engineers meet it once and file it as a rule about masonry.

The ratio between the two thresholds is √3, always #

Uplift at one edge needs $e = B/6$; overturning needs $e = B/2$. Both eccentricities are produced by the same wind, and the wind’s moment goes as the square of the height, so the two critical heights are in the ratio

$$\frac{H_{\text{tip}}}{H_{\text{lift}}} = \sqrt{\frac{B/2}{B/6}} = \sqrt{3} = 1.732$$

and there is nothing in that but the number three. It holds for a hoarding, a retaining wall, a tower crane base and a chimney; it does not depend on the weight, the width, the wind pressure or the material. For the hoarding, uplift begins at 3.54 m of height and toppling at 6.12 m.

A design at a comfortable-sounding factor of two against overturning is therefore at $\sqrt{2}$ of the height at which it tips, which is 1.41 — comfortably past 1.732/√3, and the base has been partly lifting for some time. A factor of safety against overturning of anything below three is a statement that part of the foundation is not in contact, and that is a different conversation with the ground than the one the calculation was having.

What the ground is asked for once the base lifts #

The pressure under a partly lifted base is not a small correction. It is a different problem, because the length of base in contact is itself unknown and has to be found from the requirement that the pressure block’s centroid sit under the resultant.

The kink in that curve is where a linear calculation stops being conservative and starts being wrong. Under the hoarding, with 1.2 m of eccentricity on a 2.5 m base, the contact is 0.15 m — six per cent of the foundation, carrying a peak pressure of 200 kPa where the average would have been 6 kPa.

Soil under that pressure does not stay put. It yields, the base rotates a little, the eccentricity increases, the contact shortens further, and the peak pressure rises again. That is a positive feedback, and it is why an overturning failure in the field looks nothing like the sudden rigid-body rotation the calculation imagines: the structure leans first, over minutes or seasons, and goes over when it has already lost.

The lever arm is not always half the height #

Every number so far has used a uniform pressure over the face, which puts the resultant at mid-height and gives an overturning moment of $F \cdot H/2$. That is the right assumption for a hoarding, whose face is short enough that the wind speed hardly varies over it. It is the wrong one for anything tall, because wind speed increases with height above the ground and pressure goes as its square.

The consequence for the arithmetic is a single factor and it is not a small one. A pressure that rises linearly from zero at the ground gives the same total force at two-thirds the height, so

$$M_o = F \cdot \frac{2H}{3} \quad\text{rather than}\quad F \cdot \frac{H}{2}$$

and the factor of safety falls by a quarter. The hoarding’s 1.04 becomes 0.78 under a triangular profile carrying the same total load — the difference between standing and not, produced entirely by where the load was assumed to act.

Real wind profiles are neither uniform nor triangular; the pressure varies roughly as the fifth root of the height for the lowest hundred metres, which puts the centroid at about 55% of the height for a body standing on the ground. What matters is that the lever arm is an assumption and it is the assumption the answer is most sensitive to, more so than the total force, because the total force appears once in the overturning moment and the height appears twice. Replacing a distribution by its resultant is exact for equilibrium and requires the centroid to be right, and a check that quotes a wind force without saying where it was applied has left out the half that decides the answer.

Sliding is the other failure and it is not the one to worry about #

The same free body offers a second way to fail. The horizontal force has to be transferred to the ground, and the base can only do so up to some limit — a stated capacity, from friction, from a shear key, or from the passive resistance of the soil in front of the footing.

For the hoarding, a base capacity of half the weight gives 60 kN against 48 kN of wind, a factor of 1.25. That is worse than it sounds and better than the overturning one, and the reason is visible in the sweep above: sliding resistance is independent of height, so its factor falls only as 1/H while overturning’s falls as 1/H². Whatever the numbers are at one height, overturning becomes the governing failure as the structure grows, and any tall body that satisfies overturning satisfies sliding with room to spare.

There is one exception worth naming, and it is the one that catches people. Where the resultant is outside the middle third, part of the base is not pressing on the ground, so whatever capacity depended on that pressure has gone with it. The two failures are not independent, and the check that treats them as two separate lines in a table is quietly assuming a base that is fully in contact — which the overturning check has just finished saying it is not.

The same arithmetic at four hundred times the weight #

A structure that resists nothing but its own tendency to fall over is not a special case. It is what every tall building’s foundation is doing, and the numbers scale in the way the powers say they will.

The tower is more stable than the hoarding by a factor of ten, and it is five times taller. The reason is that its weight has grown as the cube of its size while the wind force grows as the square, so the ratio improves with scale — the opposite of the direction scale usually pushes a structure, and worth stating because it is the reason nobody worries about a skyscraper blowing over and everybody worries about a hoarding.

Where towers do run out of stability is not weight but slenderness: a very tall narrow building on a small footprint reverses the arithmetic, and the check that governs its foundation is uplift on the windward piles rather than pressure on the leeward ones. A pile in tension is a different structure from a pile in compression and is designed a different way, which is the practical consequence of everything above: the moment the resultant leaves the middle third, half of the foundation changes from a bearing problem into an anchorage problem.

What the picture cannot show #

The figures above are drawn for a rigid body on rigid ground, and both halves of that are fictions.

The pressure distribution is assumed linear because the base is assumed rigid. A real footing bends, and a flexible one on stiff soil sheds load toward its middle rather than its edges, so the triangular block drawn is an idealisation whose peak is too high for a raft and about right for a small pad. The lifted length is a better-behaved number than the peak pressure and should be trusted further.

The wind pressure is not uniform over the face. The figures use one value because the argument is about the total and its lever arm; a real pressure distribution on a hoarding is higher near the edges and includes a suction on the back, and it fluctuates on a timescale of seconds. What matters for overturning is the integral, which is well behaved, but the moment about the base of a gusting pressure is not the moment of its mean, which is why a dynamic reading of the same load exists at all.

Nothing here has a time in it. Overturning is drawn as a static comparison of two moments, and a body that momentarily exceeds a factor of one does not necessarily fall — it has to rotate far enough for the weight’s lever arm to run out, and that takes energy and time the gust may not have. A rigorous treatment is an energy balance rather than a moment balance, and it gives a higher answer. The static check is used everywhere because it is conservative and because the alternative needs a wind record nobody has.

Where the ladder goes #

The obvious next rung is the one the pressure figure keeps pointing at: what happens to a base that is only partly in contact, and how a base plate or a footing is proportioned once that is admitted. The arithmetic is the same and the design question is entirely different, because the unknown moves from is it stable to how much of it is working.

The second is the load itself. Everything above took the wind as a pressure and turned it into one force; a distributed load and its resultant is the general statement, and a body under a triangular pressure has a lever arm that is not H/2 and a factor of safety that is not what the uniform assumption gives.

The third is the one this essay has been careful not to take: the ground. Everything here has treated soil as something that supplies a pressure and never moves, which is a modelling choice made for a figure rather than a fact about foundations. What the ground actually does under an edge carrying 200 kPa is a subject with its own literature and its own failure modes, and this collection stops at the underside of the base and says so.