The block that is safer for being bigger

Every dynamic model in this collection has a stiffness in it. A frame has $EI$, a spring has $k$, a cable has a pretension. Divide the stiffness by a mass, take a square root, and there is a natural period to enter a spectrum with.

A block resting on the ground has none of those. It is not attached to anything, it has no elastic restraint whatever, and when the ground moves hard enough it lifts off one corner and rocks. That is a structural system with no stiffness, no damping and no period, and it is the system a great many things are: a statue, a transformer, a filing cabinet, an unanchored tank, a masonry wall, a stack of stone.

Which free body produced the number #

The block, taken about the corner it is pivoting on.

Two moments act. The weight $mg$ acts at the centre of mass, at a horizontal distance from the corner that shrinks as the block leans — from $b$ when it is upright to zero when the centre of mass is directly above the corner. And the ground’s acceleration $a$ acts as an inertia force $ma$ at the same point, at a lever that grows.

So the restoring moment is

$$M(\theta) = mgR\sin(\alpha - \theta)$$

with $R$ the diagonal from the corner to the centre of mass and $\alpha = \arctan(b/h)$ the slenderness angle. There is no material property in it at all. There is a weight and there is a geometry, and that is the whole of the resistance.

Three facts follow immediately and each is unusual.

Uplift happens at $a/g = \tan\alpha = b/h$. Compare the two moments at $\theta = 0$: rocking begins when $ma\cdot h > mg\cdot b$. The mass cancels, so the acceleration required is a pure function of the shape — and it contains no size whatever.

The stiffness is negative. $dM/d\theta = -mgR\cos(\alpha-\theta) < 0$ everywhere. The system gets weaker as it leans, which is the opposite of every spring, and the equilibrium at $\theta = 0$ is stable only because of the jump at the origin: the moment is whatever the ground demands until uplift and then it is $mgR\sin\alpha$.

And there is a second equilibrium, at $\theta = \alpha$, where the centre of mass is over the corner and the restoring moment is zero. That is balance, and past it the block goes over.

The period that is not one #

Ask what the period of a rocking block is and there is no answer, only an answer for each amplitude.

Housner integrated the equation of motion in 1963 and got a half-period of

$$T = \frac{4}{p}\,\text{arccosh}\!\left(\frac{1}{1 - \theta_0/\alpha}\right), \qquad p = \sqrt{\frac{3g}{4R}}$$

which depends on the release angle $\theta_0$ and rises without bound as $\theta_0 \to \alpha$.

At a tenth of the toppling angle the block drawn takes about 1.6 seconds to come back; at nine tenths it takes several times that. A block set exactly at balance never returns, because the restoring moment there is zero.

That is a genuine obstacle to using any of the ordinary tools. The spectrum is not a load requires a period to enter with; a response spectrum analysis requires a stiffness to build a mode from; a ductility factor requires a yield displacement. None of them exists here, and every attempt to force the problem into that framework produces an equivalent linear system whose properties depend on the answer.

Why bigger is safer #

Now the result the essay is named for, and it is worth being careful about because half of it is size-free and half is not.

Uplift is at $\tan\alpha$, which has no size in it. Two geometrically similar blocks — one 2 m tall and one 6 m — lift off at exactly the same ground acceleration, at the same instant, in the same earthquake.

Overturning is different. To topple, the block has to rotate through the whole angle $\alpha$ before the pulse driving it ends. How fast it rotates is governed by $p = \sqrt{3g/4R}$, which falls as the block grows. A bigger block rocks more slowly, so it gets less far in the time available.

Integrate the equation for a family of blocks of one shape and the pulse amplitude required to overturn rises steeply with size. A block three times the size of the one drawn needs about four and a half times the ground motion.

This is the one result in structural dynamics where being bigger is simply and unambiguously safer, with no compensating penalty anywhere. Housner wrote it down after the Chilean earthquake of 1960, where slender elevated water tanks survived while squatter, apparently more stable ones fell — an observation that made no sense at all until the size dependence was found.

The mechanism deserves stating in words because the algebra hides how ordinary it is. Overturning is a race between the pulse’s duration and the block’s rotation. A bigger block is a slower clock, and the pulse runs out first.

What the numbers are, on an ordinary object #

It is worth putting real dimensions to it, because the conclusions sound more exotic than the objects they apply to.

A filing cabinet is 0.45 m wide and 1.8 m tall, so $b/h$ measured to the centre of mass is about 0.5 and it lifts off at half a g. That is a strong earthquake but not an unusual one, and the cabinet rocks.

A stone gatepost 0.3 m square and 2.4 m tall has $b/h = 0.125$ and lifts off at 0.125 g — which is a moderate earthquake, or a heavy lorry, or a slammed door in the right building. It rocks constantly and almost never falls over, because $\alpha$ is small and the rotation needed to topple it is large.

A transformer 1.5 m wide and 2.5 m tall lifts at 0.6 g, and if it goes over it takes the substation with it.

The ordering is worth noticing: the slender object is the one that starts rocking soonest and the one least likely to fall, because the same $\alpha$ that makes uplift easy makes toppling hard. Slenderness is a liability for uplift and an asset for overturning, which is why intuition about “stability” — which usually means “does not move” — points the wrong way for the question that matters.

The energy goes into the landings #

There is no dashpot in the model and the block still comes to rest, which needs explaining.

Rocking loses energy only at the instants the block lands on its other corner. Conservation of angular momentum about the new pivot gives a velocity ratio across the impact of

$$r = 1 - \tfrac{3}{2}\sin^2\alpha$$

so a slender block ($\alpha$ small) barely loses anything and a squat one loses most of it. The damping of a rocking block is a property of its shape, and it is the same shape that decided the uplift acceleration and the period — one geometric parameter doing three jobs.

That has an uncomfortable consequence for the slender case. A very slender block has $r$ close to 1, loses almost nothing per landing, and rocks for a long time — so the first pulse does not decide the outcome and the block can be walked over by a sequence of smaller ones. The size effect protects it and the low damping exposes it, and which wins depends on the ground motion’s duration rather than on its peak.

Where the same behaviour is designed in #

The block that rocks by accident is a hazard. The same mechanism, provided deliberately, is one of the more elegant ideas in earthquake engineering.

A rocking foundation. Let a shear wall or a bridge pier lift off its foundation — the earthquake asks for a displacement is why that is acceptable, since a system that limits its force has to accommodate a displacement instead — rather than fixing it down, and the moment it can deliver to the ground is capped at the overturning moment of its own weight. That is a force limit with no material in it, it cannot be exceeded, and it does not degrade with cycling — a much better fuse than a plastic hinge, which fatigues.

An unanchored tank. The liquid has a period of its own touches this: a tank that is allowed to lift one side of its base rather than being bolted down attracts less force, and the alternative — a rigidly anchored tank — puts its whole overturning moment into the shell as vertical compression and buckles it.

And a self-centring system. A rocking wall with a vertical post-tensioning tendon down its centre has a restoring force that grows with rotation instead of shrinking, so the negative stiffness is cancelled and the system returns to plumb. Add a replaceable energy-dissipating element and the result is a structure that survives a large earthquake with no residual drift and one component to change.

Made weaker on purpose is the general idea. Rocking is its purest instance, because the “weakness” is not a material being yielded but a connection being omitted.

Why the framework does not fit, in detail #

It is worth being explicit about which of the standard tools break and why, because the list is instructive about what those tools actually assume.

A natural frequency assumes a linear restoring force. There is none: the restoring moment is $\sin(\alpha-\theta)$, which is not proportional to $\theta$ and does not pass through the origin. The system’s “frequency” is a function of amplitude, and the function is unbounded.

A ductility factor assumes a yield displacement to normalise against. There is no yield: nothing in the block ever leaves its elastic range, and the non-linearity is entirely geometric.

A response spectrum assumes the response to a general ground motion can be assembled from the responses of linear oscillators. It cannot here, because the system is not linear and superposition does not hold.

And an equivalent linear system — the usual escape route, replacing the real system with a linear one of matched secant stiffness and matched energy — needs a secant stiffness, which for a negative-stiffness system is negative. The escape route is closed by the sign.

That leaves direct integration, which is what Housner did and what anybody doing it now does. It is not a hardship — the equation is one line and the impacts are one multiplication — and the reason it is worth noticing is that a whole discipline’s toolkit turned out to rest on assumptions that a block on the floor does not satisfy.

Where the model stops #

The block was assumed not to slide. It does, if $\mu < \tan\alpha$: a slender block on a slippery floor slides before it rocks, and the two motions can combine into a slide-rock response no two-dimensional model captures.

The impact was assumed instantaneous and central. A real corner crushes, bounces and rotates about a contact patch rather than a point, so the measured restitution is scattered around the theoretical value and generally below it.

The response is chaotic. Rocking is one of the classic sensitive systems: two nearly identical ground motions can produce survival and collapse, and the overturning boundary in the amplitude–duration plane is fractal rather than a curve. The number computed above is a boundary through a scatter.

Nothing here is about what the block is standing on. A rocking block on a stiff floor pivots about a line; on a soft one it settles into the surface and the pivot moves inward, which reduces $b$ and makes everything worse. The ground is a spring is the general effect, and here it changes a geometric constant rather than a stiffness.

And it is two-dimensional. A real block can rock about any of four edges and about its corners, wobbling round its base, which is a much harder problem and a much less stable one.

The generalisation #

The habit worth taking away is to ask, of any stability criterion, whether it has a size in it.

The period nobody chose is about a period that is a consequence of a structure’s mass and stiffness rather than a choice; here there is no period at all to be a consequence of anything, and the object still responds. Weight is the only thing resisting it is the static overturning check, and it has no size in it either: a factor of safety against overturning is a ratio of two moments, both proportional to the weight, so it is a statement about proportions alone. That is correct and it is not the whole answer, because a static check asks whether the disturbance is large enough and a dynamic one asks whether it lasts long enough. The first is size-free and the second is not.

There is a second habit, and it is about what to do when the standard framework does not fit. Rocking has no stiffness, no period and no ductility, so a spectral method cannot be applied to it — and the correct response is not to invent an equivalent linear system but to integrate the equation, which takes a few lines and produces the answer directly. The load that is over before it has moved makes the same choice for a blast: where the standard instrument is built on assumptions the problem does not satisfy, the honest route is usually the direct one, and it is usually shorter than the workaround.

The general form: a criterion about force is often size-free, and a criterion about time never is, because a time criterion has to compare the disturbance’s duration with something the object supplies — and the only thing an object of a given shape supplies is a timescale set by its size. That is why scale changes everything so often shows up as a difference between a static answer and a dynamic one on the same structure, and why an intuition trained on models is so unreliable about the buildings they are models of.