The only damping is the landing

A rocking block is safer for being bigger, which is the one result in structural dynamics where scale is simply on the designer’s side.

The hero shows why it is not the whole story. Two blocks of identical 4.4-to-1 proportion, sizes differing by a factor of three, under one 0.8 s pulse of 1.0 g. Both lift off at the same instant, because uplift depends on shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46 per cent and comes back.

Look at the kinks in the large block’s curve. Those are landings, and they are the only places in the entire model where energy leaves the system.

There is no dashpot anywhere #

A rocking block has no damping term. It is a rigid body on a rigid base under gravity; nothing in the equation of motion is proportional to velocity, and between impacts the block conserves energy exactly.

All of the dissipation happens in an instant. The block rotating about one corner reaches the vertical, its other corner strikes the ground, and it continues rotating about the new corner with less angular velocity than it had a moment before. The ratio of the two velocities is the coefficient of restitution $r$, and the energy ratio is $r^2$.

For the hero’s block, $r = 0.926$ — 14 per cent of the energy gone per landing, and nothing gone in between.

That is a completely different dissipation mechanism from anything else on this site. Viscous damping removes energy continuously and in proportion to how fast the structure is moving; this removes a fixed fraction at discrete instants whose spacing is itself amplitude-dependent. A block rocking at large amplitude has long half-cycles and therefore few impacts per second, so it damps slowly at exactly the amplitude where it most needs to damp fast.

Three figures, one pulse, one block, and the outcome changes from overturning to survival. The parameter doing that is not a load, not a geometry and not a material strength. It is a number describing what happens in the millisecond of a landing.

Which free body produced the impact #

The restitution formula is a conservation statement across an instant, and the free body is worth drawing because it explains why no force appears in it.

At the instant the far corner strikes, take the block alone. The forces on it are its weight, which is finite, and the impulsive reaction at the new corner B, which is enormous and lasts no time. Take moments about B: the weight’s moment is finite and acts over zero time, so it contributes nothing; the reaction acts through B, so its moment is zero.

Angular momentum about B is therefore conserved across the impact, and that one sentence is the whole derivation.

Before the impact the block is rotating about the old corner A with angular velocity $\omega_1$; its angular momentum about B is $I_A\omega_1$ less the term the shift of axis removes. After the impact it rotates about B with $\omega_2$ and angular momentum $I_B\omega_2$. Setting them equal and substituting the rectangle’s own inertia gives $r = 1 - \tfrac32\sin^2\alpha$.

Two consequences are worth carrying and neither is obvious.

No material property enters, because the only two quantities in the balance are moments of inertia and they are geometric. The steel block and the stone block of the same shape have the same restitution in this model, which is the strongest hint that the model is idealised.

And the loss is inevitable rather than incidental. $\sin^2\alpha$ is positive for any real block, so $r < 1$ always: the impact cannot be lossless even in principle, because the rotation axis has moved and the momentum about the new axis is genuinely smaller. That is a rare thing in an idealised model — a dissipation that survives every idealisation applied to it.

Where the classical value comes from #

The 0.926 is not measured. It is derived, and the derivation is short enough to give.

At the instant of impact, take the block as a rigid body rotating about corner A with angular velocity $\omega_1$. Its angular momentum about the new corner B is its moment of inertia about B times $\omega_1$, less the moment of the momentum that the change of centre removes. Conservation of angular momentum about B, applied across the instant, gives

$$r = \frac{\omega_2}{\omega_1} = 1 - \tfrac{3}{2}\sin^2\alpha$$

where $\alpha$ is the block’s slenderness angle — the angle from the base corner to the centre of mass, which is also the angle at which it topples.

So the restitution is a shape property with no material in it, exactly like the toppling angle and the uplift acceleration. Housner derived it in 1963, and it has been the standard value ever since.

That figure is where the shape argument bites twice.

A slender block topples at a smaller angle, which is the obvious half and is the rung below this one.

And a slender block is worse at losing energy, because $r = 1 - \tfrac32\sin^2\alpha$ goes to 1 as $\alpha$ goes to zero. A very slender block barely dissipates anything at a landing, so once it starts rocking it keeps rocking — and it has less angle to spare.

The two effects multiply and both come from the same $\alpha$. That is why slenderness is such a strong variable in this problem and why the classical formula for uplift, which contains only $\tan\alpha$, understates how much the shape matters.

What the restoring law looks like #

That law is why the impact matters so much, and the connection is worth making explicit.

A softening system has no stable large-amplitude state to settle into. In a normal oscillator, energy that is not removed shows up as a larger amplitude at which the restoring force is larger, so the system finds a balance. Here a larger amplitude means a smaller restoring moment, so nothing pulls the block back harder as it goes further out.

The only thing that stops it is the energy taken out at the landings, and if the landings do not take enough out, there is no other mechanism that will. The restitution is not a refinement to the model. It is the model’s entire stabilising term.

Which is why nothing here can be read as a resonance. A structure with no period cannot be tuned to a ground motion, and the response spectrum — the instrument the rest of earthquake engineering runs on — has no meaning for it. The block’s response to a pulse depends on the pulse’s duration against the block’s own size, and on the pulse’s shape, and there is no frequency-domain shortcut.

The size result, and where it stops #

That is the result the rung below this one is about, and this rung’s contribution is to say what it depends on.

The size effect works because a bigger block rotates more slowly — the frequency parameter goes as the inverse square root of the diagonal — so it has less time to reach the toppling angle before the pulse ends. It is a race between the pulse’s duration and the block’s own slowness, and nothing about strength or material enters it.

And the restitution decides how much of the race is left after the first swing. A block that survives the first half-cycle is still rocking, still has energy in it, and will be hit by whatever comes next in the record. Its state at that moment is set by how much the first landings took out — which brings the whole argument back to a number nobody measures.

There is one more asymmetry worth recording, because it separates this problem from every other stability question in the collection. Uplift and overturning are not the same event and they have different arguments in them. Uplift is a statics question — the ground acceleration reaches $\tan\alpha$ and the block starts to rotate — and it contains no size, no duration and no restitution. Overturning is a dynamics question and contains all three.

So a block can lift off in almost every earthquake of its life and never come close to falling, which is exactly what the tall water tanks of 1960 did. The observable event is the harmless one and the dangerous one leaves no trace until it happens, which is why rocking was for so long read as a curiosity rather than as a mechanism — a structure seen rocking looks like a structure in trouble, and it is usually a structure doing the thing that saves it.

The number nobody measures #

Housner’s restitution assumes a perfectly rigid block landing on a perfectly rigid plane along a perfectly straight line, and every one of those is wrong in a way that reduces $r$.

The corner crushes. A stone block landing on stone chips its arris; a concrete pedestal spalls; a steel base plate on grout indents. Local damage absorbs energy, and it absorbs more the harder the landing.

The base is not rigid. A block on soil transmits energy into the ground at every impact, radiating it away as a wave. That is a genuine loss and it is not in the model at all.

And the contact is not a line. A block does not land on a mathematical edge; it lands on whichever few high points are there, bounces, and lands again. Multiple micro-impacts remove more energy than one clean one.

Every departure from the ideal reduces the restitution, which makes the block safer. Housner’s value is therefore an upper bound and the model is conservative — which is fortunate, because the alternative would be a model whose one stabilising term is optimistic.

But it also means the calculation is not predictive in the ordinary sense. A rocking analysis run at $r = 0.926$ and one run at $r = 0.75$ give different answers to a yes-or-no question, and there is no test on a real base that produces the right number in advance. Experiments on rocking blocks scatter widely, and much of the scatter is exactly this.

Reading the model as a limit rather than a prediction #

Everything above says the answer depends on a number nobody can supply, which sounds like a reason to distrust the model. It is not, and the reason is which way the uncertainty runs.

Every departure from the ideal reduces $r$, and reducing $r$ makes the block safer. Crushing, radiation into the ground, multiple micro-impacts, a base that deforms — each takes out more energy than the rigid-body balance says, so the real block rocks less than the computed one.

So Housner’s value is an upper bound on the response, and a rocking analysis run at it is a conservative analysis. That is a good position to be in and it is the opposite of the usual one: most idealisations in this collection remove a mechanism and are therefore unconservative until somebody puts it back.

Which means the model should be read as a limit analysis rather than as a simulation. It answers can this block be overturned by this pulse with a defensible yes-or-no, and it does not answer how far will this block rock, because the second question needs the number the first one only bounds. The lower bound theorem’s licence is exactly this shape: a calculation that cannot say what happens can still say what cannot.

The corollary is a warning about the other direction. A rocking system designed to dissipate energy at its landings is relying on the quantity this model bounds from the wrong side. A rocking wall proportioned so that impacts take the energy out is a design whose favourable mechanism is the uncertain one, and that is why real rocking systems add a device — a damper, a dissipating bar, a friction interface — rather than trusting the corner.

What a designer does with any of this #

Three practical positions follow, and they are quite different from each other.

For an object that must not fall over — a museum case, a transformer, a hospital’s gas cylinders, a chimney — the useful output is the pulse that overturns it, and the size figure gives it directly. The response is then to change the shape rather than to strengthen anything: widen the base, lower the centre of mass, or tie it down. Nothing about the object’s strength appears in the calculation, so nothing about its strength can improve the answer.

For an object that may fall over but must not fall on anything — a parapet, a statue, a stack of stored goods — the question is displacement rather than survival, and the model is at its least trustworthy, because a block past its toppling angle is outside the range the equations describe.

And for a structure deliberately allowed to rock — a bridge pier, a rocking wall, a post-tensioned frame — the object is to get the benefits with the uncertainty removed. That means providing re-centring from something other than gravity, usually post-tensioning, and dissipation from something other than the impact, usually a yielding bar. The rocking interface then supplies the mechanism and the added devices supply the numbers.

All three share one thing worth ending on. A rocking response is a way of surviving an earthquake without any ductility demand at all: the block does not yield, does not crack, and is undamaged afterwards if it lands. The earthquake asks for a displacement, and rocking is the one mechanism in the collection that supplies the displacement by moving as a rigid body rather than by damaging itself — which is why it keeps being rediscovered, and why ancient masonry that nobody engineered is still standing in places where engineered structures are not.

What to carry away #

The impact is the whole of the damping. Between landings the block conserves energy exactly, and there is no velocity-proportional term anywhere in the model.

The classical value is a shape property. $r = 1 - \tfrac32\sin^2\alpha$, so slender blocks lose 14 per cent per landing and stocky ones 38 — and the slender block also has less angle to spare.

It decides the answer. The same block under the same pulse overturns at 0.93 and survives at 0.75.

And the restoring law is softening, so nothing else can stabilise the block. There is no larger-amplitude equilibrium to find.

Where the model stops #

The block cannot slide. Every figure here assumes enough friction to rock rather than slip — the friction cone decides which, and whether it tips or slides is a separate competition decided by the same geometry.

The motion is planar. A real block on a real earthquake wobbles about two axes and about a vertical one, and a block that has rotated in plan lands on a corner rather than an edge — which changes the restitution entirely.

One pulse is not a record. Real ground motion is a sequence of pulses of varying sign, and a block’s state when the next one arrives is the initial condition for it. That is where the restitution compounds.

And the block is rigid. A slender masonry pier is not: it deforms, it cracks where the thrust leaves the section, and the rocking interface may form partway up rather than at the base.

The ladder from here #

Later rungs on this anchor: rocking under a real record rather than a pulse, and the overturning fragility that comes out of running many of them. The rocking wall as a designed system, where the interface is detailed and post-tensioned so the restitution and the re-centring are chosen rather than inherited. Two-block and multi-block assemblies, where an impact in one is an excitation of the next — which is a masonry column. Rocking with sliding allowed, where the block walks. And the experimental literature, which is where the scatter in this parameter is actually recorded.

Housner wrote the paper in 1963 after the Chilean earthquake of 1960 left tall, slender elevated water tanks standing while shorter and apparently stronger structures beside them had failed. The explanation was the size effect in the figure above, and the mechanism he wrote down to explain it has been rediscovered repeatedly since — most recently as a design strategy rather than an accident, in walls and piers deliberately allowed to lift so that the energy goes into a landing rather than into a hinge.