The filler that knows how fast it is hit

Two buildings closer together than their earthquake movements need will hit each other, and a crushable filler in the gap turns the collision’s force from something set by the concrete’s stiffness into something set by the filler’s strength. A uniform filler promises one force to every collision that crushes it. That is its virtue and its trouble: a filler strong enough to stop the hardest collision in its crushable depth meets every small one with the same large force.

Grading it through its depth — soft at the face, strong behind — lets the small collisions stay in the soft layer. The energy then argues back. The hardest collision has to be stopped in the same depth whatever the grading, and every kilojoule the soft layer does not absorb, the strong layer must, in less depth. A 1.5 MPa outer half forced a 56 MN inner half on a 40 mm filler where a uniform one needed 31.7 MN, and no grading was gentler than the uniform filler to both a middling and the hardest collision.

That essay ended with a material that does something different: one whose crush strength rises with the speed at which it is crushed. A slow contact would meet a soft material and a fast one a hard material, at every depth, without layers. This essay follows it.

Strength set by speed #

The pair is the same: a 500 t building with a period of 0.8 s and a 300 t one at 1.2 s, 50 mm apart, with a 40 mm filler over a contact area of 5 m² that can crush through 24 mm before it is dense. The hardest collision the gap is designed for closes at 2.85 m/s with an effective mass of 187.5 t, which is 761 kJ.

The filler’s crush force at crushing speed $u$ is

$$F_c(u) = F_0\left[(1 - s) + s\left(\frac{u}{v_H}\right)^n\right]$$

— a part that does not care about speed and a part, a share $s$ of the whole at the design speed $v_H$ of 2.85 m/s, that grows as the $n$-th power of the speed. The share is chosen; $F_0$ is not. It is solved for, so that the hardest collision is stopped exactly as the filler goes dense, which is the same condition the uniform and the graded fillers were sized to.

The three force histories are the whole essay in one picture. The uniform filler’s force rises elastically to its crush force and stays there; the collision crushes through 24 mm at a level 30.5 MN. The rate-sensitive fillers’ forces rise to their peak at the same moment, but their peak is higher, and from there they fall: the crushing speed is falling as the collision slows, and the force follows it down. The purely rate-sensitive filler falls in a straight line from 63.4 MN to nothing, because a force proportional to speed decelerates a mass with a speed that falls linearly with distance.

A filler that knows the speed meets every collision hardest at its start, because the start is where the speed is highest. That is the opposite of what was hoped for. The filler was meant to be firm with the fast collisions, and it is — but within each collision it is firmest at the instant of contact and softest at the end, when the depth that is left is smallest.

Why the peak has to rise #

The energy argument that defeated the graded filler applies here in a different form. The filler absorbs the collision’s energy as the area under its force-against-depth line, and the area must be 761 kJ in 24 mm. A level line does that at 31.7 MN — the energy divided by the depth, a little less in practice because the elastic ramp at the start absorbs some. A line that starts high and falls has to start higher to enclose the same area in the same width.

For the purely rate-sensitive filler with $n$ = 1 the arithmetic is exact. A force $cu$ on a mass $m$ moving at $u$ slows it by $du/dx = -c/m$, so the speed falls linearly with distance and the force with it; stopping in depth $d$ requires $c = mv/d$, and the force at contact is $cv = mv^2/d = 2E/d$. A purely viscous filler hits the hardest collision with exactly twice the force of a uniform one, 63.4 MN against 31.7, and the integration confirms it to a tenth of a per cent.

The laws show the price from the other side. Every collision ends at no speed at all, so a rate-sensitive filler spends the end of every collision at little or none of its strength. To absorb its energy anyway it must have more strength at the speeds where the collision starts. The half-rate filler’s force at 2.85 m/s is 38.9 MN, a quarter more than the uniform filler’s; the purely rate-sensitive one’s is 66.1 MN. Below about 1 m/s both are softer than the uniform filler, which is what they were for.

Which collisions gain #

A filler is not designed for one collision. The pulse that produces the hardest collision produces lighter ones before and after it, and the useful comparison is the peak force across the whole range of approach speeds.

The four fillers rank the collisions differently. The uniform filler gives every collision that crushes it the same 30.5 MN. The graded filler gives anything that stays in its soft layer 7.5 MN — very gentle — and steps up to its strong layer for anything that does not, so a middling collision at 1.94 m/s meets 44.4 MN and the hardest 56.0. The rate-sensitive fillers rise smoothly with speed. The rate-sensitive filler ranks collisions by speed; the graded one ranks them by depth, in two steps.

That difference is worth something. The half-rate filler meets the middling collision at 32.2 MN, where the graded filler gives 44.4, and the hardest at 38.4 against 56.0. Between the four, it is the one whose worst force is closest to the uniform filler’s. What it gives up is the light collisions: at 1 m/s it is 25.7 MN against the graded filler’s 7.5, barely gentler than the uniform filler’s 30.5. Grading made the light contacts very gentle by charging the heavy ones; the rate filler charges the heavy ones less and gives the light ones less.

The share that depends on speed #

How much of the filler’s strength should follow the speed? The answer depends on which collisions matter, and the figure shows what each share costs.

The light collision gains steadily as the share rises, from 30.5 MN to 22.3. The middling collision gains nothing at all — its force rises slightly from the first share and steeply past half. The hardest collision loses from the first share and ends at twice the uniform force.

Every share on speed is paid for by the hardest collision, and the hardest collision is the one the gap is designed for. A filler with a tenth of its strength on speed costs the hardest collision a few per cent and buys the light ones a few per cent. With all of it, the light ones are a quarter gentler and the hardest is twice as hard. There is no share at which the hardest collision benefits, because no force that falls during a collision can absorb a given energy in a given depth with a lower peak than a level one can.

Gentler only when it does not matter #

The cleanest way to see the trade is to ask how fast a collision can be and still meet less force than the uniform filler would give it.

With a small share on speed, the rate-sensitive filler is gentler than the uniform one for every collision below about 1.7 m/s, and harder for every one above. As the share rises, the crossover falls, to 1.37 m/s with all of it — less than half the design speed. The energy of a collision goes as the square of its speed, so a collision at half the design speed carries a quarter of the energy, and these are the collisions a building pair’s contents and cladding notice least.

What a gap filler is for, in the end, is the hardest few collisions: it limits the force that a contact between two buildings sends into each of them, and the hardest contacts are the ones that send the most. A filler that helps every collision below half the design speed at the cost of every one above it has made the wrong trade for that purpose, whatever it does for the many light contacts.

A weaker dependence #

Real rate-sensitive materials — polymer foams, elastomers, honeycombs with air or oil in them — rarely have a strength proportional to speed. Their strength more often grows as a small power of the strain rate, a tenth or a fifth, so that doubling the speed raises the strength by a few per cent. Steel itself is stronger in a millisecond by an amount of that order.

A weak dependence costs little and buys little. A filler whose whole strength goes as the fifth root of the speed meets the hardest collision at 36.1 MN, a fifth more than the uniform filler, and the light one at 28.6 MN, a sixteenth less. As the square root, 43.5 and 25.1; in proportion, 63.4 and 22.3. The stronger the dependence, the further the trade goes in both directions, and it never reverses.

There is also a hard limit. A filler whose whole strength falls faster than its crushing speed — strength as the square of speed, say — cannot stop a collision in any depth at all. Near the end of the collision its force falls away faster than the mass’s momentum, and the mass coasts on through the filler indefinitely; it is stopped only by the concrete behind. Proportion is the steepest dependence a filler that relies entirely on speed can have, and it is already twice the uniform force at the top.

Two ways to spend twenty-four millimetres #

Set side by side, the graded filler and the rate-sensitive one are two departures from the same optimum, and the optimum is the uniform filler.

The design collision has a fixed energy, 761 kJ, and a fixed depth to spend it in, 24 mm. The filler’s force against the depth crushed is a line whose area must be that energy, and among all lines of a given width enclosing a given area, the one with the lowest peak is level. That is not a property of crushable materials; it is a property of areas, and it is why vehicle crash structures and lift buffers are designed to crush at as nearly constant a force as their materials allow. Any filler that is not level along the hardest collision’s crushing — that has a lower force somewhere — has to have a higher force somewhere else, and the highest force is the one the buildings feel.

The graded filler departs from level by depth: its force is low at the face and high behind, so the hardest collision crushes through a soft step and then a hard one. The rate-sensitive filler departs from level by time: its force is high at the start of every collision and low at the end. Each departure is chosen to favour some collisions — the graded filler the ones that stop in its soft layer, the rate-sensitive one the ones that arrive slowly — and each is paid for by the collision the gap was designed for.

So the question a designer actually faces is not which filler is gentlest, but which collisions to favour at the hardest one’s expense. A uniform filler favours none, and gives the hardest collision the least force any filler of that depth can. A graded filler favours the light contacts strongly and the middling ones not at all. A rate-sensitive one favours the light contacts mildly and the middling ones somewhat. If the gap exists to protect the buildings from the force their worst contact sends into them, the uniform filler is the answer, and the others are refinements bought with that force. If it exists to keep the cladding and the contents from being shaken by a hundred light knocks, the others have a case, and the rate-sensitive filler’s is the gentler of the two on the collision that matters most.

The viscous filler, by hand #

The purely viscous filler is the one case that needs no integration. A force $cu$ on the effective mass $m$ of 187,500 kg, closing at $v$ = 2.85 m/s, gives

$$m\,u\,\frac{du}{dx} = -cu \quad\Rightarrow\quad u(x) = v - \frac{c}{m}x$$

so the collision stops at $x = mv/c$. To stop in the crushable 24 mm, $c = mv/d = 187{,}500 \times 2.85 / 0.024 = 2.23 \times 10^7$ N·s/m, and the force at contact is $cv = 63.5$ MN — twice the 761 kJ divided by 24 mm. The force falls linearly with the depth crushed, and the area under it is half its peak times the depth, which is why the peak is twice the level filler’s.

A light collision at 1 m/s on the same filler starts at $c \times 1 = 22.3$ MN and stops in $m \times 1 / c$ = 8.4 mm. The uniform filler would meet it at its full 30.5 MN and stop it in 2.1 mm.

A single collision, a single material #

The calculation rests on choices that limit it.

The strength depends on the closing speed. A real material’s strength depends on its strain rate — the crushing speed divided by its thickness — and on its temperature, which a fast collision raises. A thin filler crushed at a given speed sees a higher strain rate than a thick one, so the share on speed is not a property of the material alone but of the material in that thickness.

The collisions are taken one at a time. In a pulse the filler is crushed by successive contacts, each starting where the last left it. A rate-sensitive filler’s crushed depth after a light contact is larger than a uniform one’s — 8.4 mm against 2.1 at 1 m/s for the purely viscous one — so it arrives at the hardest collision with less depth left, and the hardest collision then meets an even higher force. The single-collision comparison is the favourable one for the rate-sensitive filler.

The filler does not spring back. A viscous material that is not crushed but deformed recovers after the contact and is available again, which changes everything about repeated contacts; a viscoelastic pad is a damper rather than a filler, and a damper that links the buildings is a different device with a different argument.

The buildings are single masses. The force a contact produces belongs to the model of the contact as much as to the buildings, and here the model is the filler’s law; the force delivered to each building is then shared between its floors by its own stiffness, and the contact’s duration — longer for a softer, slower force — decides how much of it the structure feels as a static load and how much as a pulse it rings after.

Still open: a filler that is crushed in stages by the pulse #

The single-collision comparison favours the rate-sensitive filler, because a light collision crushes it further than it crushes a uniform filler and leaves less depth for the hardest. Through a whole pulse, with a dozen contacts of different speeds arriving in an order the earthquake chooses, the depth each filler has left when the hardest contact arrives depends on every contact before it. Whether a half-rate filler still meets the hardest contact of a real record more gently than the graded filler does — or whether its extra crushing on the light contacts uses up the depth the hardest one needed, and it ends by bottoming on the concrete where the graded filler did not — is a question about the order of collisions rather than their speeds, and it decides whether speed or depth is the better way to grade a filler that is hit more than once.