The filler that is used up before the worst

Two buildings too close together will meet in an earthquake, and the gap between two buildings is sized so that they do not. When the gap cannot be made wide enough, a crushable filler can be put in it, and the force a filler can promise is its crush strength, for as long as it has depth left to crush. Energy decides how much depth that has to be: a collision closing at 2.85 m/s between a 500 t and a 300 t building brings 761 kJ, and a filler 40 mm thick, crushable over 24 mm, must crush at 31.7 MN to absorb it.

Two variations followed. The filler that cannot be gentle twice graded the filler through its depth — a soft outer layer for the everyday contacts, a strong inner one for the hardest — and found that every kilojoule the soft layer does not take, the strong layer must, in less depth: with an outer half at 1.5 MPa the inner half has to crush at 56 MN. The filler that knows how fast it is hit made its strength follow its crushing speed instead, so that it ranks collisions by speed without layers, and found that it meets every collision hardest at its first instant: half its strength on speed takes the hardest collision from 31.7 MN to 38.4.

Every one of those comparisons met each collision on a new filler. An earthquake delivers a sequence: light contacts, middling ones, one or two hard ones, more middling ones, in an order the ground chooses, and each crushes the filler a little further. The filler that meets the hardest contact is whatever the earlier ones have left. The question here is whether that changes the comparison — whether the rate-sensitive filler’s extra crushing on the middling contacts uses up the depth the hardest contact needed, and whether a graded filler, which saves its depth for the end, comes out ahead once the whole record is run.

One pulse, and depth to spare #

Take the pair from the earlier essays — periods of 0.8 and 1.2 s, 5 per cent damping, 50 mm apart — and three fillers 40 mm thick, each made exactly strong enough to stop a 2.85 m/s collision from new: uniform, at 31.7 MN; graded, its outer half at 1.5 MPa and its inner half at 56 MN; and rate-sensitive, half its strength depending on its crushing speed. Run them first through the one-second, 0.5 g sine pulse the graded filler was first tried on, and record every contact.

This pulse is kind to a filler’s depth. Its first four contacts are light, under 0.2 m/s, and crush nothing: each filler loads elastically and lets go, and all three still have their full 24 mm when the hard contact comes at 2.74 m/s. That contact is a little slower than the one they were sized for, and each stops it with a little depth to spare — 2 mm in the uniform filler, 1 mm in the rate-sensitive one, 8 in the graded. The contact after it, at about 1.2 m/s, finds the uniform and rate-sensitive fillers nearly dense, and they bottom on the concrete — gently, at 31.8 and 27.7 MN. The graded filler never bottoms.

On this pulse the single-collision comparison holds almost exactly, because the hard contact came to a new filler. The graded filler meets it at its strong layer’s 56 MN; the rate-sensitive one at 37.7, close to the 38.4 it met the design collision at; the uniform one at its plateau. A record with more in it is a different test.

A record that spends the depth #

Real records build up. Take six one-second sine cycles of rising and then falling strength — 0.15, 0.25, 0.35, 0.5, 0.3 and 0.2 of a scale — and scale the whole record so that its hardest contact is about the speed the fillers were sized for; at a peak of 0.35 g it is 2.76 to 2.88 m/s, depending on the filler. Now the hardest contact arrives after three cycles of middling ones.

The three fillers spend their depth in three different patterns. The uniform filler crushes at 31.7 MN whatever hits it, so a contact crushes it only by the energy it brings beyond what the filler stores elastically: a 0.75 m/s contact takes a millimetre, a 1.37 m/s contact 4 mm, a 2.18 m/s contact 14. By the time the hardest contact comes it has 5 mm left of its 24, and that contact needs 23.

The rate-sensitive filler crushes further on each middling contact, because it is weaker at middling speeds than the uniform filler is — its strength falls with the speed it is crushed at, and it was made to equal the uniform filler’s energy only at the hardest speed. The same three contacts take 2, 7 and 15 mm, and it is dense before the hardest contact arrives. This is exactly what the earlier essay left open: the rate-sensitive filler’s gentleness on the middling contacts is paid for in depth, and through a whole record the depth is gone when the contact it was sized for arrives.

The graded filler’s soft outer layer goes at once, in the first middling contacts — a 0.76 m/s contact takes 7 mm of it, a 1.47 m/s contact the rest. Then the filler is at its strong layer, which crushes at 56 MN, and no middling contact reaches it: a 1.84 m/s contact brings 318 kJ, which the strong layer would absorb in a few millimetres, but the elastic compression of the filler stores most of it first. It reaches the hardest contact with 12 mm left and ends the record with 6.

So the graded filler keeps its depth, and the other two do not. That is the answer the lead expected. What it means for the forces is not.

What the spent filler meets #

Plot every contact of the record, as before.

The rate-sensitive filler meets the hardest contact dense, on the concrete, and the force is 38.7 MN. The graded filler meets its hardest with 12 mm still crushable, and the force is 56.0 MN. The filler that ran out did better than the filler that kept its depth, by a wide margin.

The reason is what the depth is made of. The graded filler’s remaining 12 mm is its strong layer, and a strong layer crushes at its strength from its first millimetre: 56 MN, the moment the soft layer is gone. Keeping that layer for the hardest contact keeps the 56 MN for it. The rate-sensitive filler, dense, transmits the contact through its own compressed body and the concrete behind it, and a dense filler’s concrete contact is a Hertzian one, whose force is set by a contact stiffness and a power rather than by a strength: here 38.7 MN at 2.88 m/s. Bottoming sounds like failure, and in the sense of the filler’s design it is. In force, it costs less than the graded filler’s strong layer, because that layer was made stronger than the concrete it protects.

That is the earlier finding carried through a record: the graded filler’s inner half at 56 MN is far above the 35 MN that bare concrete gives the design collision, and nothing in a record lowers it. A filler sized to stop the hardest collision in 24 mm has to be strong, and a graded one has to be stronger still at its back, and the record merely chooses whether the hardest contact meets that back.

The figure also shows what every filler does to the middling contacts, and it is not flattering. With no filler the buildings meet across 50 mm, rarely and softly: a 1 m/s contact on bare concrete peaks at about 10 MN. With a filler they meet across 10 mm, more often, and every contact above about 0.5 m/s reaches a filler’s plateau — 20 to 35 MN on the uniform and rate-sensitive fillers. A filler strong enough to stop the hardest contact in its depth makes the everyday contacts harsher, which is the force a filler can promise read the other way round.

How much is left, as the record grows #

Scale the record from a peak of 0.2 g to 0.4 g and read the depth each filler has left when its hardest contact arrives.

The graded filler’s line is nearly flat at 12 mm, because by the time any record is hard enough to matter its soft layer is gone and its strong layer barely touched. The other two fall steadily as the record grows, the rate-sensitive filler always below the uniform one by a few millimetres — its extra crushing on the middling contacts, accumulated. By 0.4 g both are dense before the hardest contact; the graded filler still has 11 mm.

The gap between the uniform and the rate-sensitive lines is simple arithmetic. A contact at 1.4 m/s crushes the uniform filler at 31.7 MN throughout. It crushes the rate-sensitive filler at its strength at the crushing speed, which is nine-tenths of the uniform filler’s at the contact’s first instant and falls as the contact slows, towards three-fifths at rest. Energy absorbed at a smaller force takes more depth: in the record, a 1.44 m/s contact took 7 mm from the rate-sensitive filler where a 1.37 m/s contact took 4 from the uniform one. Every middling contact costs the rate-sensitive filler more of its depth than it costs the uniform one, and it is the middling contacts that make up most of a record.

On depth alone, then, the graded filler is the one that survives a record, and the rate-sensitive one the first to be spent. Grading by depth is the better way to keep a filler for its hardest contact. Whether it is the better way to make the hardest contact gentle is a separate question, and the contacts figure has already answered it.

The order of the contacts #

The record rises and falls. Run it backwards — the 0.2 cycle first, then the 0.3, the 0.5 and so on down — so that the hardest cycle comes after two rather than three.

Each filler’s worst contact moves by less than a meganewton, and the graded filler’s not at all. The order changes which contact is hardest and how much depth is left for it — reversed, the hardest contact comes after fewer middling ones, and the uniform filler meets it with 13 mm rather than 5 and the rate-sensitive one with 10 rather than none, and both are crushed dense by it all the same — but it does not change the force, because for the two fillers that run out the worst force is a contact on concrete, set by the contact’s speed, and for the graded filler it is its strong layer’s strength, set by its design. Only bare concrete moves by more, 34.4 to 37.4 MN, and that is because reversing the record changes the buildings’ motion and so the hardest contact’s speed; no filler is there to be used up.

This is the opposite of what a question about order expects. A fatigue calculation is known to be blind to order — Miner’s rule adds damage in any sequence — and the reason to ask about order here was that a crushable filler is not: it has a memory, and each contact’s effect depends on what is left. The memory is real, and it shows in the depth. It barely shows in the force, because the force a filler can transmit depends on what its remaining depth is made of, not on how much of it there is, and once it is gone, on the concrete.

As the record grows #

Read the worst contact force against the record’s scale, with bare concrete for comparison.

The graded filler’s worst force jumps to 56 MN as soon as any contact reaches its strong layer, at about 0.3 g, and stays there. The uniform and rate-sensitive fillers rise more slowly, through their plateaus and then on the concrete once they bottom. At 0.4 g the rate-sensitive filler, dense long before its worst contact, meets its worst contact at 40.6 MN, the uniform at 44.1, the graded at 56.3 — and bare concrete at 39.5.

At no scale does any of the three improve on bare concrete. That is a statement about these fillers — each made strong enough to stop a 2.85 m/s collision in 24 mm of a 50 mm gap — and not about fillers in general: the earlier essays found that a filler earns its place when it has depth to crush slowly, which a gap this narrow does not give it. What the record adds is that the narrowness bites twice. The filler takes 40 mm of the gap, so the buildings meet more often; and its strength, chosen for one collision, is applied to every contact that reaches it.

Two buildings and a record of sines #

The buildings are two damped single oscillators on one ground, with the filler between them. That is the model every earlier essay on these fillers used, and it gives the contacts’ speeds and order without the higher modes, the torsion or the floor-by-floor contact of a floor that arrives at a column. The record is six sine cycles of a second each; a real record has a broad band of periods and its contacts are more irregular, but its hardest contacts arrive, as here, within and after its strongest cycles, so the depth spent before them is of the same kind.

The fillers were each sized for one collision from new, as they would be designed. A designer who knew the record would size them for its sequence instead — a uniform filler thick enough to keep 23 mm for the hardest contact after the middling ones have taken theirs, roughly 70 mm and most of the gap — and the comparison of depth would change. The comparison of force would not, because a uniform filler’s force is its strength, and its strength is set by the hardest contact’s energy over the depth that contact can use.

And the rate-sensitive filler’s dense contact is the same Hertzian contact as the bare concrete’s, offset by the filler’s compressed thickness. A real dense filler is a stiff, thin layer of crushed material, and whether its contact is softer or harder than bare concrete’s depends on what it is made of. The 38.7 MN is a property of the contact law assumed, which the model and not the buildings decides.

Depth is not gentleness #

A filler that is crushed in stages by a record spends its depth before the hardest contact unless something reserves depth for it, and grading reserves it: the soft layer is spent on the middling contacts, the strong layer kept. On depth the graded filler is the only one of the three that survives a record of this strength. A rate-sensitive filler spends depth fastest, because it is weakest at the middling speeds that are most of what a record delivers.

But the depth a graded filler keeps is its strong layer, and the strong layer has to be stronger than the uniform filler to make up for the soft layer’s lost energy. Reserving it for the hardest contact reserves that strength for it. The rate-sensitive and uniform fillers, spent, meet the hardest contact on the concrete, and the concrete is gentler than the graded filler’s back. The order of the contacts changes how much depth is left and almost nothing else. Speed or depth is the wrong choice to be making: in a gap this narrow, the filler’s strength is what the hardest contact meets, either directly or through the concrete it fails to cover.

Still open: the filler sized for the record #

Every filler here was sized for one collision from new, and the record then found it short of depth. A filler can instead be sized for the record — thick enough, or soft enough, that the middling contacts leave the depth the hardest one needs — and a wider gap is the obvious place for the extra thickness to go. Whether a filler sized for a record’s sequence can be weaker than one sized for its single worst collision, because the middling contacts have already slowed the buildings by the time the worst arrives; whether the buildings’ damping, increased by every crush, lowers the hardest contact’s speed enough to matter; and whether such a filler finally does better than bare concrete, is what sizing a filler for the whole earthquake, rather than for its worst moment, would ask.