The filler that cannot be gentle twice

A crushable filler in the gap between two buildings replaces a contact stiffness nobody knows with a crush strength somebody chose. Its force is the crush stress times the contact area, it is the same whether the buildings arrive slowly or fast, and it lasts for as long as the filler has thickness left to crush. In a 50 mm gap between a 500 t and a 300 t building, the earlier essay found the one choice it cannot escape: a filler soft enough to keep the everyday contacts gentle bottoms out on the hardest one, and a filler strong enough never to bottom out makes every contact firm.

It ended on the obvious way round that, the one vehicle crash structures use. Make the filler soft at its face and strong behind. The small contacts would crush only the soft layer and meet a small force; the hard one would crush through it into the strong layer and meet a force the strong layer was sized to hold. Whether a grading exists that does both within 50 mm is a question about the material’s profile through its thickness — and the answer turns out to be fixed before any profile is chosen.

A staircase in place of a level #

The filler drawn throughout is 40 mm thick on the face of the softer building, leaving 10 mm of clear movement, with a contact area of 5 m² and 60 per cent of its depth crushable before it is dense — the earlier essay’s filler in every respect but one. Instead of crushing at 4 MPa throughout, its outer 20 mm crushes at 1.5 MPa and its inner 20 mm at 6.

The uniform filler’s crush force against the depth crushed is a level: 20 MN from first crush to dense. The graded filler’s is a staircase — 7.5 MN for the first 12 mm, 30 MN for the next 12. The area under either line is the energy the filler can absorb — the same trade a structure allowed to yield under a falling weight makes, energy taken as crush rather than as force — and the area is what the buildings decide. Two buildings of 500 t and 300 t closing at a speed $v$ carry half their effective mass, 187.5 t, times $v^2$ of relative kinetic energy: 94 kJ at 1.00 m/s, 353 kJ at 1.94, 761 kJ at 2.85, the three approach speeds the earlier essay’s pulse produced.

A collision crushes the filler until the area under its line equals that energy. At 1.00 m/s the graded filler crushes 12.1 mm — nearly all of its soft layer — at 7.5 MN, where the uniform one crushes 4.7 mm at 20. At 1.94 m/s it crushes through the soft layer into the strong one, to 20.8 mm, and meets 30 MN. At 2.85 m/s neither filler has enough area: the uniform one holds 480 kJ and the graded one 450, against 761 to stop.

The soft layer serves the small collisions #

Integrated in time as single collisions of two free masses, the staircase behaves exactly as the energy says. At 1.00 m/s the collision never leaves the soft layer and peaks at 7.5 MN, against 20 on the uniform filler. That is what the grading was for, and it works.

At 1.94 m/s it fails the other way. The collision crushes through the 12 mm of soft layer, having absorbed only 90 kJ of its 353, and meets the strong layer at 30 MN — half as much again as the uniform filler’s 20, for a collision that the uniform filler held at its crush force with depth to spare. At 2.85 m/s the graded filler runs out entirely and the faces meet through it at 31.4 MN, against the uniform filler’s 23.7 when it too runs out.

The soft layer does not make the filler softer. It moves the filler’s hardness to where the larger collisions find it.

What the soft layer costs the strong one #

The trade can be written down without any integration. To stop the hardest collision without running out, the area under the staircase must be at least 761 kJ, and the depth is 24 mm whatever the grading. If the outer layer crushes at a force $F_s$ over a depth $c_s$, the inner layer must make up the rest over what is left:

$$F_{\text{inner}} \ge \frac{E - F_s\,c_s}{c - c_s}.$$

The uniform filler that just stops the hardest collision crushes at 31.7 MN throughout — 761 kJ over 24 mm. Soften the outer half and the inner half must be stronger, in exact proportion to what the outer half gave up: at 1.5 MPa outside, 56 MN inside.

The comparison that matters is with the bare concrete. Two faces meeting with no filler at all, at 2.85 m/s, peak at 35.0 MN on the contact stiffness the earlier essays used. A filler whose inner layer crushes above that has made the hardest collision worse than no filler; and an outer layer has to crush at 5.7 MPa or more for the inner layer to stay below 35 MN. At 5.7 MPa the outer layer is already firmer than the 4 MPa uniform filler that was too soft to stop the collision. In 50 mm of gap the grading has nothing to work with.

No grading is gentle to both #

The budget can be swept over every grading at once. Each dot is a two-layer filler that just stops the 2.85 m/s collision — its outer layer anywhere from a twentieth to nineteen twentieths of the thickness, at any softer stress, its inner layer as strong as the budget requires — placed by the force a 1.94 m/s collision meets on it and the force the 2.85 m/s collision meets.

The uniform filler is one point, 31.7 MN on both. Every grading that is gentler to the middling collision is harder on the hardest one, and none is below and to the left of the uniform point. The gentlest to the 1.94 m/s collision meets it at 17.5 MN and the 2.85 m/s collision at 112.6. The dots fan out from the uniform point along two branches: gradings whose soft layer is deep enough to hold the middling collision, which trade its force down against the hardest one’s up, and gradings whose soft layer is too shallow to hold it, where both collisions meet the strong layer and both are worse.

That is the shape a fixed area makes. If the hardest collision’s energy must be absorbed in a fixed depth, the least possible peak force is the uniform one — any profile that is lower somewhere is higher somewhere else, and the hardest collision, which uses the whole depth, always meets the highest step.

Through the pulse #

In the buildings’ own motion under a pulse the trade appears contact by contact. The first four contacts are light, closing at under 0.2 m/s, and the graded filler — softer at its face in stiffness as well as in crush — meets them at 4.6 to 5.8 MN against the uniform filler’s 7.5 to 9.3. Then the hardest contact arrives at 2.74 m/s, and the graded filler runs out and meets 31.0 MN against the uniform filler’s 23.0. The contacts after it, which meet whatever is left of each filler, reach 26.2 MN on the graded one — its strong layer, partly crushed — against 20.6 on the uniform one, by then dense throughout.

The pulse delivers the light contacts first and the hard one in the middle, so the soft layer is spent on the light contacts before the hard one comes. That order makes no difference to the hard contact here, because the light ones crush the soft layer by fractions of a millimetre. A pulse that delivered a run of middling contacts first — after an aftershock, say — would arrive at its hardest contact with the soft layer already crushed, and the hard contact would meet the strong layer from the start.

The light contacts never crushed anything #

There is one more thing the pulse shows, and it changes what the grading was doing even where it seemed to work.

The four light contacts meet the uniform filler at 7.5 to 9.3 MN. Its crush force is 20 MN. They never crush it at all: they are elastic contacts, pressing the filler’s face and letting it spring back, and their force is set by the filler’s stiffness before it crushes, not by its crush strength. The same is true on the graded filler, whose light contacts at 4.6 to 5.8 MN never reach its 7.5 MN soft layer either.

So the graded filler’s gentleness to the light contacts came from somewhere other than its soft layer’s crush stress. A softer foam is also a more compliant one before it crushes, as a sandwich panel’s soft core both protects its face and crushes itself, and the graded filler’s face, at a fifth of the strength, is at a fifth of the stiffness. Give the uniform filler that same compliant face — the same 4 MPa crush throughout, but the elastic modulus of the soft foam — and it meets the four light contacts at 4.7, 5.8, 5.8 and 4.6 MN, the graded filler’s own numbers to a decimal place, and the hardest contact at 22.4 MN, less than the uniform filler’s 23.0 and far less than the graded filler’s 31.0.

The everyday contacts want a soft spring; the hardest contact wants a strong crush; and they are different properties of the material. A grading of crush strength trades one collision’s force against another’s through a fixed area. A filler that is compliant before it crushes and strong when it does gives both, because the light contacts never reach the crush and the hard one never notices the compliance. That is a property of the material’s elastic range rather than of its profile through the thickness, and it is the one the grading was standing in for.

Where a grading would work #

The grading fails here because the gap is narrow and the hardest collision’s energy fills it. Two things would change that.

More depth. As a weight dropped onto a softer beam meets a smaller force for the same energy, a filler whose crushable depth is large compared with the hardest collision’s energy over the force the buildings can bear has room for a soft layer that the hardest collision passes through without needing the strong layer to make it up. A 300 mm filler, the thickest the earlier essay drew, stops the hardest collision at 4.2 MN uniform, and could carry a soft face at a fraction of that while staying far below the bare concrete. A grading is a way of spending spare depth, not a way of creating it.

A known worst collision. Vehicle crash structures grade their crush strength because the vehicle’s mass and the test speed are given, and the structure is sized so that the hardest collision it is designed for uses all of its depth at a force the occupants survive. A seismic gap has no such test: the hardest collision is whatever the earthquake and the two buildings’ periods produce, and the force belongs to the model as much as the speed does. The grading’s whole purpose — to put the strong layer only where the hardest collision reaches — needs a hardest collision that is known.

Two buildings reduced to one mass #

The free body is the two buildings’ relative motion, reduced to their effective mass closing at the approach speed, with the filler’s crush force between them. Its kinetic energy has nowhere to go but into the filler’s crush, minus the little the filler stores elastically and returns as rebound — a one-way loss of the kind a damper that ends up as a joint was meant to make on every cycle. That is the whole of the argument: a collision that is stopped by crushing has given up its energy to an area under a force–crush line, and the line’s shape only decides at which force each part of that energy was given up.

The 1.5 MPa grading by hand #

The effective mass is $500 \times 300/800 = 187.5$ t. At 2.85 m/s the energy is $\tfrac{1}{2} \times 187{,}500 \times 2.85^2 = 761$ kJ. The crushable depth is $0.6 \times 40 = 24$ mm, so a uniform filler needs $761/0.024 = 31.7$ MN.

With the outer 12 mm of crushable depth at $1.5 \times 5 = 7.5$ MN, the outer layer absorbs $7.5 \times 0.012 = 90$ kJ. The other 671 kJ must go into the inner 12 mm: $671/0.012 = 56$ MN, or 11.2 MPa over 5 m². That is the number on the budget figure, and 56 MN against 35 for the bare concrete is the whole conclusion.

What the model assumes #

A staircase with sharp steps. Real graded foams and honeycombs change strength gradually through their depth, and a densifying foam’s crush stress rises toward the end of its crush. Neither changes the area argument, which needs only that the force is somewhere higher if it is somewhere lower.

Collisions that use the filler’s area alone. The buildings’ own stiffness absorbs a little of each collision’s energy, and the earlier essay found that small. A filler that softens a collision also lengthens it, and a longer contact lets the buildings’ springs take a slightly larger share — a second-order relief that does not change the order of the forces.

And one hardest collision. The pulse’s 2.74 m/s contact is the hardest this pulse produces. A longer record, or a different pair of periods, produces a different hardest contact, and a grading tuned to one is wrong for the next.

What the bars cannot show #

They cannot show the buildings. The forces are contact forces; what the buildings suffer from them depends on where the contact lands — at a floor, or at a column between floors — and the grading changes none of that.

They cannot show replacement. A filler is crushed by every contact and kept crushed. After an earthquake, a uniform filler that ran out and a graded one that ran out are both spent, and both must be replaced; a graded one that held the light contacts in its soft layer is spent only at its face, which a gap with access to it could replace and a gap without could not.

And they cannot show the gap’s other job. The 10 mm of clear movement in front of the filler is what lets the buildings move apart and together every day without touching it. A grading does nothing for that; only width does, and the gap itself was the first decision in this whole chain.

What it comes to #

The energy fixes the area under the crush line. 761 kJ at 2.85 m/s, in 24 mm of crushable depth, whatever the grading.

So a softer face forces a stronger back. An outer half at 1.5 MPa needs an inner half at 56 MN, against 31.7 MN uniform and 35 MN for the bare concrete.

No grading is gentler than the uniform filler to both a middling and the hardest collision. The gentlest to the 1.94 m/s collision costs 112.6 MN on the 2.85 m/s one.

Through a pulse the grading softens the light contacts and hardens the worst. 4.6 to 5.8 MN against 7.5 to 9.3 for the first contacts; 31.0 against 23.0 for the hardest.

And the light contacts never crushed. A uniform filler with the soft foam’s compliant face meets them just as gently and holds the hardest at 22.4 MN.

Still open: a filler that stiffens with speed #

Every filler here crushes at a stress that depends on how far it has crushed and not on how fast. Some materials do the opposite: their crush stress rises with the rate they are crushed at, so a slow contact meets a soft material and a fast one meets a hard one, with no layers at all. A rate-sensitive filler would be gentle to the slow contacts and firm with the fast ones at every depth, rather than at different depths — and the energy argument above, which cares only about force against crushed depth, would then depend on speed as well. Whether a filler whose strength scales with its crushing rate can hold the hardest collision in 24 mm while meeting the light ones softly, or whether the fastest collision’s peak rate makes it as hard as the strong layer was, is a question about the material’s dependence on time rather than on depth.