The force a filler can promise

A pounding analysis reports two kinds of number, and they deserve different treatment. The displacements belong to the buildings: momentum fixes them, and four contact laws at two stiffnesses a hundred times apart moved them by less than a tenth. The forces belong to the model: they are a contact stiffness somebody assumed, raised to a power, and the same analyses put the largest one anywhere from 27 to 225 MN.

That essay ended on the one way the second kind of number could be made to belong to the design. The gap essay had already suggested it: a bumper of soft, replaceable material at the floor levels, so that the buildings touch through something whose behaviour is known. A crushable filler is a material with a measured force against compression, far softer than the concrete behind it, and it should set the contact by itself. Whether it can — stiff enough not to eat the gap in ordinary movement, soft enough to lengthen a collision, still there for the seventh impact, and delivering a force that can be stated as one number — is what the same pair of buildings under the same pulse can answer.

A crush force in place of a stiffness #

The filler drawn sits on the face of the softer building. It is 40 mm thick in a 50 mm gap, which leaves 10 mm of clear movement before it is touched. It is elastic up to a strain of one twentieth, which is 2 mm of its thickness, and then it crushes at 4 MPa — over a contact 5 m² in area, a steady 20 MN — and it keeps the crush when the load comes off. Sixty per cent of its thickness can crush before the material is dense, which is of the order foams and honeycombs reach; the last 16 mm is compacted material that passes the load through to the concrete behind it.

That is a very different object from a contact stiffness. The force it allows is a property somebody chose, the crush stress times the area, and it does not grow with how hard the buildings arrive. What grows is how far the filler crushes. At 1.00 m/s it crushes 4 mm and the force is 20 MN. At 1.94 m/s it crushes 17 mm and the force is still 20 MN — against 22 MN for the bare contact at that speed. The area under each loading line is the energy the filler has absorbed, and it is absorbed for good: a crushed filler hands back only the little it stored elastically before it began to crush, which is why each line comes down steeply and ends at a crush it keeps.

It lowers the force in the only way anything can. Momentum fixes the impulse a collision delivers, and a filler that absorbs the energy of the approach instead of returning it leaves almost no rebound, so at 2.85 m/s the impulse is close to the effective mass times the closing speed, 534 kN·s. Delivered at no more than 20 MN, that impulse takes at least 27 ms. The bare concrete rebounds, so it delivers more impulse, and it delivers it faster. The filler buys its lower force with a longer contact and a collision that loses its energy instead of reversing it.

At 2.85 m/s, the speed of the hardest impact the pulse produces, the filler runs out. It crushes all 24 mm it has, the faces meet through the compacted material, and the Hertz contact of the concrete takes over from 20 MN, rising to 23.7 MN before the buildings part. That is still a third less than the bare contact’s 35.0 MN, but it is no longer a number the filler chose.

Energy sets the thickness #

Whether a filler runs out is a question that can be answered before any analysis, because it is an energy balance.

Two buildings closing at a speed v carry kinetic energy in their relative motion, half the effective mass times v squared, and a filler that stops them without bottoming out absorbs almost all of it by crushing at its crush force. So the crush force times the crushable depth has to be at least that energy, and for a given thickness there is a least crush force below which the filler must bottom out. Softer fillers need to be thicker, in exact inverse proportion.

The consequence for a gap sized to keep two buildings apart is immediate. A filler filling the whole 50 mm gap, with no clearance at all, needs at least 25.4 MN to stop a collision at 2.85 m/s, against the 35.0 MN the bare contact delivers. The most any filler in that gap can offer the hardest impact is a force about a quarter lower, and it can offer that only by leaving the buildings no room for any ordinary movement. In the buildings’ own motion each building’s stiffness takes a little of the energy during the contact, so the pulse does slightly better than two free masses would: the best a 40 mm filler manages there is 22.5 MN.

The room for ordinary movement is not optional. Two buildings side by side move relative to each other every day — temperature, shrinkage and wind each take a share of any joint — and a filler touching both faces turns each of those movements into a force through its elastic range. The filler drawn reaches its crush force within 2 mm, a stiffness of 10,000 MN a metre, so a single millimetre of shared movement would put 10 MN across the joint. Its 10 mm of clearance is what keeps it out of ordinary service, and every millimetre of clearance is a millimetre it cannot crush.

The dots on the figure are the check that the balance is the whole story. A single collision integrated on a filler 8 per cent stronger than the line holds; one 8 per cent weaker bottoms out. Everything the elastic range, the unloading and the rebound do is small beside the energy of the approach.

Two buildings meeting through it #

A single collision of free masses is the idealisation. In the pulse the buildings are springs as well as masses, they meet repeatedly, and the gap the filler occupies is a gap they were going to close anyway.

The history has two surprises in it and one confirmation.

The confirmation is the displacements. The stiffer building reaches 241 mm and the softer 405 mm, against 244 and 410 with the bare gap. Momentum sets them, and changing what the buildings meet through does not change what they exchange.

The first surprise is the four contacts in the first 0.6 s, marked on the left. With the bare gap the buildings touch once in that time, at 0.39 s and 4.8 MN. The filler takes 40 of the 50 mm, so early movements that the bare gap absorbed now reach it, and they reach it four times at closing speeds under 0.2 m/s, with forces of 7.5 to 9.3 MN — elastic contacts that crush nothing, harder than the one touch they replace. A filler that is meant not to eat the gap in ordinary movement has to leave room for ordinary movement, and every millimetre of that room is a millimetre it cannot crush.

The second is what happens after the hardest impact. The filler meets it at 2.74 m/s, crushes to the end of its travel and passes 23.0 MN into the two buildings. After that it is dense.

The arithmetic of that impact is short. At 2.74 m/s the relative motion of the two buildings carries about 700 kJ. Twenty-four millimetres of crush at 20 MN absorbs 480 kJ of it. The remainder, less what the buildings’ own stiffness takes up during the contact, arrives at the concrete through the compacted layer, and it is that remainder — about a third of the energy — that lifts the force above the crush force to 23.0 MN.

The first hard impact spends it #

Laid out in time, the two sets of contacts tell the story of the filler’s one use. The bare gap gives seven impacts, one of them large. The filler turns the large one into 23.0 MN, which is what it was installed to do. Everything around that one impact is worse.

Before it, four elastic contacts replace one gentle touch. After it, every contact meets a filler with nothing left to crush. A dense filler is stiff — its elastic range reaches the crush force within 2 mm — and it still passes load at its crush stress, so each later contact carries 20.0 to 20.6 MN at closing speeds between 0.48 and 0.97 m/s. The bare gap’s later impacts, at speeds between 0.50 and 1.64 m/s, ran from 3.2 to 17.5 MN, because concrete meeting concrete is soft at small indentations. The filler has done its work once, and for the rest of the event it is a stiff spacer 16 mm thick.

That is the answer to whether a filler survives to act on the seventh impact. The seventh impact is whichever comes after the hard one, and it finds concrete behind a compacted layer. A filler sized for the hardest collision in a gap this narrow is sized to be used up by it, and replacing it after the event is part of what choosing it means.

Certainty is what it buys #

The question the filler was brought in to answer was whether a collision force can be stated as one number. That depends on whether it runs out.

With the bare gap, the two plausible stiffnesses of the concrete give 35.0 and 225.5 MN, which is the range the pounding analyses said a design would have to state. Every filler weaker than 8 MPa crushes to its end in the hard impact, and wherever it does the force still depends on what the concrete behind it is: at 4 MPa it is 23.0 MN on soft concrete and 98.0 MN on stiff. The filler has narrowed the range, sometimes a great deal, and it has not closed it.

At 8 MPa it never runs out. The hard impact crushes 19 mm of its 24, the concrete is never reached, and the largest force is 40.0 MN whichever stiffness sits behind it — exactly the crush stress times the area. That is a force that belongs to the design. It is also larger than the 35.0 MN the soft estimate of the bare contact gives.

So in a gap this narrow a filler offers a choice between two things, and not both. A weaker filler lowers the likely force and leaves it uncertain. The one filler that makes the force certain makes it certain at a value above the optimistic bare estimate. What it buys is the removal of the upper tail — the 225 MN the stiff reading allowed — and a number a slab edge and a column can be designed for. That is the same bargain as a friction damper’s: the force is not reduced so much as capped at a value somebody chose. It is a fuse in the sense the link of an eccentric brace is: the weakest part, chosen, so that everything behind it can be designed for what it passes on.

What makes the cap a design quantity also limits it. A crush stress is a material property with a scatter no single specimen shows, and a filler relied on to cap a force has to be taken at the upper end of that scatter, while one relied on never to run out has to be taken at the lower end. A 40 MN cap over 5 m² is 8 MPa of bearing, comfortably inside what concrete carries in compression. The problem it hands over is getting 40 MN out of a slab edge and into the frame behind it, which is a problem with a number in it rather than a range of orders of magnitude.

A smaller force needs a wider gap #

The energy balance says what it would take to have both: enough thickness that a weak filler never runs out.

Each point is the best a filler of that thickness can do in a gap just 10 mm wider than itself. The 40, 80 and 120 mm fillers all end dense, at 22.5, 16.9 and 12.6 MN: better with every millimetre, and still uncertain. At 200 mm, in a 210 mm gap, a filler crushing at 2 MPa stops the hardest impact with material left over, and the force is 10.0 MN and certain. At 300 mm it is 5.0 MN.

The dashed line is the reminder of what that thickness costs. A 210 mm gap is four times the one the filler was brought in to help, and the bare contact in it would still reach 33.5 MN, because the pair closes by far more than 210 mm when nothing stops it. A filler turns a gap too narrow to prevent contact into one wide enough to make contact a designed event, and the width it needs is set by the energy of the hardest approach rather than by the displacements the gap was sized for.

A viscous link across the same gap makes the opposite trade. It keeps a 50 mm gap from closing at all, carries 1.6 MN continuously rather than tens of meganewtons for a few hundredths of a second, and pays in the stiffer building’s drift. A filler leaves each building its own motion and pays in force and in width; a link leaves the gap its width and pays in drift. Neither is free, and which cost a pair can bear is a question about which of its two buildings is the weaker.

What the crushable layer leaves out #

The crush stress is one number. Real crushable materials have a plateau that rises with the rate of loading, falls with temperature and changes with age, and a specified 4 MPa is a characteristic value with scatter around it. A filler relied on for its cap needs the upper value of that scatter, and one relied on for its softness needs the lower.

The contact is square and uniform over 5 m². Two slab edges meet over part of their length, at an angle, and a filler loaded over a fraction of its area crushes deeper there and reaches its end sooner.

The filler is elastic, then plastic, then dense, with sharp corners between. Real densification is gradual, and a partly dense filler stiffens progressively rather than handing over to the concrete at one millimetre.

Each building is one mass on one spring, meeting at one level. A real pair meets at several floors, at different times, through several fillers that are used up in different orders.

One pulse is one event. A filler crushed in one earthquake is a thinner filler in the next, and the model assumes it was replaced.

And no figure here shows where the force goes. A 40 MN cap is a design quantity only if the slab edge and the column behind the filler can take 40 MN over the filler’s area, and that is a local check at the contact that no pair of oscillators can make.

The assumption every figure rests on is that the filler’s crush force is the force the buildings feel, so that nothing stiffer than the filler stands in series with it before it reaches the concrete. In a chain of springs the softest one sets the force, and a filler is chosen to be that spring; a fixing that is stiffer than the filler but weaker than its crush force would set the force instead.

Still open: whether a filler graded in strength can hold both kinds of contact #

Every filler here has one crush stress through its whole thickness, and that forces the choice between softness and certainty: the stress that keeps small contacts gentle is too low to stop the hardest one, and the stress that stops the hardest one makes every small contact firm. A filler that crushes easily at its face and harder towards the concrete would take small contacts in its soft outer layer and still hold the hard one in its strong inner layer, which is how energy absorbers in vehicles are built. Whether a grading exists that does both within a gap of 50 mm, how sensitive it is to the order in which the contacts arrive, and whether the force it delivers at its strongest layer is still a single number, is a question about the profile of the material through its thickness — and it decides whether a narrow gap can be made safe by filling it or only by widening it.