Structural Engineering & Statics
where the load goes
Dynamics

The damper that ends up as a joint

A damper across the gap between two buildings acts on exactly the motion the gap is sized for, and it can be sized for two different things. The size that takes the most energy out of the pair still lets the buildings collide. The size that keeps them apart has nearly stopped moving: it has joined them into one building, and the stiffer of the two pays for it in drift.

Assumes The gap between two buildings, The only thing that stops it and The period nobody chose.

16 min read 7 figures The load that will not hold stillWhich failure arrives firstThe load must go somewhere

The gap between two buildings listed three things a designer can change about a pair that might collide: the gap, the periods and the contact. It then named a fourth in two sentences. A damper spanning the gap resists the relative velocity of the two buildings, which is exactly the motion a collision would have turned into an impulse, and mismatched periods — the case that needs the widest gap — are the case in which such a damper has most to work on.

Both sentences are true, and neither says how large the damper should be. That turns out to be the whole question, because there are two answers and they are far apart.

The pair is the one whose collisions were computed under four contact laws: a 500 t building with a period of 0.8 s and a 300 t building with a period of 1.2 s, 50 mm apart, under one 1.0 s sine pulse of 0.5 g. With nothing between them they meet seven times. If the gap were wide enough never to close, they would approach each other by 515 mm.

A link that keeps two buildings apart has made them one. The largest closing movement between a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, and each building's largest displacement, against the size of a viscous damper joining them across the gap, from 0.01 MN·s/m to 541.3 MN·s/m. With no link they close by 515 mm, and the stiffer building moves 220 mm and the softer 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm. The least that keeps the 50 mm gap is 4.8 MN·s/m, where the stiffer building moves 269 mm and the softer 280 mm. At the largest link the two move together, 281 mm and 281 mm.
Fig. 1 The largest closing movement between the two buildings, and each building’s largest displacement, against the size of a viscous damper joining them, over five decades. With no link they close by 515 mm and move 220 and 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm; the least that keeps the 50 mm gap is 4.8 MN·s/m, where they move 269 and 280 mm. At the largest link they move together, 281 mm each.

A viscous link pushes on each building in proportion to how fast the two are moving relative to each other. It knows nothing about how fast either of them is moving over the ground. That one fact sets the shape of everything in the figure above.

Two buildings with the same period answer the same ground motion with the same history, displacement for displacement, whatever their masses — which is why the gap essay found a matched pair needs only the difference of its drifts. A link between them never moves, so it never pushes, and it does nothing at all. The pair here is mismatched by half: the softer building’s period is one and a half times the stiffer’s. Its two buildings spend the pulse out of step, and they close on each other by 515 mm when a single building of either kind moves no more than 419.

The arithmetic is the one that decided a line of cross-ties between stay cables. Two identical stays tied together have a first mode in which both move alike and the tie never stretches, so the tie does nothing to that mode at any stiffness; a link between two buildings of one period is the same statement with a dashpot in place of the spring. Whatever a connection between two structures does, it does to their difference, and where there is no difference it has nothing to do.

As the link grows, three things happen at once, and the figure has a curve for each. The closing movement falls, steadily and without exception: a larger link never lets the buildings approach further. The softer building’s displacement falls with it, because the link holds it back against the stiffer one. And the stiffer building’s displacement rises, because the same link drags it along behind the softer one. The two displacement curves meet at 281 mm. By the largest link drawn, the buildings are moving as one, and the relative motion the link acts on has gone.

The value they meet at is not a compromise the link strikes. It is the response of a single building of 800 t with both stiffnesses, whose period is 0.90 s — between the two buildings’ own — and under this pulse such a building moves 281 mm. A link stiff enough replaces the pair with that building, and each of the two inherits its displacement.

Two markers stand on that axis, a decade apart. They answer different questions, and the rest of this essay is about the distance between them.

One mode it damps, one it takes away

The modes of the pair show what the link is doing before any ground motion is applied.

One mode the link can damp, and one it removes. The damping ratio of the two modes of 500 t and 300 t buildings with periods of 0.8 and 1.2 s, with no damping of their own, against the size of a viscous damper joining them. One mode starts at 1.25 Hz, reaches 7.2 per cent of critical at 0.79 MN·s/m, and falls back as the link stiffens, ending at the locked pair's 1.11 Hz with no damping left: a link that does not move cannot dissipate anything. The other starts at 0.83 Hz — the two buildings moving against each other — and its damping climbs past 90 per cent until, beyond 2.1 MN·s/m, it no longer oscillates at all.
Fig. 2 The damping ratio of the pair’s two modes, with the buildings’ own damping removed, against the size of the link. One mode starts as the stiffer building’s own at 1.25 Hz, reaches 7.2 per cent of critical at 0.79 MN·s/m, and falls back to nothing as the link stiffens, ending at 1.11 Hz with the two buildings locked. The other starts as the softer building’s own at 0.83 Hz, becomes the two buildings moving against each other, and is damped past 90 per cent until, beyond 2.1 MN·s/m, it no longer oscillates.

With no link the pair has two modes, and they are simply the two buildings, each at its own frequency and with no damping but its own. With the link they become modes of one system, and they go two different ways.

One of them becomes the two buildings moving against each other. That is the motion the link resists directly, and its damping climbs without limit: past 90 per cent of critical, and then out of oscillation altogether. The link does not damp that mode so much as remove it.

The other becomes the two buildings moving together. Its damping rises to 7.2 per cent of critical at a link of 0.79 MN·s/m and then falls back, all the way to zero, as the link stiffens. Its frequency ends at 1.11 Hz, which is the square root of the two stiffnesses added over the two masses added — the frequency of a single building with both of their properties. A link stiff enough to do that is not moving, and a dashpot that does not move dissipates nothing.

That is the same shape as a damper near the anchorage of a stay, for the same reason. There, a damper made too large stopped moving and became a support. Here, a link made too large stops moving and becomes a joint.

It is also the reverse of a tuned mass. There a second degree of freedom is added precisely so that it can move against the first, and its damper is sized to let it. Here the second degree of freedom is a whole building that already moves against the first, and every increase in the link is an increase in how firmly that motion is stopped.

The size that takes out the most energy

Damping ratios describe free vibration. What a design needs is what happens under a ground motion, and the first answer is the energy the link takes out.

Too small a link and too large a link both take out nothing. The energy a viscous damper across the gap takes out of a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, over six seconds, against the damper's size. It peaks at 972 kJ near 0.64 MN·s/m and falls away on both sides: a small damper barely resists, and a large one barely moves. The link that keeps the gap, 4.8 MN·s/m, takes out 452 kJ and carries at most 1.6 MN. Without a link the same pair collide seven times, and a restitution of 0.65 takes 599 kJ out of them in those collisions, dashed.
Fig. 3 The energy the link takes out of the pair over six seconds, against its size. It peaks at 972 kJ at 0.64 MN·s/m and falls on both sides, because a small link barely resists and a large one barely moves. The link that keeps the gap, 4.8 MN·s/m, takes out 452 kJ. Dashed, the 599 kJ that a restitution of 0.65 takes out of the same pair in its seven collisions when nothing links them.

The curve has a clear peak, at 0.64 MN·s/m, where the link takes 972 kJ out of the two buildings in six seconds — more than half of the roughly 1,800 kJ the pulse puts into the pair, the rest going into the buildings’ own damping. That is more than half as much again as their seven collisions take out when nothing joins them, 599 kJ at a restitution of 0.65. Sized for energy, a link is a better energy sink than the collisions it was meant to prevent.

It is still not preventing them.

The same two buildings, joined across the gap. The displacement of the stiffer building and of the face of the softer one, drawn 50 mm from it, for a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, joined by a viscous damper of 0.64 MN·s/m. Dashed, the same pair with no link. The stiffer building reaches 236 mm against 220 mm alone and the softer 336 mm against 419 mm. The two close by at most 215 mm, so in a 50 mm gap they still meet, and the damper strokes 276 mm, carries at most 1.0 MN and takes out 972 kJ.
Fig. 4 The two buildings joined by the link that takes out the most energy, 0.64 MN·s/m, with the unlinked pair dashed. The stiffer building reaches 236 mm and the face of the softer 336 mm; the relative motion dies away within three seconds, but at its largest the two close by 215 mm. The link strokes 276 mm, carries at most 1.0 MN and takes out 972 kJ.

The history shows what the energy figure cannot. The relative motion dies away fast: after three seconds the two buildings never move more than 26 mm relative to each other, where the unlinked pair is still moving 252 mm. But the largest closing comes at 1.68 s, just after the pulse has ended, when the link has been at work for barely two cycles of the stiffer building — and at that instant the two still approach by 215 mm, against 515 mm at almost the same instant with no link at all. Across a 50 mm gap they collide, on the swing that matters most. The link has also had to travel 276 mm between its extremes, several times the width of the gap it spans, so a damper of this size cannot sit in the gap; it has to reach into both buildings on brackets or run obliquely between floors.

So the link that is best by the measure a damper is usually judged by does not do the job a gap does. It shortens the episode and takes the relative motion out of every swing after the largest, and it leaves the largest with most of its size.

Why the two sizes cannot be one

The reason is in what each measure asks of the relative motion. The energy a dashpot takes out is its coefficient times the square of the relative velocity, added up over time, and it is largest when the link both resists hard and is moved a long way. That needs relative motion. Keeping the gap needs the relative motion to be small at every instant. A link can take out the most energy only by allowing the motion the gap forbids, so the two optima cannot coincide, and the only question is how far apart they sit. At the energy optimum the damper strokes 276 mm; at the size that keeps the gap it strokes 59 mm and takes out less than half as much.

The size that keeps the gap

The other marker is the least link that holds the closing movement to 50 mm.

The same two buildings, joined across the gap. The displacement of the stiffer building and of the face of the softer one, drawn 50 mm from it, for a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, joined by a viscous damper of 4.8 MN·s/m. Dashed, the same pair with no link. The stiffer building reaches 269 mm against 220 mm alone and the softer 280 mm against 419 mm. The two close by at most 50 mm, inside the 50 mm gap, and the damper strokes 59 mm, carries at most 1.6 MN and takes out 452 kJ.
Fig. 5 The two buildings joined by the least link that keeps the 50 mm gap, 4.8 MN·s/m, with the unlinked pair dashed. The stiffer building reaches 269 mm against 220 alone and the softer 280 mm against 419; the two close by at most 50 mm, and the link strokes 59 mm, carries at most 1.6 MN and takes out 452 kJ.

It is 4.8 MN·s/m, seven and a half times the energy optimum, and the history looks nothing like a damped pair. The two buildings move almost in step from the start. The link strokes 59 mm in all, carries at most 1.6 MN, and takes out 452 kJ — less than half what the smaller link did, and less than the collisions did. It keeps the gap by preventing the relative motion from building up at all, which is what a stiff connection would do, and it is doing it by being nearly one. Its largest closing comes at 0.67 s, inside the pulse, and is exactly the 50 mm it was sized to allow; the two never approach so far again.

The price is on the same figure. The softer building moves 280 mm rather than 419, which is the protection a link is bought for. The stiffer building moves 269 mm rather than 220. A link that keeps the gap has not added damping to a pair; it has made the pair into something close to one building with the combined stiffness and mass of both, and a building made that way sways between the two.

The force is the part that genuinely favours the link. Across every contact law and every contact stiffness in the pounding analyses, the largest collision force ran from 27 to 225 MN, a number that belonged to the model rather than to the buildings. The link that replaces those collisions carries 1.6 MN, and that number belongs to the design: it is the damper’s coefficient times a velocity the analysis computes, and it can be specified, tested and detailed for.

It is small for a reason worth having. Even a link of 108 MN·s/m, more than twenty times as large, which holds the two buildings within 3 mm of each other, carries only 1.8 MN. The force needed to make two buildings move together supplies only the difference between what each would do on its own — the softer building’s excess swing held back, the stiffer building’s shortfall made up — and a collision delivers that same difference all at once, in a few hundredths of a second, which is where its tens of meganewtons come from.

The period ratio sets the distance between the two

One pair is one case. The same two searches across the period ratio say whether the distance between the two sizes is a feature of this pair or of links in general.

The link for energy and the link for the gap are different links. For a 500 t building with a 0.8 s period beside a 300 t building whose period is 1, 1.1, 1.25, 1.5, 1.75, 2 and 2.5 times as long, 50 mm apart under one 1.0 s pulse of 0.50 g: the size of damper that takes the most energy out of the pair, and the least size that keeps them from meeting, on a logarithmic scale. At equal periods neither exists, because the pair never moves apart. At every other ratio the second is larger, by 3.6 to 8.5 times, and the first leaves the buildings closing by 99 mm to 240 mm.
Fig. 6 The link that takes the most energy out of the pair and the least link that keeps the 50 mm gap, on a logarithmic scale, as the softer building’s period runs from 1 to 2.5 times the stiffer’s under the same pulse. At equal periods neither exists, because the pair never moves apart. At every other ratio the second is larger, by 3.6 to 8.5 times, and the first leaves the buildings closing by 99 to 240 mm.

At a period ratio of one there is nothing to draw: the two buildings never move apart, no link has anything to act on, and no gap is needed beyond the difference of the drifts. At every other ratio both sizes exist and the one that keeps the gap is the larger — 3.6 times at a ratio of 1.1, 7.5 times at 1.5, about eight times from there up to 2.5. The link that takes out the most energy never keeps a 50 mm gap anywhere on the plot; it leaves the pair closing by between 99 and 240 mm.

The two curves diverge because they are driven by different things. The energy optimum moves only gently, from 0.27 to 1.31 MN·s/m across the whole range, because it is set by the frequencies of the pair. The size that keeps the gap is set by how far the unlinked pair would close, which runs from 174 mm at a ratio of 1.1 to 600 mm at 2.5 — and not smoothly, since a single pulse favours some pairs of periods over others — and the link has to suppress all of it but 50 mm.

This is the gap essay’s observation carried one step further. Mismatch is what makes a link effective as a damper — the energy optimum and the energy at it both rise with the ratio. Mismatch is also what makes the gap large, and so what makes the link that keeps it large. The more a pair needs linking, the further apart the damper it can use and the joint it needs become.

The stiffer building pays

The last figure asks what keeping the gap does to the two buildings at every ratio.

Keeping the gap is paid for by the stiffer building. The largest displacement of each of 500 t and 300 t buildings, the first with a 0.8 s period and the second with a period between 1 and 2.5 times as long, on their own, dashed, and joined by the least viscous link that keeps a 50 mm gap closed to nothing, solid, under one 1.0 s pulse of 0.50 g. The link brings the softer building's displacement down and the stiffer building's up, toward one value, and the stiffer building pays more the more the periods differ: at 2.5 times the period it moves 303 mm instead of 220 mm, 38 per cent more, while the damper carries at most 2.9 MN.
Fig. 7 The largest displacement of each building on its own, dashed, and joined by the least link that keeps the 50 mm gap, solid, as the softer building’s period runs from 1 to 2.5 times the stiffer’s. The link brings the two toward one value. The stiffer building pays more the more the periods differ: at 2.5 times the period it moves 303 mm instead of 220, 38 per cent more, while the link carries at most 2.9 MN.

On their own the stiffer building always moves 220 mm and the softer one moves more the softer it is, up to 569 mm. Linked, the two solid curves lie almost on top of each other, because a link that keeps the gap is one that holds them nearly together. The softer building’s saving is large. The stiffer building’s cost grows with the mismatch: 4 per cent at a period ratio of 1.1, 22 per cent at 1.5, 33 per cent at 2, 38 per cent at 2.5.

That cost lands where nothing was designed for it. The stiffer building was checked for its own 220 mm, and what an earthquake asks of a structure is a displacement rather than a force, so a linked building’s columns, cladding and partitions are asked for the linked drift. A retrofit that joins an old, stiff building to a new, flexible one has moved part of the new building’s problem into the old one, and the old one is the building least likely to have been detailed for it.

The force stays small throughout — 2.9 MN at the largest ratio drawn — which is why a link is a practical proposition at all. But that force has to be delivered into each building at the level of the link and distributed through a floor, and the floor at the top of the shorter building is usually a roof.

What the pair of oscillators leaves out

One pulse is not an earthquake. A one-cycle pulse puts its worst closing in the first cycle, before any damper has time to act, which is the case least favourable to the energy optimum. A long record with many cycles gives a damper more time and may narrow the distance between the two sizes; how far is a question for records rather than for pulses.

Each building has one mode. Real buildings meet at a floor level, and a link at that level acts on the relative motion at that height, which depends on the modes each building has rather than on its roof alone.

The link is linear and has no stroke limit. Real fluid dampers resist as a fractional power of the velocity, which lowers their force at high speed, and they have a finite stroke. The energy-optimal link here needed 276 mm of it. A yielding link — a brace that yields in both directions — would cap its force instead and dissipate in hysteresis, with an optimum of its own.

The link is attached rigidly. A damper on a bracket is a dashpot in series with a spring, which has its own optimum and a lower ceiling, as the stay damper’s supports did.

The buildings stay elastic and stand on the same ground. A building that yields lengthens its period, which changes the ratio the link was sized for, and two foundations on different soils receive different motions.

And no figure here shows the connection. The 1.6 MN a link carries has to get into two floor plates, across a joint that must also accommodate every ordinary movement of the two buildings — temperature, shrinkage, wind — without transmitting it. That detail is where a link succeeds or fails, and a two-oscillator model has nothing to say about it.

The assumption every figure rests on is the one the gap essay made: both buildings receive exactly the same ground motion at the same instant. A link acts only on the difference between their responses, so any difference between their inputs is a relative motion the model has left out.

Between the energy optimum and the size that keeps the gap lies a decade of possible links, and under one pulse none of them does both jobs. Under a record with many cycles the first cycle carries less of the damage and the damper more of the protection, so a link somewhat above the energy optimum might keep a gap somewhat wider than 50 mm while still dissipating most of what it can. Whether a single size exists that does both acceptably across a range of records with different frequency content — and whether a gap of a few hundred millimetres with such a link in it is cheaper than the gap the pair would need without one — is a question about the ensemble of motions a site might receive, and it decides whether a link is a damper that happens to protect a gap or a joint that happens to dissipate energy.

Where this goes next

Buildings that sway alike need almost no gap between them. The separation two adjacent buildings need, against the ratio of their periods. The obvious answer is the sum of what each can do — 340 mm — and it is wrong, because the two peaks do not occur at the same instant. The right combination is the one modal responses use, √(u₁² + u₂² − 2ρu₁u₂), with ρ the cross-correlation coefficient of the two responses. ρ depends on the period ratio and behaves the opposite way to intuition: at a ratio of one the two buildings sway together, ρ = 1, and the gap collapses to the difference of the two, 100 mm. At the 0.57 drawn ρ is 0.029 and the gap is 248 mm — 27 per cent less than the sum, and 99 per cent of the square root of the sum of squares. The gap between two buildings Two towers side by side in an earthquake need a gap. The obvious answer is the sum of what each can move, and it is wrong — because the two peaks do not happen at the same instant. What decides the answer is the ratio of the two periods, and buildings that sway alike need almost no gap at all. One collision, four contact laws, one impulse. Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The elastic linear spring peaks at 19.7 MN and lasts 58 ms; the Hertz law peaks at 22.0 MN over 61 ms. Set to a restitution of 0.65, the spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN, a tension two faces in contact cannot carry, while the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 instead. The areas under the curves are the impulses: 600 kN·s for the spring and dashpot, which is what momentum requires at 0.65, 646 for the damped Hertz law, and 727 for both elastic laws. The force that belongs to the model When two buildings meet, momentum decides what each of them feels, and no contact law can change it. What the contact law decides is the force, and the force it reports is the contact stiffness somebody assumed, raised to a power. Four laws and a hundredfold change of stiffness move the buildings by a few per cent and the force by a factor of eight. What a damper at the anchorage can do, and the ceiling it cannot pass. Modal damping against damper size for a 200 m stay at 4500 kN, with the damper 4 m from the anchorage — 2.0 per cent of the length. Each curve is a mode, found as a complex root of the taut string with a viscous damper in it rather than from a formula. Every one of them peaks at 1.00 per cent of critical, which is x/2L exactly, and the peaks are at different damper sizes — a higher mode wants a smaller damper, because it moves faster at the same amplitude. The curves are flat near their peaks: half the optimum coefficient gives 80 per cent of the ceiling, and so does twice it. The damper that is too near the end A stay cable has almost no damping of its own, so it is given a damper — and the damper cannot go where the motion is, because the middle of a two-hundred-metre stay is a hundred metres above the road. What it can supply is then decided by one length, and no amount of damper changes it. How much a harmonic force is magnified, at four damping ratios. Displacement amplitude divided by the deflection the same force would produce if it were applied slowly, against the ratio of the forcing frequency to the structure's own, at 1%, 2%, 5%, 10% of critical damping. At the natural frequency the magnification is 50, 25, 10, 5 respectively — one over twice the damping ratio, and nothing else in the problem enters it. The only thing that stops it Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built.

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DampingEnergy dissipationModal dampingMode shapeNatural periodPoundingSeismic gapStorey driftViscous damper