The gap between two buildings

Two buildings share a boundary. Both sway in an earthquake, and if the gap between them is too small they hit each other — which is called pounding, and which has damaged more buildings in more earthquakes than almost any other single mechanism because city blocks are built to the site boundary.

The question is how wide the gap has to be, and the arithmetic looks trivial. One building moves 120 mm and the other 220, so they need 340 mm between them.

The peaks do not coincide #

Each building’s response is a time history — a sway back and forth, at its own period, driven by the same ground motion. Its maximum displacement occurs at some instant during that history, and there is no reason whatever for the other building’s maximum to occur at the same one.

What the gap has to accommodate is the maximum of the relative displacement $u_1(t) - u_2(t)$, and the maximum of a difference is not the difference of the maxima unless the two are perfectly anticorrelated.

This is not a new problem. It is exactly the problem combining modal responses solves, one level up: several responses to the same input, each with its own peak at its own instant, to be combined into one number. The answer is the same:

$$\Delta = \sqrt{u_1^2 + u_2^2 - 2\rho\,u_1u_2}$$

with $\rho$ the cross-correlation coefficient of the two responses. The only new feature is the minus sign, which is there because the quantity wanted is a difference rather than a sum.

The coefficient depends only on the period ratio #

$\rho$ has a closed form — the same one modal combination uses — and it contains the frequency ratio $r$ and the two damping ratios and nothing else:

$$\rho = \frac{8\sqrt{\zeta_1\zeta_2}\,(\zeta_1 + r\zeta_2)\,r^{3/2}}{(1-r^2)^2 + 4\zeta_1\zeta_2 r(1+r^2) + 4(\zeta_1^2+\zeta_2^2)r^2}.$$

At $r = 1$ it is exactly one. At $r = 0.57$, the pair drawn, it is 0.029 — the two responses are essentially independent. At $r$ below about 0.8 it is negligible for ordinary damping.

So the answer has two regimes and the boundary between them is sharp:

Buildings that sway alike need the difference of their drifts. Buildings that sway differently need very nearly the square root of the sum of the squares. Between the two the answer moves quickly, and it never reaches the sum.

Why that is the opposite of the intuition #

The word resonance does a lot of damage here. Two structures with the same period sound like the dangerous pair — they will get excited by the same frequencies, both reach their maxima, and surely that is worst.

Both halves of that are true and the conclusion does not follow. They do both reach their maxima. They reach them at the same instant and in the same direction, because they are being driven by the same ground motion through the same transfer function, so they are moving together and the gap between them barely changes.

That is the same effect modal combination handles inside one building: closely spaced modes are strongly correlated and combine nearly algebraically, while well-separated ones are independent and combine as a square root. Adjacent buildings are two modes of one system that happens to have a gap in the middle of it.

The mismatched pair is worse for exactly the reason it sounds safer. One is at its extreme while the other is at its middle or its other extreme, so the relative movement can approach the sum even though neither building is doing anything remarkable.

What the codes say #

Most of them say $u_1 + u_2$, the absolute sum, and some allow the square root of the sum of squares where the two structures’ dynamic properties are known.

The absolute sum is safe and expensive, in the way an envelope assembled from separate load cases is: correct at every station and corresponding to no state the structure ever occupies. For the pair drawn it demands 340 mm of gap where 248 suffices — 92 mm of building width per boundary, on both sides of every plot, on every floor, permanently. On a city block that is a material quantity of floor area given up to a combination rule.

The case where the conservatism inverts is the one worth watching, and it exists: where the two buildings are nominally identical — a pair of blocks by the same developer, same frame, same height — a designer applying the correct rule finds a very small gap is needed, and is relying on the period ratio staying at one. It will not: the two buildings have different tenants, different fit-outs, different foundation conditions on the two halves of the site, and different amounts of cracking after the first shake. A rule that gives its smallest answer where the input is least reliable is a rule to be careful with, and that is the honest argument for the absolute sum.

The edge that moves more than the building #

Every displacement so far belongs to the building. The gap belongs to its edge, and on a plan that twists at all those are different numbers.

A building whose centre of mass and centre of rigidity do not coincide rotates in plan as well as translating, and every code requires an accidental eccentricity of about 5% of the plan dimension to be assumed even where the calculated one is zero — because the mass distribution is not known, the stiffnesses are not exactly as drawn, and the ground motion is not identical across the footprint. The displacement at a point $x$ from the centre is then amplified by

$$\frac{u(x)}{u} = 1 + \frac{e\,x}{r^2}$$

with $r$ the radius of gyration of the plan. For a 30 by 20 m building, $r^2 = (30^2+20^2)/12 = 108$ m², and an accidental eccentricity of $0.05 \times 30 = 1.5$ m gives the edge at $x = 15$ m an amplification of 1.21.

That is a 21% addition to the number the gap calculation used, and it lands specifically at the edge nearest the boundary — which is the one edge a designer does not get to choose, because it is where the neighbour is.

Applied to the pair drawn, 120 and 220 mm become 145 and 266 at their facing edges, and the 248 mm the correlation formula saved becomes 300.

The refinement that removed 27% and the term that adds 21% are the same size, and only one of them is usually applied — the correlation rule appears in codes as a permitted relaxation, and the torsional amplification appears as a requirement on storey drift ratios, where it is checked against a drift limit and then not carried into the gap.

The more serious version is the case the earlier section flagged as the one to be careful with.

Two nominally identical buildings have a period ratio of one, a correlation of one, and a required gap that collapses to the difference of their drifts — nearly nothing. Their translations really are locked together: same period, same input, same phase. Their twists are not. Accidental eccentricity is accidental in each building independently — it comes from where the tenants put their filing, which way the plant room faces, which stiffnesses turned out low — so the two rotations are uncorrelated even when the two translations are identical, and they combine as a square root rather than cancelling.

At 220 mm of drift each, the torsional part is $0.21 \times 220 = 46$ mm per building, and

$$\Delta \approx \sqrt{46^2 + 46^2} = 65 \text{ mm}$$

on a pair the correlation rule says needs almost none. That is not a large gap and it is not zero, and the difference between “not large” and “zero” is the whole of whether the two buildings touch.

So the honest form of the earlier caution is quantitative rather than a warning about drift over time: the high-correlation regime cancels the translations and cannot cancel the rotations, and a pair designed on matched periods alone has a residual requirement that no amount of matching removes.

If they do meet #

The two buildings meet with a relative velocity of the order of $2\pi u/T$ from each — 1.94 m/s in total for the pair drawn. Conservation of momentum with a coefficient of restitution gives the impulse

$$J = m_{\text{eff}}\,v_{\text{rel}}\,(1+e), \qquad m_{\text{eff}} = \frac{m_1m_2}{m_1+m_2},$$

597 kN·s here. That impulse is shared equally and absorbed unequally: the 500 tonne building takes a velocity change of 1.19 m/s and the 300 tonne one takes 1.99.

The damage in recorded poundings is overwhelmingly on the lighter, more flexible building, and this is why. It is also why the worst arrangement is a stiff low building against a flexible tall one: the impact happens at the top of the low building, which is a point on the tall one’s height where nothing was designed for a horizontal point load, and the tall building’s column at that level takes a shear it has no capacity for.

Why it happens so often #

Pounding is not a rare failure. In every urban earthquake with a decent post-event survey it appears in a large fraction of the damaged buildings, and the reasons are all about how cities are built rather than about how structures behave.

Buildings are built to the boundary. Land is expensive, plots are narrow, and the gap between two buildings is floor area nobody is paid for. Every millimetre of it is argued about.

The neighbour was built first, to an older code. The gap that exists is the one the earlier building left, and it was calculated — if at all — against a smaller design event and a stiffer assumed structure.

The two buildings have different owners. So the total gap is the sum of two independently chosen setbacks, neither owner knows the other’s dynamic properties, and neither has any incentive to give up more than the minimum.

And the gap fills up. Expansion joint material, services, cladding brackets, birds’ nests and forty years of debris. A 100 mm gap that has been closed by 60 mm of accumulated fabric is a 40 mm gap, and nothing inspects it.

It is the same accounting problem a bridge joint has, where the movement to be accommodated is a sum of five terms and the joint is sized on three of them. A seismic gap is that arithmetic with the largest term in it belonging to somebody else’s building, computed to somebody else’s code, decades ago.

Which free body produced the number #

There is no free body until they touch, and that is the point. Up to the moment of contact the two structures are independent dynamic systems with no interaction at all, and the whole calculation is about a geometric condition — a distance being exceeded — rather than about a force.

That is unusual enough in this collection to be worth naming. Almost every check here is a comparison of an action with a resistance. This one is a comparison of a movement with a clearance, and the resistance side of it does not exist: there is no strength anybody can add to a building that lets it be closer to its neighbour.

The same shape of check appears at a much smaller scale in the gap nobody computed: a joint has to be wide enough for a sum of movements, the sum has terms nobody counted, and the failure is a movement exceeding a clearance rather than a stress exceeding a strength. A seismic gap is that movement budget with two whole structures in it instead of one.

The three things a designer can move #

The gap is one of them and it is the least available, because it costs floor area on every storey for ever. The other two are worth naming.

The periods. Moving one building’s period toward the other’s raises the correlation and shrinks the required gap, and it is achievable — a stiffer or softer lateral system, an added damper, a different bracing layout. It is almost never done deliberately, and it is the only intervention that makes the required gap smaller rather than making the consequence of closing it less bad.

The contact. If the buildings are going to touch, they can be made to touch well: a bumper of a soft, thick, replaceable material at every floor level, aligned floor-to-floor, sized so the contact lasts long enough for the force to stay finite. That converts an impact into a collision with a known stiffness, which is a design problem rather than an accident.

The last of those is the elegant answer and it is used: several retrofits of adjacent towers have replaced a gap that could not be widened with a set of viscous dampers across it, turning the worst feature of the pair into the mechanism that protects them.

Where the model stops #

The two buildings are on the same ground. They are not quite: two adjacent sites can have different foundation conditions, and ground with a period of its own filters the motion before either building sees it. Two buildings driven by slightly different inputs are less correlated than the formula says, which pushes the answer back toward the square root of the sum of squares.

Both structures are single-degree systems. They are not: each has several modes, the drift at the level where they touch is not the roof drift, and the relevant $u$ is the displacement at the contact level rather than at the top. For two buildings of different heights the contact is at the shorter one’s roof and the taller one’s mid-height.

The floors line up. Where they do not, the collision is floor-to-column rather than floor-to-floor, and it is much worse — a floor slab is a hard, heavy, in-plane-stiff object and a column is not. Floor levels that match is the single most important mitigating feature of an adjacent pair, and it is an architectural decision.

The displacements come from a spectrum. Which means they are already peak values of independent histories, and combining them with a correlation coefficient derived for stationary random processes is an approximation applied to an approximation.

The restitution is 0.65 and is a guess. It depends on the local damage at the contact — a concrete edge crushing is a very different collision from two steel plates meeting — and the impulse depends on it linearly.

Nothing here is inelastic. A building that has yielded has a longer effective period and a different damping, so the period ratio at the end of the shake is not the one it started with. Two buildings yielding by different amounts drift apart in period as the event proceeds, which lowers the correlation exactly when the displacements are largest — and an equal-displacement rule says the displacements themselves barely change, so the gap requirement grows during the event.

And no drawing here shows the collision. Everything above is either a peak displacement or a momentum balance, and neither of those is a picture of the event: a contact lasting a few hundredths of a second, repeated a dozen times during the shake, at whichever floor levels happen to be closest.

The number to design to #

Putting it together, the gap a designer should provide is

$$\Delta = \sqrt{u_1^2 + u_2^2 - 2\rho\,u_1u_2}$$

with two qualifications that between them recover most of the conservatism the formula removes.

Use the displacements at the contact level, not at the roofs, and take them from the design event rather than from a service one — a gap is a collapse-prevention item and there is no serviceability version of it.

Take both displacements at the facing edge, with the accidental torsional amplification in them, rather than at the centre of the plan. The two rotations are independent of each other whatever the two translations do, so they are combined as a square root and they never cancel — which puts a floor under the gap that the correlation term cannot reach.

And do not take $\rho$ above about 0.5 unless the two structures are genuinely the same structure. The high-correlation regime is the one where the formula is generous and the input is least reliable, and a period ratio that starts at one drifts away from one over the buildings’ lives.

Applied to the pair drawn, that gives 248 mm rather than 340 — a saving of 27 per cent, arrived at by asking when the peaks occur rather than only how large they are.

The ladder from here #

Later rungs on this anchor: the multi-storey case, where the contact level and the mode shapes at that level decide the relative displacement rather than the roof drifts. Floor-to-column pounding and why it is the arrangement that kills people. The contact model itself — a spring, a spring and a dashpot, or a Hertz law — and how much the answer depends on which. Retrofitting an existing pair, where the gap cannot be changed and the options are to stiffen one building toward the other’s period, to link them together deliberately, or to accept the pounding and detail for it. Linked buildings, where a damper spanning the gap is designed to dissipate exactly the energy the collision would have delivered. And the urban-scale version, which is a row of buildings in a terrace, where the end building has a neighbour on one side only and takes everything the row delivers.