The floor that arrives at a column

The gap between two buildings is a calculation about separation: how far apart two structures have to be so that their peak displacements, which do not occur at the same instant, never add up to the distance between them. It ends with a combination rule and a number of millimetres.

This is about what happens when the number is not enough, and the first thing to establish is where.

Two facts in one figure, and neither is in the separation calculation.

They touch low, and the reason is a shape #

Both buildings’ displacements grow with height, so their closure does too, and the level where the two first meet is the lowest level at which the gap is used up rather than the highest. Above it they are in contact over the whole remaining height of the shorter building.

That has a consequence for what to expect. A pair of buildings that pound do not meet once at the top; they meet along a face, at every level above the contact height, and the number of impacts per cycle is the number of floors in that region. On the pair drawn that is three storeys of the shorter building and eight metres of façade.

A check on the roof displacement is therefore checking the wrong quantity twice over: it is not where contact begins, and once contact begins the roof is not where the damage is.

Which free body produced the number #

The free body is one column of the shorter building, cut at the floor above and the floor below it, with the slab edge of the taller building pressed against it part way up.

Crossing the cuts are the ordinary things a column carries: an axial force, and the shear and moment its own frame’s sway produces. Crossing the face is something no design case contains — a transverse point load, applied at a fraction $a$ of the storey height, for a few milliseconds.

Resolve it. A column restrained at both ends against rotation, carrying a point load at $a$ of its height, has end moments $Fa(1-a)^2h$ and $Fa^2(1-a)h$, and the shear either side of the load is $F(1-a)$ and $Fa$. At mid-height that is $Fh/8$ at each end and $F/2$ of shear.

Compare that with the same force arriving at a floor. The slab takes it in plane, which is what a diaphragm is for — the stiffest and strongest element in the building, sized for exactly this kind of in-plane force and continuous all the way to whatever resists it. The column above and below sees almost nothing.

So the two cases are not a matter of degree. One delivers an impact into a member designed to carry in-plane force; the other delivers it into a member with no capacity for a transverse point load at all, at a point with the minimum transverse reinforcement its detailing rules require and no more.

Three quarters of a metre #

The difference between those two figures is the shorter building having four storeys in 15 m rather than five. It is a decision made by an architect for reasons of use, and on the pounding problem it is the whole answer.

Storey heights differ for ordinary reasons: an office beside a residential block, a retail ground floor beside a warehouse, an old building beside a new one. Two buildings of the same height in metres and different heights in storeys is the normal case, not the pathological one.

What makes it dangerous is that the misalignment is largest in the middle of the contact region. If the two buildings’ floors coincide at the ground and their storey heights differ by $\delta$, the $n$-th floors are $n\delta$ apart — so the first few floors nearly align, the middle ones land at mid-height, and further up they come back into alignment for a floor before drifting again. The worst level is where $n\delta$ is half a storey, and on the pair drawn that is the fourth and fifth floors of the taller building.

What the impact is worth #

The demand is an impulse rather than a force, and turning one into the other is where the numbers get soft — which is worth saying plainly rather than presenting a false precision.

The momentum exchanged in an impact is fixed by the two masses, their relative velocity and the coefficient of restitution: that much is mechanics and the rung below computes it. Turning it into a force needs a contact duration, and the contact duration depends on the local stiffness of two concrete edges meeting — which is a Hertzian contact problem between two objects whose surfaces are cover concrete, cladding, and whatever was in the joint. It is the same difficulty a dropped weight presents, with the added complication that neither body is rigid.

Published estimates run from a few milliseconds to a few tens, which is a factor of ten in the force for the same impulse. So the honest statement is a comparison rather than a value: the force is very large and its magnitude is not calculable to better than an order of magnitude, and design proceeds by avoiding the arrangement rather than by checking it.

That is an unusual position for a structural problem to be in and it explains the shape of every code provision on the subject. They are all geometrical — a required separation, a requirement that floors align, a prohibition on building against an existing structure without a gap — and none of them is a load case.

Which is why the remedies are geometrical #

Given a force that cannot be computed, the interventions available are about arranging matters so that the impact does not happen or does not matter, and there are four of them.

Widen the gap, which is available in a new building and never in an existing pair. The required separation is not small: on the pair here it is over 100 mm and on tall flexible buildings it is several hundred, which is a real cost in plan area and a difficult detail to seal and maintain.

Align the floors, which is available surprisingly often and costs almost nothing at design stage. It converts the failure mode from a column strike to a slab strike, and slab strikes damage cladding, joints and finishes rather than the frame.

Fill the gap with something crushable, which turns an impact into a slower transfer. The intervention is a soft collision rather than no collision, and its purpose is to lengthen the contact duration — the one variable that decides the force for a given impulse.

Or tie the two buildings together on purpose, so that they move as one and there is no relative displacement to close. That is a large intervention with a large consequence — two buildings tied together are one structure with a period neither of them had — and it is done, most often with dampers spanning the gap so that the connection dissipates the energy the collision would have.

The four are in increasing order of intrusiveness and decreasing order of availability, and the second is the one this page exists to argue for.

Why the older building is the one that suffers #

There is an asymmetry in a pounding pair that follows from the mechanics and is visible in every survey of earthquake damage.

The building that is stiffer takes the impact at a smaller displacement of its own, so the impact arrives when it is nearer its own equilibrium and it is not the one moving fastest. The building that is more flexible is moving faster at the instant of contact, and the momentum exchange decelerates it hard.

And the building that is shorter is struck at a level that is a larger fraction of its height, so the impact enters its structure at a place where it has fewer storeys above to distribute it and more of its total shear to redistribute.

An older, shorter, stiffer building beside a taller flexible new one is therefore the worst arrangement available, and it is the commonest one in a city that has been built twice. The pair drawn is exactly that shape, and the level of first contact — half way up the shorter building — is where the older frame’s own storey shear was already largest.

The mid-height column, in the company it keeps #

A column struck part way up its height is not a new failure mode. It is the same failure as a short column, arriving by a different route, and putting the two beside each other says why it is so dangerous.

A short column is one whose clear height has been reduced by something — an infill panel to sill level, a ramp beam, a mezzanine — so the storey’s drift is forced into a shorter length and the shear rises in inverse proportion. A column struck at mid-height has the same thing done to it by an impact rather than by a panel: half the column is asked to accept a transverse force with the other half restraining it, and the shear demand in that half has no relationship to the shear the frame analysis gave it.

Both fail in shear, brittly, at a location with the minimum links their detailing rules allow. Both are visible in earthquake photographs as an X of diagonal cracks in the middle of a column, and both belong to the family of failures that arise because a member’s real boundary conditions were not the ones drawn.

That kinship is the useful part. A designer who has understood why a captive column is dangerous has already understood floor-to-column pounding, and the intervention is the same one: give the member the transverse reinforcement for a shear it might be asked for rather than for the shear the analysis produced. It is capacity design applied to a demand that has no analysis behind it at all.

Where the model stops #

Both buildings are given a first mode and nothing else. A structure has more than one period, and the closure profile of a real pair contains the second mode of each — which changes the shape near the base, where the first mode has almost nothing and the second has a good deal.

The contact is treated as kinematic. Once the two touch they interact: the impact changes both velocities, the buildings separate, and they may meet again in the same half-cycle. The full problem is a nonlinear time history with a contact element in it, and its answer depends on the contact model to a degree that makes the results of different studies hard to compare.

Torsion is absent. Two buildings touching along one face are being pushed apart eccentrically, which twists both of them in plan and changes the closure along the length of the face. The corner is where contact begins.

And the buildings are assumed to survive to the point of contact. The displacements used are elastic ones scaled to the design event; a building that has yielded has larger displacements and a longer period, and what an earthquake asks for is a displacement that grows as the structure softens.

What the pictures cannot show #

The sequence. Every figure here is a snapshot of a peak, and pounding is a sequence of events over twenty or thirty seconds — contact, separation, contact again, with the two buildings’ phase relationship drifting as each of them softens at a different rate.

They also cannot show the cladding, which is what is actually between the two buildings. A 50 mm gap between structural frames is not a 50 mm gap between buildings: there is a façade on each side, and the real clearance is whatever is left. Pounding damage very often begins with the cladding closing a gap the frames were given, which is a failure of a component nobody analysed to protect a component nobody expected to be hit.

The assumption the figure rests on #

That the two buildings’ floors are at fixed levels and their displacement profiles are smooth.

The second is the interesting one. A frame’s first mode is a smooth curve because its storeys are alike, and the closure figure inherits that smoothness. A building with a soft storey has a mode that is nearly a step — almost all the displacement happens in one storey — and the closure profile then has a step in it too.

That changes the conclusion in a specific way. The contact level of a pair including a soft-storey building is the soft storey, whatever the heights, and the impact lands at whatever height that storey’s neighbour happens to have. It is the combination of the two commonest deficiencies in older reinforced-concrete buildings, and it is worth naming as the case where both of the arguments on this page point at the same storey.

What to carry away #

Three sentences, and the last is the one that transfers past this pair of buildings.

Two buildings that pound touch at the lowest level where their closure exceeds the gap, and remain in contact over everything above it — so the number of impacts is a number of storeys, and the roof is neither where it starts nor where the damage is.

What the impact does is decided by whether the floors align. A slab meeting a slab is a diaphragm problem and a slab meeting a column is a shear problem, and only the second is a collapse mechanism.

And the variable that decides the failure mode appears in none of the calculations. The separation check produces millimetres, the response analysis produces displacements, and the thing that separates a damaged façade from a lost column is a storey height that was set for reasons of headroom. That is a general hazard of a check that reduces a configuration to a number: an envelope is not a structure, and a separation is not an interface.

The evidence, which is a photograph rather than a calculation #

Pounding is one of the few subjects in this collection whose literature is mostly forensic, and the reason is the one already given: the force cannot be computed, so what is known about it comes from looking at buildings afterwards.

Mexico City in 1985 is the survey everybody cites. Of some 330 collapsed or severely damaged buildings, over 40 per cent showed evidence of pounding and around 15 per cent had it as the apparent primary cause — in a city built to a rectilinear grid with buildings hard against one another and storey heights that changed with every rebuilding. The finding that came out of it, repeated after Loma Prieta in 1989 and Kobe in 1995, is that the damage is concentrated in the buildings whose floors did not align, and that mid-height column failure is the signature.

What that record cannot supply is a design number, and eighty years of pounding research has not produced one either. What it produces instead is the geometrical rules already listed, and a persistent recommendation that reads oddly in a code full of load cases: align the floors of adjacent buildings.

A code provision that is an instruction about architecture rather than a check on a member is a sign that the mechanics has been understood and cannot be usefully computed — which is a category worth recognising, because the temptation is always to replace such a rule with an analysis that looks more rigorous and is not.

The ladder from here #

Later rungs on this anchor: the contact model itself — a spring, a spring and a dashpot, or a Hertz law — and how much the answer depends on which. Multi-storey pounding as a time history, where the relative velocity at contact is what the analysis exists to produce. 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, or to accept the pounding and detail for it. Linked buildings with dampers in the gap, where the connection is designed to dissipate exactly the energy the collision would have delivered. And the same argument at the scale of a component: a stair, a bridge deck at an abutment, and an item of plant on a floor are all objects with a gap and a neighbour, and all of them fail the same way.