The stair that empties three floors at a time

The column under the escape corridor took the imposed load of an office building as the load that stays and the load that visits had drawn it — a sustained part that stays until the tenancy changes, and visits that come and go floor by floor — and added what a corridor carries: a crowd at every evacuation, about one a year, on every floor’s corridor at once, because an evacuation empties every floor down the same stairs. A column whose tributary area was all office needed 0.26 of its twenty floors’ summed characteristic loads. A quarter corridor, 0.49. Half corridor — the column at a stair core — 0.90, more than the 0.73 the storey rule would have let it take.

That essay emptied every floor at once. Tall buildings are often evacuated otherwise. A phased evacuation sends the floor where the alarm started and the floors either side down the stairs first, and the rest only if the fire spreads, precisely so that the stairs are never asked to carry every floor’s occupants together. The question it left was whether the column under such a stair earns back the reduction a simultaneous evacuation took from it — and whether the full evacuation the phased strategy exists to avoid is, in the end, the event that sets the column’s load after all.

A phase is three floors in twenty #

The model keeps everything from the earlier essay and changes one thing. The column’s tributary area is still the rectangle the column given more than its rectangle warned is only a first estimate, and the office load on it is unchanged. Each alarm starts on a floor drawn evenly from the building’s twenty, and empties that floor and the floor above and below. The column carries the top n floors; a phase loads the corridors of whichever of its floors fall in the three, and a phase near the bottom of the building loads none of a short column’s floors at all. The crowd’s density is the earlier essay’s: a mean of 1.0 kN/m² with a coefficient of variation of a half, drawn once per alarm and laid on every corridor the alarm empties.

Each figure is 4,000 simulated fifty-year lives at each point. A column’s factor is the 95th percentile of its floors’ largest total over n times the 95th percentile of one floor’s — the reduction the averaging pays for, at the same fractile.

The dashed line is the earlier essay’s stair-core column, rising to 0.90 at twenty floors because every floor’s corridor peaks in the same hour. Phased, the same column needs 0.29 — barely above the office column’s 0.26 — because a phase never loads more than three of its floors at once, and three in twenty is close enough to one in twenty for the corridor crowd to behave like one more independent visit. A stair evacuated in phases gives its column back the whole of the reduction that a simultaneous evacuation took away.

The fourth line is the same phased stair with one alarm in fifty escalating to a full evacuation. It runs with the phased line for short columns and leaves it as the column lengthens: at twenty floors it needs 0.67. Escalating two alarms in a hundred has taken back two-thirds of what phasing gave.

How many floors a phase can empty #

The size of the phase is the fire strategy’s choice, and the column’s factor follows it smoothly.

For the quarter-corridor column the phase hardly matters until it reaches several floors: one floor or three both give 0.26, the office column’s value, because a quarter of a single floor’s crowd is small beside the column’s other loads. For the stair-core column the factor climbs steadily with the phase, from 0.25 for one floor at a time through 0.29 for three to 0.54 for ten. The largest phase stays below the dotted line because a phase near the top or bottom of the building empties fewer floors than its nominal size: a ten-floor phase starting on the top floor loads six.

None of this is surprising. It is the earlier essays’ independence argument with a block size: the floors that fill up together found that a tenancy of k floors made the column’s reduction behave as if it carried n/k blocks, and a phase of three is a block of three that moves around. What is surprising is what happens when the blocks are occasionally the whole building.

One alarm in a few hundred #

The horizontal axis is logarithmic, and it has to be. For the stair-core column, one alarm in five hundred escalating — 0.2 per cent — takes the factor from 0.29 to 0.39. One in a hundred, to 0.59. One in twenty, to 0.75, past the storey rule. Escalating every alarm returns the earlier essay’s 0.90. The quarter-corridor column follows the same shape at a third of the height, because a quarter of a crowd is a smaller share of its load.

The column’s load is set by the rare full evacuation, not by the plan. A fire strategy that escalates one alarm in a hundred is a phased strategy working as intended — almost every alarm stays on its three floors, and the stairs are spared — but the column’s characteristic load is two-thirds of the way back to what a simultaneous strategy would have demanded.

Why so rare an event sets the fractile #

A characteristic value is a fractile of the largest load in a life, and the life is fifty years long — the definition that the envelope is not a structure found it necessary to keep separate from any single arrangement of the load. With one alarm a year there are fifty alarms in it. If each escalates with probability p, the chance that a life contains at least one full evacuation is $1 - (1-p)^{50}$: 64 per cent at p = 0.02, 39 per cent at 0.01, 9.5 per cent at 0.002, and 4.9 per cent at 0.001.

The light histogram is the phased stair alone, every life alike: a narrow peak at about 0.45 kN/m² averaged over the twenty floors, with its 95th percentile at 0.53. The solid one is the same stair with 2 per cent of alarms escalating, and it has split in two. A third of its lives never have a full evacuation and sit in the same narrow peak; the rest do, and they spread out to the right, each placed by how dense the crowd was on the one day the building emptied. The 95th percentile, 1.22, lies in the right-hand tail.

That is the mechanism, and it gives a threshold. Once more than one life in twenty contains a full evacuation, the 95th percentile is a full evacuation’s, and the column’s characteristic is drawn from the event a phased strategy is designed to avoid. At fifty alarms a life, that happens above about one escalation in a thousand alarms. Below it the column is a phased column; above it the column is a simultaneous column whose full evacuation merely comes less often, and the factor climbs towards 0.90 as the chance grows.

The same arithmetic says why it climbs so slowly near the top. The 95th percentile of all lives, when 64 per cent of them contain a full evacuation, is about the 92nd percentile of those lives alone — and when every life contains several, it is the 95th percentile of the largest of several crowds. More full evacuations make the column’s characteristic a slightly worse crowd, not a different kind of load.

What it does to the storey rule #

The storey rule — two floors at their characteristic and the rest at $\psi_0$ = 0.7 — gives 0.73 at twenty floors, and the earlier essay found the stair-core column exceeding it under a simultaneous evacuation in 39 per cent of lives. Phasing removes that, until escalation brings it back.

With no escalation no life exceeds the rule: phased, the stair-core column is comfortably inside it. At one alarm in a hundred escalating, 1.5 per cent of lives exceed it; at one in twenty, 6.1 per cent, more than the twentieth a characteristic value allows; at one in five, 16 per cent. The rule is then unsafe for this column by the same measure the earlier essay used, and for the same reason — a load that reaches every floor together — arriving through a door the fire strategy left open on purpose.

The quarter-corridor column never reaches the rule in any of these lives. Its crowd is a quarter of its tributary area, and even a full evacuation cannot lift it past 0.49. The stair-core column, with half its tributary area corridor, is the one that depends on how often the building empties.

An alarm every five years #

Everything so far assumes one alarm a year, which is a busy building: kitchens, contractors, a detector that trips in a dusty winter. A quiet office might see one every five years. The life’s chance of a full evacuation depends on how many alarms it holds, so the same escalation rate means much less.

With ten alarms in a life instead of fifty, the curves shift a factor of five to the right. One escalation in five hundred now does nothing to the stair-core column — 0.29 against 0.28 — and one in a hundred takes it to 0.41 rather than 0.59; one in twenty gives 0.62, still under the rule. Even every alarm in full gives 0.85 rather than 0.90, because ten crowds in a life have a smaller largest one than fifty.

The quantity that sets the column is the number of full evacuations in a life, not the share of alarms that become one. The threshold falls where the expected number of full evacuations in fifty years is about a twentieth: one escalation in a thousand alarms at one alarm a year, one in two hundred at one every five years. A drill run as a full evacuation every year puts fifty in a life, whatever the alarm policy is.

A fire strategy is a structural assumption #

A phased evacuation strategy is written by a fire engineer for the people in the building, and it is right to be: three floors at a time down a stair sized for three floors is safer than twenty floors at once down a stair sized for everyone. The strategy also contains, without saying so, a statement about loads. It says how often the building empties at once, and that number is the input that sets the stair-core column’s characteristic load.

The trouble is that the number is not under the fire strategy’s control. A phased alarm escalates when the fire spreads, when a second detector trips on another floor, when smoke reaches the stair, when the fire service on arrival decides to clear the building, and in drills that the building’s management chooses to run as full evacuations. Some of those are rare. Some are habitual — a building that runs one full drill a year has a full evacuation in every year of its life, and its stair column is a simultaneous column whatever the alarm policy says. A crowd that behaves differently because of the structure is a familiar trap — the bridge that was pushed by its own sway is the famous one — and a crowd that behaves differently because of a procedure is the same trap one step further from the drawings. And none of them appears in a structural calculation unless someone asks.

The column’s designer can therefore take one of two positions. Assume nothing about the fire strategy and design for the simultaneous evacuation, which is the earlier essay’s 0.90 for a half-corridor column. Or take the phased reduction and make the assumption explicit — that full evacuations, drills included, are rarer than one alarm in a few hundred — and write it into the building’s operating constraints, the same way a floor’s imposed load is written onto a notice. The visits that come to every floor at once found the same choice for an office whose tenants are synchronised by their use; here the synchronising event is the building’s own procedure.

It helps that the decision is a small one. The quarter-corridor column barely moves whatever the strategy — 0.26 phased, 0.49 simultaneous, and never past the rule — and the office columns not at all. Only the columns at the stair cores, with half or more of their tributary area in corridor and landing, carry a load that the fire strategy decides, and there are two or four of them on a typical floor plate. Designing those few for a simultaneous evacuation costs a little concrete in a handful of columns; taking the phased reduction in them costs an operating constraint that has to survive every change of management for fifty years. The first is cheap and needs no one to remember anything, which is usually the better kind of conservatism.

By hand, for the threshold #

The rule of thumb can be checked without the simulation. For a column with a share $c$ of its tributary area corridor, carrying n floors, the full evacuation puts $c \cdot \bar q_{crowd}$ on every floor; the phase puts it on three. The column’s 95th-percentile load per floor, with a full evacuation in the life, is roughly the office’s own plus $c$ times the crowd’s density at the percentile the full-evacuation lives require. At $c = 0.5$, a crowd with mean 1.0 kN/m² and a coefficient of variation of a half has its 90th percentile near 1.7 kN/m², so the column’s corridor share contributes about 0.85 kN/m² per floor, on top of an office characteristic per floor near 0.4: about 1.25 kN/m², against the simulated 1.22.

Phased alone, three floors of crowd among twenty contribute at most $3/20 \times 0.5 \times 2.5 \approx 0.19$ kN/m² per floor even for the largest crowd of a life, so the column sits at its office load plus a little: the simulated 0.53. The difference between those two numbers is the difference between 0.29 and 0.67, and the only thing that moves the column from one to the other is whether a full evacuation happened in its life.

Where the simulation stops #

The crowd is the same on every corridor an alarm empties, and it stands still. A descending crowd on a stair landing is also a dynamic load, which the crowd that is also the structure treated separately; here it is counted only by its weight. In a phased evacuation the stair carries only three floors’ occupants, so its flow is faster and the corridors queue less than in a full evacuation, when every floor waits its turn at the stair door. That would make the phased crowd lighter than the full one, and the full evacuation’s dominance stronger than the figures show.

The alarm floor is equally likely anywhere. A building whose kitchens, plant rooms or laboratories are concentrated on some floors has its alarms concentrated there too, and a column carrying those floors sees more phases than the even distribution gives it. And the column carries the top n floors of a twenty-floor building; in a taller building a column carrying twenty floors would sit lower down, and see phases from above and below it.

The life is fifty years. A characteristic value is the fractile of a life’s largest load, and every number here would move with the length of the life — a longer one holds more alarms and more chances to empty the building. That dependence is not peculiar to this column; the load that is never all there at once found every fractile of an imposed load decided by the life it is a fractile over.

Still open: the stair that is also the refuge #

Every crowd here leaves the building. Tall buildings increasingly hold some of their occupants instead — refuge floors where people wait, protected lobbies at each stair where those who cannot use the stair wait for rescue, and stair landings designed for a wheelchair space beside the flow. A refuge floor is loaded at evacuation density for as long as the evacuation lasts, which in a phased strategy can be hours, and it is loaded every time the building empties, whichever floor the alarm began on. Whether a column under a refuge floor carries a load that is phased like the stair’s or simultaneous like the earlier essay’s — or worse than both, because every phase delivers its occupants to the same floor — is the question that a building which keeps its people inside puts to the columns under the place it keeps them.