The column under the escape corridor

The visits that come to every floor at once took the office imposed load that the load that stays and the load that visits had split into a sustained part and a few intermittent visits, and let a share of those visits be made by the whole building at once. Nothing changed on any one floor, and the column carrying twenty floors lost most of its reduction: a tenth of the visits shared took the factor on its summed loads from 0.26 to 0.51, and a third used up the whole of what the storey rule allows. That essay named the plainest case of a shared visit and then left it, because it does not belong to a building’s use but to a part of its plan. An evacuation puts every occupant into the corridors and stairs at once. The columns carrying those routes are loaded by every floor together, by construction.

What that essay did not ask is what a column carries when only part of its tributary area is a route. A column under the corridor’s edge carries some office and some corridor; a column in the stair core carries almost nothing else; the perimeter column beside them carries none. All three carry the same number of floors and get the same reduction. This essay asks whether they should.

Office load and the crowd in the corridor #

The office part of every floor is loaded exactly as in the two earlier essays, so that whatever changes here is the corridor and nothing else. A sustained load — furniture, partitions, a tenant’s settled use — has a mean of 0.5 kN/m² and is drawn afresh when the tenancy changes, every five years on average; on top of it, visits — a party, a move, a delivery of paper — arrive on each floor about once every three years, last three days, and have a mean of 0.2 kN/m² with a long tail. Each floor’s history is its own.

The corridor part carries none of that. Escape corridors and stair landings hold no furniture and no tenant’s filing; nobody holds a party in them. What they carry is a crowd, at every evacuation, and the crowd is there on every floor at once because every floor empties down the same stairs at the same time. The crowd’s load is drawn once per evacuation, from a distribution with a mean of 1.0 kN/m² — about one and a third people per square metre, a corridor that is full and moving — and a coefficient of variation of a half, so that the worst of fifty evacuations is typically a dense, slow-moving crowd of about 2.4 kN/m². It lasts about an hour, and there is one evacuation a year, drills included.

A column’s tributary area is a share corridor and the rest office, and both parts are carried on every floor the column supports. Each of 4,000 fifty-year lives is simulated with every floor’s sustained load, every visit and every evacuation in time, and the largest total on n floors is recorded. The column’s factor is the 95th percentile of that largest over n times the 95th percentile of one floor’s largest — the reduction the averaging earns at the same fractile, as in the earlier essays.

Why the reduction depends on the plan #

The hero figure is five columns carrying the same floors. The office column, with no corridor, is the earlier essays’ column: its factor falls quickly with the number of floors, to 0.26 at twenty, because the twenty floors’ worst visits fall on different days and the column sees their average. A column whose area is all corridor hardly falls at all, to 0.98, because every crowd it carries is on every floor and twenty floors of crowd are twenty times one floor’s. The columns between them divide according to their share of corridor: a tenth, 0.30; a quarter, 0.49; a half, 0.90.

The next figure shows where those numbers come from, in one life.

The office column’s twenty-floor average wanders slowly as tenancies change and barely registers any one visit; its fifty-year largest is 0.85 kN/m² per floor. The half-corridor column carries half as much of that wandering, and on top of it a spike at every evacuation that reaches every floor together. None of those spikes averages out: the twenty floors’ average at an evacuation is half the crowd’s load, whatever it was that day, and the largest of fifty-six of them is 1.46 kN/m² per floor. The corridor makes the column’s load a single event repeated on every floor, and averaging over floors does nothing to a single event.

The one-floor characteristic does something less expected. Adding corridor to the area makes one floor’s characteristic smaller, not larger: from 3.20 kN/m² with no corridor to 2.45 with a quarter and 2.09 with a half. A floor’s worst moment is either a heavy office visit or a dense crowd, rarely both, and a floor whose area is half each never sees either at full size over its whole area. The column’s twenty-floor load meanwhile rises, because the crowds add on it and the visits did not. The factor is the ratio of the two, and it is pushed up from both ends.

How much corridor uses up the reduction #

The storey rule allows a column carrying twenty floors a factor of 0.73 on its summed characteristic loads. The next figure asks how much corridor a column’s area can hold before it needs more.

A little corridor costs almost nothing. With a tenth of the area corridor the factor moves from 0.26 to 0.30, because the crowd is a small part of one floor’s characteristic and the office visits still dominate it. From about a fifth the curve steepens, as the crowds start to set the column’s worst day, and at about 38 per cent corridor the factor reaches the rule’s 0.73. Beyond it the rule is no longer conservative by the measure the earlier essays used, and at half corridor the rule’s load is exceeded in 39 per cent of fifty-year lives, where a characteristic value is meant to be exceeded in one life in twenty.

The curve is a reading of a share of the plan, not of the building. A building’s use, its number of floors and its occupancy class are the same for every column in it, and the rule reads only those. What changes between columns is how much of the floor each one gathers is route rather than room, which is a fact about the plan’s geometry that the rule never asks for.

Three columns on one floor #

A double-loaded office floor makes the point concrete.

Office bays 6 m deep run either side of a 2.4 m corridor along the building’s spine, with columns on four lines every 7.2 m. A perimeter column gathers half an office bay and no corridor, and its factor is the office column’s 0.26. A column on the corridor’s edge gathers half an office bay and half the corridor, 29 per cent of its area, and its factor is 0.56. A column at the stair’s corner, its area all landing and flight, has a factor of 0.98. The three carry identical floors and the rule gives them all 0.73: nearly three times what the perimeter column needs, a third more than the corridor-edge column needs, and a quarter less than the core column does.

The column given more than its rectangle found that the area a column actually gathers is not the rectangle drawn round it, because continuity moves load towards the interior. On the corridor’s edge that works slightly in the column’s favour. A slab continuous over 6 m office bays and a 2.4 m corridor hands the corridor-edge column about five-eighths of the office bay rather than half, while the short corridor span gives it roughly its half and no more. The corridor’s share of what the column actually carries is then nearer a quarter than 29 per cent, and its factor about 0.48 rather than 0.56 — still nearly twice the perimeter column’s, and still a reading of the plan that the rule does not make.

Dense crowds matter more than frequent ones #

Two numbers in the corridor’s load are uncertain, and the next figure varies the more important.

A crowd’s load depends on its density, and density in an evacuation depends on how the building empties: a corridor that feeds a stair faster than the stair can take it fills to a standing crowd, and a corridor that never queues stays a stream. At a quarter corridor the factor is 0.34 with a thin crowd of 0.5 kN/m², 0.49 with 1, and 0.77 with a dense one of 2 — past the rule. At half corridor even the thin crowd gives 0.64, close to the rule, and anything denser takes the column to almost no reduction at all.

How often the building is evacuated matters much less. With a quarter of the area corridor and a crowd of 1 kN/m², one evacuation a decade gives a factor of 0.41, one a year 0.49 and one a month 0.54. More evacuations make the worst one worse, but they make it worse on one floor and on the column together, because every one of them is on every floor; the factor moves only through the office part, which the extra evacuations do not touch. A fire drill programme does not change how much reduction a corridor column has; the stair’s width, which sets how densely the corridor queues, does.

Against the loads a designer writes down #

The factor is a ratio, measured against the column’s own characteristic, and it is what the earlier essays used. A designer does not compute that characteristic. They take a nominal load for the occupancy — 3.0 kN/m² for offices is common, with corridors in some national rules at 4.0 — and multiply its sum over the floors by the rule’s factor. The last figure asks what that leaves.

The nominal load is generous for a mixed floor. One floor’s characteristic, the dotted line, dips to 2.09 kN/m² at about half corridor, far below 3.0, for the reason the history gave: a floor half office and half corridor never sees either part’s worst over its whole area. So the rule’s reduced load, $0.73 \times 3.0 = 2.19$ kN/m² per floor, stands well above the twenty floors’ real characteristic for every column up to about 60 per cent corridor: 0.85 for an office column, 1.32 for the corridor-edge column of the plan, 1.89 at half corridor. The margin that the factor says has gone is supplied by the nominal load’s surplus over a mixed floor’s real one.

Beyond about 60 per cent the twenty floors’ characteristic passes the reduced nominal load, and an all-corridor column reaches 3.26 kN/m² per floor against 2.19 — half as much again. With corridors loaded at 4.0 kN/m² the crossing moves to about 83 per cent, and the all-corridor column still exceeds the reduced load, 3.26 against 2.92. The columns that run out are the ones whose areas are mostly route: stair cores, landings, the columns of a lift lobby that every floor’s escape passes through. In the plan drawn, that is the column at the stair’s corner and no other.

So the storey rule’s error and its danger are in different places. It is most wrong for the perimeter column, which it over-designs threefold, and that is merely waste. It is unsafe only for the core column, whose factor it understates by a quarter and whose nominal load does not make up the difference, and those are few, heavily loaded and, in most buildings, already among the columns most carefully designed.

A reduction read from the plan #

The load that is never all there at once found the storey rule’s justification in averaging: n floors that are loaded independently carry, at the same fractile, much less than n times one floor’s worst, because their worst days do not coincide. Every result above is that reasoning applied honestly to a mixed area. The office part of a column’s area averages over floors as the rule assumes; the corridor part does not average at all. A reduction that respected both would be computed separately for each, the office part reduced by the number of floors and the corridor part not reduced, and the two added.

For the plan drawn, that is a small change in practice. The corridor-edge column’s area is 29 per cent corridor; reducing the remaining 71 per cent by the rule’s 0.73 and the corridor by nothing gives an effective factor of $0.71 \times 0.73 + 0.29 = 0.81$, which is larger than its simulated 0.56 and so safe by a wide margin. The stair-core column gets a factor of one, which is what the simulation says it needs. The perimeter column keeps the rule’s 0.73, which is still nearly three times its need, because the rule was written to be safe for the office column in every building and not to be economical for any one of them.

That is the same kind of split that the load a beam is given is a decision found in a beam’s line load: the tributary area is not a property of the column but a choice about what part of the floor is assumed to reach it, and once the area is chosen, what it contains is a second choice made silently. Most design offices already draw their columns’ tributary areas. Few of them write down how much of each is route.

The American loading standard comes closest by another door: it refuses live-load reduction outright for assembly occupancies, recognising by category that a gathering crowd does not average. The corridor column is an assembly column inside a building of another category, and the column is identified by its plan, not by the building’s name. The crowd that pushed a bridge by its own sway was synchronised by the structure’s movement; the crowd in an evacuation is synchronised by the alarm, and its synchrony is just as complete.

The core column by hand #

An all-corridor column carries crowds and nothing else, and every crowd is on every floor, so its twenty-floor load is exactly twenty times one floor’s at every instant. The column’s characteristic is then twenty times one floor’s, and its factor is one. The simulation’s 0.98 is one less a sampling wobble: the two characteristics are 95th percentiles of the same fifty-year largest crowd, read from two different sets of 4,000 lives.

What that largest crowd is can be bracketed by hand. Fifty evacuations, each a gamma variable of mean 1.0 kN/m² and coefficient of variation 0.5: the largest of fifty such draws has a median near 2.4 kN/m², and its 95th percentile over many lives is about 3.2 — the simulation’s characteristic of 3.32 with the office’s nothing and the evacuations’ own randomness in number added. Against the rule’s reduced nominal load of 2.19 kN/m², twenty floors of corridor carry half as much again as the column was sized for, and no choice of a reasonable nominal corridor load short of about 4.5 kN/m² closes the gap once the storey factor has been applied to it.

One crowd on every floor #

The crowd is the same on every floor. In an evacuation the floors nearest the stair’s discharge empty first, and the upper floors’ corridors may queue longer and more densely than the lower ones’; a column under a stair carries the stair’s own load, which grows towards the bottom as the descending flow accumulates. Both make a core column’s load more than the model’s, not less.

The office and the corridor are independent. In an evacuation the offices empty into the corridors, so the office part of the floor is at its lightest at the moment the corridor is at its heaviest. The model adds the sustained office load at its current value, which is right for furniture, and ignores the people who left it, who are a small part of an office’s characteristic.

The corridor carries nothing else. A corridor used for moving furniture, a trolley route in a hospital, or a lobby where people wait for lifts carries visits of its own, independent between floors, which would lower the corridor column’s factor slightly. And the crowd that is also the structure is the dynamic case that this static load leaves aside: a crowd walking in step on a stair landing loads it at its own frequency, which a column carries as a static sum only if the landing is stiff.

Still open: the floors that are evacuated in phases #

Everything here empties every floor at once, which is how most buildings below a certain height are evacuated. Tall buildings are often evacuated in phases instead — the floor with the alarm and the floors either side first, the rest only if the fire spreads — precisely so that the stairs are not asked to carry every floor’s occupants together. A phased evacuation loads three floors’ corridors at a time, not twenty, and moves the corridor column back from a shared load towards an independent one, in steps of three. Whether a column under a phased-evacuation stair earns back the reduction a simultaneous one loses — and whether the rare full evacuation, the one a phased strategy exists to avoid, is then the event that sets its characteristic after all — is the question that the floors that fill up together asked of tenancies and that a building’s fire strategy now asks of its columns.