The visits that come to every floor at once

The load that stays and the load that visits split an office floor’s imposed load into a sustained part, which arrives with a tenant and stays until the next, and intermittent visits — a party, a move, a crowd — that load one floor for a few days. Simulated over fifty years, a floor’s characteristic load turned out to be set mostly by its worst visit, and because visits to different floors almost never coincide, a column carrying twenty floors let one by one needed only 0.27 of their summed characteristic loads, against the storey rule’s 0.73.

That result rests on one assumption the essay named and did not test: the visits are independent between floors. Each floor’s party is its own. Some buildings are not like that. An office building during a fire drill, a school at the end of the day, a conference centre between sessions, a stadium’s concourses at half-time: in each, the event that sets the floor’s worst day reaches every floor together. This essay keeps everything else and lets a share of the visits be made by the whole building at once.

The same building, its visits made together #

The model is the earlier one, unchanged. Each floor’s sustained load is a gamma variable with a mean of 0.5 kN/m² and a coefficient of variation of 0.7, renewed when its tenancy changes, every five years on average. Visits arrive on each floor 0.3 times a year and last three days, each a gamma variable with a mean of 0.2 kN/m² and a coefficient of variation of 2 — most visits are trivial, a few are heavy. Over a fifty-year life the largest total a column carries is recorded, and 4,000 lives give its 95th percentile, the characteristic value — a fractile of a scatter, with all the caution the strength no specimen had attached to reading one.

The one change is the share. A proportion of each floor’s visits is not the floor’s own: it arrives on every floor at the same moment, with the same size per square metre on each. The rate and the size distribution are untouched, so a floor that is shared a quarter of its visits still has 0.3 visits a year, of the same sizes, and lasting as long. What changes is only which visits happen together.

Over twenty floors, the floors’ own visits average out: a heavy party on one floor is a twentieth of the column’s load, and the average per floor wanders by a few tenths of a kilonewton per square metre as tenants come and go. A shared visit does not average out. It arrives on all twenty floors at once, so the average carries the whole of it, and in this life one shared visit more than doubles the largest average load the column ever sees.

One floor cannot tell #

The first thing to establish is that the share is invisible from any one floor, because that is what makes it dangerous.

One floor’s curves are the same, because a shared visit is still, from the floor’s point of view, one visit of the usual size at the usual rate. A load survey that recorded each floor of a building separately, and estimated a characteristic floor load from it, would find the same 3.2 kN/m² whether the visits were shared or not. The design load per square metre in a code is exactly that kind of number, a single floor’s characteristic.

The twenty-floor average does not lie on its curve. With a quarter of the visits shared, the average’s characteristic value is two and a half times what it is with none. Everything a column’s reduction depends on is in the relation between the floors’ histories, and none of it is in any single floor’s.

The reduction stops at a few floors #

The column’s factor is the ratio the reduction rules are written in: the characteristic load on n floors over n times one floor’s characteristic.

With independent visits the factor keeps falling as floors are added, to 0.26 at twenty, because each floor added brings a sustained load that is mostly average and a visit that almost never coincides with another floor’s. With a share of the visits made together, the factor falls for the first few floors and then stops. Above four or five floors, adding floors adds sustained load, which averages, and shared visits, which do not; the shared visits dominate, and the factor levels off at a value set by the share. A tenth of the visits shared holds the factor at about half; a quarter holds it at about 0.7; all of them hold it at nearly one.

The storey rule is a curve of a different shape. It falls with the number of floors, from 1 at two floors towards its limit of 0.7, and it falls slowly. It sits above the independent-visit curve by a factor of nearly three and below the all-shared curve at every count. The rule is not calibrated on any particular share; it is calibrated on an assumption about how floors add, and the share is what decides whether that assumption holds.

A quarter of the visits uses up the reduction #

The factor at twenty floors, against the share, is the figure a designer needs.

The factor rises steeply at first. A tenth of the visits shared doubles it, from 0.26 to 0.51; a quarter takes it to 0.68; at about a third it reaches the storey rule’s 0.73, and beyond that the rule is unsafe. For a building let to a single tenant, whose sustained loads are already correlated, the rule is reached at about a fifth, and with every visit shared the tenancy hardly matters — 0.96 and 0.97 — because the shared visits are now the whole of what the column sees.

The earlier essay found that letting a building to one tenant correlates its sustained loads and raises the factor from 0.27 to 0.60, still well inside the rule. Synchronised visits do something the tenancy cannot: they correlate the part of the load that sets the floor’s characteristic. A tenancy changes the sustained part, which for an office is the smaller part of the characteristic; a shared visit changes the visiting part, which is most of it.

How often the rule is exceeded #

A factor above the rule’s means the rule’s load is exceeded more often than intended, and the intended rate is one life in twenty.

With independent visits the rule is almost never exceeded; it is conservative by a wide margin, which is the earlier essay’s finding seen from the other side. With a quarter of the visits shared it is exceeded in about one life in twenty-five let singly and one in sixteen let to one tenant — at or past the intended rate. With every visit shared it is exceeded in nearly a fifth of lives. The column of such a building, designed by the rule, has roughly three times the intended chance of seeing more than its design imposed load in its life.

That is not a collapse. The design load is factored, the materials have their own margins, and the imposed load is one part of a column’s load among several. But the rule’s reduction is a claim about how often the summed load reaches a value, and in a building whose visits are synchronised the claim is not true.

Shared visits matter most where the load mostly visits #

The last figure asks which buildings the share matters in, by varying the sustained load the floors carry.

The tenancy effect in the earlier essay mattered most where the sustained load was large — archives and libraries, whose settled use dominates the floor’s characteristic. Synchronised visits are the mirror image: they matter most where the sustained load is small and the floor’s characteristic is set by its visits. At a sustained mean of a quarter of a kilonewton per square metre, the share nearly quadruples the factor. At two kilonewtons, where the visits are a small part of the floor’s worst day, it raises the factor by a quarter.

The buildings in which visits are both large and synchronised are the ones whose use is gathering: schools, conference and exhibition buildings, places of worship, stadiums, transport interchanges. Their sustained loads are light — chairs, tables, a few fittings — and their characteristic loads are crowds. They are exactly the buildings where a reduction based on independent floors is least justified.

Why the share and not the size #

It would have been easy to make the point a different way, by giving the building-wide events larger loads than ordinary visits — a crowd is heavier than a party — and showing that the column then carries more. That would be true and would prove nothing about reduction, because it would also raise each floor’s characteristic load, and a column designed on the higher floor load with the same reduction would be as safe as before.

Keeping the size and the rate the same isolates what reduction is about. A floor’s history is a sequence of visits whose sizes and timing are statistically the same in every version of the building drawn here. The only thing the share changes is the joint timing across floors, and the only quantity that responds to joint timing is the sum. A design method that starts from a single floor’s characteristic load and applies a factor for the number of floors is applying a factor that encodes an assumption about joint timing, and it has no input through which a building’s use could tell it otherwise.

That is also why the effect is so steep at first. A shared visit’s contribution to the column is twenty times its size, while an own visit’s is one — the same arithmetic as the envelope that is not a structure, where the worst case is the arrangement that puts every load where it does most harm at once — so even a small share puts most of the column’s worst day into the shared visits. A tenth of the visits shared is one or two shared visits in a fifty-year life; each of them arrives on the column twenty times over, and the larger of them is already worth as much as the whole of the averaged remainder. The factor doubles.

What a designer can do with it #

Three responses follow, in increasing order of effort. The first is the one the American standard takes: deny the reduction to any column that carries floors whose use synchronises crowds, whatever the building is called. It is safe and it is blunt, and it costs most in the buildings whose floors are mixed.

The second is to split the load as the model does and reduce only what averages. Carry the shared visits unreduced — the column takes $n$ times a shared visit’s characteristic — and reduce the sustained part and the floors’ own visits as the earlier essay’s independent case allows. For twenty floors with a quarter of the visits shared, the shared visits’ characteristic per floor is about 1.7 kN/m², so they bring 34 kN/m² of floor-equivalent; the sustained loads bring about 15.5. The sum, 49.5 against 64, is a factor of 0.77, above the simulation’s 0.68 and therefore safe, because it adds two characteristic values that rarely arrive together.

The third is to ask, for a particular building, what share of its floors’ worst days are shared. That is a question about the building’s use that an owner can usually answer and a designer seldom asks, and its answer moves the column’s design load by a factor of nearly four between the extremes.

What the codes already do, and what they leave #

The American loading standard does not permit live load reduction in assembly occupancies at all, and limits it for heavy loads and for garages; the European standard allows its storey reduction for most categories and leaves the details to each country. Both, in effect, recognise the assembly case by category: the reduction is denied to a building whose use is gathering.

The simulation says the category is the right instinct and the wrong test. What the reduction depends on is the share of the floors’ worst visits that come together, and that is a property of how a building is used rather than of what it is called. An office building with a large atrium where all-staff meetings gather, a hotel’s conference floors stacked over its bedrooms, a school building classified as offices because it is let to an education trust, a residential tower whose every resident gathers in the lobby during a fire alarm — each may carry visits that are synchronised for part of the time and loaded by a column whose reduction assumes they never are.

The fire drill is the purest case, and the crowd that pushed a bridge by its own sway is its dynamic cousin — a crowd synchronised by the structure rather than by a timetable. The drill is worth one sentence more: an evacuation puts every occupant of the building into the stair cores and corridors at once, and for the columns and beams carrying those routes the visits are shared by construction.

The twenty-floor factor by hand #

The plateaux can be bracketed without a simulation. With all visits shared, the column’s visiting load on twenty floors is twenty times one floor’s, exactly, because every visit is on every floor. The largest of a floor’s fifteen or so visits in fifty years is about 2.6 kN/m² at the 95th percentile, so the shared visits alone bring $20 \times 2.6 = 52$ kN/m² of floor-equivalent; the twenty independent tenancies, as the earlier essay found, bring about 15.5 at their own 95th percentile. Added, $52 + 15.5 = 67.5$ against a summed characteristic of $20 \times 3.2 = 64$ is a factor of 1.05 — an upper bound, because the worst visit and the heaviest tenancies rarely coincide. The simulation’s 0.96 sits just under it.

With a quarter shared, the shared visits form their own process at a quarter of the rate. Over fifty years each floor has about fifteen visits, of which about four are shared; the largest of four draws from the long-tailed visit distribution, at the 95th percentile, is about 1.7 kN/m², so the column carries about $20 \times 1.7 = 34$ from shared visits, plus about 10 from the sustained loads and a little from the largest own visit: about 45 against 64, a factor near 0.7. The rule’s 0.73 is crossed when the shared visits’ largest, multiplied by twenty, approaches three quarters of the summed characteristic, which happens at around a third of the visits shared.

What the simulation leaves out #

A shared visit here has the same size on every floor, the same duration, and the same distribution as a floor’s own visits. Real synchronised events are probably larger than typical visits — a fire drill fills a floor’s escape routes to a density no party does — and they are concentrated on parts of a floor rather than spread over it, which affects the columns and beams carrying those parts more than the rest. The share is a single number for the whole building; a real building may have events that reach some floors and not others, which lies between the cases drawn. The parameters are of the order load surveys give for offices, and as in the earlier essay the conclusions are in the comparisons, not in the third digit; a strength read from a short test programme carries the same caution about a tail that was never sampled.

The floors that fill up together treated correlation between floors as one number applied to one load. The split model locates it: a tenancy correlates the load that stays, and a building’s use correlates the load that visits. The load that is never all there at once began this line of argument with the observation that the reduction is a statement about averaging; whether the averaging happens is decided by when the visits come, not by how many floors there are.

Still open: the floor that carries the corridor #

Every load here is spread over a floor’s whole area. A column under a building’s escape corridors and stair landings carries a part of each floor that is loaded by every evacuation and by little else, while the column beside it carries offices. Whether the reduction a column is allowed should depend on what its tributary area contains — corridor or office, route or destination — rather than on how many floors it carries, and whether the columns of an escape core are, in effect, assembly columns in an office building, is the question synchronised visits put to the load path drawn through a building’s plan.