The load that stays and the load that visits

The floors that fill up together questioned the independence behind live load reduction. A column carrying twenty floors may be designed for less than the sum of their characteristic imposed loads because the floors do not reach their worst day together, and that argument needs their loads to be independent. Give any two floors a correlation and the reduction the averaging can pay for shrinks; at a correlation of 0.16 a very tall column needs exactly the 0.7 the storey rule of EN 1991-1-1 falls to. One tenant on five floors makes twenty floors behave, statistically, like four.

Every load there was one number with one spread. The essay ended on the observation that imposed load is not one number. Load surveys of offices have long described it as two: a sustained load — furniture, files, partitions, the settled density of a tenant’s use — that arrives with a tenancy and stays until the tenancy changes, and intermittent loads — a party, a move, a training day, a crowd at a window — that arrive on one floor for a day or three and go. The sustained part is what a tenancy ties floors together through. The intermittent part has no reason to. The two should be reduced by different amounts, and the question was whether splitting them confirms the rule, or shows it to be generous where tenancies are largest.

Two loads, followed for fifty years #

The load model is the simplest that keeps the two parts distinct. Each floor’s sustained load is drawn afresh whenever its tenancy changes, from a distribution with a mean of 0.5 kN/m² and a coefficient of variation of 0.7, and every floor of one tenancy carries the same draw. Tenancies change on average every five years, independently of one another. Each floor also receives intermittent loads, 0.3 times a year on average, each lasting three days and each drawn from a distribution with a mean of 0.2 kN/m² and a coefficient of variation of 2, so that most visits are trivial and a few are large. Visits that overlap in time on different floors add.

Those proportions are typical of the office load surveys behind probabilistic load models, and one check says they are not far from the truth: the largest load a single floor sees in fifty years, at the 95th percentile, comes out at 3.24 kN/m². That is the range offices are designed for. A sustained load whose mean is a sixth of that is not a mistake. It is the reason imposed loads are reduced at all.

Each of 3,000 seeded fifty-year lives is followed for one floor and for columns of up to twenty floors, and the column’s factor is the 95th percentile of the floors’ largest total over the sum of their individual 95th percentiles: the same fractile on both sides.

A single floor’s load in one life is a staircase with spikes on it. The steps are its tenancies, each a level held for a few years; the spikes are visits, mostly small, occasionally doubling the floor’s load for a few days. The floor’s largest load over fifty years is whichever is larger: its heaviest tenancy, or a large visit on top of whatever tenancy it came during. In this life it was a tenancy; across 3,000 lives the 95th percentile of one floor’s worst day is set mostly by visits.

The average of twenty floors is a different object. Its steps are smaller, because twenty tenancies are never all heavy at once, and its spikes are a twentieth of a single floor’s, because a visit is to one floor and is shared, on the column, among twenty. One floor’s worst days are its visitors; the column’s are its tenants.

Why a floor’s worst day is a visit #

A single floor’s characteristic load is a statement about an extreme: the 95th percentile, across many lives, of the largest load the floor sees in one. For the sustained part the largest in a life is the heaviest of about ten tenancies, each drawn from a moderately spread distribution, and the heaviest of ten is not far above the mean: 1.89 kN/m² against a mean of 0.5. For the visits it is the largest of about fifteen draws from a distribution so long-tailed that most visits are a tenth of a kilonewton a square metre and a few are ten times that, and the largest of fifteen sits far out in that tail: 2.6. A characteristic value is a property of the whole population, its tail especially, and the visits’ population has the longer tail.

That is the step the one-number model could not take. Given one spread for the whole floor load, it attributed the floor’s extreme to the same variability that produces its everyday value, and so it let the tenancy’s correlation reach the extreme. Split, the extreme belongs to the visits and the tenancy reaches only the everyday part. The tail that decides each floor’s design load is the part that averages away on the column.

Visits do not add #

Taken alone, the intermittent load’s factor falls as fast as anything can: almost as $1/n$, the dotted line, which is the factor for a column that only ever carries one floor’s worth. Twenty floors’ visits take 0.09 of their summed characteristic values. A visit lasts three days in fifty years, and twenty floors each visited 0.3 times a year spend almost all their lives with at most one floor visited. Two large visits on the same three days, on floors of the same column, is so rare that it barely enters the 95th percentile.

The sustained load’s factor falls much more slowly. Floors let one by one have independent tenancies, and the column averages them, but each tenancy is held for years and the column’s worst tenancy mix lasts long enough to coincide with everything else: twenty floors take 0.41. In tenancies of five there are only four independent loads to average and the factor is 0.61.

The rule cannot see this division, because it reduces one number. The one-number model of the earlier essay could not see it either. It gave every floor’s load one spread and one correlation, and so applied the tenancy’s correlation to the visits as well as to the tenants.

The column needs far less than the rule gives #

Put the two parts together and the column needs far less than either the rule or the one-number model. Twenty floors let one by one need 0.27 of their summed characteristic loads; in tenancies of five, 0.37; let to one tenant, 0.60. The one-number model with tenancies of five said 0.75, and the rule takes 0.73.

The reason is in where the single floor’s characteristic comes from. It is 3.24 kN/m², and most of that is a visit’s peak. The column’s characteristic is mostly its tenants’ sustained load plus one floor’s visit, so dividing it by twenty floors’ characteristic values divides a sustained-load number by a visit-load number. The averaging the rule allows for is mostly the averaging of visits, and visits average almost completely.

Even a building let to one tenant, every floor’s sustained load tied together, needs only 0.60. The tenancy correlates the floors’ sustained loads perfectly, and for an office the sustained load is a small fraction of what each floor was designed for.

Where the load stays #

That is the office. The rule is applied to more than offices, and the proportions are what decide.

As the sustained load grows, the floor’s characteristic is set more by its tenancy and less by its visits, and the column’s factor rises towards whatever the tenancies allow. Let singly, it reaches only 0.39 even with a sustained mean of 2 kN/m², because twenty independent tenancies still average. In tenancies of five it reaches 0.59. With one tenant it reaches 1.01: the whole column is one tenancy, and there is nothing left to average. The rule is unsafe only where the load stays and one occupier has it all.

Read the other way, the rule’s column load is exceeded almost never in a building let singly or in tenancies of five, whatever the sustained load. With one tenant it is exceeded in 1.4% of lives for an office, below the intended 5%; in 13.6% with a sustained mean of 1 kN/m²; and in 24.9% with 2. A building whose floors carry their settled use at 2 kN/m² — a library’s stacks, an archive’s shelving, a records store — let to one occupier, has a column designed for a load it will exceed in one life in four.

That is why the rules exclude it. EN 1991-1-1 allows no storey reduction for its storage category, and the other codes exclude storage and assembly uses in similar words. The split model shows what the exclusion is protecting against, and that the protection belongs to a kind of load rather than a kind of building: an office let to a single firm that fills its floors with compactus shelving has become a storage building without changing its category.

The tenancy matters only where the load stays #

The earlier essay’s finding — that a tenant on five floors makes twenty behave like four — is true of the sustained load, and the split shows how much of the column it reaches. For an office the factor rises from 0.27 let singly to 0.61 with one tenant: the tenancy more than doubles it, from a level so low that doubling still leaves it below the rule. For a heavier sustained load of 1.5 kN/m² it rises from 0.38 to 0.96, crossing the rule at a tenancy of about ten floors.

So the earlier essay’s suspicion was right in mechanism and wrong in where it bites. Large tenancies do correlate floors, and the rule does not know about them. For offices the correlated part is small enough that the rule’s margin covers it many times over. Where the sustained load is large, the tenancy’s correlation is the whole story, and the rule has no margin to give.

Refits, moves and the renewal that is not independent #

Tenancies here change independently, on average every five years. A building’s whole occupancy sometimes changes at once: a single occupier leaves and the building is refitted for the next, or a landlord refurbishes floor by floor in one campaign. A simultaneous renewal does not correlate the floors’ sustained loads any more than a single tenant does — the new tenants may be several and unrelated — but it does make the building’s heaviest tenancy mix and its moving day coincide. A move is the visit most likely to be heavy, stacked crates on a floor whose new partitions are already up and whose old furniture has not yet gone, and in a building-wide move it happens on every floor in the same weeks.

That is a correlation of the visits, the one part of the load the split found to average away completely. For a column it is the single largest exception to everything above, and a designer who knows a building will be fitted out or vacated in one operation knows that its columns’ reduction depends on the contractor’s programme. The load model cannot know it; the rule cannot either. The averaging that reduces a column’s load is an averaging over chance, and a move scheduled for the whole building is not chance; it is the one coincidence a programme arranges and no fractile covers.

By hand, for twenty office floors #

The single floor’s characteristic, 3.24 kN/m², is close to the largest of about fifteen visits in fifty years ($0.3 \times 50$), on top of a typical tenancy. Each visit is drawn from a distribution of mean 0.2 and standard deviation 0.4, so long-tailed that the largest of fifteen, at the 95th percentile, is near 2.6 kN/m². The heaviest tenancy at the 95th percentile, of about ten in fifty years, is near 1.9.

The column of twenty floors let singly: its sustained total at the 95th percentile over fifty years is about twenty times the mean, 10 kN/m² of floor-equivalent, plus a margin for the spread of twenty independent tenancies that averages to about half again: 0.41 of twenty floors’ sustained characteristic, 0.41 × 20 × 1.89 = 15.5. Its worst visit adds about one floor’s, 2.5. Together, 18 against twenty floors’ characteristic of 65: a factor of 0.27.

With one tenant, the sustained total is twenty times one tenancy, and its 95th percentile is twenty times one floor’s: 37.8. One floor’s visit adds 2.5, and the column carries 40 against 65, a factor of 0.62 — the 0.60 the simulation gives.

Two parts, a few parameters #

The parameters are typical, not measured. The sustained mean and spread, the renewal period, the visits’ rate, duration and size are of the order load surveys give for offices. They differ between countries, decades and kinds of office, and the conclusions here are in the comparison between the split and the one-number reading and in where the rule stops being safe. They are not in the third digit.

Visits are independent between floors. A building-wide event — an evacuation drill that crowds the stairs, a firm’s move that loads every floor’s corridors at once — correlates visits, and it is exactly the event that would make them add on a column. The model has none of those; a building with a planned all-floor event has a column load the averaging does not cover.

Spatial variation within a floor is in the spread. A floor’s load is not uniform over the area a column collects from; the sustained load’s coefficient of variation includes that, and the column’s area changes it. Like a movement joint’s budget, the column’s load is a sum whose largest term is often the one nobody computes carefully — here the visits, which a design load absorbs without naming.

Fifty years, one fractile. Other lives and fractiles move the numbers together and the comparison very little. A characteristic value read from a short survey carries the same caution as a strength read from a short test programme: its tail is known less well than its mean, and it is the tail that sets it.

What the rule is really calibrated on #

The load that is never all there at once found that the storey rule is roughly what independent floors would justify, and the floors that fill up together found that a modest correlation removes most of that justification. The split puts both in their place. The rule is calibrated on a floor whose characteristic load is mostly visits and whose column load is mostly tenants, and for such a floor it is conservative by a factor of two or more whatever the tenancy. Its safety does not come from independence between floors. It comes from the difference between a load that stays and a load that visits, and it is lost exactly where that difference is lost.

That is a statement about loads that have to be known before they can be found as much as about reduction. The sustained part behaves like a permanent load whose value is uncertain; the intermittent part behaves like a variable load in the strict sense. A combination rule that separated them, reducing the second freely and the first only by the number of independent tenancies, would be closer to the truth than a storey count. It would also ask the designer to know who will occupy the building.

Still open: the floor whose visitors all come at once #

The visits here are independent: each floor’s party, move or crowd is its own. Some buildings’ worst loads are not. A stadium’s concourse, a conference centre, a school at the end of a day, an office building during an evacuation, are loaded by one event that reaches every floor together, and for them the intermittent part is the correlated one. Whether a building whose visits are synchronised by its use behaves like the one-tenant building here — or worse, because the synchronised part is the larger one — and whether the assembly category that the rules already exclude covers every building that has such a day, is the question of what happens to reduction when the load that visits stops visiting one floor at a time.