The floors that fill up together

A column at the bottom of twenty storeys may be designed for less than twenty floors’ imposed load, and the reason is the central limit theorem. Each floor’s design load is a fractile — a value that floor exceeds only rarely, in the way a material’s design strength is a value no specimen need have had — and twenty floors do not all exceed their own fractiles on the same day. The total has a mean of twenty floors’ means and a spread of only $\sqrt{20}$ floors’ spreads, so the fractile of the total is well below the sum of the floors’ fractiles. With a coefficient of variation of 0.6 and the 95th percentile, the factor is $(1 + zv/\sqrt{n})/(1 + zv)$: 0.61 at twenty floors, falling toward 0.503 and never below it.

The whole of that argument is the $\sqrt{n}$, and the $\sqrt{n}$ is the word independently. The earlier essay named the assumption in its last section — independence is an assumption about occupancy, not a fact — and left it there. This essay prices it.

Floors that move together #

Independence means that knowing one floor is heavily loaded today tells nothing about the floor above. That is a good description of a building let floor by floor to unrelated tenants, and a poor description of almost anything else. A single company’s floors fill and empty together; a building’s use changes all at once when it is refitted; an archive, a library or a call centre puts the same load on every floor it takes.

The simplest way to say that is a correlation $\rho$ between the loads of any two floors. The total of $n$ floors then has a variance of $n$ floors’ variances plus every covariance between them — $n(n-1)$ of them, each $\rho$ times a floor’s variance — so its spread is

$$s\sqrt{n\,(1 + (n-1)\rho)}$$

where the independent floors’ was $s\sqrt{n}$. The factor that keeps the column’s load at the same fractile becomes

$$\alpha(n, \rho) = \frac{1 + zv\sqrt{(1 + (n-1)\rho)/n}}{1 + zv}.$$

With $\rho = 0$ it is the earlier essay’s factor. With $\rho = 1$ it is one: floors that always carry the same load cannot be averaged, and the column needs the full sum.

The five curves are the same building at five degrees of togetherness. At twenty floors, independent floors can be reduced to 0.61. A correlation of only 0.1 raises that to 0.69, 0.3 to 0.79, 0.6 to 0.89. A modest correlation takes back most of what the averaging gave, because the covariances outnumber the variances: of the 400 terms in twenty floors’ total variance, 380 are covariances, and a correlation of 0.1 on each is worth nearly twice the twenty variances together.

The dashed line is the storey rule of EN 1991-1-1, $(2 + (n-2)\psi_0)/n$ with $\psi_0 = 0.7$ for offices. At twenty floors it takes 0.73. It sits between the independent curve and the correlated ones — far more cautious than independence and less cautious than a correlation of 0.3.

The floor under the reduction #

The earlier essay found that the independent factor never falls below $1/(1+zv)$, because the mean of every floor is always there. Correlation raises that floor. As $n$ grows the factor tends to

$$\alpha(\infty, \rho) = \frac{1 + zv\sqrt{\rho}}{1 + zv}$$

and the correlated part of every floor’s load is a part the averaging can never remove, however many floors there are.

The curve rises steeply from the origin, because it goes as $\sqrt{\rho}$: a correlation of 0.05 already lifts the floor from 0.503 to 0.61. The storey rule’s own floor is $\psi_0 = 0.7$, and the two cross at a correlation of 0.157. A building whose floors are correlated more than that is one where the rule’s reduction for a very tall column is larger than the averaging can pay for.

That is the number this essay is about. A rule calibrated as though floors were independent has a margin for correlation, and the margin is a correlation of about 0.16 — not large, for a quantity that a single tenant, a single refit or a single use can push to one.

How much correlation each column can stand #

The crossing at 0.157 is for a column of unlimited height. A shorter column has less averaging to lose and the rule has more room.

Three floors tolerate a correlation of 0.46; five, 0.26; ten, 0.19; twenty, 0.17; and the tolerance approaches 0.157 from above. The taller the column, the more independence the rule is quietly assuming. That is the opposite of where the uncertainty about independence is greatest: a tall building is more likely to be let in large tenancies, and its lowest columns carry exactly the floors those tenancies fill.

One tenant on five floors #

A common correlation between every pair of floors is the simple model. The more realistic one is blocks: floors let together are loaded alike, and floors in different tenancies are independent.

The arithmetic is clean. A tenancy of $k$ floors loaded alike is one load of $k$ times a floor’s, so twenty floors in tenancies of five are four independent loads, each five floors heavy. Their total’s spread is $\sqrt{4} \times 5 = 10$ floors’ spreads, against $\sqrt{20} = 4.5$ for independent floors. A tenant on five floors makes twenty floors behave, statistically, like four.

The factor follows: 0.61 when every floor is let separately, 0.66 in pairs, 0.75 in fives, 0.85 in tens. The rule’s 0.73 is safe for pairs and not for fives. And the rule cannot know, because it counts storeys and the averaging counts independent loads; the difference between them is a lease.

The curves have a staircase in them. Each one stays at one while the column is still inside the first tenancy — five floors of one tenant are one load, and there is nothing to average — and falls fastest each time the column reaches a new tenancy and gains one more independent load.

What fractile the rule actually gives #

The other way to read the same arithmetic is to hold the rule fixed and ask what it delivers.

With independent floors the rule is far safer than intended: the load it gives a ten-storey column is exceeded 0.36 per cent of the time, and a twenty-storey column’s 0.039 per cent — the 99.96th percentile, where every floor was designed at the 95th. That is the earlier essay’s observation that refusing the reduction would make columns uneven in their safety, seen from the other side: the rule itself already does.

At a correlation of 0.15 the rule stays below the intended 5 per cent at every height, approaching it from beneath as the column grows. At 0.3 it is the other side of it: 8.1 per cent at ten floors, 9.7 per cent at twenty — the 90th percentile, where the 95th was meant. The rule does not move between those three buildings. The probability of exceeding its load moves by a factor of 250.

The column, storey by storey #

In kilonewtons the three readings of one building part company steadily down the column. A column taking 36 m² of 5 kN/m² from each of twenty floors carries 3,600 kN unreduced — and 36 m² is itself a rectangle the column may be given more than, which is a second, separate error in the same number. The rule gives it 2,628. The averaging gives it 2,212 if the floors are independent and 2,847 if they are correlated by 0.3 — a spread of 635 kN, a sixth of the unreduced load, between two equally reasonable descriptions of who works in the building.

The rule sits closer to the correlated reading than to the independent one, which is some evidence that it was drawn with the possibility in mind. But it sits below the correlated reading, and nothing in the rule says when that is acceptable.

Where a correlation comes from #

A correlation between floors is not a mysterious property. It is what is left when part of every floor’s load is set by something the floors share.

Write each floor’s load as the sum of two parts: a building-wide part, the same on every floor — how busy the building is this year, what kind of business occupies it, whether it is being refitted — and a floor’s own part, independent from floor to floor. The correlation between any two floors is then simply the share of a floor’s variance that the building-wide part accounts for. If a quarter of the scatter in a floor’s load is the building’s and three quarters the floor’s own, $\rho = 0.25$, and the floor under the reduction is 0.75.

Put that way, a correlation of 0.16 is a small claim. It says that about a sixth of the variation in what a floor carries is explained by which building it is in. Load surveys that compare floors across many buildings, rather than floors within one, would see exactly that share as the difference between buildings — and the more a survey’s buildings differ in use, the larger it would be. The independence that the reduction rests on is the statement that the building explains nothing.

The block model is the same idea with a sharper edge. A tenancy is a part of the building that shares a load-setting decision completely and shares nothing with the next tenancy. Within it the correlation is one; across tenancies it is zero. Most buildings are somewhere between the two pictures, with a little shared across the whole building and a lot shared within each tenancy. The same structure appears where people move together rather than merely sit together: a crowd jumping to one beat is a set of loads whose correlation comes from the music, and it is the correlation, not the number of people, that decides what the stand feels.

What the rule would need #

The storey rule has one parameter, $\psi_0$, and the question can be turned round: what $\psi_0$ would make the rule deliver the 95th percentile for a given building?

For independent floors, about 0.57 at every height from ten floors up — the rule’s 0.7 is generous, by the margin the earlier essay found. For floors correlated by 0.3, 0.76 at ten floors, 0.77 at twenty and at forty. For tenancies of five floors, 0.82 at ten floors, where the column has only two tenancies to average, 0.72 at twenty and 0.66 at forty. For tenancies of ten, the rule would need 0.84 at twenty floors and 0.74 at forty.

The value the rule uses, 0.7, sits inside every one of those ranges. It is too large for independent floors and too small for correlated ones and for large tenancies in all but very tall buildings. That is what a single number calibrated across many buildings looks like from inside any one of them, and the earlier essay’s suspicion that the rule was fitted rather than derived is borne out: no one model of occupancy gives 0.7 at every height.

It also suggests what a more careful rule would read. The storey count is a stand-in for the number of independent loads, and the number of independent loads is the storey count divided by the size of a tenancy. A designer who knows a building will be let to one tenant on every five floors can count four loads where the rule counts twenty.

The area rule already knows #

EN 1991-1-1’s second reduction, by tributary area, makes the same assumption inside one floor — that the load on one square metre tells nothing about the next — and inside one floor that assumption is plainly false. One tenant furnishes the whole floor, sets its density and decides where its files go, and the bays of a floor are correlated far more strongly than floors are.

The area rule survives that because it was not derived from independence. The load surveys behind it measured how the scatter of the average load over an area falls as the area grows, and the model fitted to them has two terms: one that falls in proportion to one over the area, and one that does not fall at all. The second is the part of the load that the whole floor shares — the correlated part, measured and kept. The area rule’s floor of $\tfrac{5}{7}\psi_0$, a half for offices, is that measured constant, and the correlation is already in it.

The storey rule, read as the averaging argument the earlier essay reconstructed, has no such term: it sums floors as independent loads, and a term that does not fall with the number of floors is exactly what a correlation adds. That is the asymmetry this essay turns on — the reduction across a floor keeps a part that never averages away, and the reduction down a building, argued from independence, does not.

The column cut above its footing #

The free body is the column cut just above its footing, with the floors above it as the load — the running total that a beam’s load per metre starts and nobody checks at the bottom. The only thing statistics changes is which number is put on that free body: not a sum of characteristic values but a characteristic value of the sum. Equilibrium is exactly the same at every fractile; the reduction is a statement about which fractile, and the correlation is a statement about how the floors’ loads combine into the sum.

That is also why a correlation cannot be argued away by structural reasoning. No stiffness, no load path and no arrangement of the floors changes it. It is a property of the occupancy, and the column’s free body inherits it whole.

The twenty-storey column by hand #

With $v = 0.6$ and $z = 1.645$, $zv = 0.987$ and $1 + zv = 1.987$.

Independent: $\sqrt{(1 + 19 \times 0)/20} = 0.224$, so $\alpha = (1 + 0.987 \times 0.224)/1.987 = 0.614$.

Correlated by 0.3: $\sqrt{(1 + 19 \times 0.3)/20} = \sqrt{6.7/20} = 0.579$, so $\alpha = (1 + 0.987 \times 0.579)/1.987 = 0.791$.

Tenancies of five: four loads of five floors, a spread of $\sqrt{4} \times 5 = 10$ floors’ worth against twenty floors’ mean, $10/20 = 0.5$ in place of the 0.224, so $\alpha = (1 + 0.987 \times 0.5)/1.987 = 0.752$.

The storey rule: $(2 + 18 \times 0.7)/20 = 0.73$.

What the model assumes #

Normal loads. Imposed loads are skewed, not normal, and an average of skewed quantities is not a safe stand-in for them — a floor is far more often lightly loaded than heavily — and the fractile of a skewed total is further out than the normal one. The direction of every comparison above survives, because skew affects the independent and the correlated totals alike, but the numbers are those of the normal model.

A single correlation. Real floors are correlated more strongly with their neighbours than with floors ten storeys away, and the block model above is the opposite extreme from a common correlation. Neither is measured here; both are stated so that a survey which did measure one could be read against them.

And a single load. The imposed load in a design is two loads — a sustained part that stays for years and an extraordinary part that comes for a day, a meeting or a move. The sustained part is where tenancies correlate floors; the extraordinary part is closer to independent. A rule that treated the two separately would reduce the second freely and the first hardly at all, and it would be doing for imposed load what the arrangement of a load across spans already does for where it stands: asking which combinations can occur together.

What the curves cannot show #

They cannot show the correlation. Every curve takes $\rho$ as given, and the one number a designer would need to use them is the one no survey of the building being designed can supply before it is occupied.

They cannot show a change of use. A building let floor by floor today can be let to a single tenant in ten years, and the correlation its columns see then is not the one they were designed for. The rule’s indifference to occupancy is also its robustness to this: it gives the same answer whatever happens to the leases.

And they do not cover the area rule. EN 1991-1-1 offers a second reduction by tributary area, for the same reason and with the same assumption — that the load on one square metre is independent of the load on the next. Correlation within a floor is if anything stronger than between floors, and the same arithmetic would price it.

What it comes to #

The reduction is an average, and correlation is what an average cannot remove. With a common correlation $\rho$, $n$ floors’ total spreads as $\sqrt{n(1 + (n-1)\rho)}$, not $\sqrt{n}$.

The floor under the reduction rises from 0.503 to $(1 + zv\sqrt{\rho})/(1 + zv)$. It reaches the storey rule’s 0.7 at a correlation of 0.157.

So the rule is safe for a tall column only if its floors are nearly independent. Ten floors tolerate a correlation of 0.19, three floors 0.46.

A tenant on k floors divides the floors by k. Twenty floors in tenancies of five allow 0.75, against the rule’s 0.73.

Still open: the floor that is partly permanent #

Every load here has one coefficient of variation and one correlation. Real imposed load is a sustained part — furniture, files, partitions, a tenant’s settled use, closer in character to the dead load that has to be known before it can be found — and an intermittent part that arrives for a day. The sustained part is where tenancies tie floors together and the intermittent part is where they do not, so the two should be reduced by different factors: the first barely, the second freely. Whether splitting the load that way recovers the storey rule’s numbers for a building let in large tenancies — or whether, as the earlier essay suspected of a rule calibrated on offices in the nineteen-seventies, it shows the rule to be generous exactly where tenancies are largest — is a question about the composition of a load that the rule treats as one number.