The load that is never all there at once

A beam on the eighteenth floor of an office building is designed for an imposed load of five kilonewtons per square metre. So is the beam on the seventeenth, and the sixteenth, and every one below them. The column that carries all twenty of them is designed for the sum, and the sum is a number nobody expects to see.

Not because the five is wrong. It is a characteristic value — a load with a small probability of being exceeded on a floor over the building’s life — and it is roughly right for each floor taken alone. The trouble is what happens when twenty of those are added. Each was drawn at, say, the ninety-fifth percentile of its own distribution; the total of twenty is not the ninety-fifth percentile of the total, because twenty floors would all have to be having their worst afternoon at once.

The arithmetic, which is one line #

Let each bay carry a load with mean $m$ and standard deviation $s$, and let the bays be independent. The characteristic value for one bay is the fractile

$$q_k = m + z s,$$

with $z = 1.645$ for the ninety-fifth percentile. Now put $n$ of them on one column. A sum of independent variables has mean $nm$ and standard deviation $s\sqrt{n}$ — the variances add, not the deviations — so the total’s own fractile is

$$Q_k = nm + z s \sqrt{n}.$$

Divide the second by $n$ times the first and the reduction factor falls out:

$$\alpha(n) = \frac{nm + zs\sqrt n}{n\,(m + zs)} = \frac{1 + zv/\sqrt n}{1 + zv},$$

with $v = s/m$ the coefficient of variation. That is the whole derivation. There is nothing in it about buildings, floors or occupancy; it is the central limit theorem written for a column.

Two features of it are worth pausing on. The first is that the factor depends on the product $zv$ and not on $z$ and $v$ separately, so a rarer fractile and a more variable load are interchangeable as far as the reduction is concerned. The second is that the numerator and denominator differ only in that $\sqrt n$: at $n = 1$ the two are identical and the factor is exactly one, which is not a boundary condition imposed on the formula but the formula agreeing that one bay is one bay.

The floor it never goes below #

The interesting part of that expression is not how fast it falls but where it stops. As $n$ grows the $1/\sqrt n$ term vanishes and

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

a floor with no number of storeys in it at all. For an imposed load with a coefficient of variation of 0.6 read at the ninety-fifth percentile it is 0.503. A hundred floors and a thousand floors get the same answer.

The reason is worth stating plainly, because the mistake it prevents is common. Averaging does not remove the load; it removes the variability. Every bay really does carry its mean, and no amount of independence makes the mean go away — what disappears is the pretence that all of them are simultaneously at the top of their own distributions. The reduction is bounded below by the ratio of the mean to the characteristic value, and that ratio is a property of how variable the loading is and of how far out the fractile was drawn.

That makes the floor a useful diagnostic. A load with a small coefficient of variation has almost nothing to reduce: at $v = 0.3$ the limit is 0.670. A wildly variable one has a great deal: at $v = 1.0$ it is 0.378. How much reduction a load class deserves is a statement about its scatter and about nothing else, which is why the rules distinguish offices from storage and not steel from concrete.

What the codes say, and where they came from #

Two rules are in general use and they are the same argument in different clothes.

The storey rule reduces on the number of floors: $\alpha_n = \big(2 + (n-2)\psi_0\big)/n$, capped at one, with $\psi_0$ the combination factor — 0.7 for an office. It gives no reduction at all for one or two storeys, 0.90 at three, 0.80 at six and 0.76 at ten, and its own floor is $\psi_0 = 0.70$.

The area rule reduces on the area collected: $\alpha_A = \tfrac57 \psi_0 + A_0/A$, capped at one, with $A_0 = 10$ m². Its floor is $\tfrac57\psi_0$, which for $\psi_0 = 0.7$ is exactly a half.

Set the two against the independence result and something worth noticing appears. At ten storeys the storey rule gives 0.760 and the independence argument gives 0.660: the code is the more conservative of the two, and it stays that way at every count. So the rule is not the statistical argument transcribed — it is the statistical argument with something held back, which is the right way to write a rule whose central assumption is one nobody can check.

The square metre worth half a square metre #

The area rule has a property that is not obvious from its form and is the most practically useful thing about it.

The load the column is designed for is $A \times q \times \alpha_A$, and substituting the rule gives

$$A\,q\left(\tfrac57\psi_0 + \frac{A_0}{A}\right) = q\left(\tfrac57\psi_0 A + A_0\right).$$

The area cancels out of the second term. The design load is linear in the area collected, with a slope of $\tfrac57\psi_0$ — a half, for an office. Past the point where the reduction bites, every additional square metre of floor handed to a column contributes half of its own load and no more, for ever.

That is a strong statement about where the rule pays. A floor beam collecting 30 m² gets 0.78 — barely worth the arithmetic. A column collecting 360 m² gets 0.528. A transfer structure collecting the load of an entire tower gets the floor and nothing better. The rule is not a small correction applied uniformly; it is a correction that grows with exactly the members whose design is hardest.

Which free body produced the number #

Every quantity in this collection comes from cutting something and insisting the sums cancel, and it is worth being explicit about what has been cut here, because the free body is not the usual one.

Take a horizontal cut through the column immediately below the topmost floor it carries, and draw the free body above the cut: $n$ floors, each carrying whatever it happens to be carrying at that instant. Vertical equilibrium says the axial force in the column is the sum of those $n$ floor loads. That is exactly the running total the tributary rule builds, and nothing about it is in question.

What is in question is which value of each floor load goes into the sum. The free body is drawn at an instant; the characteristic value is a statement about a distribution over decades. Putting twenty characteristic values into one instantaneous equilibrium equation is a category error, and the reduction factor is the correction for it. The equilibrium is exact and the loading is a hypothesis, which is the shape of most of the honest difficulty in this subject.

The dead load is not reduced, and the reason is the whole argument #

Nobody reduces the self-weight, and it is worth asking why not, because the answer confirms the mechanism rather than merely stating an exception.

Self-weight is not independent between floors. If the screed on floor three came out twenty millimetres thick instead of fifteen, the screed on floors four to twenty very probably did too — same contractor, same specification, same misunderstanding. The variables are strongly correlated, the variances do not simply add, and $\sqrt n$ is replaced by something much closer to $n$. In the limit of perfect correlation the reduction factor is exactly one.

So the rule’s real content is not live loads are reduced and dead loads are not. It is loads that vary independently between the bays being summed may be reduced, and loads that do not may not, and the familiar version is a shorthand that happens to be right for the usual case. It also says immediately where the shorthand fails.

The same argument, pointing the other way #

There is a second place in this collection where a structural quantity is decided by many independent draws rather than by one, and it is worth setting beside this one because the two point in opposite directions for the same reason.

A larger specimen is weaker than a small one. Strength is set by the worst flaw the volume happens to contain, so a big volume is a large sample and a large sample has a worse worst member. The governing operation is a minimum over many draws, and a minimum gets worse as the number of draws grows.

Load works the other way. A column’s demand is set by the sum of many draws, and a sum concentrates: its coefficient of variation falls as $1/\sqrt n$ while its mean grows as $n$, so the sum becomes more predictable in relative terms and its characteristic value falls below the sum of the characteristic values.

That is not a coincidence of arithmetic. It is the two ways a large structure differs from a small one, and a designer who knows only the first has half of it.

What it does to a foundation #

The place the reduction is largest is the place where the consequence of getting it wrong is hardest to inspect.

A pad or a pile group under a tall building’s column collects the whole of that column’s reduced load, and its size follows almost linearly from it: at the area rule’s floor the foundation is being asked for a little over half of what the unreduced sum would demand. On a large raft or a heavily loaded pile group that is the difference between one arrangement and another.

But settlement is not decided by the characteristic load; it is decided by something much closer to the mean, sustained for years, and the two questions want different numbers out of the same distribution. A foundation checked for bearing capacity at the reduced characteristic load and for settlement at the quasi-permanent one is using two different points on one curve, which is correct and is often not what happens.

Where it must not be applied #

The exceptions are not a list to be memorised; each of them is a place where the independence assumption is visibly false.

Storage. A warehouse is loaded to its rated capacity on every floor by the same operation, on purpose, and the loads are close to perfectly correlated. No reduction.

Plant rooms and equipment. The load is a machine that is either there or not there, and it is there on every floor of the plant level simultaneously. It is closer to a dead load than to an imposed one.

Car parks. The correlation is unfortunate rather than physical: everyone arrives at the same time.

Anything already reduced. The storey rule and the area rule are two estimates of the same effect and must not be multiplied together. A designer who applies both to one column has taken the reduction twice, and the second application has no argument behind it at all.

Loads whose arrangement is the point. Pattern loading asks what happens when some bays are loaded and others are not, which is a question about arrangement rather than about magnitude. The two are independent decisions and both apply — but a reduction taken on a pattern case has to be taken on the intensity, not on the number of bays chosen to be loaded, or the two arguments collide.

The rule is a fractile correction, not a saving #

The most common misreading is that the reduction is margin being given away, and it is worth refuting directly because the correct reading changes what a designer should feel about it.

Take twenty floors at the ninety-fifth percentile each. If the loads really are independent, the sum of twenty characteristic values sits at roughly the 99.9th percentile of the total distribution — a return period of centuries rather than of decades. That is not the reliability the design was aiming at; it is a much higher one, arrived at accidentally, applied only to the columns, and different for every column depending on how many floors it happens to carry.

The reduction restores the intended fractile. A designer who declines it has not built in extra safety in any principled sense — they have built in an unknown and uneven amount of it, concentrated on the members where material is most expensive, and they have made the structure’s reliability a function of its height. The characteristic value exists precisely so that this does not happen; refusing to apply it consistently discards the thing it was for.

What it looks like when it is wrong #

Two failure modes, and both are quiet.

Taking the reduction on a correlated load. The arithmetic still produces a number and the number is smaller, so nothing complains. The error is invisible in the calculation and shows up, if at all, as a column that is more heavily loaded than its neighbours expected. A structure whose imposed load is genuinely correlated between floors and which has been reduced as though it were not has had a real margin removed.

Taking it twice. A spreadsheet that applies the area rule at the beam and the storey rule at the column has multiplied 0.78 by 0.76 and arrived at 0.59 for a column the rules would each have given about 0.76. Nothing in either rule refuses this, because each is written as though it were the only one.

The second is worth a check that could fail: sum the reduced column loads over a floor plate and compare with the reduced total of that floor. If the reductions have been applied consistently, the two agree; if a column has been reduced twice, the sum of the parts falls below the whole by exactly the extra factor. A statement that can be tested against a total is worth more than one that can only be read, and this one costs a single line in a schedule.

Where the model stops #

Independence is an assumption about occupancy, not a fact. It is a good one for offices and a poor one for buildings where one tenant fills several floors and moves in over a weekend. Nothing in the calculation knows the difference.

The fractile drifts with the number of bays being summed. The derivation above holds the fractile fixed and asks what the factor must be. Real rules were calibrated against reliability targets on a small number of representative buildings, which is not the same operation and does not give the same answer to more than two figures.

The coefficient of variation is not measured on the building being designed. It comes from surveys, most of them decades old, most of them of offices, and it is the single number the whole floor depends on. A change from 0.6 to 0.8 moves the limiting factor from 0.503 to 0.432 — a fifteen per cent change in the load a tall building’s columns carry, from a statistic nobody on the project will ever see.

The rule says nothing about arrangement. Reducing the intensity leaves the question of which bays are loaded entirely open, and on a continuous structure that question is often the one that governs.

And the whole apparatus is about the ultimate limit state. A deflection check, a vibration check and a crack-width check each want a different point on the same distribution, and none of them wants the reduced characteristic value. Carrying one reduced number through every check is the commonest way this rule is misused, and it is misused in the direction of optimism on exactly the checks — the serviceability ones — that most often decide the member.

Where the ladder goes #

Later rungs on this anchor: the reliability calculation behind the combination factors, and what target it was calibrated to. Correlated imposed loads and how the factor changes when the correlation is not zero. Reduction rules for members other than columns — walls, transfer beams, foundations — and why they differ. The interaction between reduction and load arrangement, where the two are usually applied together and rarely reconciled. Imposed load surveys and what they actually measured. The treatment of loads that are partly permanent, such as partitions. Reduction in seismic mass, where the same argument is made about a completely different quantity. And the question underneath all of it: whether a rule calibrated on offices in the nineteen-seventies describes a building that is now half meeting rooms and half server racks.