The survey that arrives after the floors

The perimeter columns of a tall concrete building carry about twice the stress of its core, so they shorten twice as much, and every floor that spans between the two is pulled down at its column end. The essay that followed the shortening in time did the arithmetic for a forty-storey building cast a floor a week: the worst floor is level 31, its column end drops 26 mm against the core over thirty years, and setting each floor high when it is cast by about four-fifths of its predicted drop keeps every floor within 11.0 mm of level over its whole life, with the creep anywhere within ±30 per cent of the prediction. The four-fifths is a hedge. With the creep known exactly the best share is 0.70 and the worst error 7.9 mm.

That essay ended on an obvious improvement. The building is not cast in one day. By the time the twentieth floor is set, the first has been under load for twenty weeks and has done a large part of its early creep, in full view of a surveyor with a level. Measure what the lower storeys have actually done, fit the creep to it, and set the upper floors for the creep the building has rather than the one it was designed for. The method is used, under the name of compensation updating, on most supertall buildings, and it is a genuinely good idea. This essay is about how much it can buy, and about the two reasons the answer is much less than it looks.

What the lower floors have done when they are surveyed #

A survey reads, for each floor already set, how far its column end has dropped against the core since the day it was cast. The prediction of that drop has two parts. One is elastic, the instant shortening under each new floor’s weight; it does not depend on the creep. The other is creep, and if the building’s creep is the design creep times a scale $s$, the predicted drop of level $k$ at the survey date is

$$D_k = a_k + s\,b_k,$$

with $a_k$ the elastic part and $b_k$ the creep part at the design creep. Fitting $s$ to the readings $y_k$ is a least-squares problem in one unknown, and it has a closed answer:

$$\hat s = \frac{\sum b_k (y_k - a_k)}{\sum b_k^2},\qquad \operatorname{se}(\hat s) = \frac{\sigma}{\sqrt{\sum b_k^2}}$$

where $\sigma$ is the standard error of one reading. The second formula is the whole of the problem. The precision of the estimate is the survey’s precision divided by how much creep there is to see.

After ten floors the largest drop anywhere in the building is 0.59 mm. That is not an error in the model; it is what seventy days of a 6 N/mm² stress difference does to concrete in a few storeys, and it sits well inside the noise of any survey carried out on a construction site, where the reference is a core that is itself moving and the instrument is a level read across a floor plate full of formwork. A survey read to a millimetre has nothing to measure. After twenty floors the largest drop is 2.4 mm, about the size of two survey errors. After thirty it is 5.5 mm and the shape is clear. After forty, with the frame complete, it is 9.9 mm — and there are no floors left to cast.

How fast the doubt falls #

A survey’s result should not replace the design’s prediction; it should be combined with it. The design says the creep scale is one, give or take thirty per cent at ninety-five per cent confidence — a standard deviation of 0.15, treated as a scatter rather than a value, the way a material strength is. The survey says $\hat s$ with its own standard error. Two independent estimates of one quantity combine by their inverse variances, and the combined doubt is always smaller than either alone. Every floor cast after a survey can then be set high by the share of its predicted drop that keeps it nearest level for whatever doubt is left: 0.79 when the doubt is the design’s, falling towards 0.70 as the doubt goes to nothing.

The figure is two lines racing, and the wrong one wins. The doubt from the survey alone is off the top of the chart for the first fifteen floors, falls steeply between floors fifteen and thirty as the creep grows into view, and crosses the design’s ±30 per cent at floor 25. By then fifteen floors remain. The combined doubt hardly moves before floor 20, and it halves — to ±15 per cent — only after floor 31, with nine floors left to set. A survey every ten floors therefore has three chances to act: after floor 10, when it knows nothing the design did not; after floor 20, when it knows a little; and after floor 30, when it knows the creep well and has ten floors to apply it to.

The information arrives at the rate the creep does, and the creep is slow by definition. That is the whole of the first limitation, and no choice of survey interval escapes it. It is the same arithmetic that lets an arch left with its hinges open shed its fastest creep by waiting: the early creep is fast enough to be waited out but too small, at the dates that matter here, to be measured. Surveying every five floors, or every floor, changes how often the estimate is updated but not how much creep there is to see at any date.

The shares the surveys set #

Each survey resets the share for the floors cast after it, and only for those. The floors below are set and stay set: there is no going back to a slab that has been cast at a level.

The survey after floor 10 changes nothing — the three buildings take the design’s 0.79 through floor 20, because a reading of a fraction of a millimetre cannot move a prior of ±30 per cent. The survey after floor 20 separates them a little, and the one after floor 30 separates them properly: a building that creeps less than predicted sets its top ten floors at 0.65 of their prediction, one that creeps more at 0.84. If the creep is exactly as designed, the share does not fall to 0.70, because the doubt left after floor 30 is still ±15 per cent and the hedge against it is worth keeping.

What a survey buys, and what nothing can buy #

The test of a compensation is the worst error any floor shows over its life — standing high when it is fitted out, or low after thirty years — with the creep anywhere in its range. The fixed share’s answer, from the essay this one continues, is 11.0 mm.

The two horizontal lines are the bounds. Below the dotted one nothing can go, because it is what the best compensation achieves when the creep is known exactly from the start, and the gap between the lines — 0.9 mm — is everything that knowing the creep is worth at the worst case. A perfect survey reaches the lower line: its readings are exact, the scale is recovered exactly by floor 20, and the top twenty floors are set as though the creep had been known. A survey read to a quarter of a millimetre gets most of the way. A survey read to a millimetre, taken at the unlucky end of its error, ends 0.8 mm above the fixed share: the readings after floors 10 and 20 move the share in the wrong direction by more than the reading after floor 30 can recover.

Why is the worst case so insensitive to knowing the creep? Because the worst case is the building that creeps most, and the error left in that building is not doubt. The essay before this one found that even a perfect prediction leaves a floor high at its fit-out and low at thirty years, because it moves between the two dates and can only be set once. The amount it moves is proportional to the creep. At 1.3 times the design creep, the floor at level 33 moves enough between those dates that the best it can do, set perfectly, is 10.1 mm. The fixed share was already within a millimetre of that, because its 0.79 is the share that the high-creep building wants.

The surveys earn their keep elsewhere, and the same calculation shows where. Split the worst case into the three buildings. If the creep is 0.7 times the design’s, the fixed share sets every floor too high and the roof stands 11.0 mm high at its fit-out; set perfectly for the creep it has, the worst floor is 5.8 mm out. If the creep is as designed, the fixed share’s worst is 10.2 and the perfect one’s 7.9. If it is 1.3 times, 11.0 and 10.1. A survey is worth five millimetres on the building that creeps less than predicted and one on the building that creeps more. That is not the building the hedge was worried about, and a client paying for an updating programme should know which of the two it is buying protection for.

The creep a survey cannot see #

There is a second limitation, and it is worse than the first because it points the wrong way. The scale fitted from a survey is the scale of creep under loads that are weeks or months old. The scale that decides where a floor ends after thirty years is the scale of creep under loads that are decades old. A single multiplier assumes the two are the same number, and the evidence on real concretes is that they are not. A member’s creep belongs to its own size, humidity and age at loading, and a creep model can fit the first year of a test and miss its tenth year by a large fraction, because the early creep is dominated by the paste’s response to fresh load and the late creep by drying, ageing and the aggregate — the same split that makes a beam’s deflection arrive three years late. Let the building’s creep be as designed for a load a few months old, and a different multiple of the design long after, blended over a thousand days.

When the late creep exceeds the design’s, every compensation does worse, and the surveyed ones do worst. The surveys read a creep close to the design’s — because that is what the early creep is — and so they lower the share towards the 0.70 that suits a known design creep, removing exactly the hedge that would have protected the building against the late creep that then arrives. At a late creep of 1.3 times the design’s the fixed share leaves its worst floor 12.3 mm out; perfect surveys leave it 14.3 mm out, and surveys read to a millimetre 14.6. The measurement has made the building worse, not by being inaccurate but by being accurate about the wrong thing. It is the trap a jack set to the at-once moments falls into from the other side: a correction applied early is carried by the creep that follows it, and here the creep that follows is the part nobody measured.

When the late creep is smaller than the design’s, the picture reverses: the surveys see a building creeping as designed, the floors end higher than the fixed share would leave them, and a perfect survey’s worst error falls to 7.9 mm. The asymmetry is the same one as before. The survey helps the building that creeps less.

The profile shows where the damage is. The two compensations are identical through floor 20, because the first two surveys change almost nothing. They part at floor 21 and again at floor 31, where the surveyed compensation steps down: those are the floors it set from readings that showed a creep as designed. At thirty years it is those floors, the top twenty, that end lowest, a millimetre below the fixed share’s. At the fit-out the surveyed floors stand a little lower too, which is the half of the trade a client sees and likes. The other half arrives over the following decades.

The survey’s precision, by hand #

The numbers in the doubt figure come straight from the formula for the standard error. At the survey after floor 30 the creep parts of the thirty readings, $b_k$, are the drops each floor would have shown at the design creep less their elastic parts; squared and summed they give $\sqrt{\sum b_k^2} = 11.4$ mm. A reading error of 1 mm then gives a standard error of $1/11.4 = 0.088$ in the scale, or ±18 per cent at 95 per cent confidence. The design’s own standard deviation is 0.15. Combining the two by inverse variances,

$$\sigma_{\text{both}} = \left(\frac{1}{0.15^2} + \frac{1}{0.088^2}\right)^{-1/2} = 0.076,$$

which is ±15 per cent — half the design’s ±30. After floor 20 the same sum is only 4.0 mm, the scale’s standard error is 0.25 and the combined doubt is ±26 per cent, barely changed. After floor 10 the sum is 0.66 mm and the standard error 1.5: a survey that reports the creep as anything between nothing and four times the design’s.

The floor under the noise figure can be checked the same way. The best any compensation can do for a floor that has moved $M_1$ between its setting and its fit-out and $M_2$ by thirty years is to stand $\tfrac12(M_2 - M_1)$ high at the fit-out and the same amount low at the end. The floor that moves most between those two dates is the roof, because it is set last and carries the least of its creep before the finishes go on: 15.2 mm at the design creep and 19.7 mm at 1.3 times it. Half of 19.7 is 9.9 mm, and the 10.1 mm on the figure is that plus the small cost of setting every floor by one share rather than each by its own. No survey changes $M_2 - M_1$. It is the movement of a floor after its finishes are on, and only a different construction sequence — fitting out later, or a floor laid on adjustable pedestals that can be reset — reaches it.

The cut column that makes the fit one-unknown #

Every number above is a sum of storey shortenings, and each storey’s shortening comes from one free body: a column storey cut at its top and bottom, carrying the axial force of every floor cast above it, each arriving at a different age. The core storey beside it carries its own share. The column and the core are not connected by any stiffness in this model — the floors are taken to span between them without restraining the difference — so the difference is purely the two cut columns’ strains multiplied by their heights and summed down to the foundation.

That is the assumption that makes the survey’s estimate a one-unknown problem: if the floors restrained the difference, a survey would read the floors’ stiffness as well as the creep, and the scale would be confounded with how hard the slabs pull the columns up. In a building with outriggers or deep transfer floors that coupling is large — an outrigger connected late so the drop can happen first is exactly a stiffness of that kind — and the essay that measured how much shorter the columns are than the core has the stresses that would set it. Here the cut is clean, and so is the estimate.

Where a one-unknown fit is too kind to the survey #

The survey errors are independent from floor to floor. A real survey is read against a reference that moves with the building, and an error in the reference moves every reading together. A common error is worse than an independent one of the same size, because averaging thirty readings does not reduce it. The noise figure is therefore an optimistic account of a survey’s precision.

The creep has two scales and nothing else. The real uncertainty in a creep prediction is a whole function of time, of which the early and late multipliers are the crudest description that can show the second limitation. A concrete whose drying shrinkage is the uncertain part, rather than its creep, gives a third scale with a different time shape, and a survey sees it differently again.

The fit-out date is fixed. A building whose fit-out can be delayed, or whose finishes include a levelling screed laid at the fit-out from a survey made then, has a second chance to compensate that this one does not, and it is aimed exactly at the movement that the noise figure says no survey during construction can reach.

The building is uniform. Every storey has the same stress difference and every floor the same cycle. A real tower changes column size, concrete grade and floor cycle up its height, and the elastic part $a_k$ of each reading has its own uncertainty — the modulus of a concrete placed on a given day is itself a variable — which the one-unknown fit assigns to the creep.

Still open: the floor that is reset at its fit-out #

Everything here is decided at the moment a floor is cast: the slab’s level is fixed then, and the only freedom is how high. A floor finished with a screed, a raised access floor or a levelling compound has a second level, set months later at its fit-out, and that one can be set from a survey made with the frame complete, when the doubt about the early creep has fallen to a few per cent. Whether a screed laid to a survey at the fit-out can recover the millimetres the construction-stage surveys could not — the floor that ends high on the low-creep building, and the top floors that a misleading early reading set too low — and how thick a screed has to be allowed for, on every floor, to make room for a correction that only some of them will need, is the question that moves the compensation from the structure to the finishes, where the movement that matters is happening.