The floor that is level on one date

The columns are shorter than the core found that every column in a tall building shortens as the building is built on it, that the core beside it shortens by a different amount, and that what a finished building shows is not the column’s 127 mm of total shortening but the 20 mm difference that arrives after each floor is set level. Its remedy was compensation: build each floor high by the amount it is predicted to drop, and the building arrives level. Its honest weakness was the prediction. Creep carries an uncertainty of perhaps ±30 per cent, and compensation carries that uncertainty into the built levels.

That essay counted creep as a single multiplier, one plus the coefficient on every strain, and counted only the load that arrives after a floor is set. Neither is true of a real building, and both matter to the question it left open: whether to compensate for elastic movement only, for some of the creep, or for the whole prediction. The answer depends on when the building is looked at, and on how wrong the creep can be.

The same building, built in time #

The building is the earlier essay’s: forty storeys of 3.6 m, perimeter columns at 12 N/mm² under the finished load and a core at 6, a concrete modulus of 34,000 N/mm² and a creep coefficient of 2.2 for a load applied at 28 days. Now it is built. A floor is cast every seven days, so the frame takes 280 days. Each new floor adds its share of the load to every storey below it, at whatever age that storey has reached: the storey under level 30 is a week old when level 31 lands on it and seven months old when the roof does. Three quarters of the load arrives that way, with the frame; the remaining quarter — screeds, cladding, services, partitions — arrives at the fit-out, ninety days after the frame is complete. Each load creeps from the age it was applied by Eurocode 2’s form, which makes a load on young concrete creep more than the same load on old.

Each level is set level on the day it is cast. What happens to it afterwards is the shortening of every storey beneath it from that day on — the column’s, less the core’s.

Two things change. The worst floor moves up, from level 20 to level 31. And the roof, which the earlier count said could not move after it was set because nothing is built above it, drops 24.1 mm against the core. At completion the profile still looks like the familiar parabola, peaking half way up at 10 mm; by thirty years it has leaned over, and its top half is where the movement is.

The creep that was already on its way #

The earlier count said that a level is moved only by load that arrives after it. That is true of the elastic shortening and false of the creep, because creep is a strain that keeps growing under a load already applied.

When the roof is cast, the storeys beneath it are carrying three quarters of their final load, the oldest of it applied at a week’s age, and all of it has years of creep ahead. None of that creep has happened when the roof is levelled, and all of it happens afterwards. Low in the building the effect is small: a level near the bottom is set when almost nothing is on the storeys below it. High in the building it dominates: at the roof it is 8.5 mm, a third of the roof’s whole movement, and the rest is the finishes, which arrive after every level has been set and so move every level, the roof included.

That is why the worst floor rises. The earlier count’s parabola was the product of two things, how much load is still to come and how many storeys are beneath to carry it, and it peaked where the product did. Add to it the creep of everything already on, which grows with height, and the finishes, which move every level, and the peak slides up the building. It is the same effect the arm connected late met in an outrigger: a connection made after the frame is complete still collects everything the frame’s creep has to come.

Half of it is still to come #

The worst floor has dropped 6.9 mm when the frame is complete and 14.1 mm at the fit-out — just over half of its thirty-year movement. Two thirds of the rest arrives in the first year after the fit-out and almost all of it within five: 25.1 of the 26.4 mm has happened by then. That is the familiar shape of creep, the deflection that arrives three years late, applied to a column instead of a beam. The creep coefficient’s uncertainty fans the curves out as time goes on: at thirty years the floor may have dropped anywhere from 21.0 to 31.9 mm.

This is the shape of the problem compensation has to solve, and it already shows why compensating for everything is not obviously right. A floor set high by its whole thirty-year prediction is level only at thirty years. Before that it stands high, and the day it stands highest is the day the building is handed over, when the occupants and their partitions arrive.

Compensating for some of it #

Compensation sets each level high, when it is cast, by a chosen share of its predicted thirty-year drop. A share of nothing leaves every floor to sink below level as the building ages; a share of one leaves every floor high at handover and level at thirty years, if the creep is what was predicted. The question is which share keeps every floor nearest to level over the whole of its life, from the day it is fitted out, when the creep may be anything within ±30 per cent of the prediction.

The two errors pull in opposite directions. The amount a floor can stand high grows with the share: it is set by the floor that has moved least when it is first seen — the roof, at its fit-out, under the least creep. The amount a floor can stand low shrinks with the share: it is set by the floor that moves most — around level 33, at thirty years, under the most creep. They cross at 0.79, where the worst error is 11.0 mm. Uncompensated the worst is 31.9 mm; compensated in full, 16.0. The best compensation is about four fifths of the prediction, and it is half as wrong as the full one at its worst.

The share hardly depends on the floor cycle: it is 0.79 on a four-day cycle, 0.79 on seven and 0.80 on fourteen, with worst errors of 12.4, 11.0 and 9.2 mm. The share is not the elastic part either. The elastic shortening after a floor is set is about a third of its thirty-year movement here; compensating for that alone, a share of about a third, leaves a worst error of 23.1 mm. Nor is the answer simply a hedge against the creep’s uncertainty: with the creep known exactly the best share is 0.70 and the worst error 7.9 mm, because even a perfect prediction cannot make a floor level at handover and at thirty years at once. The uncertainty raises the best share — at ±50 per cent it is 0.86 — because a floor that may creep half as much again as predicted can end far lower than one that creeps half as much can stand high.

Level on one date, or near level on all of them #

The worst floor’s own history shows the trade. Set high by its whole prediction it starts 12.4 mm high, sinks for thirty years, and arrives level only if the creep was exactly what was predicted. Set high by four fifths, it starts 6.9 mm high, passes through level within a year at the design creep, and spends the rest of its life a few millimetres low. The full compensation is aimed at one date and hits it under one assumption; the partial one accepts a small error at the end to halve the error at the beginning, when the building is most closely looked at and its tenants are fitting it out.

Across a 9 m bay the numbers are slopes. The uncompensated worst floor ends 31.9 mm low at the column, one in 280; the fully compensated building starts with floors one in 560 high; the best share keeps every floor flatter than one in 820, which most floor finishes and partitions accept without complaint.

A faster floor cycle, a larger drop #

The seven-day cycle is a choice, and it enters through the age at which each storey is loaded. A load on concrete a week old creeps more than the same load a month later — the effect the day taken off the floor cycle priced in the strength of a young slab, met here in its stiffness over time. On a four-day cycle the frame is complete in 160 days, every storey carries the floors above it younger, and the worst floor’s thirty-year drop rises from 26.4 to 29.0 mm, at level 32. On a fourteen-day cycle it falls to 23.5 mm, at level 30. The worst floor moves up the building as the cycle quickens, because the young storeys are the ones high in the frame, loaded within days of being cast by the floors that follow them.

None of that changes the compensation’s logic. The best share stays near four fifths, because it is a balance between the movement before the fit-out and the movement after, and a faster cycle increases both. What it does change is the size of everything: a programme that saves three days a floor buys about a tenth more differential, which is the same arithmetic the creep that belongs to the member found for a single loaded member, accumulated up a building.

The movement compensation cannot touch #

There is a second question hidden in the first, and the limit that depends on a date is its general form. A level error is a property of the floor; it is what a spirit level on the slab reads. But the damage — a partition cracking, a door frame racking, a façade joint closing — is done by the movement that happens after the partition, the frame or the joint was built.

Setting a floor high moves where its history starts and where it ends by the same amount, and changes nothing in between. A partition built at the fit-out sees the 12.4 mm the worst floor drops afterwards whether the floor was set high by nothing, by four fifths or by everything. Compensation chooses the date a floor is level; it does not choose how far the floor moves after anything is built on it. That movement is set by when the finishes arrive. Fitted out as the frame finishes, they see 15.7 mm; three months later, 12.4; a year later, 8.9. The finishes’ own weight is part of what moves them — a quarter of the load arrives with them, and its creep is in every one of those numbers — so waiting reduces the movement but cannot remove it.

The date of the fit-out also moves the best compensation, which is the one place the two questions touch. A building fitted out as its frame finishes is first seen with less of its movement behind it, so its floors would stand higher if fully compensated, and the best share falls to 0.71 with a worst error of 13.6 mm. Fitted out a year later, more of the movement is done before anyone looks, the best share rises to 0.84 and the worst error falls to 9.2 mm, against 12.8 for full compensation. A later fit-out is the one decision that helps the level and the movement together, and it is a decision about the programme, not about the structure.

The design responses are therefore two different kinds of thing. Compensation is a surveyor’s instruction, applied floor by floor as the frame goes up, and it answers the question of level. Movement joints at the head of partitions, slotted connections in the façade and a programme that leaves time between the frame and the finishes answer the question of movement, and no compensation helps them. The same split runs through every case where a structure is built to the wrong shape on purpose: a cambered beam arrives level under its dead load and still deflects by every millimetre of the load that follows, and the settlement that matters is the difference between two footings rather than how far either has gone.

The worst floor by hand #

The thirty-year movement of level 31 can be checked against two pieces. Its elastic part, from load arriving after it is set — the nine floors above it and the finishes — is about $(12 - 6)/34{,}000$ times the storey height times the load still to come, summed over the thirty-one storeys below: roughly 8 mm, which is the figure the creep-free building gives for the level’s peak. The creep multiplies that, by a coefficient larger than 2.2 because much of the load lands on storeys a few weeks old, and adds the creep of the three quarters of the frame’s load already on the storeys when level 31 is cast. The two together make the 26.4 mm. The first piece is what the instantaneous model counts with a multiplier; the second is what it leaves out.

The best share follows from two numbers the figures give. The roof at its fit-out under the least creep has dropped 34 per cent of its predicted 24.1 mm, so a share $c$ leaves it standing high by $(c - 0.34) \times 24.1$ mm; level 33 at thirty years under the most creep has dropped 1.21 times its predicted 26.3 mm, so it stands low by $(1.21 - c) \times 26.3$ mm. The two are equal when $c(24.1 + 26.3) = 0.34 \times 24.1 + 1.21 \times 26.3$, which is $c = 0.79$, and either side is then 11.0 mm. The best share is the point where the floor that has moved least when first seen and the floor that will move most are wrong by the same amount in opposite directions — a balance between two floors in two different states, which no single floor’s prediction could have found.

Plain concrete and one creep law #

The columns and the core are plain concrete. Reinforcement carries a growing share of a column’s load as the concrete creeps, and a heavily reinforced column shortens noticeably less than its concrete alone would; the core, usually more lightly reinforced in proportion, less so. The differential then shrinks, and the share worth compensating moves with it.

Shrinkage is left out, because the earlier essay found its differential depends on the two members’ different drying rather than on the load, and it peaks at the roof. Added, it would push the worst floor higher still.

The creep’s uncertainty is one number for the whole building. In reality the column and the core may creep differently from their predictions, and in opposite directions; the error in a differential can then be larger than the error in either, which widens every band here.

The floors are perfect. A floor is cast with its own tolerance of several millimetres, set by a surveyor reading an instrument on a moving structure; the compensation is applied on top of whatever that tolerance leaves. The gap nobody computed found that the construction tolerance is usually the largest term in a movement budget, and here it may be comparable to the error the best share saves.

The slab between column and core #

They cannot show the floor’s slab, which spans from column to core and is bent by the difference as well as tilted by it, nor the beams framing into the core, which are the members that turn a differential into a force. They cannot show measurements, which are the one thing that could tell a designer which way the creep is going: a building instrumented as it rises can have its compensation corrected floor by floor, and the share chosen here is the one to use before anything has been measured.

Still open: the compensation that is corrected as the building rises #

Every share here is fixed before the first floor is cast. A frame rises over months, and the shortening of its lower storeys can be measured long before its upper floors are set — a survey of the column and core levels every few floors gives the creep the building actually has, not the one it was designed for. Whether a compensation recomputed every ten floors from what the lower storeys have already done can bring the upper floors’ share back towards one, or whether the creep that matters most — the decades after the fit-out — is exactly the part no measurement during construction can see, is the question of how much an instrumented building can know about itself while it is still being built.