The columns are shorter than the core

A 40-storey concrete column at a working stress of 12 N/mm² shortens by about 26 mm elastically over its height. Creep takes that to 83 mm, and shrinkage adds another 43, so the column is about 127 mm shorter than the length of concrete that was cast into it.

That number is quoted often and it is almost never the number that matters. Every floor was built to its correct level on the day it was built, so the accumulated shortening beneath it had already happened and was already levelled out. What a finished building shows is only the part that arrived after each floor was set — and that part is much smaller, differently distributed, and shaped like nothing an intuition about accumulation would suggest.

Why the middle and not the top #

Take a floor at level $k$ of $N$. It is built, levelled, and then the building above it continues.

What shortens the columns beneath it afterwards is the load from levels $k+1$ to $N$, which is a fraction $(N-k)/N$ of the final stress. That falls to zero as $k$ approaches the roof.

How many storeys of column are beneath it to do the shortening is $k$, which rises from zero at the base.

The post-installation shortening at level $k$ is the product:

$$\delta(k) \;\propto\; k\,(N-k)$$

which is a parabola, zero at both ends and maximum at $k = N/2$.

Both ends being zero is worth sitting with, because both are surprising for opposite reasons.

Zero at the base because there is no column beneath the ground floor to shorten. Obvious once stated.

Zero at the roof because nothing is built above the roof. The columns beneath the top floor do continue to creep and shrink, but the top floor was set at the very end of construction, and by then the elastic part of the movement had already happened. The top of a tall building is the one place where differential shortening is not a problem.

The difference, not the movement #

The parabola above is the column’s post-installation shortening. The core does the same thing at its own stress, and the floor spans between them, so what matters is the difference.

For the building drawn, the column works at 12 N/mm² and the core at 6 — cores are sized for wind and lateral stiffness rather than for gravity, so they are habitually at half the stress of the columns they stand beside. That is the stiffest path taking the load inverted: the stiffest element here is deliberately the one carrying least. The differences run:

The peak differential is 20.3 mm, over a 9 m bay from the core to the column. That is a floor out of level by one in 443, which is about twice the tilt a person can detect underfoot and roughly the tolerance a raised floor system is set to.

Which free body produced the number #

Every quantity above is $\sigma L/E$, applied storey by storey, and the only difficulty is bookkeeping.

Take one storey of column, of height $h$, at a stress $\sigma_i$ once the building is complete. Its total elastic shortening is $\sigma_i h/E$. Multiply by $(1 + \phi)$ for creep and add $\varepsilon_{sh} h$ for shrinkage, both of which act on the whole storey whatever the load history.

Then apply the sequence: for a floor at level $k$, only the load applied after level $k$ was set contributes. The stress from that load is $\sigma_0 (N-k)/N$ for every storey below, and summing over the $k$ storeys gives the parabola.

The arithmetic is elementary and the failure mode is not: using the total rather than the post-installation part overstates the problem by a factor of six for this building, and using the column’s movement rather than the difference from the core overstates it by a factor of two on top of that.

Shrinkage points the other way #

The second curve on the hero figure is a different mechanism with a different shape.

Shrinkage does not care what is built above. A concrete element loses moisture and gets shorter on its own schedule, so its contribution accumulates with the number of storeys and is largest at the top:

$$\delta_{sh}(k) = \varepsilon_{sh}\,h\,k$$

which is a straight line, zero at the base and largest at the roof.

When the column and the core are both concrete this cancels almost entirely — both shrink, both by roughly the same strain, and the difference is small. When they are different materials it does not. A steel perimeter column against a concrete core has no shrinkage at all on one side and 300 microstrain on the other, so the core gets shorter than the column by 43 mm at the roof, and the tilt reverses direction on the way up.

That is the crossing visible on the first figure: the load-driven part makes the column shorter, peaking in the middle; the shrinkage-driven part makes the core shorter, peaking at the top; and somewhere around two thirds of the height the sum passes through zero and changes sign. A floor that tilts one way at level 15 tilts the other way at level 35.

All three of the mechanisms on this page belong to one family. Creep, shrinkage and temperature are movements nobody applied — the structure is asked to change length by something that is not a load, no equilibrium equation contains them, and what happens next depends entirely on what is restraining the change rather than on how large it is. That is why the two curves above can be added without any structural analysis at all, and why the section after this one is about a connection rather than about a member.

What it does to the structure #

The differential is a support settlement as far as the floor beams are concerned: one end goes down relative to the other, by 20 mm over 9 m, and an indeterminate floor system responds with forces.

Which limit it threatens is worth stating once. Nothing on this page is a strength problem — the forces induced are modest and they relax by creep — but everything on it is a fit problem, and fit has no reserve factor: a partition either meets the slab above it or it does not.

The consequences that actually arise are all about things that are not structure:

Cladding. A curtain wall is fixed to the slab edges and has movement joints sized for a stated storey-to-storey change. A differential of 20 mm distributed over the height as a parabola means the joints at mid-height close and the ones near the top do not.

Lifts. A lift guide rail is fixed to the core and must stay straight and plumb over the full height. The core moving relative to the floors that brace the shaft is exactly the movement the rail brackets have to absorb.

Services. A riser fixed to the core and branching into a floor supported by columns crosses the differential at every level.

And finishes. Partitions, screeds and level thresholds all record a tilt that developed after they were installed.

Whether it is a tilt or a force is decided by one connection #

Compensation deals with the level. There is a second response, and it decides whether the differential produces any force at all — because 20 mm over 9 m is an imposed support movement, and what an imposed movement does depends entirely on what is restraining it.

If the floor beam is simply supported, it accommodates the whole differential by tilting. Its two ends rotate relative to one another by 20/9000 = 0.0022 radian — an eighth of a degree — and it carries not one kilonewton-metre more than it did before. A determinate member cannot feel a support movement.

If the floor beam is moment-connected at both ends, it cannot tilt without bending. A relative end settlement $\delta$ on a fixed-ended beam produces end moments of $6EI\delta/L^2$, and for a floor beam of $EI = 94{,}000$ kN·m² over a 9 m bay with 20 mm of differential that is

$$\frac{6 \times 94{,}000 \times 0.020}{81} = 139\ \text{kN·m}$$

which is not a correction. It is comparable with the moment the beam was designed to carry, arriving from a movement nobody applied, in a member whose load case list contains nothing like it.

So the connection detail at the core is the whole of the difference between a fit problem and a strength problem, and it is chosen for other reasons. That is the argument for the near-universal practice of connecting floor beams to a core with a pinned detail — a fin plate, a shear plate, a slotted connection — rather than a moment connection. It is not primarily about the frame’s stability, which the core is supplying anyway; it is about refusing to carry an incompatibility that is guaranteed to arrive.

Two qualifications keep it honest. The connection has to be genuinely able to rotate through 0.0022 radian, which is easy, and it has to keep being able to after fireproofing, cladding brackets and a screed have been fitted around it — which is a detailing matter that nobody checks against a rotation. And creep relieves whatever force does arise, because the movement it is responding to is slow: a moment induced over five years in a concrete structure is relaxed by the same creep that caused the movement, so the elastic 139 kN·m is an upper bound and the sustained value is a fraction of it.

That relief is exactly what is not available where the connection is deliberately stiff and the movement is deliberately resisted — which is why the members that genuinely have to be designed for this are the ones tying a core to its columns on purpose.

There is a general rule in this worth taking away from the arithmetic. The differential itself is fixed by the stresses, the materials and the height, and no detail changes it: the column will be 20 mm shorter than the core whatever anybody does. What the detail chooses is where that 20 mm is absorbed — as a rotation in a connection, as bending in a beam, or as a crack in a partition. It is going somewhere, and the only decision available is which of the three is asked to take it. A design that has not decided has chosen the third by default, because a finish is the softest thing in the load path and an incompatibility finds the softest thing.

Compensation is the answer, and it is applied during construction #

Nothing here is repaired by analysis. The remedy is to build each floor higher than its nominal level, by the amount it is predicted to drop after it is set.

It is the same instruction a cambered beam carries — fabricate the member in the wrong shape so that the right shape arrives with the load — applied to a level rather than to a profile, and issued to a surveyor rather than to a fabricator.

The compensation follows the parabola, so it is largest in the middle of the building and zero at both ends — floor levels near the top are set nominally, floors at mid-height are set 20 mm high, and the building arrives level. It is applied by the surveyor rather than by the designer, from a table the designer produces, and it depends on a prediction of creep and shrinkage made years before the movements finish — the same prediction a cambered beam depends on, and just as hard to check afterwards.

Which is the honest weakness of the whole exercise: the compensation is exact and the prediction is not. Creep coefficients carry an uncertainty of perhaps ±30%, and the compensation carries that uncertainty into the built levels.

It gets worse with height, linearly #

The parabola’s shape is fixed by the sequence and its size is fixed by the height, so the whole height argument can be made by drawing the same picture at three heights and reading the peak off each.

Half a millimetre per storey, for these stresses and this bay. That is why differential shortening is a subject for tall buildings and a footnote for short ones.

At twelve storeys the differential is inside the tolerance the building was set out to; at eighty it is 41 mm and outside everything.

And it is one of the few problems in this collection that gets worse in proportion to the height rather than to some power of it. Span to the fourth is the usual shape of a serviceability limit, and this one is linear. Wind moments grow as the square of the height, overturning as the cube of it, and drift as the fourth power. Differential shortening grows linearly — which means that at some height it stops being the governing issue, and below that height it is a nuisance rather than a design driver.

Why the core and the columns differ at all #

The whole subject exists because two vertical elements in the same building work at different stresses, and it is worth saying why they do rather than treating it as an accident.

A core is sized for lateral stiffness. Its walls are thick and long because the building has to resist wind and to keep its drift inside a limit, and what a tall building’s lateral system is for has almost nothing to do with how much gravity load the core carries. Having been sized that way, it has a great deal of area and a modest share of the weight, so its working stress is low.

Every wall in that core was sized by a lateral-drift calculation, and its gravity stress is whatever is left over once the wind has decided the thickness.

A column is sized for the gravity load it carries and nothing else. It is made as small as the load allows, because floor area is the product being sold and a column occupies some. So its working stress is as high as the code permits.

The ratio between the two is therefore not a design choice — it is a consequence of two elements being sized against two different criteria, and it is stable across buildings at around two to one. Which is why the differential is so reliably present, and why it is a core-against-column problem rather than a general one about columns.

Two ways of removing it exist and both cost something. Sizing the column for stiffness rather than strength wastes floor area on every storey. Sizing the core down until its stress matches the columns’ costs lateral stiffness, which is the thing the core was there for. Nobody does either, which is why compensation is the standard answer.

Where the model stops #

The stress profile is assumed linear with height. Real columns are sized in groups, so the stress steps rather than tapers, and the steps put kinks in the parabola. The shape survives; the smoothness does not.

Creep is modelled by a single multiplier. Real creep depends on the age at loading, the member’s size, the humidity and the load history — and every storey is loaded at a different age. A proper calculation is a step-by-step integration with an ageing coefficient, and it gives a different number, usually larger near the base.

Reinforcement has been ignored. Steel in a concrete column does not creep or shrink, so it restrains the concrete around it and picks up load over time — which reduces the column’s shortening and increases the stress in the bars. A heavily reinforced column shortens appreciably less than this calculation says.

The core is treated as a single element at one stress. A real core has walls at different stresses, coupling beams between them, and openings, and its own shortening varies across its plan.

And nothing here is a temperature. A perimeter column exposed to outside air and a core in the middle of a heated building differ in temperature by tens of degrees, and $\alpha \Delta T$ over 144 m is comparable with everything computed above.

What the pictures cannot show #

The shortening figures plot millimetres against metres — 20 mm of differential against 144 m of height, which is a ratio of 1 in 7,000. Every one of them is drawn with the horizontal axis exaggerated by three orders of magnitude, and the honest statement is that none of this movement is visible in a building.

Nor can they show time. Every curve is the final state, and the movements arrive over five to ten years with the elastic part on the day and the creep part asymptotically. A building handed over at eighteen months has completed perhaps 60% of what these figures show.

And the parabola is drawn as though every floor were built on the same schedule. Construction pauses, floors are built out of order, and a slab cast in winter shrinks differently from one cast in summer.

The ladder from here #

Later rungs on this anchor: step-by-step creep analysis with an ageing coefficient, and how much it moves the answer. The effect of reinforcement, and the long-term redistribution from concrete to steel inside a column. Outrigger structures, where the differential between core and columns is not merely a fit problem but a force — the outrigger is exactly the member that objects to it, and its forces can double over the life of the building. Compensation strategy: whether to compensate for elastic movement only, for elastic plus a fraction of creep, or for the predicted total. Composite megacolumns, where a steel core inside a concrete column changes the shortening and the fire rating at once. And the measurement question: what a building actually does, which is known from very few instrumented towers and differs from prediction by more than anybody would like.