The limit that depends on a date

A deflection limit looks like a property of a beam. Span over two hundred and fifty, span over three hundred and sixty, span over five hundred: a number to divide the span by and a number to compare with what the beam does.

The one that usually governs is not of that kind at all. It is a limit on the deflection occurring after the brittle finishes are built — after the plaster, the blockwork partitions, the glazing, the stone cladding — and it is therefore a limit on a difference between two times, not on a deflection. The same beam, in the same building, under the same loads, passes or fails depending on the day the partitions went up.

Which free body produced the number #

There is no free body here; there is a timeline, and it is the thing the calculation is about.

The beam is cast or erected at some age. Its own weight comes on immediately and starts creeping. The finishes go on at some later date, at which point the deflection so far is set into the building as its datum — the partition was cut to fit the floor as it stood that day. From then on, every further millimetre is an increment that the partition has to accommodate, and it accommodates it by cracking.

So the quantity to compute is

$$\delta_{inc} = \delta_{total} - \delta(t_1)$$

where $t_1$ is the day the finishes went on, and both terms include creep measured from each load’s own age at loading. That is the only way the programme can enter the answer at all, and it is the reason the answer is a date.

The shape of the creep function does the rest. A creep coefficient reaches roughly half its final value in the first three months and takes years over the remainder, so a partition built at four weeks inherits about 56% of the creep still to come and one built at a year inherits about 40%. The increment falls from 32 mm to 14 as $t_1$ runs from a week to a year, on a beam whose total moves by a fifth of that.

Why camber cannot touch it #

Camber is the standard answer to a total-deflection problem: build the beam bent upward by what it is going to drop, and the finished floor is flat. Built to the wrong shape on purpose is the operation, and it works.

It does nothing for the increment, and the reason is one line of arithmetic. Camber is subtracted from the deflection at every time, including both of the times in the difference:

$$(\delta_{total} - c) - (\delta(t_1) - c) = \delta_{total} - \delta(t_1)$$

The camber cancels exactly — not approximately, not to within the accuracy of the creep model, but identically. A cambered beam and an uncambered one of the same section have the same incremental deflection to the last decimal place.

That is worth stating as flatly as possible because the mistake it prevents is expensive and common: a beam is found to fail its total deflection check, camber is added, the total check passes, the drawings go out, and the partitions crack anyway. The check that camber fixed was not the check that was going to break anything.

The difference of two large numbers #

There is a second reason the incremental check deserves care, and it is about precision rather than about programme.

The total is 35 mm and the pre-finishes deflection is 12, so the increment is 23. A ten per cent error in the first term is 3.5 mm and a ten per cent error in the second is 1.2 — and if they happen to have opposite signs the increment is out by 4.7 mm, which is twenty per cent of it.

The relative uncertainty in a difference is larger than the relative uncertainty in either term, by the ratio of the terms to the difference — here about 1.5. And the terms are not precise: they contain a creep coefficient known to perhaps ±30%, a modulus known to ±15%, a cracked stiffness that depends on the load history, and tension stiffening that decays.

So the honest position is that an incremental deflection is a number with a large uncertainty compared against a limit chosen without much evidence. Which is an argument for designing to it with margin rather than to three significant figures, and for spending effort on the one variable that is both large and controllable: the date.

Propping, which moves the two checks in opposite directions #

A composite steel-and-concrete beam can be propped during the pour or left unpropped, and the choice does something to these two checks that nothing else in the calculation does: it improves one and worsens the other.

Unpropped. The wet concrete is carried by the bare steel beam. That deflection happens during construction, before any finish exists, and belongs to neither check — it is simply somewhere the floor is. The composite section then carries everything afterwards.

Propped. The wet concrete is carried by the props, and when they are struck the composite section takes the whole of it. That deflection happens after the slab has hardened, so it counts toward the total; it also happens before the finishes, so it does not count toward the increment.

The result is that the unpropped beam has a larger total deflection in absolute terms and a smaller one in the checks, and a smaller increment as well. On the beam drawn: propped, 0.72 on the total check and 0.96 on the increment; unpropped, 0.55 and 0.76.

Which inverts the usual instinct — propping feels like the careful option — and it means that a decision made on site for reasons of access and programme has decided a serviceability check nobody re-ran. The section that changed while it was being loaded is the same sequence problem for stress; this is it for deflection.

The arithmetic on one beam #

It is worth following one member through, because the shape of the calculation is unfamiliar even to people comfortable with the individual pieces.

A 12 m span, $EI = 7.7\times10^{14}$ Nmm². Permanent load 14 kN/m, finishes 6, imposed 12. Cast at seven days; finishes at twenty-eight; fifty years of service.

The instantaneous deflections are $5wL^4/384EI$ for each: 4.9 mm for the permanent load, 2.1 for the finishes, 4.2 for the imposed.

At twenty-eight days the permanent load has been on for twenty-one days and has a creep coefficient of 1.36, so the deflection then is $4.9\times2.36 = 11.6$ mm. That is the datum.

At fifty years the permanent load’s creep coefficient is 3.15, so it contributes $4.9\times4.15 = 20.4$. The finishes have been on since day twenty-eight with a coefficient of 2.39, contributing $2.1\times3.39 = 7.1$. The quasi-permanent part of the imposed load — 30% of it — creeps too, contributing $4.2\times0.3\times3.39 = 4.3$. The remaining 70% of the imposed load acts instantaneously, adding 2.9. Total 34.7 mm.

The increment is $34.7 - 11.6 = 23.1$ mm. The limits are $12000/250 = 48$ and $12000/500 = 24$. The beam is at 72% of the first and 96% of the second, and the second is the one nobody would have got from a table.

Move the finishes to day one hundred and the increment falls to 18 mm, 76%. Move them to day seven and it is 32 mm, and the beam fails.

What the limits are protecting #

It is worth knowing what the two numbers are for, because they protect different things and only one of them protects the structure at all.

Span/250 on total is about appearance and about drainage. A floor that has sagged by more than a two-hundred-and-fiftieth of its span looks sagged; a roof that has done so may pond, which is a load that increases the deflection that increases the load — the water that will not run off is that instability, and it is the one deflection limit in this collection with a collapse behind it.

Span/500 on the increment is about the finishes, and the number has no derivation behind it at all — it is roughly the deflection at which blockwork built tight to a soffit begins to crack, established by observation and rounded. A blockwork partition built tight to a soffit cracks at a deflection of a few millimetres over its length, glazing binds in its frame, a stone façade panel loses its joint. None of that endangers anything and all of it is expensive.

There is a third limit that is not a length at all: the vibration criterion, which is about frequency and response rather than displacement, and which governs a great many long-span floors that pass both deflection checks comfortably. The floor that is strong and unusable is that one.

The details that make the check unnecessary #

There is a response to all of this that is better than any calculation, and it is worth putting beside the arithmetic because a designer’s first instinct is to make the beam deeper.

Do not build the partition tight to the soffit. A head detail with a compressible joint of twenty millimetres, a sliding channel, or a deflection head accommodates the increment without cracking, and it costs a fraction of the steel that a deeper beam costs. The limit exists because the detail is usually a hard one, not because the deflection is intolerable.

Build the finishes late. The figure’s whole message is that the date is a free variable worth tens of per cent, and on a programme where the frame is topped out months before fit-out begins, most of the benefit is already there and simply needs to be claimed.

Or make the increment somebody else’s problem. A demountable partition, a raised floor with a tolerance in its pedestals, a façade on brackets with slotted holes: each of them converts a structural limit into a detailing allowance, and the allowance is the cheaper purchase every time.

That is a general move worth naming and it recurs throughout serviceability. A serviceability limit is a contract between the structure and something attached to it, and either party can be the one that changes. The gap nobody computed makes the same argument about movement joints, and where the structure is allowed to move makes it into a scheme. Strength limits have no such option; there is nobody to negotiate with.

Where the model stops #

Cracking was not tracked. A reinforced concrete member’s stiffness depends on how much of it has cracked, which depends on the largest moment it has ever seen — so a beam that was overloaded during construction is permanently softer, and the increment computed on an uncracked or fully cracked section brackets the real answer rather than giving it. Stiffer than its cracked section says is the term in between, and it decays.

Shrinkage was ignored. A concrete member with more reinforcement in the bottom than the top shrinks unevenly and curves upward or downward with no load at all, and that curvature arrives on the same timescale as the creep. The curvature nobody applied is a real component of the increment and is often ten to twenty per cent of it.

The quasi-permanent load was a guess. The fraction of the imposed load treated as sustained — 0.3 for an office, 0.6 for storage — decides how much of the live load creeps, and it is a convention rather than a measurement.

Span-to-depth ratios hide all of this. The usual first sizing of a member is a span-to-depth rule, which is a strength-and-deflection rule for a beam under a typical loading with a typical programme. It is a good rule and it contains none of the variables in this essay — depth is the cheapest strength is why it works at all, and the reason it fails is nearly always a programme or a finish it did not anticipate.

And the programme was assumed known. It is not, at design stage. The honest response is either to design for the earliest credible date, or to state the date the design assumed on the drawings so that somebody can object to it — and the second is rare enough to be worth recommending.

The generalisation #

The habit to take from this is to ask, of any limit: is this a limit on a quantity or on a change in a quantity? Because the two behave completely differently under everything a designer can do.

A limit on a quantity responds to anything that reduces the quantity. A limit on a change is blind to anything that shifts both ends of it equally — camber here, but also a datum shift, a pre-set, a lock-off force, an initial reading. Those operations are extremely useful against the first kind of limit and completely useless against the second, and the failure is always the same shape: the operation is applied, the check it was aimed at passes, and the check that was going to bite is untouched.

The same distinction sorts a lot of serviceability. The settlement that matters is the difference — a building can go down a hundred millimetres uniformly and be fine, and twenty differentially and not be. Drift is limited over a storey rather than over a building, because it is the change that racks the cladding. Crack width is a limit on a width and not on a strain. Temperature movement is limited by what a joint can take, which is a change and not a position.

And there is one more member of the family worth naming, because it is the one nobody checks: the change during a repair or an alteration. A structure that has already taken its creep and settled into its finishes has a new datum, and any operation that moves it — removing a wall, adding a load, jacking a support — starts a fresh increment against finishes that are already there. The limit is the incremental one, and the clock starts again — against finishes that have already used up their allowance once. A structure’s second life is checked against a datum that is not on any drawing, and nobody measured it before the work started.