Stiffer than the model said

Measure the natural frequency of a finished floor and compare it with the number computed for it during design. The measured value is higher, by between ten and sixty per cent, and it is higher on nearly every floor anybody has ever measured.

That is a very unusual kind of discrepancy. Most modelling errors scatter around zero: some models are too stiff, some too soft, and the mean is not far off. This one is one-sided.

Which free body produced the number #

The model’s, and the point is that the model’s free body is smaller than the building’s.

A floor model contains beams, a slab and columns. The building contains all of those plus a facade fixed to the slab edge, partitions standing on the slab and touching the one above, a raised floor, a ceiling, services hung from the soffit, a connection detail that was drawn as a pin and behaves as a spring, and concrete whose modulus is higher at thirty years than the value assumed at twenty-eight days.

Every one of those adds stiffness. None of them adds a comparable amount of mass, because they are light. So the frequency, which goes as the square root of stiffness over mass, moves up.

The four contributions #

Cladding and partitions. A facade fixed at every floor is a shear panel of some stiffness — the effect the floor is a beam lying down relies on deliberately, arriving here by accident — and a partition standing on one slab and touching the next is a very stiff spring in a direction the model has nothing at all in. Between them they are worth 5 to 25 per cent of the stiffness of an ordinary floor, and the number depends on a fit-out that will change three times during the building’s life.

Partial fixity at nominally pinned joints. Neither pinned nor rigid is the argument in full: a beam-to-column connection detailed as a shear-only fin plate has a rotational stiffness that is not zero, and a stiffness that is 10 per cent of the beam’s is worth 5 to 20 per cent on the beam’s own deflection.

Unintended composite action. A steel beam with a concrete slab cast on it acts compositely to some degree whether or not shear connectors were provided, because friction and bond across the interface carry some of the shear flow. Two beams or one is the mechanism; the consequence here is that a non-composite beam designed as non-composite is 10 to 40 per cent stiffer than the calculation.

Concrete stiffer than assumed. Between cracks a reinforced member is uncracked, so its effective stiffness is above the fully cracked value — stiffer than its cracked section says. And the modulus rises with age: a mix specified at 30 N/mm² is routinely at 40 by the time anybody measures it, and the modulus goes as roughly the cube root, so that is another 10 per cent. A floor also vibrates at very small strains, where concrete’s tangent modulus is higher than the secant value a deflection calculation uses.

Multiply the four together and the stiffness ratio is around 1.8, which is a frequency ratio of 1.35. That is a floor. On a whole building the same list applies to the lateral system, with cladding and partitions contributing a great deal more, and measured sway periods of tall buildings come out shorter than computed by a similar margin — which matters for a structure has more than one period and for every seismic force derived from one.

Why the same factor looks small on one axis #

Here is the arithmetic that makes the whole effect easy to miss.

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

so a factor $r$ on stiffness is $\sqrt{r}$ on frequency and $1/r$ on deflection. A stiffness ratio of 1.8 is:

• a deflection 44% smaller than computed — a large and obvious discrepancy;
• a frequency 35% higher than computed — noticeable;
• a period 26% shorter — modest.

The square root is a compressor. It takes a factor that would be glaring in a deflection and makes it look like ordinary scatter in a frequency, which is one reason the effect went unremarked for so long even though it is measured every time anybody bothers.

The measurement, and why it is trustworthy #

It is worth saying why the discrepancy is believed, because “the building is stiffer than the model” is the kind of claim that could easily be an artefact of how frequencies are measured.

It is not, and the reason is that a natural frequency is one of the very few structural quantities that can be measured without loading the structure at all. Put an accelerometer on a floor, record ambient vibration for twenty minutes, take a spectrum, and the peaks are the natural frequencies. No load has to be applied, no reference has to be established, and the answer does not depend on knowing the mass, the damping or anything else — the frequency is where the peak is.

That makes the comparison unusually clean. A deflection measurement needs a known load, a datum and a way of separating the structural movement from the support’s; a strain measurement needs a gauge factor and a zero. A frequency needs a spectrum and a peak, and both sides of the comparison — measured and computed — are the same quantity.

Which check the asymmetry spoils #

Now the consequence, and it is the reason this essay exists.

For deflection, leaving stiffness out is conservative. The model predicts more movement than the building will have, the member is sized on the larger number, and the building is better than predicted. That is the safe direction and it is why nobody objects.

For vibration, leaving stiffness out is unconservative in the direction that matters. A floor’s vibration performance is judged partly on whether its fundamental frequency is above the range that footfall excites — around 4 to 5 Hz for a walking pace and its harmonics. A model that under-predicts the frequency places the floor closer to that range than it will be, so the check fails on floors that would have passed.

That sounds harmless — a floor rejected that would have been fine is an expensive mistake rather than a dangerous one. Two things make it worse than that.

It is not always in the safe direction. A floor whose computed frequency is just above the first harmonic and whose real frequency is a third higher may have moved onto the second, and the response at a higher harmonic is smaller but not negligible. The mapping from frequency to acceptability is not monotonic, so a shift of a third can go either way.

And a floor is often stiffened to pass a check it did not need. The remedy for a failed footfall check is a deeper beam or an extra column, both of which cost money and space, and both of which are being bought against a prediction that is known to be biased.

A worked case, and what it costs #

Take an office floor: a 9 m composite beam grid, a computed fundamental frequency of 4.4 Hz, and a criterion that wants 5.2 Hz to avoid the second harmonic of a brisk walking pace.

The model fails by 0.8 Hz. The response is one of two things. Deepen the beams by 15% — which raises the frequency by about 7% and costs steel, floor-to-floor height and, on a thirty-storey building, an entire floor over the height of the tower. Or add columns, halving the span, which raises it by 40% and costs a column grid nobody wanted.

Now put the four sources of stiffness back. The stiffness ratio is about 1.8, the frequency ratio about 1.35, and the real floor is at 5.9 Hz — comfortably past the criterion, on the beams that were originally drawn.

The remedy was bought against a bias. And there is no way to know, from the calculation alone, which of the two situations is in front of the designer: a floor that genuinely fails, or a floor whose model does. Only a measurement on a comparable completed building distinguishes them, and by then it is somebody else’s building.

That is the practical shape of the problem, and it is why the interesting engineering response is not a better model. It is a feedback loop: measure the floors that get built, compare them with what was predicted, and use the ratio to calibrate the next prediction. That is ordinary practice in aeronautics and almost unknown in buildings, for the plain reason that nobody owns the measurement — the designer has left, the contractor has been paid, and the occupier has no instrument.

Why the correction cannot simply be made #

The obvious response is to include the four sources and stop under-predicting. It is not available, and the reason is the distinction this collection keeps drawing between what is there and what may be relied on.

Partitions are demountable. Cladding is replaced at forty years. A fin plate connection may be built exactly as drawn on the one job where it matters. Composite action without connectors is real and is not permitted to be counted for strength. Every one of the four is stiffness that exists and that no engineer may design against, because the building may be stripped back to the model on any Monday.

So the profession has arrived, without ever quite saying so, at a position where the model is deliberately wrong and everybody knows by roughly how much. Which would be tolerable if the compensation were explicit, and it is not: there is no factor anywhere in a footfall calculation labelled “the building will be 35% stiffer than this”.

The nearest thing to a resolution in current practice is to do the calculation twice: a lower-bound model for strength and deflection, and an upper-bound one, with the non-structural stiffness included, for the vibration check. Two models of one building, each deliberately wrong in a different direction, each used for the questions where its error is safe.

The same asymmetry in three other places #

Once the shape of the argument is visible it turns up repeatedly, and always for the same reason: a simplification made for one check being read by another.

Seismic period. A code’s period formula is deliberately short, because a short period gives a large spectral acceleration and therefore a conservative force. It also gives a small spectral displacement, which is unconservative for a drift check and for anything that depends on how far the building moves — a separation gap, a pounding check, a cladding connection. The earthquake asks for a displacement is the essay about that inversion, and it is the same one-sided assumption read two ways.

Cracked section properties. Taking a concrete frame as fully cracked is conservative for deflection and for the force attracted to it, and unconservative for the force attracted to everything else — because the stiffest path takes the load, and softening one element in a model hands its load to the others.

Damping. A low assumed damping is conservative for a resonant response and unconservative for anything where damping limits a movement being relied on, such as a tuned mass system.

In all three the assumption is defensible, standard, and correct for the check it was chosen for. What goes wrong is that a model is built once and asked many questions, and the sign of its error changes between them without anything in the model recording that it has.

Where the model stops #

Damping was ignored and it is the larger uncertainty. A floor’s response goes as one over the damping ratio, and the ratio ranges from 0.5% for a bare structure to 3% for a fitted-out one — a factor of six on the answer, against a factor of 1.35 on frequency. The non-structural elements that stiffen a floor damp it far more.

The four sources were combined as independent multipliers. They are not: a facade that stiffens a slab edge also changes where the mode’s amplitude is, and composite action changes both the stiffness and the mass distribution.

Only the first mode was considered. A floor’s response to footfall involves several modes, and the higher ones are affected differently by the same non-structural stiffness.

The mass was assumed known. It is not, quite: a floor’s superimposed dead load is an allowance, its live load during a measurement is whatever happens to be in the room, and both go into the denominator. A 10 per cent error in mass is 5 per cent in frequency, in the same direction as the stiffness error rather than against it — the design mass allowance is generous, so the real mass is lower, so the real frequency is higher again.

And the measured data is biased in its own way. Buildings get measured when somebody is worried about them, or when a researcher can get access, and neither sample is random.

The generalisation #

The habit is to ask, of every deliberate simplification, which direction it errs in for each check it feeds.

A conservative assumption is conservative for a purpose. Neglecting stiffness is conservative for deflection, unconservative for frequency, unconservative for a seismic force computed from a period, and conservative for a seismic displacement. One assumption, four checks, four different signs — and the assumption is usually made once, at the top of the model, by somebody thinking about one of them.

There is a companion habit for the moment a simplification is made, and it is cheap. Write down, next to it, which way it errs — one word, safe or unsafe, for each of the checks the model will be used for. Most simplifications are made once and used a dozen times, and the one-word annotation is the only record that the person who made it had thought about the twelfth use. The envelope is not a structure makes the same point about load cases: a single set of results serves many questions, and the assumptions inside it were made with one of them in mind.

The second reading is about square roots and about how much a factor looks like. The load that will not hold still is full of quantities that go as a square root: a frequency in a stiffness, a period in a mass, a wave speed, a rocking block’s overturning acceleration. In every one of them a large error in the underlying quantity presents as a modest error in the observable, and the temptation is to conclude that the underlying quantity is well known. It is not. It is being viewed through a function that flatters it.