Structural Engineering & Statics
where the load goes
Dynamics

The floor that is strong and unusable

A floor can satisfy every strength check, deflect less than the limit, and still be rejected by the people who work on it — because somebody walking across it at two steps a second happens to be exciting it at exactly the rate it likes to move.

Assumes The period nobody chose and The only thing that stops it.

17 min read 8 figures The load that will not hold stillWhich failure arrives first

The most common serviceability failure in modern buildings is not a floor that sags or cracks. It is a floor that bounces, on which nobody can be persuaded that anything is wrong by any calculation, because every calculation says the floor is fine.

Which floor frequencies a 2 Hz pace punishesThe response factor of a floor of 12 tonnes modal mass and 3.0% damping, against its own natural frequency, under a walker at 2 steps per second. The peaks are at 2, 4, 6, 8 Hz — the harmonics of the pace — and they fall away sharply: four peaks, each smaller than the one below it, because the harmonics of walking get smaller. A floor at 2 Hz reaches R = 31 and one at 8 Hz reaches 6. Between them the floor is quiet.2345678910051015202530floor frequency (Hz)response factor1× pace2× pace3× pace4× pace4 Hz — R = 106 Hz — R = 7
Fig. 1 The same floor — 12 tonnes of participating mass, 3% damping — at every natural frequency from 1.4 to 10 Hz, walked on at two steps a second. The response is not a curve that falls as the floor gets stiffer. It is a set of spikes at 2, 4, 6 and 8 Hz, each smaller than the last, with quiet ground between them. A floor at 4 Hz reaches a response factor of 11; a floor at 5 Hz reaches 2.4.

Nothing in a strength calculation contains the number 4. Nothing in a deflection check contains a pace. The whole of this problem lives in a coincidence between two frequencies, one of which is a property of a person.

Walking is four sine waves

A person walking puts down a force that repeats at the pace — about two steps a second, with a range from perhaps 1.5 to 2.5. Because it repeats, it can be written as a mean plus harmonics, and the harmonics are what matter.

Walking, as a force: four harmonics of the paceThe dynamic part of the force a walker puts into a floor at 2 steps per second, as a fraction of body weight, over three steps. It is four sine waves at 2, 4, 6, 8 Hz with amplitudes 0.40, 0.10, 0.06, 0.05 of body weight — Kerr's coefficients, measured from instrumented walking. The static weight is not drawn: it deflects the floor and does not shake it.00.20.40.60.811.21.4-0.6-0.4-0.20.20.40.6time (s)force ÷ body weight2 steps per secondthe sum — what the floor feelsharmonic 1 at 2 Hz, 0.40 of body weightharmonic 2 at 4 Hz, 0.10 of body weightharmonic 3 at 6 Hz, 0.06 of body weightharmonic 4 at 8 Hz, 0.05 of body weight
Fig. 2 The dynamic part of the force from a 700 N walker at two steps a second, as a fraction of body weight: four sine waves at 2, 4, 6 and 8 Hz with amplitudes 0.40, 0.10, 0.06 and 0.05 of body weight. The static weight is not drawn, because it deflects the floor and does not shake it. Everything a floor does about footfall is a response to one of these four components.

The coefficients are measurements — from instrumented walking on force plates — rather than derivations, and they are the one piece of pure empiricism in the calculation. What they establish is the shape of the problem: the first harmonic is four times the second and eight times the fourth, so which harmonic lands on the floor decides the answer as much as whether one does.

That is the whole content of the spike pattern. A floor at 2 Hz is resonant with a 0.40-of-body-weight force and reaches a response factor of 36. A floor at 4 Hz is resonant with 0.10 and reaches 11. A floor at 8 Hz is resonant with 0.05 and reaches 6.3. The peaks fall away because the harmonics do.

What “response factor” means

The unit of the vertical axis needs stating, because it is not a stress, a force or a deflection.

Human perception of vertical vibration is roughly constant in acceleration over the frequency range that floors occupy, and the threshold at which a standing person begins to notice is about 0.005 m/s² root-mean-square. A response factor is simply the multiple of that: R = 1 is the threshold of perception, R = 8 is a common limit for an ordinary office, R = 4 for a residence at night, R = 1 or below for an operating theatre or an electron microscope.

So a floor with R = 11 is producing accelerations eleven times what a person can just detect. It is not shaking violently — 0.057 m/s² rms is under a hundredth of gravity — and it is entirely obvious to anyone standing on it.

A 4 Hz floor under a walker at 2 steps per secondAcceleration against time for a floor of 4 Hz, 3.0% damping and 12 tonnes of modal mass, under a walker at 2 steps per second. Harmonic 2 of the pace falls at 4 Hz — 1.00 times the floor's frequency — and the response builds over several seconds to a peak of 0.11 m/s², an rms of 0.06 m/s², which is a response factor of 11 against the 0.005 m/s² threshold of perception.012345678-0.1-0.050.050.1time (s)acceleration (m/s²)4 Hz floor · harmonic 2 at 4 Hzresponse factor 11
Fig. 3 The acceleration history of the 4 Hz floor while a person walks across it. The response builds over several steps as the second harmonic of the pace drives it at its own frequency, peaks at 0.11 m/s², and decays when the walker stops. This is resonance in an office, produced by an entirely ordinary person doing nothing unusual.

Three ways to fix it, and their exchange rates

The response at resonance is the harmonic force divided by the modal mass, divided by twice the damping. Every term is a design variable and they are not equally useful.

Damping. Trebling it from 1% to 3% takes the response factor from 21.9 to 11.3, and to 5% takes it to 7.5 — inversely proportional, exactly as the resonance essay says it must be. This is the most reliable lever and the least available: the damping of a bare structure is what it is, and the additional damping from partitions and finishes is a bonus that cannot be counted on before the fit-out is designed.

Mass. Doubling the participating mass from 12 to 24 tonnes halves the response factor from 11.3 to 5.7. Mass is a genuine fix and it is unusual in this subject — everywhere else on this site, adding weight is the thing to be avoided. Here it is the answer, because the excitation is a fixed force rather than a fixed acceleration, and the person walking does not push harder because the floor is heavier.

Frequency. Not a lever at all, but a lottery. Moving a floor from 4 Hz to 5 Hz drops the response factor from 11.3 to 2.4. Moving it from 3 Hz to 4 Hz raises it from 3.4 to 11.3. The variable is not “higher is better”; it is “land between the harmonics”, and the harmonics of a pace that varies between 1.5 and 2.5 steps a second are bands rather than lines.

Which floor frequencies a 2 Hz pace punishesThe response factor of a floor of 12 tonnes modal mass and 1.0% damping, against its own natural frequency, under a walker at 2 steps per second. The peaks are at 2, 4, 6, 8 Hz — the harmonics of the pace — and they fall away sharply: four peaks, each smaller than the one below it, because the harmonics of walking get smaller. A floor at 2 Hz reaches R = 44 and one at 8 Hz reaches 13. Between them the floor is quiet.2345678910010203040floor frequency (Hz)response factor1× pace2× pace3× pace4× pace2 Hz — R = 444 Hz — R = 18
Fig. 4 The same sweep for a floor with 1% damping instead of 3% — a bare composite deck before any partitions or ceilings are installed. Every peak is three times higher: the 2 Hz spike reaches beyond 100 and the 4 Hz spike reaches 22. The shape is identical and the scale is not, which is the whole of what damping does.

Why long spans are worse for two reasons at once

A long-span floor has a lower frequency — the span-squared law from the period essay — which walks it down the spike pattern towards the larger harmonics. That much is obvious.

The second reason is less so and is usually the larger. A long span has a smaller participating mass relative to the walker, because the mode is confined to that bay and the person’s own weight is a bigger fraction of what is moving. A 6 m bay and an 12 m bay both put one walker on the floor, but the second one has that walker exciting a mode whose modal mass may be no larger, since the mass per square metre is fixed and the mode shape is what it is.

That combination is why the problem arrived with a change in construction rather than with a change in loading. Long-span composite floors with few internal partitions are lighter, more flexible and less damped than the short-span, heavily partitioned floors that preceded them — three variables all moving the wrong way, none of which appears in a strength calculation.

Knowing the frequency before anything is built

None of this is usable unless the floor’s frequency can be estimated at the design stage, and it can — from a number the deflection check already produced.

Rayleigh's method: the frequency read off the deflection that was computed anywayA simply supported beam of 12 m, sagging 27.09 mm under its own weight, sampled at 21 stations. Rayleigh's quotient over those deflections gives 3.42 Hz against the exact 3.41 Hz — 0.119% high, and high rather than low because an assumed shape is a constraint and a constraint stiffens. The rule of thumb, 18 divided by the square root of the deflection in millimetres, gives 3.46 Hz.12 m simple span · sag 27.09 mmthe static shape, not an eigenvectorω² = g · Σ(w·δ) ⁄ Σ(w·δ²)Σ(w·δ) = 1839 · Σ(w·δ²) = 39.17Rayleigh, from the static shape: 3.415 Hzexact, from the characteristic equation: 3.411 Hz18/√δ with δ in mm: 3.458 Hz
Fig. 5 A 12 m floor beam sagging 27.1 mm under its own weight and its permanent load. Rayleigh’s quotient over that shape gives 3.415 Hz; the exact solution gives 3.411. The rule of thumb — 18 divided by the square root of the deflection in millimetres — gives 3.458. All three agree, and all three come out of a calculation that was done for the span/360 check.

Inverting the rule of thumb gives the sentence a designer actually needs: a floor sagging about 20 mm is a 4 Hz floor, and one sagging 50 mm is a 2.5 Hz floor. Since 20 mm on a 12 m span is L/600 — comfortably inside every deflection limit ever written — the whole problem can be restated as the observation that the deflection limits were never about this.

A beam's frequency against its span, which falls as one over the squareThe fundamental frequency of a simply supported beam of fixed section and fixed mass per metre, against its span, from 2 to 16 m. At 12 m it is 3.41 Hz. Doubling the span quarters the frequency, because the frequency goes as βL squared over the span squared and the flexibility it is competing with goes as the fourth power.246810121416020406080100120span (m)fundamental frequency (Hz)12 m — 3.41 Hz
Fig. 6 The same floor section and mass per metre, across the spans it might be used at. The frequency falls as the square of the span, so the 6 m version is at 13.6 Hz — above every harmonic of walking — and the 14 m version is at 2.51 Hz, close enough to the first harmonic of a slow pace to be in the worst part of the pattern. The whole of the modern floor-vibration problem is contained in that curve and in the fact that spans got longer.

The curve crosses the danger band rather than approaching it, which is why the problem is a modern one in a specific sense. Floors spanning 5 or 6 m are naturally above 10 Hz and were never at risk. Floors spanning 12 to 15 m land between 2 and 3.5 Hz, on the two largest harmonics. The change in construction that produced the complaints was not a change in the loading, the material or the analysis — it was an increase in span, made possible by composite action and by the strength calculations getting better.

The pace is a distribution, not a number

Which floor frequencies a 1.8 Hz pace punishesThe response factor of a floor of 12 tonnes modal mass and 3.0% damping, against its own natural frequency, under a walker at 1.8 steps per second. The peaks are at 1.8, 3.6, 5.4, 7.2 Hz — the harmonics of the pace — and they fall away sharply: four peaks, each smaller than the one below it, because the harmonics of walking get smaller. A floor at 1.8 Hz reaches R = 30 and one at 7.2 Hz reaches 6. Between them the floor is quiet.2345678910051015202530floor frequency (Hz)response factor1× pace2× pace3× pace4× pace3.6 Hz — R = 105.4 Hz — R = 7
Fig. 7 The same floor walked on at 1.8 steps a second instead of 2.0. The whole spike pattern moves: the peaks are now at 1.8, 3.6, 5.4 and 7.2 Hz, and the 4 Hz floor that reached a response factor of 11.3 at the faster pace now reaches 3.9. The floor did not change. The walker did.

Which raises the practical question of what to design for. Real walkers cover a range, and a floor is walked on by many people at many paces over a working day, so the honest statement is that the response factor is itself a distribution and the design value is a high percentile of it.

The consequence is a design rule rather different from most on this site: a floor is designed to be outside a band rather than above a threshold. The usual practice is to keep the fundamental frequency above about 4 Hz for a floor with ordinary walking — placing it above the second harmonic of the fastest realistic pace — or, if that cannot be achieved economically, to accept a low-frequency floor and check the response directly. Both are respectable. What is not respectable is a floor at 4.1 Hz sized on a rule that says “above 4”, because the rule was a proxy for the response and the response has a spike a hundredth of an octave away.

How much a harmonic force is magnified, at three damping ratiosDisplacement amplitude divided by the deflection the same force would produce if it were applied slowly, against the ratio of the forcing frequency to the structure's own, at 1%, 3%, 5% of critical damping. At the natural frequency the magnification is 50, 16.67, 10 respectively — one over twice the damping ratio, and nothing else in the problem enters it.00.511.522.50510152025forcing frequency ÷ natural frequencyamplitude ÷ static deflection1% damping — 50× at the peak3% damping — 16.67× at the peak5% damping — 10.01× at the peak
Fig. 8 The multiplier underneath the whole essay, at the three damping ratios a floor might have: a bare deck at 1%, a fitted-out office at 3%, a heavily partitioned floor at 5%. A harmonic exactly on the floor’s frequency is magnified 50, 16.7 or 10 times. Everything a floor designer can do is either to move off the peak or to move down the family of curves.

The complaint is about people, and the calculation is about a beam

Two features of this problem separate it from everything else on this site, and both are worth stating because they change what a good answer looks like.

The limit state is a judgement rather than a capacity. Every other check here compares a demand with a resistance that the material supplies: a stress against a yield stress, a load against a buckling load, a deflection against a span/360 that at least protects the finishes. The response factor compares an acceleration with a threshold of annoyance, and the threshold was established by asking people. There is no material property anywhere in it.

The failure is reversible and the remedy is not. A floor that is judged too lively does not have to be replaced; it has to be modified, and the available modifications — adding mass, adding damping, adding an intermediate support — are all things that are far cheaper before the building exists than after. So the calculation’s real job is to be done early and approximately rather than late and exactly, which is why the Rayleigh estimate above matters more in practice than any finite element model.

That combination — a soft limit and an expensive remedy — is the signature of a serviceability problem, and it is the same shape as ponding and as long-term deflection: nothing fails, and something has to be paid for anyway.

What the picture cannot show

The mode is not the bay. All of the above treats the floor as one oscillator, and a real floor plate has many modes with shapes that span several bays. A walker excites whichever ones their path crosses, and the response at a given point is a sum. The single-degree-of-freedom treatment is a good screening tool and it is not an analysis.

Walking is not steady. The four-harmonic model assumes a person walking indefinitely at a constant pace in one place. A real walker crosses the floor in a few seconds, which is often not long enough to build up to resonance — the build-up rate needs roughly 1/(2πζ)1/(2\pi\zeta) cycles, about five at 3% damping, which at 4 Hz is a little over a second, so on a long floor there is time and on a short one there is not.

Perception is not a number. The response factor pretends that a person’s judgement of a floor is a single scalar. It is not: the same acceleration is judged differently standing and sitting, differently when the source is visible, and very differently when the person expected the floor to be still. A great deal of the literature on this subject is about the psychology, and the mechanics is the easy half.

The high-frequency floor, which is a different problem

A floor above about 10 Hz cannot be resonant with any harmonic of walking, because the harmonics have run out. It is not therefore quiet.

Each footstep is then an impulse — a load too short for the floor to follow, in exactly the sense the shock spectrum gives — and the floor responds by ringing at its own frequency and decaying before the next step arrives. The response is a series of decaying transients rather than a steady build-up, and it is assessed by peak velocity rather than by an rms acceleration.

The distinction matters because the two regimes want opposite things. In the low-frequency regime, the fix is to avoid the harmonic. In the high-frequency regime, the fix is to increase the damping so each transient dies sooner, and moving the frequency achieves nothing at all. Getting the regime wrong means applying the right fix to the wrong problem.

Where the ladder goes

The same apparatus answers three neighbouring questions, and each is its own rung.

A footbridge is a floor with a lower frequency and a crowd on it, and when the crowd is walking sideways rather than up and down, the excitation stops being independent of the response — which is a different mechanism and a much more dangerous one.

A grandstand is a floor with people jumping in time to music, at two to three hertz, with coefficients an order of magnitude larger than walking’s.

And a floor with a machine on it is the same calculation with the excitation known exactly rather than statistically, which turns out to make the problem easier to solve and harder to solve well.

What all three share with this one is the observation the essay opened with, and it is the reason the field is worth a foundation phase of its own: the structure was never in doubt. Every member here is stressed to a fraction of its capacity, every deflection is inside every limit, and the building is unusable. Strength is not the only question a structure has to answer, and it is not even reliably the first one.

Where this goes next

8 m simple span · sag 13.08 mmthe static shape, not an eigenvectorω² = g · Σ(w·δ) ⁄ Σ(w·δ²)Σ(w·δ) = 197.3 · Σ(w·δ²) = 2.030Rayleigh, from the static shape: 4.915 Hzexact, from the characteristic equation: 4.909 Hz18/√δ with δ in mm: 4.977 Hz The period nobody chose Every structure has a natural period, it decides the answer to every question in this field, and no drawing anywhere records it. It is a consequence of a mass picked for one reason and a stiffness picked for another — and it has already been computed, by the serviceability check. 00.511.522.530510152025forcing frequency ÷ natural frequencyamplitude ÷ static deflection1% damping — 50× at the peak2% damping — 25.01× at the peak5% damping — 10.01× at the peak10% damping — 5.03× at the peak The only thing that stops it Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built. 0.60.811.21.41.61.8200.511.5span, relative to the firstthey cross heredeflection runs out at 1.40strength runs out at 1.54the limitstrengthdeflection Stiffness is not strength, and usually it is the one that governs A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all. 11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth Span to the fourth, which is why spans are short Doubling a span multiplies its deflection by sixteen. No other relationship in ordinary structural work is that steep, and it is the reason long spans are always a different kind of structure. 00.511.522.533.540510152025forcing frequency ÷ natural frequencyforce out ÷ force in√2 — nothing gained, at any damping2% damping5% damping20% damping The machine that shakes the building Put a machine on springs to keep its vibration out of the floor, and below a frequency ratio of root two the springs make things worse. Every transmissibility curve ever drawn passes through exactly one at that ratio, whatever the damping — so a soft mount either works well or fails badly, with nothing in between.

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DampingFloor vibrationFootfallModal massNatural frequencyResonanceResponse factorServiceability