Most of the mass moves together

An eight-storey frame has eight modes. Analysing all eight is not difficult, but the question of how many are needed has to have an answer, because the same question about a real building has ten thousand modes in it and no computer answers that one for free.

The answer is a mass count.

That figure is the reason structural dynamics is practical. Eight coupled equations became one equation carrying 86% of the answer, plus a second carrying most of the rest.

What “the mass a mode moves” means #

The phrase needs care, because every floor’s mass is in every mode. What differs is how effectively each mode is driven by a motion of the ground.

When the ground moves sideways, the force it applies to a mode is proportional to how much that mode’s shape resembles a rigid sideways translation. The first mode leans the whole building one way, which resembles it strongly. The second mode moves the lower half one way and the upper half the other, so the pushes largely cancel — the ground is trying to shift the building and this mode is shaped like a shuffle. That resemblance is the participation factor, and its square gives the effective modal mass:

$$M_{\text{eff},i} = \frac{\left(\{\phi_i\}^{\mathsf T}[M]\{1\}\right)^2}{\{\phi_i\}^{\mathsf T}[M]\{\phi_i\}}$$

with $\{1\}$ the vector that is 1 at every floor — the rigid translation. Modes that look like it get most of the mass; modes that do not get very little.

And the sum over all modes is exactly the total mass. That is a completeness statement: the modes span every possible motion, so between them they must be able to represent a rigid translation, and the shares in which they represent it must add to the whole. Nothing has been normalised to make it true. It is the property that turns “how many modes is enough” from a judgement into an arithmetic test — take modes until the cumulative mass passes 90%, and it is known exactly what has been left out.

The share barely depends on the height #

Three storeys give 91%, eight give 86%, sixteen give 84%, twenty give 83%. The distribution converges rather than degrading, and the reason is that it is set by the shape of the modes rather than by their number — the first mode of a tall uniform building looks much like the first mode of a short one, stretched.

That stability is what allows rules of thumb to survive across a whole class of buildings, and it is worth contrasting with what happens if the building is not uniform. Softening one storey, changing the mass on one floor, or setting the core off-centre redistributes the shares immediately, and a structure whose first mode carries only 60% of the mass is telling the designer something about itself before any load has been applied.

The number is different for a force and for an acceleration #

Here is the part that gets missed, and it is the reason this rung exists rather than being a paragraph in the last one.

The modal mass says how much of the building each mode moves. It does not say how much of the answer each mode supplies, because the answer also depends on how hard each mode is driven — and each mode is driven at its own period, where the ground motion has its own intensity.

Put the two together for the sixteen-storey frame and the modes rank differently depending on what is being asked.

For the base shear — the total horizontal force the foundations must resist — each mode contributes its effective mass times its spectral acceleration. Mode 1 gives 5,301 kN, mode 2 gives 1,776 kN, mode 3 gives 1,102 kN, and combined they give 5,722 kN. The first mode alone is 92.6% of the answer, so the hand calculation that uses one mode is right to within a tenth.

For the acceleration at roof level — which is what an item of plant, a water tank or a person on the top floor experiences — each mode contributes its participation factor times its shape at the roof times its spectral acceleration. Mode 1 gives 1.68 m/s², mode 2 gives −1.68, mode 3 gives 1.73, and combined they give 3.23 m/s². The first mode is 52% of that answer.

Same building, same earthquake, same three modes. For a force, two modes are plenty. For an acceleration, the modes carrying 12% of the mass supply half the demand, and the reason is entirely that they live at short periods where the ground shakes hardest. It is the same asymmetry a leaning frame shows in a different guise: the quantity that governs the members and the quantity that governs everything attached to them are not the same quantity, and a check on one is silent about the other.

Why the two questions separate #

The mechanism is worth stating in one sentence, because it generalises well beyond earthquakes: a force is an integral over the structure and an acceleration is a local value.

Base shear sums the inertia forces of every floor, and in the higher modes those forces alternate in sign up the building, so they cancel in the sum. Roof acceleration takes a single point, where nothing cancels, and every mode with any amplitude there contributes at full value.

So any quantity that is a sum over the structure — base shear, overturning moment, the force in a foundation — is dominated by the first mode. Any quantity that is local — a floor acceleration, a storey drift near the top, the force in one brace — is not.

That distinction is the same one that runs through influence lines: the load position that maximises a total is not the load position that maximises a local effect, and a design check on the wrong one of the two is not conservative, it is simply about something else.

That picture is the whole mechanism, and it is worth reading twice, because it explains a fact about earthquake damage that otherwise looks arbitrary: buildings that survive with their frames intact routinely lose their rooftop plant, their lift motor rooms and their parapets. The modes that shake those things hardest are precisely the modes that the structural check found negligible, and they were negligible for the structural check.

A building that does not share nicely #

That pairing is worth dwelling on because it inverts the usual reading of the mass table. A high first-mode participation is generally taken as a sign of a well-behaved structure, and here it is a symptom: the building is behaving like a single mass on a single spring because one soft layer is doing all the deforming, and everything above it is going along for the ride.

The general statement is that the modal mass distribution measures how uniformly a structure deforms, not how good it is. A structure that concentrates its deformation gets a cleaner modal decomposition and a worse failure mode, and the number that would have warned about it is not in the participation table at all — it is in the mode shape, where the storey drift is.

The tail, and why it converges #

The convergence is fast enough that the practical rule — take modes until 90% of the mass is accounted for — usually stops at two or three however tall the building is. The rule survives because of the shape argument rather than because of any property of buildings, which is why it also holds for structures that are nothing like frames: a chimney, a mast and a dam all put most of their mass in their first mode for the same reason.

Where it fails is where the shapes stop alternating cleanly. A structure with a heavy mass at the top — a water tower, a building with a plant floor at roof level — has a first mode that resembles the mass swinging on the structure below it, and the participation can drop sharply. So can a building whose stiffness changes abruptly partway up, which is the setback problem: two structures stacked, each with its own preference, and modes that belong to one or the other rather than to the whole.

The diagnosis in every one of those cases is the same, and it is available before any load case is run: look at the participation table, and if the first mode is not taking most of the mass, find out which part of the structure the modes have decided to belong to. It is the dynamic equivalent of asking which supports a redundant beam is actually leaning on — the structure has an opinion, and it is better to read it than to impose one.

Which mass, though? #

Everything above computes with “the mass”, and deciding what that is turns out to be one of the larger uncertainties in a seismic calculation.

The structure’s own weight is known well. What is on the floors is not: an office is designed for an imposed load that is not present most of the time, and the load that will be there during the earthquake is a statistical question rather than a design one. Codes take a fraction of the imposed load into the seismic mass — a quarter or a third is typical, a much higher fraction for storage — and that fraction moves both the period and the force. It is the same difficulty a settlement calculation has from the other end: the number that decides the answer is a fact about the world rather than about the structure, and no amount of analysis improves it.

The direction of the error is not obvious, which is why it is worth spelling out. Extra mass raises the demand, because the force is mass times acceleration. It also lengthens the period, which usually moves the structure to a part of the spectrum where the acceleration is smaller. The two effects fight, and which wins depends on where on the spectrum the building sits — the flat plateau or the falling tail.

The first mode as one oscillator #

That figure is the point of the whole decomposition. Once the mass, the period and the shape of a mode are known, the mode is an oscillator, and every result on this site about oscillators transfers to it without modification. The modal analysis of a sixteen-storey building is sixteen of these, three of which are worth running.

Combining them, and why the answers are not added #

The modal peaks cannot simply be summed, and the reason is in the numbers above: mode 1 contributes +1.68 m/s² at the roof and mode 2 contributes −1.68. Adding them gives zero, which is certainly wrong, and adding their magnitudes gives 3.36, which is too large because the two peaks do not occur at the same instant.

The standard treatment takes the square root of the sum of the squares, which assumes the modal peaks are statistically independent, and gives 3.23 here. It is a statistical statement rather than a mechanical one, and it fails in a specific and known case: when two modes have nearly the same period, their responses are correlated rather than independent, and the root-sum-square underestimates. That is why closely spaced modes — which a tall uniform frame produces at its short-period end, and which torsionally flexible buildings produce at their fundamental — get a more careful combination rule.

The general lesson is one this site keeps meeting: a rule that is derived under an assumption of independence fails exactly where the structure is symmetric enough to make two things happen at once, and symmetric structures are the ones people build.

What the picture cannot show #

The bar chart ranks modes by mass and says nothing about three things that decide whether a mode matters.

It says nothing about the period, which is where the driving intensity comes from — the whole of the acceleration argument above is invisible in the ranking.

It says nothing about direction. A real building has modes in two horizontal directions and in torsion, and the mass participation is computed separately for each. A mode with 80% of the mass in the y direction contributes nothing at all to an x-direction analysis, so the chart is really three charts.

It says nothing about what is attached. A rooftop mast, a water tank or a plant room with its own period close to one of the building’s will amplify that mode’s motion locally by a factor set by its own damping, and the mass chart for the building says nothing whatever about it. Secondary systems on flexible supports are a separate analysis and a common source of earthquake damage without any structural failure at all.

Where the ladder goes #

The next question is where the spectral accelerations came from — the curve that multiplied each modal mass to make a force. It is the most used and most misunderstood object in the subject, and the misunderstanding is in its name: it is not a load, and it is not a property of an earthquake.

After that comes the question this essay has quietly assumed away. Every number here treated the structure as elastic, so that modes exist at all. Under a design-level earthquake it will not be, and what happens then is not a correction to this calculation but a different one.