The modes that were left out

A structure has as many modes as it has freedoms, a model of a twenty-storey frame has thousands, and no analysis uses them. Six is a common number. Twelve is a careful one.

The rule for how many is a mass count: keep enough modes to account for ninety per cent of the structure’s mass, and the rest may be discarded. It is in every seismic code, it is easy to check, and the effective modal mass it counts is a real and well-defined quantity.

It is also written in the one thing that converges fastest.

Why base shear is the easy quantity #

Effective modal mass is defined so that the modal masses add up to the total mass, and so that a mode’s base shear is its effective mass times its spectral acceleration. That definition is what makes the ninety per cent rule sound like a statement about accuracy.

It is a statement about accuracy of the base shear, and only approximately of that, because the modes are combined by square root of the sum of squares rather than added. A mode carrying nine per cent of the mass contributes nine per cent of the base shear to a sum and about half a per cent to a square root of a sum of squares. The rule is conservative in the quantity it is written in and says nothing about any other.

So one mode, on the frame above, gives 97.6 per cent of a base shear while carrying 84.8 per cent of a mass. That is not an error in the rule; it is the rule working, in the direction that flatters it.

Where the higher modes actually are #

The quantity that is short is short for a reason that a picture of the modes makes obvious.

A higher mode’s contribution to the base shear is small twice over: its effective mass is small, and the shear it does produce is partly cancelled by the reversals in its own shape. Its contribution to the forces at the top is neither of those things. The top of the building is where every mode is large.

That profile is the design consequence, and it is worse than it looks, because the upper storeys are where the forces are smallest in absolute terms and where a designer’s attention is least. A twenty per cent shortfall at the roof lands on a member that was governed by something else anyway — until the roof carries plant, or a mast, or a parapet, and the acceleration there is what those are designed for.

What a mode is worth, mode by mode #

The numbers behind those two figures are worth setting out, because the pattern in them is the essay.

On the ten-storey frame the effective modal masses run 84.8, 9.1, 3.1, 1.4 and 0.7 per cent — a sequence falling roughly as the square of the mode number, which is what a regular structure gives and is why two modes reach ninety.

The roof force runs 79.8, 93.7, 97.6, 99.1 and 99.7 per cent of the exact answer. Compare the two sequences and the arithmetic is visible: mode two carries nine per cent of the mass and supplies fourteen points of the roof force. Mode three carries three per cent of the mass and supplies four points. Every higher mode is worth more at the roof than its mass share suggests, by a factor of about one and a half, and the factor is the ratio of the mode’s amplitude at the roof to its amplitude averaged over the building.

That ratio is a property of the shape and it has a name in every other field that uses modal decomposition. It is why an acoustic room mode is loudest at a wall, why a plate’s higher modes are found by tapping near an edge, and why the frequency response of anything measured at one point is not the frequency response of the object.

The sum and the square root of the sum of squares #

One more piece of arithmetic explains why the base shear flatters the rule, and it is worth a paragraph because it is often stated the other way round.

Effective modal masses add: the ten of this frame sum to the building’s whole mass, exactly, and that is what makes a mass count a count. Modal responses do not add — they are combined by square root of the sum of squares, because the modes reach their peaks at different moments and adding them would assume they did not.

Those two facts pull in opposite directions and the second wins. A mode carrying nine per cent of the mass contributes nine points to the mass count and about half a point to a root-sum-square base shear, because $\sqrt{1^2 + 0.09^2}$ is 1.004 rather than 1.09. The combination rule suppresses small contributions quadratically and the mass count treats them linearly, so the mass count is systematically the more demanding of the two — on the base shear.

It is not more demanding on anything else, and that is the whole of the trouble. At the roof the higher modes are not small contributions: mode two’s roof amplitude is comparable to mode one’s, so the quadratic suppression does almost nothing and the linear count has no bearing on it at all.

The building where it is worst #

Make the frame taller and the gap widens, because a taller building’s higher modes have periods that sit further up the spectrum’s plateau.

Two modes, a rule satisfied, and a roof force a fifth short. That is not a marginal case; it is a twenty-storey building analysed by the ordinary procedure.

The irregular building, where the rule asks for less #

Which brings the essay to the result that is genuinely uncomfortable.

Softening one storey concentrates the first mode’s shape into that storey and the rest of the building rides on top of it as a nearly rigid block. A nearly rigid block moving as one has almost all of the participation, so the first mode’s effective mass goes up — from 84.8 per cent to 93.5.

The rule therefore demands fewer modes of the more irregular structure. It is not a perverse rule; it is a rule about mass participation, and mass participation genuinely does concentrate. What it is not is a rule about how many modes the answer needs, and on this structure those two things point opposite ways.

The general form of that observation is worth carrying past this essay. A criterion that measures one property of a solution is a proxy for accuracy only while the two remain correlated, and the cases where they decorrelate are systematically the awkward ones — because what makes a structure awkward is exactly what breaks the correlation.

What the correction puts back #

There is a standard repair, and it is a good one, and it does not fix this.

The missing-mass correction takes the mass the retained modes did not account for and applies it as a static force. The reasoning is exact rather than approximate: the modes that were dropped were dropped for being fast, their periods are far below the ground motion’s own, and a structure much stiffer than its excitation follows it without amplification. Following without amplification is what a static response is.

Three tenths of one per cent on the quantity that was already right to within three tenths of one per cent.

The correction is a base-shear correction, and it is worth having for that: it removes the last of the shortfall in the quantity a code’s force check is written in, and it does so exactly rather than by adding a mode. What it does not do is supply the higher modes’ contribution to the shape of the response, because that contribution is dynamic and the correction is static by construction.

The honest description is that the missing-mass method fixes the amount of force and not its distribution. That is a real distinction and it is not usually drawn, because the amount of force is what the rule was about and the distribution is what the structure feels.

What to do instead, which is not more modes #

The obvious response — run more modes — is right and is not the interesting answer, because more modes cost nothing on a computer and the number is already generous in most offices.

The useful response is to check a quantity that converges slowly and see whether it has. Three candidates, in order of how much they say:

The force in the top storey, or equivalently the roof acceleration. It is the slowest-converging quantity in the analysis and it is one number.

The storey shear at the level where the mode shapes have a node. The first mode contributes nothing there by definition, so whatever shear appears is entirely higher-mode, and its convergence is the convergence of the modes nobody counted.

The overturning moment against the base shear. The two converge at different rates because the higher modes’ reversals cancel in the moment and not in the shear, so the ratio between them is a free diagnostic that costs nothing to read.

Each of those is a check on the analysis rather than on the structure, and each takes a minute. A convergence study of one quantity is worth more than a mass count, and it is the check a numerical analyst would have asked for first.

The check nobody runs, written out #

It takes about a minute and it needs nothing that is not already on the screen.

Run the analysis at the mode count chosen. Note the roof-level force, or the roof acceleration, or the force in whatever is mounted up there. Then run it again with twice as many modes and note the same number.

If it has moved by more than a per cent or two, the truncation was not converged for that quantity — and the base shear will have moved by almost nothing, which is the diagnostic rather than a reassurance. On the twenty-storey frame above, going from two modes to four moves the base shear by 1.2 per cent and the roof force by 18.

Two runs and one subtraction. What makes it worth stating as a procedure is that nothing in the ordinary workflow prompts it: the mass table is printed automatically, the ninety per cent is checked automatically, and there is no line in any output that says this quantity is still moving.

It is also the check that generalises. Every numerical method in this collection has a discretisation somewhere in it — a mesh, a step size, a number of terms — and the honest test of any of them is the same one: change it and see what moves. A criterion that can be satisfied without changing anything is not that test.

A shear building’s modes are as simple as modes get #

A shear building has no rotational freedoms. Every model here has one horizontal displacement per floor, so its modes are pure lateral shapes and its higher modes are as simple as they can be. A real frame has joint rotations, local floor modes and vertical modes, most of which have negligible participation and some of which matter for equipment.

The combination is square root of the sum of squares. That assumes the modes are well separated in period, which they are here. Closely spaced modes are correlated in time and their contributions do not combine that way — the complete quadratic combination exists for exactly that case, and it changes the numbers above in the direction of more modes mattering rather than fewer.

The spectrum is a design shape and not a record. Its high-frequency end is a plateau falling to the ground acceleration, which is what makes the static correction exact. A real record has structure up there, and a mode whose period happens to sit on a peak in it is not responding statically at all.

And the whole calculation is linear. Once a structure yields the modes change while the motion is happening, and none of the arithmetic above survives it. Truncation is a question about a linear analysis and the answer says nothing about what the structure does past first yield.

Where the same trap is set elsewhere #

A criterion measuring one property of an answer, standing in for accuracy, and decorrelating exactly on the awkward cases — this collection has met that shape several times and it is worth naming the family.

A pushover analysis uses one load pattern, usually triangular, chosen to resemble the first mode. Its target displacement is calibrated on the base shear and the roof displacement, which are the two quantities the first mode gives well; the storey drifts, which is what the analysis exists to produce, are the ones the pattern gets wrong — and it gets them most wrong on the irregular frames the method is used for.

The portal method assumes a point of contraflexure at every mid-height and gets the storey shears right on average while putting the contraflexure in the wrong fifth of the member.

A mesh refined until the displacements stop moving has been refined until the quantity that converges fastest has converged; the stresses at a re-entrant corner are still moving and will keep moving.

In each case the criterion is not wrong and is not lazy. It measures something real, it is cheap, and it was chosen because it is checkable. What it cannot do is know which quantity the answer is for, and that is a question about the design rather than about the analysis — which is why it is the one part of this that cannot be automated.

A mode count is a limit on attention, not on arithmetic #

Every figure here plots a quantity against a mode count, and a mode count is not a cost. The reason nobody runs fifty modes is not that fifty is expensive; it is that the results have to be looked at, and a designer reading fifty modal contributions is reading a table nobody can hold. The truncation is a limit on attention rather than on arithmetic, and no plot against mode number can show that.

The other absence is time. Every number above is a peak — the largest value a quantity reaches, combined statistically across modes — and none of these figures shows the modes arriving at different moments. The first mode’s peak and the third mode’s peak do not happen together, which is the entire justification for the square-root combination and is the thing the combination hides.

The assumption underneath the mass rule #

The ninety per cent rule assumes that mass participation is a proxy for importance, and mass participation is a proxy for one thing: how strongly a mode is driven by a uniform ground motion.

That is exactly right for a base shear and it is exactly the wrong question for anything mounted on the structure. A floor spectrum — what an item of plant on the roof actually feels — is dominated by the modes at the frequencies that item cares about, and those are usually not the modes with the mass.

So a truncation that is correct for the building can be badly wrong for the things inside it, and the two calculations are done by the same model with the same modes. The mode count that satisfies the structure’s own check is not the mode count its contents need, and there is nothing in the rule that says so.

Where ninety per cent came from, and what has changed since #

Later rungs on this anchor: closely spaced modes and the complete quadratic combination, where the assumption of independence fails and the correlation has to be computed. Non-classical damping, where two very different damping levels in one structure mean no set of real modes describes it at all. Modal analysis of a structure with a rigid diaphragm, where translation and torsion pair into modes that are neither. And the residual-mode methods that go past the static correction — the ones that put back a shape rather than a force.

The rule itself has a history worth a sentence. Ninety per cent was chosen in the 1970s, when a modal analysis was expensive and the number of modes was a real budget, and it was calibrated on the base shear because the base shear was what a code checked. Both of those conditions have gone. The number survives because it is easy to demonstrate compliance with, which is a good reason for a rule to exist and not a reason to believe it.