The twist the combination rule invents

Orthogonality is what lets a structure with many modes be analysed as a set of oscillators that do not talk to each other, and the essay that introduced mode shapes was careful to say what it separates and what it does not. It separates the equations. Each mode is solved alone and returns its own peak response. It does not separate the answers, because those peaks happen at different instants, and putting them back together requires a statement about how the instants relate.

The statement nearly everyone uses is the square root of the sum of squares. It is exact if the modal peaks are independent random quantities, and the same essay worked out what happens when they are not: two equal modes responding exactly in step should add to twice their size, the rule gives 1.41 times, and so it returns 71 per cent of the answer. That is the version usually quoted, and it makes the rule sound as though it errs in one direction.

It does not. The number that decides which way it errs is the sign of a product the rule never looks at, and the building that shows this most clearly is a floor that twists.

A floor whose stiffness is a metre off its mass #

Take a single square floor 24 m across, weighing 600 tonnes, held laterally by walls or frames whose combined stiffness gives it a period of 0.8 s in one direction. The mass is spread evenly, so it sits at the centre and the floor’s radius of gyration in plan is $a/\sqrt 6 = 9.80$ m. The stiffness does not quite sit at the centre: the walls on one side are a little stiffer than those on the other, and the centre of stiffness is 0.98 m off the centre of mass — a tenth of the radius of gyration. And the resisting elements are arranged so that, if the floor were forced to twist without moving, it would do so at the same period as it moves without twisting.

That last condition sounds special and is not. It is what a square plan with its walls on the perimeter tends toward, and it is the arrangement where plan torsion is worst.

Measured at the centre of mass, the floor has two freedoms: a movement $u$ and a rotation $\theta$, which it is convenient to multiply by the radius of gyration so both are lengths. In those coordinates the mass matrix is simply the floor’s mass times the identity, and the stiffness matrix, divided by the same mass and the uncoupled frequency squared, is

$$\begin{bmatrix} 1 & \varepsilon \\ \varepsilon & \Omega^2 + \varepsilon^2 \end{bmatrix}$$

with $\varepsilon = 0.1$ the eccentricity over the radius of gyration and $\Omega = 1$ the ratio of the two uncoupled frequencies. The off-diagonal term is the coupling: pushing the floor sideways through its centre of mass also twists it, because the push does not pass through the stiffness centre.

The two eigenvalues are $1 + \varepsilon^2/2 \pm \varepsilon\sqrt{1 + \varepsilon^2/4}$, so at $\Omega = 1$ the two frequencies differ by almost exactly $\varepsilon$ — ten per cent, here — and each mode is very nearly half translation and half twist. They carry 52.5 and 47.5 per cent of the mass. Neither of them is “the sway mode” or “the torsion mode”: the building has no such modes, only two mixtures, and the whole of what follows is about the sign on the twist in each.

Why the two torques are always equal and opposite #

Each mode moves the floor along the ground motion — the translation component of both is positive — and orthogonality then forces the twist components to have opposite signs, because two vectors in a plane with positive first components can only be perpendicular if their second components disagree.

That is visible in the figure. What is less visible is how much it constrains.

The peak base torque a mode contributes is proportional to the product of its translation component and its twist component: the translation decides how hard the ground motion drives the mode, and the twist decides how much of the resulting inertia force acts as a couple. For two unit vectors in a plane at right angles, that product is exactly equal and opposite between them. On a spectrum that gives both modes the same acceleration — which for two periods within a few per cent of each other is what any spectrum does — the two modal base torques are equal in size and opposite in sign, for every eccentricity and every frequency ratio. On this floor they are $\pm 8{,}640$ kNm.

So the torque the floor actually carries is not in either mode. It is the amount by which the two fail to cancel, and that depends entirely on how their peaks line up in time — which is the one thing a combination rule has to supply and the square root of the sum of squares assumes away.

The base shear is the opposite case. Each mode’s shear is its effective mass times the acceleration, both positive, and they add to the floor’s whole mass times the acceleration: 927 and 839 kN, which is 1,766 kN in total. The drift at each edge is a mixture: at the flexible edge, one mode’s translation and twist add and the other’s subtract, and the two modal contributions come out as 60 mm and −6 mm.

The cross term, and why it has a sign #

The complete quadratic combination writes the peak of a response made of two modal contributions $R_1$ and $R_2$ as

$$R = \sqrt{R_1^2 + R_2^2 + 2\rho R_1 R_2}$$

with $\rho$ a correlation coefficient between the two modal responses: one if they rise and fall together, nought if they are unrelated. At $\rho = 0$ it is the square root of the sum of squares. At $\rho = 1$ it is $|R_1 + R_2|$ — the algebraic sum, with signs.

The cross term is $2\rho R_1R_2$, and it is the whole subject of this essay. Its sign is the sign of $R_1 R_2$, not of $\rho$. Where the two modal contributions agree it adds, and the root-sum-square, which drops it, is too small. Where they oppose it subtracts, and the root-sum-square is too large.

The hero figure and this one put the finding in one picture. At a correlation of one half, the root-sum-square understates the base shear by 18 per cent and overstates the base torque by 41 per cent, from the same two modes of the same building under the same ground motion. It has the edge drifts nearly right, because each is dominated by one mode and the cross term of a large number with a small one is small.

For the torque, the arithmetic reduces to something tidy. With $R_2 = -R_1$, the full combination is $|R_1|\sqrt{2(1-\rho)}$ and the root-sum-square is $|R_1|\sqrt 2$, so their ratio is

$$\frac{\text{SRSS}}{\text{CQC}} = \frac{1}{\sqrt{1-\rho}}.$$

At $\rho = 0.5$ that is 1.41. At $\rho = 0.96$ it is five. As $\rho$ approaches one it grows without limit, because the torque the building carries goes to nothing and the root-sum-square’s does not.

The mass count is satisfied and the answer is still wrong #

It is worth noticing what the usual safeguards say about this floor, because every one of them passes.

The rule for how many modes to keep is a mass count: include modes until ninety per cent of the mass is accounted for. Here the first mode carries 52.5 per cent and the second 47.5, so the rule demands both, both are included, and the analysis accounts for all of the mass. Nothing is missing. The error is entirely in how two correctly computed, correctly included modal answers are put back together, and a mass count, which adds effective masses linearly and never asks about signs, has nothing to say about it.

The modal periods are right, the mode shapes are right, the participation is right and the spectrum is right. Each modal torque of 8,640 kNm is a correct statement about what that mode alone would do. The only wrong step is the last one, and it is the step that looks most like bookkeeping.

Where the correlation comes from #

The correlation coefficient is not a fudge factor, and it is worth being precise about what it is and what it is exact for.

Drive two damped oscillators with the same broadband ground motion and record their responses. Each oscillator’s response is concentrated in a band around its own frequency, of a width set by its damping. If the two bands overlap, the two responses share the part of the input that falls in the overlap and move together while it lasts; if they do not, the responses are driven by different parts of the input and are unrelated. The correlation is the overlap of the two bands, measured properly — the integral of one oscillator’s frequency response against the other’s, divided by the size of each.

For an input with the same intensity at every frequency, and equal damping $\zeta$ in both modes, that integral has a closed form due to Der Kiureghian:

$$\rho = \frac{8\zeta^2(1+\beta)\beta^{3/2}}{(1-\beta^2)^2 + 4\zeta^2\beta(1+\beta)^2}$$

where $\beta$ is the ratio of the two frequencies.

Two things in the figure are worth carrying.

The coefficient falls off fast. At 5 per cent damping, modes 20 per cent apart are correlated only 0.23, and 40 per cent apart 0.08. The root-sum-square’s assumption is a good one for most pairs of modes in most buildings, which is why it has survived.

Damping widens the bands, so more damping means more correlation. At 2 per cent, the floor’s two modes are correlated 0.14; at 10 per cent, 0.80. A heavily damped structure — one with added dampers, or a base-isolated one — has correlated modes at separations where a lightly damped one does not, which is the opposite of the intuition that damping makes dynamics simpler.

The dots are there because the closed form is a claim and the integral is its definition. They agree to four decimal places, which says the formula is being evaluated correctly. It does not say the formula is right for an earthquake. It is exact for a stationary white-noise input, and an earthquake record is neither stationary nor white, and a peak is not a root-mean-square. The complete quadratic combination is therefore itself an approximation to what a response history would show — a much better one than the root-sum-square for close modes, and not a proof that the numbers here are what a time-history analysis would return.

What the rule does to a building that barely twists #

Move the stiffness centre closer to the mass, and the argument sharpens into something that looks like a paradox.

At an eccentricity of 0.20 m — a fiftieth of the radius of gyration, well inside the tolerance to which anybody knows where a building’s stiffness is — the floor’s two modes are 2 per cent apart and correlated 0.96. They are still each half translation and half twist, because at $\Omega = 1$ any eccentricity at all mixes them completely; only the separation depends on its size.

The full combination gives this floor a base torque of 2,401 kNm. The root-sum-square gives 12,233 kNm, five times as much, for a building that is almost exactly symmetric, while giving it 1,249 kN of base shear against a combined 1,749 — 71 per cent. The floor moves as a nearly rigid translation, and the rule reports a structure that twists hard and sways weakly, because it has treated two modes that are really one motion, seen in a skewed basis, as two independent events.

That is the invention in the title. As the eccentricity goes to zero, the true torque goes to zero, the true shear goes to the uncoupled building’s 1,766 kN, and the root-sum-square goes to a torque of about $1.4 \times 8{,}650$ and a shear of about $0.71 \times 1{,}766$ — a finite twist in a building with no eccentricity at all, and the shear of a building with only half its mass.

The sweep says where the danger is, and it is not where the discussion of closely spaced modes usually places it. The dangerous building is not the badly irregular one: at an eccentricity of half the plan’s radius of gyration, 4.9 m, the modes are far enough apart that the rule is within 2 per cent on everything, and even at a quarter of it the torque is only 8 per cent long. It is the one that is nearly regular — symmetric by intention, off-centre by the inevitable few tens of centimetres of real construction — and whose analysis, being modal, has converted a small eccentricity into two fully mixed modes.

What the full combination costs, and what fixes the problem without it #

The fix is not difficult. The complete quadratic combination has been the default in most analysis programs for decades, and it reduces to the root-sum-square wherever the modes are well separated, so there is little reason to use anything else. The value of understanding the sign is in reading the results.

Damping moves the problem, it does not remove it. Drop the floor’s damping to 2 per cent and the same 10 per cent separation gives a correlation of only 0.14.

The modal contributions are identical to the 5 per cent case — the spectrum was held fixed so that only the correlation could move — and the combination is not. A lightly damped steel frame and a heavily damped concrete one of the same stiffness and plan would be given different combined torques by a correct rule, and the same ones by the root-sum-square. For a structure fitted with viscous dampers — added precisely to raise the damping — the correlation between its coupled modes rises with the damping ratio, and the rule’s error grows with it.

Separating the frequencies removes it. The error exists because the torsional and translational frequencies are equal. Give the floor a torsional frequency half as high again as its translational one, by putting its stiffest walls on the perimeter, and at the same 0.98 m eccentricity the two modes are 51 per cent apart, the first carries 99.4 per cent of the mass, the correlation is 0.05, and the root-sum-square agrees with the full combination to within three per cent on every response. The coupling is still there; it has stopped mattering because the second mode carries almost none of the response.

That is also the design advice plan torsion arrives at from statics, reached from the other end. A torsionally stiff plan does not only reduce the corner displacements; it separates the modes, and separated modes are the ones every combination rule handles correctly.

What the numbers leave out #

One storey. The floor has two modes. A real building with several storeys has pairs of closely spaced coupled modes beside each of the translational modes in turn, and responses at the upper storeys mix all of them, so the torque of a given storey is the uncancelled remainder of several opposed pairs.

One direction of ground motion. Real ground shakes in both horizontal directions at once, and combining the response to each direction is a second combination problem with its own correlation. The usual rules for that — thirty per cent of one direction added to the other, or a square root of squares across directions — sit on top of everything here.

A flat spectrum across the two periods. For modes a few per cent apart this is exact enough to be invisible. For the separated cases in the sweep the two periods fall at different spectral accelerations and the modal contributions are no longer in the ratios drawn; the correlations, which do not depend on the spectrum, are unchanged.

Correlation from white noise. The coefficient is exact for a stationary broadband input and approximate for anything else. Records with a strong narrow-band component — soft-soil sites, or near-fault pulses — can correlate two modes more or less than the formula says, and a response-history analysis is the only calculation that does not rely on the assumption.

And an eccentricity known to within its own size. The figure where the rule is worst is the one where the eccentricity is smallest, and the eccentricity of a real building is not known to the nearest twenty centimetres. What the argument establishes is the direction and the scale of the combination error for a nearly symmetric plan, not the torque of any particular building. A design code’s accidental eccentricity exists precisely because the true one is unknown, and applying it changes which of the sweep’s regimes the analysis sits in.

Where the model stops #

Linear elastic. Modes exist only while stiffness is constant. A floor whose perimeter walls crack on one side has a stiffness centre that moves during the earthquake, and the modal picture is a description of the start of the motion rather than of what governs.

Rigid diaphragm. The floor is assumed to translate and rotate as a body. A floor flexible in its own plane has further modes in which it bends between the walls, and the diaphragm is then part of the problem rather than its assumption.

Classical damping. The same modal damping ratio is assigned to both modes, and the damping matrix is assumed to be one the mode shapes diagonalise. A building whose two sides are damped very differently — a stiff concrete core on one side, a lightly damped steel frame on the other — has modes that do not exist in real form at all, and that is the next question.

Peaks rather than histories. Every number here is a combined peak, and a response history is the calculation that does not need one. The combination answers how large, not when, and a design that needs the torque and the shear at the same instant — to check a wall under both — cannot read that from two separately combined peaks.

Still open: when the damping cannot be split into modes #

Everything above assumed that each mode could be given its own damping ratio and left alone, which is what makes a mode a mode. That assumption holds when the damping is spread through the structure in the same proportions as its mass or stiffness. It fails when one part of a structure is damped very differently from another: an isolated building on high-damping bearings with a lightly damped superstructure above, a soil-structure system radiating energy at its base, a frame with viscous dampers in one bay. Then the equations do not uncouple in real modes, each mode’s shape changes during its own cycle, and the question of what a mode shape even means has a less comfortable answer than the one drawn here.