Modal combination: SRSS and CQC

From a spectrum reading to storey forces

For mode n, read the spectral acceleration at its period and damping. The peak modal response then follows:

Modal displacement at each floor:

uₙ = Γₙ φₙ Sdₙ

Equivalent static storey forces:

fₙ = Γₙ M φₙ Saₙ

These are the forces which, applied statically, would produce the modal displacements — which is what makes member forces recoverable from a spectrum analysis at all.

Modal base shear, by summing the storey forces:

Vₙ = M*ₙ Saₙ

The effective modal mass appears directly, which is what makes it the natural quantity — and note that it is scale-independent, so this base shear does not depend on how the modes were normalised.

Why the peaks cannot be added

Suppose mode 1 gives a base shear of 4.39 MN and mode 2 gives 0.36 MN. Adding gives 4.75 MN.

That answer assumes both modes reach their peaks at the same instant AND in the same direction. Mode 1 has a period of 0.365 s and mode 2 of 0.130 s — they are running at quite different rates and there is no reason for their peaks to coincide.

Adding them is not conservatism. It is the answer to a different question: what would happen if the peaks were simultaneous, which they are not.

SRSS

The standard treatment is the square root of the sum of squares:

R = √(ΣRₙ²)

It has a probabilistic justification — for responses that peak at random, uncorrelated times, this is the expected maximum of the combination. It is always less than the arithmetic sum and greater than the largest single contribution.

Its assumption is that the modes are well separated in frequency. Two modes at very different frequencies really do peak at essentially unrelated instants, and SRSS is then a good estimate.

When SRSS fails, and in which direction

When two modes have nearly the same frequency, their responses are not independent — they rise and fall nearly together, so their peaks tend to coincide. SRSS then UNDER-estimates, and it does so silently.

This is not a rare case. A symmetric building has two lateral modes at nearly the same frequency by construction, and a torsionally coupled building has translational and torsional modes that can be very close.

CQC

The complete quadratic combination adds the cross terms that SRSS omits:

R = √(ΣᵢΣⱼ ρᵢⱼ Rᵢ Rⱼ)

with a correlation coefficient ρᵢⱼ that is 1 when the frequencies coincide and falls away as they separate:

ρᵢⱼ = 8√(ζᵢζⱼ)(ζᵢ + rζⱼ)r^(3/2) / [(1−r²)² + 4ζᵢζⱼr(1+r²) + 4(ζᵢ²+ζⱼ²)r²], r = ωᵢ/ωⱼ

When the frequencies are well separated every ρᵢⱼ is nearly zero for i ≠ j, and CQC reduces to SRSS. So CQC is never worse and is sometimes essential.

Note that the signs of the modal responses matter in CQC, because cross terms can be negative. They do not matter in SRSS, where everything is squared.

Use CQC whenever any two modal periods are within about 10% of each other. Many engineers use it always, on the grounds that it costs nothing extra and removes a judgement.

Directional combination

An earthquake has components in both horizontal directions, and they too peak at different instants. Two conventions are in use:

The 100/30 rule. Take the full effect in one direction with 30% of the other, then swap, and take the worse.

Directional SRSS. √(Rx² + Ry²).

Both are approximations to the same underlying fact, and which is required is a matter for the governing design standard rather than for this course.