Modal Combination: SRSS or CQC?

1. Modes do not peak at the same time

In the modal method the contribution of each natural mode is computed separately: a spectral acceleration is read from each mode's period, and that mode's storey forces, base shear and displacements are found. But these modal maxima do not occur at the same time during an earthquake. Adding them directly (absolute sum) is safe but needlessly large; taking only the largest is unsafe. TBDY 2018 requires them to be combined statistically (Annex 4B.2.4).

2. An example: a two-storey building

A two-storey shear building with storey masses of 100 t and storey stiffnesses of 20 000 kN/m; S_DS = 1.0, S_D1 = 0.4, R = 8, D = 3. The site's engine (verified against the official TBDY text):

The first mode carries 95% of the mass. Because the second mode's period is below T_B = 0.4 s, the load reduction factor is not the full R/I but, by Eq. (4.2), R_a = 3 + 5·(0.2746/0.4) = 6.43; short-period modes are reduced less.

3. SRSS: when modes are well separated

The square-root-of-sum-of-squares rule (SRSS, KTKK in TBDY) r = √(Σ r_j²) assumes the modes are independent. TBDY Annex 4B.2.4(d) accepts it on one condition only: β_mn = T_m/T_n < 0.8 for every pair of modes considered. In the two-storey building β = 0.38; the condition holds and the two rules give almost the same result (first row of the table below).

4. CQC and the cross-correlation coefficient

The complete quadratic combination (CQC, TKB in TBDY) accounts for the correlation between pairs of modes through ρ_mn: r = √(Σ Σ r_m ρ_mn r_n). With equal damping in all modes, TBDY Eq. (4B.5b):

ρ_mn = 8ξ²(1 + β_mn)β_mn^3/2 / [(1 − β_mn²)² + 4ξ²β_mn(1 + β_mn)²]

Below a period ratio of 0.7 the correlation is negligible; beyond 0.8 it grows quickly and reaches 0.79 at 0.95. The code's 0.8 limit sits exactly at this knee.

5. With close modes SRSS is wrong — in both directions

With close modes (β = 0.9) SRSS underestimates the result by about 16% for same-sign contributions and overestimates it by about 31% for opposite-sign ones. Opposite signs arise, for example, in torsionally sensitive buildings where the translational and torsional modes contribute in opposite directions to the displacement at a corner. SRSS ignores sign and cannot tell these cases apart; even the direction of its error cannot be known in advance. That is why CQC is mandatory in asymmetric buildings with close translational and torsional periods.

6. Last step: scaling the base shear

If the base shear from modal combination V_tx is below a given fraction of the equivalent-lateral-force base shear V_tE, all internal forces and displacements are scaled up (TBDY §4.8.4.1, Eq. 4.31): β_tE = γ_E·V_tE/V_tx ≥ 1. If at least one of the A1, B2 or B3 irregularities exists, γ_E = 0.90; if none of the irregularities in Table 3.6 exist, 0.80.

7. In practice

• Check the period ratio of every pair of modes, not only consecutive ones. If any pair has β above 0.8, use CQC.
• Keep the signs. CQC's advantage comes from using the signs of the modal contributions; do not take absolute values before combining.
• Know your software's default. Many programs default to CQC; if SRSS is chosen, the condition above must be checked separately.