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Structural Dynamics / Response-spectrum analysis

Modal combination: SRSS and CQC

Each mode's peak is known and the instants at which they occur are not — so they cannot be added. What replaces addition, and when the simpler rule stops working.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.
  • Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
  • Verify current governing spectrum, modal/directional combination and modal-mass rules. SRSS/CQC are approximations, not guaranteed record-specific bounds.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Why this matters

Modal analysis gives each mode's peak response. A spectrum gives each mode's peak from a single reading. Both are straightforward.

The difficulty is putting them together. Mode 1 peaks at some instant, mode 2 at another, and the spectrum — having thrown away time — cannot say whether those instants coincide. Adding the peaks assumes they do, which need not be true. The absolute sum gives an upper bound on true modal histories but can be excessively conservative.

This lesson is about what to do instead.

By the end of this lesson you should be able to

  • Compute modal storey forces and base shear from a spectrum
  • Explain the upper-bound meaning and possible conservatism of the absolute modal sum
  • Apply SRSS and state the condition under which it is valid
  • Apply CQC and recognise when it is required

Modal response and combination

For mode n, un=Γnφ_nSd_n. Equivalent elastic forces are fn=ΓnMφ_nPSa_n, and the corresponding modal base shear is M*_nPSa_n. Here PSa=ωn²Sd, not necessarily peak absolute acceleration.

The absolute sum of modal peak magnitudes is an upper bound on the sum of their time histories, but usually overstates the coincident peak. SRSS estimates the combined response as √ΣRn² when modal responses are sufficiently uncorrelated. It is an approximation, not the exact maximum of a given record.

CQC includes correlations: R=√ΣiΣ_jρ_ijR_iR_j. Preserve the signs of modal responses. Closely spaced modes can increase or decrease the result relative to SRSS, depending on those signs. With equal damping, coincident frequencies giveρ=1; for unequal damping the coefficient need not equal1. Check the governing procedure and damping assumptions when selecting a combination rule.

A10%frequency-spacing guide is not a universal acceptance rule. Directional combinations, including100/30and directional SRSS, likewise follow the applicable design standard.

Worked example

Modal response-spectrum analysis of a three-storey building

Given

  • The uniform three-storey building: 400 t per floor, 600 MN/m per storey
  • Periods 0.365 s, 0.130 s, 0.0900 s (Module 11)
  • Effective masses 1 097 t, 89.9 t, 13.2 t (Module 12)
  • A design spectrum whose plateau covers all three periods, at Sa = 4.0 m/s²

Find

The modal base shears and their correct combination.

Assumptions

  • All three periods fall on the spectrum plateau, so the same Sa applies to each — chosen to make the arithmetic transparent
  • 5% damping in every mode

    Predict first

    Two modes have periods1.24s and1.19s, equal5%damping and positive signed contributions to the response being combined. How does SRSS compare with CQC?

    Practice

    Modal base shears are 3.2 MN, 0.9 MN and 0.4 MN. What is the SRSS combination, in MN?

    Practice

    Two modes have frequencies of 5.0 and 5.5 rad/s, both with 5% damping. What is the CQC correlation coefficient ρ between them?

    Practice

    Responses of 100 and 60 units act in two orthogonal directions. What is the 100/30 directional combination?

    Check yourself

    When does CQC reduce to SRSS?

    Check yourself

    Why is the pseudo-acceleration, rather than the true peak acceleration, tabulated in design spectra?

    Worked example

    Combining three modal responses

    Given

    • Modal base shears of 148, 62 and 25 kN
    • The three modal periods are well separated

    Find

    The combined base shear, and what the alternatives would give

      Worked example

      When SRSS is unsafe

      Given

      • Two modes with responses of 100 and 90 kN
      • Their circular frequencies are 10.0 and 10.5 rad/s
      • Damping is 5 % in both

      Find

      Whether SRSS is appropriate

        Summary

        • Modal storey forces are fₙ = ΓₙMφₙSaₙ; modal base shear is Vₙ = M*ₙSaₙ
        • Modal peaks occur at different instants, so they cannot be added
        • The absolute sum bounds the sum of modal histories; it can be much more conservative than a justified modal combination
        • SRSS neglects correlation; CQC includes signed cross terms that can increase or decrease the result
        • For equal damping, CQC correlation tends to1 at coincident frequencies and becomes small when well separated
        • Use a modal combination justified by mode spacing, damping and the governing procedure
        • Directional combination — 100/30 or SRSS — handles the two horizontal components