Structural Dynamics / module 14
Response-spectrum analysis
How a response spectrum summarises peak oscillator responses to a ground motion, and what information is lost when the time history is condensed.
What you will understand
- Construct a response spectrum from a ground motion by solving many oscillators
- Relate spectral displacement, pseudo-velocity and pseudo-acceleration, and say why 'pseudo'
- State the two asymptotes every spectrum must satisfy and use them as checks
- Explain the assumptions behind SRSS and CQC modal combination and choose an appropriate rule.
- Explain the assumptions behind SRSS and CQC, and why the absolute modal sum can be overly conservative
- Distinguish a response spectrum from a design spectrum
Lessons
Constructing a response spectrum
Solve one oscillator, record its peak, change the period, repeat. The definition is the algorithm, and the two ends of the curve are fixed by physics.
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.
Module summary
- A spectrum is the peak response of every possible SDOF structure to one record, at one damping
- PSv = ωSd and PSa = ω²Sd — 'pseudo' because PSa is exactly the elastic force over the mass
- As T → 0, PSa → PGA. As T → ∞, Sd → PGD. Both are free checks on a computed spectrum
- A spectrum contains no information about WHEN each peak occurred
- Modal base shear is M*ₙSaₙ — effective mass times spectral acceleration
- The absolute sum bounds the combined modal histories, but often overestimates their coincident peak
- SRSS assumes well-separated modes; CQC is needed when they are close
- A design spectrum envelopes many records and is not the response of anything
Teaching basis and review status
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.
Read the lesson register and outstanding review notesCheck your understanding
Practice
An oscillator of period 0.8 s has a spectral displacement of 65 mm. What is its pseudo-acceleration, in m/s²?
Practice
Modal base shears are 6.0, 2.5, 1.2 and 0.8 MN. What is the SRSS combination, in MN?
Check yourself
A single-mode check with the specified spectral ordinate0.28g reports V/W=2.8. What should be investigated first?
Practice
Two modes have a frequency ratio r = 0.90 and 5% damping each. What is the CQC correlation coefficient?
Check yourself
What information does a response spectrum deliberately discard?