Structural Dynamics / Response-spectrum analysis
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.
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.
Read the lesson register and outstanding review notesWhy this matters
A time-history analysis gives everything: the whole response, instant by instant. It is also expensive, it needs a record chosen and scaled, and its answer applies to one structure and one record only.
A response spectrum trades all that detail for a single curve that applies to EVERY structure. That trade is what makes routine seismic design possible — and understanding exactly what was given up is what stops the method being misused.
By the end of this lesson you should be able to
- State the definition of a response spectrum and follow its construction
- Relate the three spectral quantities and explain the word 'pseudo'
- Use the short- and long-period asymptotes as checks
- Say precisely what information a spectrum does not contain
What you should already know
- Base excitation (Module 13)
- Newmark integration (Module 8)
- Natural period and damping ratio (Module 4)
Constructing a response spectrum
Choose an oscillator period T and damping ratio ζ. Integrate its response to a specified ground-motion record, record the peak, then repeat over the period range of interest. The resulting spectrum applies to that record and damping model.
The main quantities are:
- Spectral displacement Sd: peak relative displacement.
- Pseudo-velocity PSv=ωSd.
- Pseudo-acceleration PSa=ω²Sd.
For the linear spring, peak elastic force is kSd=mPSa. True peak absolute acceleration Sa includes the damping-force contribution and is a different quantity. At zero damping, Sa=PSa. At nonzero damping they need not coincide; compute the quantity required rather than assuming a fixed difference. Check how the governing standard defines its spectral ordinates and force recovery.
Limiting behaviour
For a suitably represented input and converged integration, PSa approaches PGA as T tends to zero. At fixed damping ratio and over a consistent finite response window, Sd approaches PGD as T tends to infinity, provided ground displacement and initial conditions are defined consistently. A finite period is not automatically close enough to either limit.
Use these limits to investigate units, integration and record-processing problems. Do not turn an arbitrary period or percentage discrepancy into a universal pass/fail rule. Long-period displacement is especially sensitive to baseline correction and the analysis window.
Time step and input bandwidth
A record sampled at0.01s has a50HzNyquist limit, but this does not establish usable physical content up to50Hz. Review filtering and sampling metadata. An oscillator at0.03s may still respond dynamically to the input; reducing its period to0.02s does not guarantee a particular percentage agreement with PGA.
Subdividing the integration step using a documented input interpolation can improve numerical accuracy. It cannot recover physical frequencies absent from the original record. Check both the input representation and integration convergence.
Information the spectrum omits
The spectrum records peaks rather than their timing. Peaks of different modal responses cannot generally be assumed simultaneous. Their absolute sum is an upper bound when the individual modal peaks are known, while SRSS and CQC provide estimates under their respective assumptions.
Try it
Building a response spectrum, one oscillator at a time
A spectrum is the peak response of every possible single-degree-of-freedom structure to one record. This tool computes exactly that, and lets you inspect any one of them.
Record
Plot
Compare another record
- ζ = 5%
- Smoothed
Show the numbers behind this plot
The oscillator at T = 0.60 s, solved in full — its peak is the single point marked on the spectrum above.
- Sd at this period
- 25.3 mm
- PSv
- 0.264 m/s
- PSa
- 2.76 m/s² (0.28 g)
- Peak elastic force per tonne
- 2.76 kN/tonne
- True peak absolute acceleration Sa
- 2.78 m/s²
- Peak occurred at
- 4.61 s
- Short-period check: PSa → PGA
- 0.1% error
- Long-period trend: Sd → PGD
- 40.6% from PGD
f = m·PSa — one multiplication, which is why codes tabulate spectral acceleration.
0.6% from PSa — they coincide only at zero damping.
The spectrum does not record this. It is what a spectrum throws away.
A sharp check: PSa must reach the PGA. Anything above a few per cent here is a units or integration error.
A weak check, and deliberately not scored. The asymptote is only approached when the period far exceeds the record's DURATION, so a large gap here is expected rather than wrong.
Both ends of the spectrum are fixed by physics. As T → 0 the structure is rigid, rides with the ground, and PSa must approach the peak ground acceleration of 2.4 m/s². As T → ∞ the mass stays still and Sd must approach the peak ground displacement of 109 mm. A computed spectrum that misses either asymptote has a units or integration error in it.
A response spectrum is not a design spectrum
- This curve is jagged because a single earthquake happens to have more energy at some periods than others, by accident of that particular rupture and path.
- A design spectrum is smooth because it envelopes many records and includes a chosen probability of exceedance. It is a different kind of object and it is never the response of anything.
- Two structures whose periods differ by 10% can have spectral demands differing by a factor of two on a single-record spectrum. On a design spectrum they will not. That difference is the whole reason design spectra are smoothed.
- These records are synthetic and generated for teaching. Nothing here is a design value.
What this shows: A response spectrum is a summary of what one earthquake does to every possible SDOF structure. It contains no information about WHEN each peak occurred — which is precisely why modal peaks have to be combined statistically.
From first principles
The spectral relations and their asymptotes
We want to show: Establish PSv = ωSd and PSa = ω²Sd, show that PSa is exactly the elastic force per unit mass, and derive both limiting values of the spectrum.
For a lightly damped oscillator the response near its peak is close to a sinusoid at its own frequency. For a sinusoid, velocity and acceleration amplitudes are simply ω and ω² times the displacement amplitude — so once the peak displacement is known, two other quantities follow with no extra work. The word 'pseudo' records that these are defined quantities rather than measured ones, and that one of them turns out to be exactly a force.
A response spectrum is not a design spectrum
These are different objects and the course keeps them apart deliberately.
| Response spectrum | Design spectrum | |
|---|---|---|
| Comes from | One recorded or simulated motion | Many records, statistically enveloped |
| Shape | Jagged, with sharp peaks and troughs | Smooth, with straight branches |
| Meaning | What that earthquake did | What a code requires to be designed for |
| Includes | Nothing but the record | A hazard level, a return period, site effects, importance |
| Is the response of | One specific event | Nothing at all |
A single-record spectrum is jagged because that particular rupture happened to have more energy at some periods than others. Two structures whose periods differ by 10% can show demands differing by a factor of two on it.
A design spectrum is smooth precisely because that accident has been averaged out across many records. It is a requirement, not a prediction, and it is not the response of any structure to any event.
Reading a design spectrum as though it were an earthquake — or a single-record spectrum as though it were a design requirement — are both category errors, and both are common.
This course teaches how a response spectrum is built and what it means. The rules for constructing design spectra belong to the current design standards and must be taken from them directly, not from any textbook and not from here.
Worked example
Reading a spectrum, and checking that it is credible
Given
- A synthetic rock-site record with PGA = 2.40 m/s² and PGD = 0.109 m
- A response spectrum computed at 5% damping over 0.03 s to 4 s
- A structure with a first period of 0.62 s and a seismic mass of 3 200 tonnes
Find
The design force, and whether the spectrum can be trusted.
Assumptions
- Elastic response; single-mode estimate for this illustration
Predict first
A computed response spectrum tends to 0.24 m/s² at very short period, while the record's PGA is 2.4 m/s². What has happened?
Practice
An oscillator with a period of 0.5 s has a spectral displacement of 42 mm. What is its pseudo-acceleration, in m/s²?
Practice
A structure of mass 5 000 tonnes has a spectral acceleration of 3.4 m/s². What is the peak elastic base shear, in MN?
Practice
For that structure, the base shear coefficient V/W is what value?
Check yourself
Why is pseudo-acceleration called 'pseudo'?
Summary
- A spectrum is the peak response of every possible SDOF structure to one record
- The definition is the algorithm: build an oscillator, integrate, record the peak, repeat
- PSv = ωSd and PSa = ω²Sd, and PSa = fmax/m exactly for an elastic system
- Sa=PSa at zero damping; at nonzero damping they are distinct measures that can coincide in particular cases
- PSa → PGA as T → 0 and Sd → PGD as T → ∞ — check both, every time
- For the single-mode idealisation, V/W=PSa/g; assess its value against the actual input spectrum
- A design spectrum envelopes many records and is not the response of anything