Constructing a response spectrum

The definition is the algorithm

A response spectrum is a plot of the peak response of a single-degree-of-freedom oscillator against its natural period, for one ground motion and one damping ratio.

The construction follows directly:

1. 1.Choose a period T and a damping ratio ζ.
2. 2.Build an oscillator with those properties.
3. 3.Integrate it under the ground motion — Newmark, from Module 8.
4. 4.Record the largest response it reaches.
5. 5.Change T and repeat, perhaps sixty times.
6. 6.Plot the peaks against period.

That is genuinely all of it. The course library does exactly this in responseSpectrum(), and the generator lab performs it visibly so that one oscillator can be inspected and its peak seen becoming one point on the curve.

Three quantities, one curve

From each oscillator, three numbers are conventionally taken.

Spectral displacement, Sd — the peak RELATIVE displacement. What deforms the structure.

Pseudo-velocity, PSv = ωSd.

Pseudo-acceleration, PSa = ω²Sd.

Why 'pseudo' — and why it is not a hedge

PSa is not an approximation to the acceleration. It is an exact statement about FORCE.

The peak elastic force in the oscillator is

f_max = k · Sd = mω² · Sd = m · PSa

So PSa is exactly the peak elastic force divided by the mass. That is why design codes tabulate spectral acceleration: it converts to a force with one multiplication and no further thought.

What PSa is NOT is the true peak absolute acceleration, Sa. The two coincide only at zero damping and diverge as damping rises — by a few per cent at 5%, more at 20%. When a code says 'spectral acceleration' it means PSa, and when a time-history analysis reports peak floor acceleration it means Sa. They are different quantities with confusingly similar names.

The two asymptotes, which are free checks

Both ends of a spectrum are fixed by physics and neither depends on the record's details.

Short period, T → 0. The structure is rigid. It moves exactly with the ground, so it deforms not at all and feels precisely the ground acceleration:

Sd → 0, PSa → PGA

Long period, T → ∞. The mass is effectively unattached. It stays where it is while the ground moves under it, so the relative displacement is the ground displacement:

Sd → PGD, PSa → 0

A computed spectrum that does not approach the PGA at its short-period end has a units error or an integration problem. Check it every time — it costs nothing and it catches the errors that matter most.

The two checks are not equally useful, and it is worth being honest about why.

The short-period check is SHARP. A rigid oscillator follows the ground immediately, so PSa reaches the PGA as soon as the period is short compared with the record's frequency content. Anything more than a few per cent out is an error.

The long-period check is WEAK. Reaching that asymptote requires the period to far exceed the record's DURATION, not merely to be long — an oscillator with a 10 s period, under a 20 s record, has completed only two cycles and its peak is still dominated by transient behaviour rather than by the 'mass stays still' limit. On the teaching records here, Sd sits about 40% above PGD at T = 4 s, 24% above at 10 s and 18% above at 20 s, and it does not approach monotonically.

So treat the short-period value as a pass/fail test and the long-period value as an order-of-magnitude sanity check only. A spectrum that is 40% above PGD at 4 s is behaving normally; one that is ten times PGD is not.

What a spectrum throws away

Time.

The spectrum knows the largest value each oscillator reached. It does not know when, and it therefore cannot know whether two different oscillators reached their peaks at the same instant.

That single omission is the reason modal peaks cannot simply be added, and it is why the combination rules of the next lesson exist. Everything difficult about response-spectrum analysis traces back to this one discarded piece of information.

Two cautions about the short-period end

Sub-sampling. A record sampled at 0.01 s gives Δt/T = 1 for a 0.01 s oscillator, which is meaningless. Real spectrum routines sub-sample the record for short periods, and the course library does so automatically in resampleFor(). A spectrum computed from a coarsely sampled record without resampling is untrustworthy exactly where the asymptote check would reveal it.

And the asymptote check only works if you go short enough. 'Short period' means short compared with the record's own frequency content, not merely a small number. A record sampled at 0.01 s has a Nyquist frequency of 50 Hz, so it contains real energy up to that frequency. An oscillator at T = 0.03 s — that is, 33 Hz — is still being genuinely excited by that content and will read perhaps 20% ABOVE the peak ground acceleration.

That is not a failed check. It is a spectrum that has not been taken far enough to the left for the asymptote to apply. Take the same record down to T = 0.02 s and PSa lands within a fraction of a per cent of the PGA.

Before treating a short-period discrepancy as an error, confirm that the shortest period plotted is genuinely above the record's frequency content. If it is not, the check has not been performed.

A response spectrum is not a design spectrum

These are different objects and the course keeps them apart deliberately.

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.