The spectrum is not a load

The curve a seismic calculation begins with is drawn like a load: a smooth line with acceleration up the side and period along the bottom, from which a number is read off and multiplied by a mass. It behaves like a load, in the sense that a bigger one gives a bigger answer.

It is not a load, and almost every misuse of it follows from reading it as one.

The definition, which is the argument #

A response spectrum is constructed like this. Take a record of ground acceleration. Take an oscillator of some period and some damping ratio. Integrate its equation of motion through the whole record — the same Newmark integration that drew the suddenly applied load — and note the largest displacement it ever reaches. That is one point.

Change the period. Do it again. Forty times.

That is the whole definition, and three consequences follow directly from it.

A point on the curve is not a moment in time. The peak for the 0.3 s structure happened at one instant and the peak for the 1.8 s structure at a different one, several seconds apart. Nothing on the curve happens simultaneously, so nothing on the curve can be used as a set of loads applied at once — which is why the modal responses of a multi-storey building have to be combined statistically rather than added.

The curve carries a damping ratio. Every point on it assumed one, and a curve for 2% is a different curve.

The curve is not a property of the earthquake. It is a property of the earthquake and of the family of structures it was evaluated against. Two spectra from the same record with different damping are two different curves describing one event.

Why it takes the shape it takes #

The two ends are fixed by physics rather than by convention, and knowing why is what makes the middle interpretable.

At zero period the structure is infinitely stiff. It cannot deform relative to the ground, so it goes where the ground goes and its acceleration is the ground’s acceleration. The curve therefore starts at the peak ground acceleration, exactly, for every damping ratio — 3.5 m/s² here.

At infinite period the structure is infinitely flexible. The mass stands still in space while the ground moves underneath it, so the displacement of the structure relative to its base is the ground’s own displacement, and its acceleration tends to zero.

In between there is amplification, and it peaks where the structure’s period matches the dominant period of the shaking. The amplification here is 2.02 at 5% damping. For real records it is commonly between 2 and 3, which is a much smaller number than the resonance essay’s $1/2\zeta$ — 10 at 5% — and the difference is entirely because an earthquake is not a sine wave. It contains a band of frequencies, none of them for long, so no structure gets the sustained forcing that a steady-state resonance needs.

Two buildings, one earthquake #

Those two figures are the argument of this essay in the form a reader can check. The spectrum is a summary of exactly this: forty-four such histories, one number kept from each. It compresses a great deal and the compression is the point — nobody can design from forty-four time histories — but what is being compressed is a set of structural responses, not a set of facts about the ground.

Pseudo-acceleration, and why the prefix is there #

The vertical axis is not the peak acceleration the structure experiences. It is $\omega^2 S_d$ — the peak displacement multiplied by the square of the natural frequency.

The reason for defining it that way is exact rather than approximate: at the instant of peak displacement, the force in the structure is $k S_d$, and since $k = m\omega^2$, that force is $m \cdot \omega^2 S_d$. So mass times pseudo-acceleration is exactly the elastic force, which is the number a designer wants, and it is exactly that at the instant it matters.

The true peak total acceleration is slightly different, because at the instant of peak displacement the velocity is zero and the damping force vanishes, while the true peak acceleration occurs at a different instant with damping contributing. At 5% damping the two agree to a fraction of a per cent; at 20% they do not. The distinction is worth keeping because the two quantities have different names in the literature and the same symbol on most drawings.

Every record gives a different curve #

That variability is the fundamental difficulty of the subject and it is not reducible by better analysis. A spectrum from a single record is jagged, and the jaggedness is real — it records the accidents of how that particular rupture happened to load that particular period. Another earthquake of the same size at the same site produces peaks in different places.

The response is to average. A design spectrum is a smooth envelope over many records, scaled to a chosen intensity, and it is deliberately not the spectrum of anything. It is smooth because the peaks of individual records are not repeatable, so designing a building to fit into a trough would be designing for an accident.

That distinction — the difference between a response spectrum, which is a measurement, and a design spectrum, which is a statistical construction — is the single most useful thing to keep straight in this subject, and the fact that both are drawn with the same axes is why it is so often lost.

Biot’s pendulums, and why the spectrum exists at all #

The idea is Maurice Biot’s, from the early 1930s, and its original form is worth knowing because it explains the shape of the object.

Biot proposed measuring the response of a set of pendulums of different periods to a recorded ground motion, and built a mechanical analogue to do it — a rack of torsional pendulums, each tuned to a period, each with a way of recording how far it swung. Run the record past the rack and read off the swings, and the result is a response spectrum obtained without any computation at all.

That is the concept in its purest form: the spectrum is what a shelf of structures did. George Housner then developed it into a design tool through the 1940s and 50s, averaging over the small number of strong-motion records that existed and producing the first smoothed design spectra — and it was that step, the averaging, that turned a measurement into a specification.

The history explains an oddity in modern practice. Integrating a structure through a record is now a few milliseconds of work, and integrating it through twenty records is a few seconds; the computational saving the spectrum was invented to provide has evaporated entirely. It survives because it is a communicable summary of a seismic hazard — a curve that can be published, argued about in committee, mapped over a country and written into a regulation — and because a profession’s methods outlive the constraints that produced them.

What is lost in the smoothing #

Three things, and each has caused trouble.

The duration is gone. A spectrum records the largest excursion and says nothing about how many large excursions there were. Two records with identical spectra can demand a handful of cycles or fifty, and for a structure that yields the number of cycles is what decides whether it survives, because damage accumulates.

The sequence is gone. Whether the largest pulse arrived first, when the structure was undamaged, or last, when it was not, changes the outcome and does not change the spectrum.

Near-fault pulses are averaged away. A record taken close to a rupture can contain a single large velocity pulse, which loads a structure very differently from a long shaking of the same spectral content — the structure is given one shove rather than many, which is the impulse regime of the shock spectrum rather than the resonant one.

Displacement, which is the quantity the structure cares about #

The acceleration and displacement spectra contain identical information — one is $\omega^2$ times the other — and they lead to opposite intuitions. Read as acceleration, a tall flexible building looks safe: the force is small. Read as displacement, it looks alarming: the deformation is large. Both are true, and which one governs depends on whether the structure fails by breaking or by moving too far.

For most of the twentieth century seismic design was force-based, and read the first curve. The modern tendency is towards displacement-based design, which reads the second, and the reason for the shift is the subject of the next essay: a real structure does not stay elastic, and once it yields the force stops being the interesting variable.

What the free body is #

It is worth restating, because the spectrum’s abstraction hides it.

The equation being integrated at every point is the free body of the mass, with the ground moving underneath. In coordinates relative to the ground, the effective force on the mass is $-m\ddot u_g$ — the mass times the ground’s acceleration, applied backwards. There is no reaction to that force anywhere. It is an inertial force, and the only thing holding it is the structure’s own stiffness.

That inertial force is the same object that appears in every free body drawn in this field, and it is the one that troubles people most, because it is the only force on the diagram with nothing on the other end of it. A wind pushes and the air is pushed back; a weight pulls and the earth pulls back. Here there is nothing: the term exists because the coordinate system is accelerating, and drawing it is what allows the sums to cancel as though nothing were moving.

That is why an earthquake is different in kind from every other load on this site. A wind load has a source and a reaction; a self-weight has a source and a reaction. This one has neither: the ground is not pushing the building sideways, it is moving out from underneath it, and everything the structure experiences is a consequence of its own unwillingness to follow.

Which also explains the field’s most counter-intuitive rule of thumb: a heavier building is worse. Everywhere else on this site, mass is a nuisance in proportion to itself; here the load is the mass, so halving the weight of a building halves the earthquake it has to resist. This is why timber and light steel structures perform well and why a masonry building is a difficult one — and it inverts the usual reading of what a structure’s own weight costs it, where mass is a burden to be carried rather than the thing doing the shaking.

Reading a spectrum honestly #

Three habits follow from everything above, and they are what separates using a spectrum from being used by one.

Ask which damping it was drawn for, and whether the structure has it. A spectrum drawn at 5% applied to a welded steel chimney at 0.4% understates the response by a factor of three, and nothing in the arithmetic complains.

Ask where on the curve the structure sits, and how far it can move. A building at the peak is at the worst point of a curve that has a peak because of an accident of that record; a building on the falling tail is somewhere the curve is smooth and the answer is robust. The uncertainty in a computed period — ±20%, from everything the period essay lists — is a horizontal band on this plot, and the honest reading takes the worst value inside it.

Ask what the structure will do when the force arrives. The spectral acceleration multiplied by the mass gives the force an elastic structure would attract, and no ordinary building is designed to resist that force elastically. What is done instead is the subject of the next essay, and it is not a refinement of this calculation.

Where the ladder goes #

The spectrum as used above assumes the structure stays elastic, so that a period exists and modes can be superposed. Under a design-level earthquake it will not stay elastic, and the whole apparatus needs a second idea to survive the transition: the reason a weaker structure can be a safer one, and why what the earthquake actually asks for is a displacement.

Two other directions lead out. One is the question of what a spectrum should be for a site where the seismic hazard comes from several sources — an argument about probability rather than mechanics. The other is the observation, already visible in the history above, that the spectrum was invented to make a time-history problem into a table look-up, at a moment when integrating a record for every structure was impossible. That is no longer true, and the survival of the spectrum in an age of cheap computation is a fact about how engineering knowledge is transmitted rather than about what is calculable.