The ground has a period of its own

An earthquake is recorded on rock and experienced on soil, and the two are not the same motion.

Between them is a layer that is, structurally, a very familiar object: a column, fixed at its base to the bedrock, free at its top surface, with a mass and a stiffness of its own. It has a fundamental period. It has higher modes. It has damping. And the motion arriving at the base of a building is the rock motion filtered by that column.

Which free body produced the number #

A vertical column of soil of unit plan area, carrying a shear wave travelling upward.

The wave equation for that column is $\rho\,\partial^2u/\partial t^2 = G\,\partial^2u/\partial z^2$, whose solutions are waves travelling at $v_s = \sqrt{G/\rho}$. The boundary conditions are a free surface at the top — no shear stress — and the bedrock at the bottom.

If the bedrock were rigid, the column would be exactly a shear beam fixed at one end and free at the other. Its modes are quarter-wavelength, three-quarter-wavelength, five-quarter, and so on, so

$$T_1 = \frac{4H}{v_s}, \qquad f_n = \frac{(2n-1)v_s}{4H}$$

The period contains nothing but the layer — no earthquake, no building, no density, exactly as the period nobody chose contains only the structure’s own mass and stiffness. Thirty-eight metres at 75 m/s gives 2.03 seconds. Thirty metres at 400 gives 0.30. Neither the earthquake nor the building nor the density appears in it.

And the resonances are the odd harmonics — one, three, five, seven — which is the signature of a fixed-free column and of nothing else. A layer between two rigid boundaries would give one, two, three; a free-free layer would too. The odd series is a fingerprint, and it is visible in real recordings.

What limits the amplification #

If the bedrock were rigid the amplification at resonance would be limited only by the soil’s own damping, and would be $1/(\pi\xi/2)$ for the first mode — about 12.7 at 5% damping, and unbounded as the damping goes to zero.

It is not rigid, and that changes the answer qualitatively. Some of the wave energy arriving at the surface reflects back down, reaches the bedrock, and passes into it rather than reflecting again. That is radiation damping, and it is governed by the impedance ratio

$$\alpha = \frac{\rho_s v_s}{\rho_r v_r}$$

— the product of density and velocity in the soil over the same product in the rock. For the lake bed above, $\alpha = (1300\times75)/(2200\times800) = 0.055$.

The peak amplification is then

$$A_n = \frac{1}{\alpha + \frac{\pi}{2}(2n-1)\xi}$$

and with $\alpha = 0.055$ and $\xi = 0.05$ the two terms are 0.055 and 0.0785 — comparable, with the impedance term a substantial share. Assume rigid rock and the answer is 12.7; include the contrast and it is 7.5.

The clean way to see that the impedance term is doing the work is to change only the rock.

The soil did not become more resonant. The exit did, and the exit narrowed. Harder rock is a stiffer, more reflective boundary, so a larger share of each downgoing wave comes back up rather than passing into the half-space, and the column rings for longer at every one of its periods.

There is a design reading of that and it is uncomfortable. A soft layer sitting directly on very hard rock is the worst arrangement of the two ingredients — the long period comes from the soft layer and the sharpness comes from the contrast — and it is the arrangement a shallow soil basin over granite naturally produces. A soft layer over merely firm ground is a gentler filter, because the contrast that made the period interesting is the same contrast that lets the energy leave.

The thing that keeps the surface motion finite is that the energy can leave downward, and a model that assumes rigid bedrock has removed the only escape. That is a general hazard with rigid boundary conditions in dynamics: a rigid support does no work and dissipates nothing, and a structure connected to one has lost a damping mechanism it really has. The ground is a spring makes the same point about a foundation, where the radiation damping into the half-space is often larger than the structure’s own.

The selectivity, which inverts the intuition #

The transfer function is not a broad hump. It is a peaked filter, and what it does to a short period is quite different from what it does to a long one.

This site amplifies a 2.03-second motion by 7.5 and a 0.3-second motion by 1.4 — a selectivity of 5.4. Soft ground amplifies long periods a great deal and short periods hardly at all, and it can even de-amplify them once the soil’s strain-dependent damping is included.

So the received idea that soft ground is dangerous is half of the truth. It is dangerous for buildings whose period is near the site’s, which for a soft site means tall or flexible ones, and it is comparatively benign for stiff low buildings. A two-storey masonry building on a lake bed and a fifteen-storey frame on the same lake bed are in completely different situations.

Mexico City, 1985 #

The standing case, and it is worth setting out because every element of the argument appears in it.

The earthquake was a magnitude 8.0 subduction event on the Pacific coast, 350 km away. At that distance the short-period content had been attenuated and what arrived at the valley was already long-period motion of modest amplitude — the rock sites in the city recorded peak accelerations around 0.03 g, which is nothing.

The historic centre stands on the bed of a drained lake: 30 to 50 m of very soft high-plasticity clay with a shear wave velocity around 75 m/s, over much stiffer material. $4H/v_s$ puts its period near two seconds — and the arriving motion was rich at two seconds.

The surface record at SCT showed a peak acceleration of about 0.17 g, five times the rock value, in a very narrow band around two seconds, with a nearly harmonic appearance lasting for a minute. The soil column did what the transfer function says it does: it took a small, long-period, broadband rock motion and returned a large, narrow-band, sustained one.

The damage followed the filter. Buildings of six to fifteen storeys were destroyed in large numbers; taller and shorter ones very largely survived. A frame of that height has an elastic period of perhaps 0.6 to 1.5 seconds, below the site period — and as it damaged, its stiffness fell and its period lengthened into resonance. The earthquake asks for a displacement is the demand side of that, and the period-lengthening is why a building can walk into the worst possible situation as it is being shaken.

Why the peak acceleration understates it #

The Mexico City figures above are the ones usually quoted — 0.03 g on rock, 0.17 g on the lake bed, a factor of about five — and that factor is the wrong measure of what happened, by a wide margin.

Peak ground acceleration is the zero-period ordinate of a response spectrum. It is the acceleration felt by a structure so stiff that it rides the ground exactly, which is to say by no structure at all. And zero period is precisely where the transfer function drawn at the top of this essay is at its smallest: this site amplifies at 2.03 s by 7.5 and at very short periods by about 1.4, so a summary statistic taken at the short end is being read at the one place on the curve where the site does least.

What a building receives is the ordinate at its period, and there the ratio is far larger. At two seconds the lake-bed spectrum reached a pseudo-acceleration near 1 g while the rock spectrum at the same period was a few hundredths — a ratio nearer twenty than five. The soil did not multiply the motion by five; it multiplied one narrow band by twenty and left the rest roughly alone, and the peak acceleration, being a mixture of everything with the high frequencies dominating, averaged that away.

Two consequences follow, and both are about instruments rather than about ground.

A site factor calibrated on peak acceleration is calibrated on the least sensitive part of the signal. It will be a small number, it will look like a modest correction, and it will be wrong by the selectivity — which for a soft site is a factor of five on top of it.

And the ratio of two peak accelerations is not the ratio of two demands. Comparing recordings that way is comparing the two records’ high-frequency content, on a phenomenon whose entire mechanism is at low frequency. The comparison that means anything is between spectra, at the period of the thing being asked about.

Measuring it, which is easier than most site work #

A useful feature of this problem is that its two governing quantities can be measured rather than inferred, and comparatively cheaply.

The shear wave velocity is measured directly — by a downhole or crosshole survey, by a seismic cone, or by surface-wave methods that need no borehole at all. It is one of the few soil properties that can be obtained without disturbing the ground, because a shear wave at small strain samples the soil in the state it is in.

The site period can be measured with nothing but a sensitive seismometer and patience. Ambient ground noise — traffic, wind, the sea — excites the soil column continuously, and the ratio of the horizontal to the vertical component of that noise peaks at the site’s fundamental frequency. The technique is thirty years old, needs one instrument and twenty minutes, and gives $T_1$ directly.

That is worth contrasting with almost everything else geotechnical. A friction angle is inferred from a penetration test through a correlation; a stiffness is inferred from a settlement or from a correlation with the same penetration test; a $K_0$ is guessed from geological history. Here the property that governs is a wave speed, and wave speeds are measurable.

Which raises an obvious question about practice: given that the site period is cheap to measure and that it is the quantity the whole response turns on, why is design done from a thirty-metre average velocity? The answer is that the design spectrum has to be codified for a whole country before any particular site is visited — and a site-specific study, which is exactly the measurement above, is what a significant project does instead.

What that measurement produces, though, is a property of the place rather than a load. A ground motion record is a specific history and a site period is a filter; neither is a thing a designer can apply to a structure until the two have been put together and passed through a family of oscillators. That is why the sequence in this essay runs rock motion, filter, spectrum, and never the other way round.

What a code does with this #

A design spectrum is a rock spectrum times a set of site factors, and knowing what the factors are approximating is worth the paragraph.

The classification is by $v_{s,30}$ — the average shear wave velocity over the top 30 m — which is a sensible proxy because it correlates with both terms of the answer. A low $v_s$ gives a long period and a low impedance ratio, so a soft site is shifted right and amplified.

Two limitations follow immediately from the physics above.

Thirty metres is not the layer. The period is $4H/v_s$ with $H$ the depth to the impedance contrast, which may be 10 m or 300. Two sites with identical $v_{s,30}$ and different depths to bedrock have different periods, and the classification cannot tell them apart. Mexico City’s lake bed at 38 m and a deep alluvial basin at 120 m are the same class and are two seconds apart.

A factor is not a filter. The site factors broaden and raise a spectrum; the real transfer function is peaked. A design spectrum is deliberately smooth because a designer cannot know a building’s period to better than 20% — but it means the code’s answer at the site period is lower than reality and its answer away from it is higher, which is a redistribution rather than a conservatism.

Two sites, one earthquake, and the design that follows #

It is worth putting the whole argument on a decision, because the conclusion is not “avoid soft ground” and it is not “add a factor”.

Take one rock motion and two sites: 30 m of stiff soil at 400 m/s, and the 38 m lake bed at 75.

Two things in that comparison are worth separating. The period moved by a factor of seven because $4H/v_s$ moved by a factor of seven. The sharpness collapsed for an entirely different reason: a site whose soil is not much softer than its rock has almost no boundary to reflect off, so it is barely a filter at all. A stiff site is not a quiet version of a soft one; it is a different kind of object, and the two changes are independent.

Now put a building on each.

A four-storey frame, period about 0.4 s. On the stiff site it is close to the site period and receives nearly the full 1.9. On the lake bed it receives about 1.5, because it is far below the peak. The soft site is better for it.

A twelve-storey frame, period about 1.2 s and lengthening as it damages. On the stiff site it receives almost nothing above the rock motion. On the lake bed it receives 4 to 7 and rising. The soft site is very much worse for it.

Put the two selectivity numbers side by side and the whole essay is in them. A soft site multiplies one narrow band by seven and leaves the rest; a stiff site multiplies everything by about two. Only the first of those can be got right or wrong by a choice of structural scheme, and only the first punishes a wrong choice by a factor of five.

The design response is therefore not a factor but a choice of period, and the levers are the ones a scheme has anyway: a braced or walled structure is stiff and short-period, a moment frame is flexible and long-period, and an isolated structure is deliberately very long-period. On a soft site the isolation move — which works by moving a building’s period away from a short-period peak — has nowhere to go, and made weaker on purpose has to be argued differently or abandoned.

A site’s period should be one of the first numbers on a project, and it decides the structural scheme rather than a coefficient in it. That is unusual: most site information affects the foundations and leaves the superstructure alone.

Where the model stops #

The soil was linear. It is not, at strong shaking. The shear modulus falls with strain — by a factor of two or three at the strains an earthquake produces — and the damping rises to 15 or 20%. Both effects lengthen the site period and reduce the amplification, so a linear analysis of a soft site over-predicts the peak and under-predicts the period. Equivalent-linear analysis, which iterates on a strain-compatible modulus and damping, is the standard response.

The layer was uniform. Real profiles have stiffness increasing with depth, which changes the mode shapes and moves the harmonics off the odd integers. A layered analysis is a straightforward extension of the same transfer function and gives peaks at frequencies no simple formula predicts.

The wave was vertical and one-dimensional. A basin does two things this model cannot represent: it traps waves laterally, so the motion rings for far longer than a one-dimensional column would; and it generates surface waves at its edges. Both were present at Mexico City and both are why the shaking lasted a minute.

Vertical motion was ignored throughout. The same column filters vertically travelling compression waves at $4H/v_p$, and $v_p$ in a saturated soil is the speed of sound in water — around 1,500 m/s — so the vertical site period is very short and the vertical amplification is small. That is why vertical response is usually a short-period problem and horizontal is not, and it is a consequence of the two wave speeds being so different in a soil with water in it.

And the building was absent. A heavy building on soft soil changes the soil’s own response beneath it, and the system that oscillates is the building and its foundation together — the ground is a spring is that interaction, and it lengthens the period and adds radiation damping, both usually beneficial. The exception is a soft site, where lengthening the period moves a stiff building toward the site period rather than away from it — the same trap made weaker on purpose has to avoid, where an isolator’s whole benefit depends on the site period being short.

The generalisation #

The idea worth carrying is that a load path can contain a filter, and a filter has a period of its own that nobody designed.

The soil column is the clearest instance because it is large and its period is easy to compute. There are others in this collection with the same structure. A floor supporting a machine filters the machine’s excitation before it reaches the building — the machine that shakes the building is a filter deliberately inserted. A secondary structure mounted on a primary one receives the primary’s response rather than the ground’s, which is why equipment on the tenth floor is designed for a floor spectrum and not a ground one. A cladding panel on brackets receives the frame’s motion through the brackets’ own stiffness. And a tuned mass damper is a filter built deliberately and tuned on purpose — the mass that helps by being late is the same physics used as a tool rather than met as a hazard.

In every case the same two questions settle it: what is the filter’s period, and how sharp is it? The period comes from the filter’s own mass and stiffness and is usually easy. The sharpness comes from its damping and, more often than people expect, from whether energy can leave the system altogether — which is the term this essay’s model would have got badly wrong by assuming a rigid boundary, and which is the term most often left out.