The ground is a spring

Every dynamic result on this site so far has begun with a structure fixed at its base — a period computed from a stiffness and a mass, with the ground treated as a rigid boundary that the structure is attached to and that does not participate.

Nothing is attached to a rigid boundary. A foundation on soil can translate horizontally and it can rock, and both are flexibilities in series with the structure’s own.

Which free body produced the number #

The structure, its foundation, and the half-space beneath, modelled as a mass on three springs in series: the structure’s own lateral stiffness $k$, a swaying spring $k_x$, and a rocking spring $k_\theta$ acting through the height $h$ of the effective mass.

Adding flexibilities:

$$\frac{\tilde{T}}{T} = \sqrt{1 + \frac{k}{k_x} + \frac{k h^2}{k_\theta}}$$

The foundation impedances come from Gazetas’s solutions for a rigid footing on a homogeneous half-space, converted from a rectangular plan to an equivalent circular one by matching area for sliding and second moment for rocking:

$$k_x = \frac{8Gr_x}{2-\nu}, \qquad k_\theta = \frac{8Gr_\theta^3}{3(1-\nu)}$$

with $G = \rho V_s^2$. Everything about the soil enters through $V_s$, the shear wave velocity — which is measurable, and is the one soil parameter in this whole calculation that is. That is a better position than a modulus of subgrade reaction leaves the static problem in, where the governing number is taken from a table with a range of three.

For a 1,200 tonne building of 0.6 s period on an 8 by 8 m footing, with its effective mass 12 m up, on ground of $V_s = 200$ m/s:

Rocking supplies 89% of it and sliding 8%. The mechanism is a rotation, and it is one that no drawing of a building on a footing shows.

Why it belongs to tall buildings #

The sliding term $k/k_x$ has no height in it at all. It is a fixed number for a given structure and footing, and it is small — 0.079 here.

The rocking term carries $h^2$. So the effect is negligible for anything squat and grows as a square, which is why soil-structure interaction is described as a tall-building problem and why it is often left out of low-rise design without consequence.

The usual screening parameter is $h/(V_s T)$, which is 0.10 here — right at the threshold below which codes permit the effect to be ignored. Soften the ground to $V_s = 120$ m/s and the parameter goes to 0.17, the period ratio to 1.73, and the effect stops being optional.

Force down, drift up #

Now put the lengthened period through a response spectrum, which is where the asymmetry appears.

A code spectrum is flat below its corner period and falls roughly as $1/T$ above it. A longer period therefore means less acceleration, so:

$$V_{\text{base}} \text{ falls by } 12\% \qquad\qquad \delta \text{ rises by } 51\%$$

Both of those follow from the same period shift. The base shear is an acceleration times a mass and it falls. The displacement is an acceleration times $T^2/4\pi^2$ and the $T^2$ beats the falling acceleration comfortably.

Which produces the trap this essay exists for. A fixed-base model is conservative for force and unconservative for drift. The check that is easy to do — and that almost every design does — is the one that was already safe. The one that gets worse is the drift, and drift is what breaks the cladding, the services, the lift guides, and the gap to the building next door.

On soft ground the numbers become uncomfortable:

And the damping goes the wrong way #

Radiation damping is the part of this subject that is most often stated and least often checked.

A foundation pushing on a half-space sends waves away that never come back, which is a dashpot with no material damping in it at all. It is usually described as a bonus that soil adds — and for a slender building on ordinary ground it is not.

The reason is that the structure’s own damping is diluted. Only part of the system’s deformation is now happening in the part that dissipates, and the standard result is a cube:

$$\tilde{\zeta} = \frac{\zeta}{(\tilde{T}/T)^3} + \zeta_{\text{foundation}}$$

At a ratio of 1.31 that divides 5% by 2.25, leaving 2.22%. The foundation returns 0.28%. The net is 2.50% against the 5% the fixed-base model assumed — the building has half the damping it was designed with.

Why so little back? Because the mode that did the lengthening is the one that radiates least. A rocking foundation pushes the ground down on one side and up on the other, and the two nearly cancel in the far field; the rocking dashpot carries a frequency factor of $0.3a_0^2/(1+a_0^2)$ with $a_0 = \omega r/V_s$, and $a_0$ here is 0.18. That factor is 0.0096 — the rocking dashpot is one per cent of what a frequency-independent reading would give.

Sliding radiates properly, at essentially its high-frequency limit. But sliding supplied only 8% of the flexibility, so it gets only 8% of the weighting.

The case where the textbook sentence is right #

Turn the building around and the conclusion turns with it.

A squat building — 6 m to the effective mass, on a 20 by 20 m raft, 0.3 s period, on $V_s = 150$ m/s ground — has its flexibility dominated by sliding rather than rocking: 76% against 24%. Sliding radiates efficiently, so the foundation supplies 12.5% of damping and the total reaches 14.9% against the structure’s own 5%.

That is the case the textbook sentence describes, and it is a real and large effect: a squat building on soft ground genuinely does gain a great deal of damping from its foundation.

So the honest general statement is not “soil-structure interaction adds damping” and not “it removes damping”. It is:

The mode that supplies the flexibility decides whether damping is gained or lost. Sliding-dominated systems gain; rocking-dominated ones lose, because rocking is quiet.

And which one dominates is settled by $h^2$ against $r_\theta^3$ — a height against a footing size — which is a question about proportions and is answerable before any dynamics is done.

The other structure this looks like #

The pattern — lengthen the period, cut the force, pay in displacement — is exactly base isolation, and the comparison is worth making because it says what soil-structure interaction is and is not.

An isolation system lengthens the period on purpose, by a factor of three or four rather than 1.3, and it pays for the displacement with a device designed to accommodate it: a bearing with a 300 mm stroke and a moat around the building. The trade is made deliberately, the displacement is put where it can be tolerated, and damping is added at the isolation plane precisely because the dilution described above would otherwise leave the building with almost none.

Soil-structure interaction is the same trade made accidentally, at a smaller factor, with the displacement appearing in the structure rather than in a device, and with no compensating damping. It is base isolation without the bearings, the moat, or the dampers.

Read that way it is clear which parts of the phenomenon are benign. The force reduction is real and it is small. The displacement increase is real and it lands somewhere nothing was designed to accommodate it. And the damping loss is the part an isolation designer would have addressed and a building designer does not know is happening.

What embedment does #

A footing buried in the ground is stiffer than one sitting on it, in both modes, because the sides of the excavation participate.

Embedding this footing 6 m raises the sway stiffness by 73% and the rocking stiffness by more, taking the period ratio from 1.311 to 1.119 and the effective damping from 2.50% to 3.72%. So a basement moves the whole problem towards the fixed-base assumption — which is one reason the effect is less often noticed in buildings with basements, and it is not the only one.

The other is that a basement is usually a very large footing. $k_\theta$ goes as $r_\theta^3$, so doubling the plan dimension of the foundation multiplies the rocking stiffness by sixteen, and a building on a full-footprint raft has a rocking flexibility a fraction of one on isolated pads.

The design variable, where there is one, is the foundation’s plan size rather than anything about the soil — which is a second moment argument about a footing, with the same cube in it that any other section has.

What rocking does that the period does not report #

There is a consequence of a rocking foundation that the period ratio conceals entirely, and it is the one a façade engineer notices first.

A fixed-base building’s top displacement is all deformation: every millimetre of it is a storey drifting relative to the one below, and the cladding, the partitions and the lift guides all have to accommodate their share.

A building on a rocking foundation has two contributions. Part of the top displacement is deformation, as before. The rest is a rigid-body rotation of the whole building about its base, and that part produces no interstorey drift at all — the building leans as a unit and nothing racks.

So the two outputs move apart. The top displacement rises by 51% while the drift — the quantity that damages things — rises by considerably less, because a growing fraction of the total is rotation rather than racking.

That is a genuine mercy and it is the reason the drift numbers in the table above should be read as an upper bound on the damage. It also creates its own problem: the rotation is a tilt, it does not recover if the soil yields asymmetrically, and a building that ends an earthquake half a degree out of plumb has suffered damage that no drift limit was written about.

Where this model stops #

One mode, one mass. A structure has more than one period and the modes do not shift together. The whole treatment above is a single-degree-of-freedom idealisation with an “effective mass” at an “effective height”. A real building has several modes, they are affected differently — higher modes are shorter-period and see less lengthening — and the effective height for the first mode is not the effective height for the second.

A homogeneous half-space. Like every foundation model on this site, real ground is layered, and a stiff layer at depth reflects the waves the radiation model assumed were leaving. A shallow bedrock can remove most of the radiation damping the half-space solution predicts, which makes the loss described above worse rather than better.

Linear soil. At the strains a design earthquake produces, $G$ falls to a fraction of its small-strain value and the soil’s own hysteretic damping rises — both of which are handled by iterating on a strain-compatible modulus, and neither of which the impedances above contain.

And kinematic interaction is left out entirely. A foundation of finite size averages the ground motion over its footprint, so what it feels is not what a free-field instrument records — an effect that reduces the input, particularly at short periods, and which acts before any of the above.

What the picture cannot show #

The curves plot ratios against a soil stiffness, which makes $V_s$ look like a design parameter. It is a measurement, it varies with depth, and it is usually known from a handful of boreholes over a site whose properties change between them.

More importantly, the figure shows a smooth trade of force for displacement and cannot show that the two are checked by different people against different criteria. The base shear goes to the frame designer and is compared with a capacity. The drift goes to the façade engineer, the lift supplier and the party-wall surveyor, and is compared with a movement joint. When a fixed-base analysis is handed to all of them, the first is being given a conservative number and the rest are not — and nothing in the output labels which is which. It is the serviceability limit arriving first, in a calculation everybody read as a strength calculation.

Why the effect was ignored for so long #

Soil-structure interaction has been understood since the 1970s and appears in every seismic code, and it is still routinely omitted. Three reasons, and only the first is technical.

The force reduction is what the codes permit and it is small. Most codes allow a reduction in base shear for soil-structure interaction, cap it at something like 30%, and require the analysis to be done to claim it. A designer who does not want the reduction — and on most buildings 12% of the base shear is not worth an extra analysis — simply does not do the calculation, and the code is satisfied. The permission is framed as a benefit to be claimed, so declining it looks conservative.

The drift consequence is not framed as a requirement. Nothing in the permission says that without the analysis the computed drifts may be half of the real ones. The asymmetry described above is a property of the physics that the regulatory framing does not make visible.

And it needs a number nobody has. $V_s$ is measured on major projects and estimated on ordinary ones, and an estimated $V_s$ carries a factor of two, which carries a factor of four in $G$ and a large range in the answer. A designer facing that will reasonably prefer the assumption that is at least definite.

The result is a standing gap between what is known and what is done, which is not unusual and is worth naming for what it is: a case where the safe-looking simplification is safe for the quantity the code checks and not for the quantity the building’s occupants will notice.

The generalisation #

The habit worth carrying is a suspicion of the word conservative.

A model is not conservative; a model is conservative for a particular output. The fixed-base assumption is conservative for force and unconservative for displacement, and the two are the same analysis read at two different points.

That pattern recurs across this site. Ignoring the twist in a plate is safe for the collapse load and wrong about the corners. A lower-bound thrust line proves a masonry arch stands and says nothing about how it moves. Neglecting joint flexibility is conservative for one member and unconservative for another in the same frame.

In each case the useful question is not whether the simplification is safe but what it is safe about, and the answer is nearly always narrower than the word suggests.