The damping that comes through the sides

A machine block whose mass sits above its base cannot sway without rocking, and the two separate checks a foundation is usually designed with describe neither of the modes it actually has. That essay worked with a block on the surface of the ground, where the only thing joining sway to rocking is the block’s own inertia. It ended on the case most real blocks are: set into the ground, with soil against their sides as well as under their base. Its closing guess was that side resistance acting below the centre of mass would work with the inertial coupling and push the modes further apart, while resistance acting near the centre of mass might remove the coupling altogether.

The half of that guess about removing the coupling is close to right and the half about reinforcing it is backwards, and both halves turn out to matter less than something the guess did not mention.

The block here is the awkward one from that essay: 150 tonnes, 3.2 m across and 3.2 m deep, standing on ground with a shear modulus of 60 MPa. On the surface its two checks are 9.0 Hz for sway and 6.5 Hz for rocking, its modes are 5.5 and 17.4 Hz, and a compressor running at 5.5 Hz sits on its lower resonance, moving 191 µm for every kilonewton of unbalanced force.

Two things the sides do at once #

Soil against the sides of a block resists it the way soil under a base does, with a spring and a dashpot, and the standard way of putting numbers on both is Novak’s side-layer model. It treats each metre of embedded depth as a thin slice of soil pushing back against the block’s sides: horizontally with a stiffness of 4.1 times the side soil’s shear modulus, in rocking with 2.5 times the modulus times the square of the block’s radius, and with dashpots for the waves that slice sends outward. The constants are fitted to elastic solutions for a Poisson’s ratio near 0.4, and they assume the soil stays in contact with the block over the whole embedded height.

The two parts of that model do different things to the block, and the figure above separates them by drawing the embedded block twice.

With both parts, the block set half its depth into the ground is transformed. Its resonance moves from 5.5 to 8.2 Hz and its peak falls from 192 to 11 µm per kN, a seventeenth. That is the result embedment is usually credited with.

With the springs and no side dashpots, the resonance moves just as far, to 8.4 Hz, and the peak falls only to 169 µm per kN. The compressor at 5.5 Hz still benefits almost as much either way — 8.0 µm per kN with radiating sides and 9.5 with silent ones — because its resonance has moved away from it. But the resonance has not been tamed. It has been moved, and all of its taming came through the side dashpots. The rest of this essay is those two parts taken one at a time.

Where the ground pushes back from #

The coupling the earlier essay found has a height in it, and embedment changes that height.

A side spring at a height z above the base resists the block’s sliding and its tilting together, because a point at that height moves by the base’s displacement plus z times the rotation. Added up over the embedded depth, the side springs put a term into the stiffness matrix that the surface block did not have: a stiffness joining sway to rocking directly, equal to the side stiffness per metre times half the square of the depth.

That term has a reading that decides the whole question. Every horizontal stiffness has a height at which it acts — its centre of stiffness — and writing the block’s motion about that height instead of about its base makes the stiffness matrix uncoupled. What is left coupled is the mass, and the mass is coupled by its own mass times the distance from its centre of mass down to the centre of stiffness. On the surface the ground pushes only at the base, the centre of stiffness is at the base, and that distance is the full height of the centre of mass above it. That is the inertial coupling the frequency below both checks was about, seen from a different reference.

The centre of stiffness is found the way the centroid of a section made of pieces is: each spring’s stiffness times the height it acts at, added up, over the total stiffness. The base’s spring contributes its stiffness at a height of nothing, and each slice of side soil contributes its stiffness at its own height, so the more of the total stiffness the sides supply, the higher the centre goes.

Side soil pushes from above the base, so it lifts the centre of stiffness. Wherever on the sides it acts, it closes the distance to the centre of mass. Embedment weakens the coupling. For it to strengthen the coupling, the ground would have to push from below the base.

The figure measures how far the lift goes. With backfill as stiff as the ground and the block embedded its whole 3.2 m, the centre of stiffness rises from the base to 1.00 m, and the distance that couples the two motions falls from 1.60 m to 0.60 m. Stiffer backfill lifts it further and weaker backfill less.

It never gets there. A resistance spread evenly over the embedded height acts at the middle of that height, and the base’s own spring, still acting at the base, pulls the combined centre below the middle. Only backfill infinitely stiffer than the ground under the base would bring the centre of stiffness to mid-embedment, and for a uniform block embedded its full depth mid-embedment is precisely its centre of mass. The coupling can be made small. A uniform block cannot be embedded out of it, because the height it would need is the one limit the side soil approaches without reaching.

A block could be. One with its mass concentrated low — a heavy machine inside a pit rather than on a pedestal — has its centre of mass below mid-depth, and there the centre of stiffness can pass it. So can one whose backfill is stiffer near the top than near the base, since that lifts where the side resistance acts. The decoupling the earlier essay hoped for exists, but it is a property of where the mass is and how the backfill varies, not of how deep the block is set.

The checks and the modes move up together #

Embedment stiffens everything, so every frequency on the figure rises. The rocking check rises fastest, from 6.5 to 14.3 Hz, because the side soil resists tilting with the cube of the depth in it, a slice’s lever arm counted twice; the sway check rises from 9.0 to 14.6. The two coupled modes rise with them and, at every depth drawn, still straddle both: the lower below both checks and the upper above both, as the surface block’s were.

What changes is how far. On the surface the lower mode sits 15 per cent below the lower check. Fully embedded it sits 8 per cent below both — closer, because the coupling distance has shrunk, and not gone, because it has not.

The fully embedded block lands exactly where the depth figure of the earlier essay warned a designer to look hardest. Its two checks now agree to within 2 per cent, 14.6 and 14.3 Hz, so a design reading them sees one comfortable frequency near 14.5 Hz. The block’s lower mode is at 13.2. Embedment has moved the block toward the case in which the separate checks look most convincing and are still wrong.

The damping comes through the sides #

The frequencies are the stiffness’s half of the story, and they are the half the model is most sure of. The damping is the other half.

The tall block’s lower mode is mostly a rocking, and rocking radiates little, which is why it had only 3.2 per cent of critical damping on the surface. The side soil changes that more than anything else embedment does. A slice of soil pushed sideways by the block sends shear waves away through the ground on both sides, and summed over the embedded depth that is a great deal of radiation.

It matters that the mode is only mostly a rocking. The lower mode turns about a point below the base — about a metre below it on the surface, nearly two metres below it fully embedded — so the block’s sides do not merely tilt in it; they translate, the whole embedded height moving the same way at once. A side slice radiates in proportion to how far and how fast it is pushed, and a mode that swings its sides bodily through the backfill is a mode that pumps waves into it along the whole depth. At half depth the lower mode has 27 per cent of critical damping; fully embedded, 75.

Take the side radiation away and the damping goes the other way: 1.6 per cent at half depth. The base’s dashpots are exactly what they were, and they now act on a block whose springs are stiffer, and a damping ratio is a dashpot over the square root of stiffness times mass. Embedment with silent sides makes the block less damped than it was on the surface. The peak of a lightly damped resonance goes as one over twice its damping ratio, so halving the damping doubles the peak.

That is the figure an embedment decision should be made from. With the side radiation the model gives, embedding the block half its depth takes its peak amplification from 15.7 to 2.0, and embedding it fully leaves no resonance worth the name. With half that radiation the peak is 3.6 at half depth. With none it is 30.8, twice what it was on the surface, at a higher frequency. The benefit of embedment to the peak is entirely radiation, and the stiffness that moved the frequency contributed nothing to the height.

A little backfill looks like a cure #

Radiation and stiffness do not grow with the soil’s modulus at the same rate, and the difference is where a designer’s confidence can outrun the ground.

A side dashpot goes as the square root of the backfill’s shear modulus, and a side spring goes as the modulus itself. So a backfill a tenth as stiff as the ground supplies a tenth of the springs but nearly a third of the dashpots, and on this block that is already enough to take the peak from 15.7 to 1.2. With radiating sides the curve falls almost vertically out of the surface value and is flat thereafter: nearly all of the damping embedment can give arrives with the first tenth of the backfill’s stiffness.

That shape is the reason to distrust it. Backfill is soil placed against a block after the block is cast, compacted to whatever the site managed, and it is the one part of the ground nobody tests after it is placed. Its modulus is uncertain by more than a factor of two, as the soil’s own was, and its contact with the block is uncertain in a way the model does not represent at all: a block that rocks pushes its backfill away near the top on every cycle, and dry soil shrinks away from a concrete face. Wherever the contact is lost, the side slice neither pushes nor radiates. The curve says a little contact buys almost everything, and the ground says a little contact is exactly what is hardest to guarantee.

A building’s foundation is usually argued the other way round, and for a reason. Embedment in the calculation of a building’s period is counted for its stiffness, which moves the period toward the fixed-base value, and the radiation is a small correction to a structure whose own damping is already several per cent. A machine block is the opposite case: its frequencies are ten times a building’s, its own material damping is negligible beside what the ground radiates, and radiation from its sides is not a correction but most of the damping it has. The same backfill, doing the same thing, is a minor term in one design and the governing term in the other.

Where the resonance goes decides who benefits #

The design reading is not that embedment is useless. It is that its two effects have to be taken separately, because only one of them can be counted on.

For the compressor at 5.5 Hz, embedment is a reliable cure. The resonance was at its running speed and embedment moves it away, so the displacement falls from 191 to about 2 µm per kN whether the sides radiate or not. The benefit comes from the stiffness, which is the part of the model that is well founded, and the radiation is a bonus the design never needed.

For a machine running at 8.4 Hz on the same block, embedment is a hazard. On the surface that machine sits between the block’s modes and moves 8.9 µm per kN. Set the block 1.6 m into the ground and the lower resonance arrives at 8.4 Hz: with the side radiation the model gives, the displacement rises only to 16 µm per kN on the way through, and with silent sides it reaches 170. Deeper still, the resonance passes on upward and the machine is clear of it again.

That is the lesson of widening the footing in the earlier essay, arriving from a different repair. Anything that stiffens a foundation moves its resonance, and a resonance moved away from one machine can be moved onto another. The coupled frequencies have to be recomputed after the change, and the height of the peak they produce should be read with the side radiation discounted, because that is the number the repair’s success most depends on and the ground most easily withholds.

Where neither a change of depth nor a change of footprint can move the resonance clear of the running speed, the remedies are the ones that do not depend on the soil at all: a mass tuned to the lower mode, or springs under the machine that keep its force off the block.

What the side-layer model leaves out #

The constants are single numbers. The side layer’s stiffness and radiation are fitted to elastic solutions over the frequency range machines usually run at, for one Poisson’s ratio. Outside that range, and in a soil whose ratio is different, they change.

The side slices do not talk to each other or to the base. Each metre of embedment pushes back as though the soil above and below it were not there, and the base’s spring is unaffected by the soil beside the block. Both are approximations that work best for a block deeper than it is wide.

The contact is perfect over the whole depth. A rocking block separates from its backfill near the top, first on one side and then on the other, and the loss starts where the side resistance matters most. It lowers the centre of stiffness, which strengthens the coupling again, and it removes the most effective radiation. Even where contact holds, a thin soft zone at the concrete face — disturbed soil, a membrane, a layer of drainage board — sits in series with the side soil, and in a chain of springs the softest one decides how much of either the stiffness or the damping reaches the block.

The block is uniform and the machine has no mass. A heavy machine on top raises the centre of mass and lengthens the coupling distance; a machine set low in a pit shortens it, and can bring the centre of mass below mid-embedment.

The response is linear and in one plane. Vertical motion and torsion are left out, and so is any softening of the soil under large strains.

And no figure here shows the backfill. Every line on every figure is the side soil as the model assumes it — placed, compacted and in contact — and the actual backfill around a real block is the thing the figures most depend on and cannot represent.

The assumption the figures rest on is that the soil against the block’s sides stays in full contact with it over the whole embedded height, at the stiffness assumed, for as long as the machine runs.

Still open: whether the sides stay in contact with a block that rocks #

Every damping ratio above belongs to side soil that stays pressed against the block, and a block whose lower mode is mostly a rocking pushes its backfill away near the top on every cycle and draws away from it on the next. Soil does not generally follow a face that retreats from it; what fills the gap, if anything does, arrives slowly. So a block embedded to damp its rocking may, after a season of running, have a loosened collar of backfill around its upper metre, where the side resistance did the most to lift the centre of stiffness and radiated the most energy away. Whether that collar re-forms under the machine’s own vibration or opens progressively, and how much of the radiation a design counted on is still there once it has, is a question about the backfill’s history under the load it was placed to resist — and it decides whether an embedded block’s quiet is a property of its design or of its first year.