The footing the storms press down

The ends the ground was holding down took Winkler’s springs and stopped them pulling. A footing strip with a moment on its column then lifts off at one end instead of being held down there, and on an 8 m strip under a 1,000 kN column a moment of 1,000 kN·m lifts about a metre and a half. Reverse the moment and the other end lifts. A footing under a frame carries a column load that is nearly steady and a wind or earthquake moment that reverses, so its contact rocks between two positions, and the soil under each end is loaded only half the time.

That essay’s springs had no memory. Lifted off and pressed again, they gave back exactly what they had given before. Soil does not. Ground pressed further than it has ever been pressed compacts, and when the pressure comes off, the compaction stays: it comes back up only a little, along a much stiffer line than the one it went down along, and pressed again it climbs that stiffer line to where it had been before it yields any more. The fear was that a footing rocking on such ground would compact a shallow bowl under each end, then sit in it with its ends hanging over the hollow, and that its contact, and so its moments, would change from storm to storm.

A spring that keeps what it was given #

The ground under each slice of the strip is one spring with three branches. Pressed beyond the furthest it has ever been, it follows its loading stiffness $k$ — 50 MN/m² here, as in the earlier strips. Unloaded, it comes back along a stiffer line, $k_{\text{ur}}$, four times $k$ in most of what follows: unloading-reloading moduli of three to five times the loading modulus are the ordinary range for sands and stiff clays. It stops at a permanent set, $w_p = w_{\max}(1 - k/k_{\text{ur}})$, three-quarters of the furthest it went. Below that set it is a gap: the strip has to come down to where the ground now is before it is carried at all. Pressed again from there, it climbs the stiff line back to $w_{\max}$, and only past it does it yield further along $k$.

A storm is the moment on the column going from nothing to $+M$, back to nothing, to $-M$ and back to nothing, in small steps, with the column’s 1,000 kN on throughout. At every step the strip is solved as the tensionless strip was, its springs on whatever branch their settlement puts them, and the step is repeated until no spring changes branch; then each spring’s memory is updated before the next step. The ground under the strip before the first storm has already carried the column, so the middle of the strip starts on ground that has been loaded once.

Why the first storm already finds the ground stiff #

The plainest result is the one the fear did not anticipate. A strip on remembering ground rocks less than a strip on springs that forget.

On springs without memory the column turns 1.42 milliradians at the peak of the storm. On the remembering ground it turns 1.12 in the first storm, already a fifth less, and 0.87 by the third. The reason is in what rocking does. A moment on the column presses one end of the strip down and lets the other end up. The end that comes up is unloading ground that has carried the column, and ground unloads along its stiff line: it pushes back on the rising end four times as firmly, per millimetre, as the elastic spring would. Half the strip is working on the stiff branch from the start.

The first storm’s loop is also open where the elastic one is closed. The column comes back from the peak along a different path than it went out along, and at zero moment it is not where it started. The down-going end pressed ground further than the column ever had, and that ground kept the compaction. The second half of the storm — the moment reversed — presses the other end the same way, and the strip comes to rest on ground that has been loaded hard at both ends.

From the third storm on, the loop is closed again and much stiffer. A footing on real ground is stiffer in rocking after its first storm than before it, and stiffer before it than its loading modulus suggests.

The bowl, and the strip that fills it #

The ground under the strip does keep the storms, and where it keeps them is the bowl that was feared.

Before any storm the permanent set follows the column’s own settlement: 3.19 mm under the column, falling to almost nothing at the ends. One storm adds most of a millimetre everywhere, and at the ends it adds something from nothing, 0.81 mm, because each end in turn was the one pressed down. Three storms add a little more under the column and nothing more at the ends. The ground does remember, and its memory is a bowl: deepest under the column, the whole strip’s length wide, with its rim at the ends set 0.8 mm below where it was.

The strip does not hang over it.

At rest after the first storm the column has settled 5.16 mm, against 4.25 before it. The whole strip has gone down into the bowl, and at rest it is on the ground over its whole length, every slice in contact. The second storm adds 0.17 mm, the third 0.03, and from the fourth the settlement no longer moves: 5.37 mm. The storms do not dig a hollow for the strip to hang over; they press the strip down into the hollow they dig, by about a quarter of its original settlement, and then stop.

Where the ground reloads less stiffly relative to its loading, the effect is smaller: a ratio of two adds six-tenths of a millimetre. Where it reloads more stiffly, larger: a ratio of eight adds 1.7 mm over six storms and has not quite stopped. With no memory at all, nothing changes. The extra settlement is not a property of the storm or of the strip. It is the soil’s ratio of unloading to loading stiffness, written into the column’s settlement by the rocking that exercises it.

The settlement also stops. Storm by storm the increments shrink, and after a few storms of the same size the ground has been pressed, at every slice, as far as any storm of that size will press it: from then on every storm is carried on the stiff branches alone. That is shakedown, in the same sense the frame that settles down used the word: a structure that yields under the first few cycles of a repeated load, and is elastic under every one after.

Why the middle goes down #

The bowl is deepest under the column, not at the ends, which looks backwards for a footing whose ends are the ones that rock.

At the top of a storm one end of the strip has come up and is carrying little or nothing, and the column’s load, and the moment’s couple, are carried by the other two-thirds of the strip. That two-thirds includes the middle. Under the column and towards the down-going end, the ground is pressed harder than the column alone ever pressed it, and yields. Then the moment reverses, and the middle is pressed hard again from the other side. The middle is the only part of the strip loaded beyond its rest pressure in both halves of every storm, and it compacts most.

The ends are loaded hard only half the time, and they start from nothing. Each end is pressed down once per storm, compacts by that storm’s amount, and is lifted or unloaded the rest of the time. After the first storm, pressing an end down again to the same depth does no more.

Pressure at the top of the storm #

What the memory does change is where the strip presses when the storm is at its height.

On springs without memory the strip at the top of the storm has lifted off over its first metre and a half and presses hardest a little past the column, 235 kN/m². In the third storm on the remembering ground it has not lifted at all — its rising end still carries 13 kN/m², because it is on the stiff branch of ground that the strip is now seated in — but its down-going end carries much less than the elastic strip’s, 32 kN/m² against 75. That end is pressing down into the hollow the earlier storms left. It has to close its 0.8 mm of permanent set before the ground takes any load, and by the time it does, the middle has taken it instead. The pressure gathers towards the column: 277 kN/m², 18% above the elastic peak.

So the ground’s memory does shorten the footing, but not the way the fear supposed. It is not the rising end that loses contact. It is the falling end that arrives late, because it falls into a hollow and has to cross it before it is carried. The contact stays the whole length while the pressure concentrates in the middle.

The moment that hardly moves #

The question was whether the changing contact changes the strip’s moments, and the answer is almost not. The largest bending moment anywhere along the strip at the top of the storm is 1,234 kN·m on elastic springs, 1,181 in the first storm on remembering ground and 1,205 from the third storm on. At rest under the column alone it is 698 kN·m before any storm and 705 after six. The changes are a few per cent.

The reason is that a footing’s bending moment is mostly fixed by equilibrium. The column’s load and its moment have to be carried by the ground pressure, and the moment in the strip under the column is the moment of that pressure about the column, whatever its distribution. Moving pressure from the far end towards the column reduces the lever arm of some of it and increases the intensity of the rest. For a strip whose characteristic length is a good fraction of its own, the two nearly cancel. The beam that sits on the ground found its moments controlled by that length, and the ground’s memory does not change it, because the stiffness that sets it is the loading stiffness on the yielding branch, which the memory leaves alone.

A larger storm #

A run of equal storms shakes down. A storm larger than any before it starts again.

The run of equal storms settles to 5.37 mm. Make the third storm half as large again and the column goes from 5.33 to 5.81 in that one storm, more than twice what the second and third equal storms added together, because every slice of the strip is pressed further than it has been. The storms after it add a hundredth of a millimetre and rock within what the larger storm left: their rotation is the same as on the evenly stormed ground, because the stiff reloading branch that carries them is the same stiffness, whatever height it starts from.

The ground’s memory is a record of the largest storm, not of the number of storms. That is the honest limit of a model whose springs remember only their deepest settlement. Real soils do more: under many cycles of the same storm, sands densify a little more each time and clays build up pore pressure, and the settlement under repeated loading creeps upwards roughly with the logarithm of the number of cycles, long after a memory of the largest load alone would have stopped it. The springs here describe the first few storms well and the thousandth not at all.

What it means for a dynamic model #

The settlement is the thing a footing’s designer checks. The rocking stiffness is the thing that changes the building.

A building’s natural periods, and the share of an earthquake’s motion its foundation takes up by rocking, depend on how stiff its footings are in rotation, and the ground is a spring in series with the frame that sits on it. That spring is usually derived from the same modulus that gives the footing’s settlement under its column, and the settlement modulus is the modulus of ground being loaded for the first time. Rocking is half loading and half unloading. On ground that unloads four times as stiffly, the footing is 29% stiffer in rocking than the settlement modulus says in the first storm and 64% stiffer from the third on.

A stiffer foundation shortens the building’s period. It is one more entry, and an entirely physical one, on the list of reasons finished buildings are stiffer than the models said — not an error in the model of the frame but in the model of what it stands on, which was calibrated on a load that the ground then spends its life being relieved of. And it is a stiffness that changes with the building’s history: softer before the first storm, and softer again in a storm larger than any before it.

The strip by hand #

The strip is 8 m long, with $EI = 540\ \text{MN}\cdot\text{m}^2$, on 50 MN/m² of ground, under a 1,000 kN column. On springs that forget, the column settles 4.25 mm, and at the top of a storm of 1,000 kN·m the strip has lifted over 1.6 m of one end and turns 1.42 milliradians.

The permanent set the column leaves is three-quarters of its settlement, $4.25 \times (1 - 1/4) = 3.19$ mm, since the ground springs back along a line four times as stiff as the one it went down along. At the end of the first storm each end has been pressed down about a millimetre, once, and keeps three-quarters of it — 0.81 mm on average between the two. A spring pressed from 3.19 mm of set back to its deepest settlement on the stiff branch moves only a quarter as far, for the same pressure, as it did the first time: that factor of four is all the stiffening, and it is diluted to 29% in the first storm and 64% later because only part of the strip is on the stiff branch at any one time.

Springs that remember one thing #

Each spring remembers only its deepest settlement. That is the simplest memory a soil can have, and it misses the slow accumulation of settlement under many repeated cycles. A storm repeated a thousand times is not described.

The springs do not share. Winkler’s springs carry only the slice above them. Real ground spreads a load sideways, so the compaction under one end of the strip is felt a little at the other, and the edges of the strip load the ground beside them, which the strip’s own earlier essay also left out.

The loading is slow. The storm is a sequence of static solutions. A storm’s real moment arrives in gusts and an earthquake’s in cycles of a second or less, and saturated clay loaded that fast has no time to drain: it does not compact during the storm at all, only afterwards, as the ground that arrives after the weight does.

One strip and one column. A footing under a frame is one of several, and the frame redistributes the moment between them as their rotational stiffnesses change. A footing stiffened by its storms attracts more of the next storm’s moment, which presses it further.

The strip is weightless unless it is given weight. A strip with its own weight sits deeper into its bowl from the start; its numbers move by a few per cent and its conclusions not at all.

What the ground’s memory is for #

The settlement that matters is the difference, and a footing that settles a further quarter under its storms has settled evenly, under its own column, by an amount set by the soil’s unloading ratio. A row of identical footings under a row of identical columns, each rocked by the same storms, settles together and does the building no harm. A row whose footings see different storms — the corner columns of a frame, which carry the frame’s overturning, against its inner columns, which carry little — settles differently, by the soil’s memory of each footing’s own storms, and that difference is new.

Still open: the footing whose storms are not the same at both ends #

Every storm here reverses symmetrically: as far one way as the other. A footing under the corner column of a frame in a prevailing wind is rocked more one way than the other, storm after storm, and its ground is pressed further under one end. The memory it builds is then lopsided: a deeper set under the leeward end, a shallower one under the windward end, and a strip that settles into a tilted bowl. Whether that tilt grows with each storm, as a lopsided memory sends the next storm’s pressure further towards the end already deeper, or stops as the symmetric bowl stopped, is the question of whether a footing in a prevailing wind leans, and by how much, before anyone notices.