The footing a prevailing wind leans

The footing the storms press down rocked a footing strip on ground that remembers. Each slice of ground is a spring that yields along its loading stiffness when it is pressed further than ever before, comes back along a stiffer line when the pressure comes off, keeps a permanent set, and climbs the stiff line again before it yields any more. A storm pushed the column’s moment to one side and then the same distance to the other, and the strip pressed a shallow bowl under each end in its first storm or two and then rocked inside it without pressing any further: the ground shook down.

Every storm in that essay reversed symmetrically. A footing under the corner column of a frame in a prevailing wind does not see that. The storms that matter come mostly from one quarter, and the moment they put on the column is larger in one direction than in the other: the gust pushes the frame over, and the swing back is the frame’s rebound and the lulls between gusts, which are smaller. The essay ended by asking whether a footing rocked unevenly leans, whether the lean grows with every storm, and by how much before anyone notices.

Storms that push harder than they swing back #

The strip is the one that the earlier essays drew, so that only the storms are new. It is 8 m long, with a flexural rigidity of $540\ \text{MN}\cdot\text{m}^2$, and sits on ground with a loading stiffness of 50 MN/m² per metre of strip under a 1,000 kN column at its middle. The ground unloads and reloads four times as stiffly as it first loads — the ordinary range for sands and stiff clays — and keeps three quarters of the furthest it has been pressed as a permanent set, below which the strip has to come down before it is carried at all.

A storm now pushes the column’s moment from nothing to 1,000 kN·m, back to nothing, and then the other way to a fraction of that — all the way back for the symmetric storms of the earlier essay, half-way for a storm that swings back half as far as it pushes, not at all for one that only pushes. The strip is solved at every step with each spring on whatever branch its settlement puts it, the step is repeated until no spring changes branch, and the springs’ memories are updated before the next step. Between storms the strip is solved under its column alone, on whatever ground the storms have left, and the column’s rotation then is the footing’s lean.

The lean that stops #

The hero figure is the whole answer to the first question. Storms that reverse fully leave a lean of 0.011 milliradians — the trace of the fact that each storm pushes first one way and then the other, so the first push meets fresh ground and the second meets ground the first has partly prepared. Storms that swing back three quarters of their push lean the footing 0.064 milliradians; half, 0.117; a quarter, 0.171; storms that only push, 0.218.

Every line is flat after the third storm. With half the swing back, the first storm leans the footing 0.099 milliradians, the second adds 0.015, the third 0.003, and the fourth to the eighth add nothing that the third figure of a milliradian can show. The footing leans, and then it stops leaning, at a tilt set by how unequal the storms are and not by how many there have been.

The reason is the same as the reason the symmetric bowl stopped. Each spring’s yield is set by the furthest it has been pressed, and a storm no larger than the last presses no spring further than the last did. The first storm does almost all the pressing; the second presses a little more at the pushed end, because the strip comes back from the first storm resting slightly differently and starts the second from a slightly different place; by the third the strip starts every storm from where it started the one before, and the ground under it is no longer being asked for anything it has not already given. The difference between the pushed end and the other end is then fixed, and the lean with it.

The bowl, tilted #

The lean is what a tilted bowl does to a strip resting in it, and the next figure draws the bowl.

Most of the bowl is the column’s own load: the ground under the middle of the strip is set about 4 mm by the column, before and after any storm, and the storms add only at the ends. They add at the pushed end in the same way whatever happens on the swing back — 0.91 mm there in all three cases — because every storm pushes that end down by the same 1,000 kN·m. What differs is the other end. Storms that swing back fully press it 0.72 mm, almost as far; storms that swing back half press it 0.30; storms that only push leave it at 0.03, barely touched since the column was first put on it.

The strip at rest sits in that bowl under its column alone. The column’s load is the same and its pressure must still balance about the column, so the strip finds a position in which both ends bear — and with the right-hand ground set lower, that position is turned towards the right. The ends’ permanent sets differ by 0.61 mm, which spread over the strip’s 8 m would be 0.076 milliradians, and the column at rest turns 0.117 — more, because the rims of the bowl do not set the rotation. A flexible strip bears hardest near its column, where the bowl is deepest and the ground’s reloading line is steepest, and the rotation it settles at is the one that balances the column’s load on the whole shape of the bowl.

Stiffer towards the lean #

A footing that leans towards one side might be expected to give way more easily in that direction. The next figure says the opposite.

In the first storm the push turns the column 1.12 milliradians at its peak, and by the sixth only 0.97. The ground under the pushed end has been pressed to that depth before, so the next push climbs its stiff reloading line, four times as stiff as its first loading; the footing rocks towards its lean on stiffened ground. The swing back presses the other end, whose ground has been pressed much less — only by the swings back, which are half the pushes — and stays nearly as soft as it was: 0.33 milliradians in the first storm, 0.31 in the sixth.

This is the earlier essay’s result made lopsided. That essay found the symmetric footing stiffer in rocking after its first storm than before; this one finds the lopsided footing stiffer in the direction it leans than in the other, by about a sixth here. A frame on such footings has a lateral stiffness that differs with the wind’s direction, and a frame analysed with one rocking stiffness for both directions is wrong in opposite senses for the two. The prevailing wind finds the frame stiffer than calculated; the occasional wind from the other quarter finds it about as calculated, or softer than the stiffened value someone might have measured in a load test from the prevailing side.

The lean grows faster than the storm #

The size of the storm sets the lean as well as its asymmetry, and the relation is not proportional.

Up to about 1,000 kN·m the lean is nearly proportional to the storm: 0.058 milliradians at 500 kN·m, 0.117 at 1,000. Beyond that it grows faster: 0.335 at 2,000, nearly three times the lean at 1,000. The change is lift-off. At 1,000 kN·m the moment on the column’s 1,000 kN load is an eccentricity of a metre, enough to lift the strip’s far end clear of the ground for some of its length, as the ends the ground was holding down found; past it, the contact shortens and every further kilonewton-metre is carried by less ground, pressed further. The pushed end’s set grows faster than the storm, and the lean with it.

The lean is roughly a tenth of the storm’s own peak rotation throughout — 11 per cent at 500 kN·m, 10 at 1,000, 9 at 2,000 — because the storm’s rotation grows faster still once the strip lifts. That gives a quick estimate. A footing that rocks a milliradian in its design storm, from a prevailing quarter with half the swing back, leans about a tenth of a milliradian for good; one that rocks three or four milliradians leans about a third.

Fifty years, and the storm that beats the rest #

Real storms are not all of one size, and the next figure follows the footing through fifty years of them.

The lean is a staircase. In the first year it jumps to 0.24 milliradians; in the fourth, whose storm beat the first, to 0.31; then nothing for thirty-six years except a few thousandths, until the forty-first year’s storm, the largest of the fifty, takes it to 0.53. Of the 0.544 milliradians the footing has leaned after fifty years, 94 per cent arrived in those three years, and the rest is the second and third storm after each record shaking the new bowl down.

The footing’s lean is a record of the largest storm it has met, not of how many it has met. That is the property of every memory of this kind: the ground keeps only its furthest excursion, so only a storm that goes further than any before can add to what it keeps. The number of records in a sequence of independent years grows only as the logarithm of the number of years — about four or five in fifty, five or six in a hundred — so the lean of a footing in a prevailing wind grows in a few steps that get rarer as the building ages, and a footing that has not leaned further for twenty years has not stopped; it has not yet met its storm.

The same structure turned up at the top of the same kind of column, in the nut turned after the storm: a preloaded base’s bolts stretched once, in the first storm, and never again, because a bolt, like a spring of ground, keeps only its furthest excursion. That essay found that retightening the bolts after every storm turned a base that settled into one that ratcheted, by handing each storm back the preload it had taken. The footing has no nut to turn; its ground cannot be given back what a storm took from it, and so it only ever settles.

The order of the storms matters here only through which of them is the record. Reverse the fifty years, and the same three storms are not the records any more — the forty-first year’s storm comes tenth and is the record at once — but the lean at the end is nearly the same, 0.566 milliradians against 0.544, because it is set mostly by the largest storm from the prevailing side; the few hundredths between them are the storms that came after the record in one order and before it in the other, shaking the bowl down a little further. A fatigue crack is the opposite: the record played backwards found the same cycles in a different order failing at a different life, because a crack grows from whatever the last block left it. The ground’s memory is shorter and simpler than a crack’s. It holds one number per slice, and only a larger number can replace it.

That is a different kind of history from the one the load it can carry once described, in which a beam loaded cyclically beyond its shakedown limit adds rotation every cycle. The footing here never ratchets cycle by cycle. Between records it is in shakedown; each record moves it to a new shakedown state; and the steps between those states are set by the storm’s size alone.

How much the ground remembers #

The last figure varies the property that makes all of this happen, the reloading stiffness.

On elastic ground, which reloads exactly as it loaded, there is no permanent set and no lean, whatever the storms do. Ground that reloads twice as stiffly keeps half of every excursion and leans the footing 0.075 milliradians; four times, 0.117; sixteen times, 0.159. The lean rises with the ground’s memory and flattens, and it cannot exceed what the pushed end was actually pressed by the storm: a spring that keeps everything keeps no more than the excursion it was given.

That bound is a useful one in practice, because the storm’s own rotation is the number a designer computes anyway. The settled lean in this model never exceeded about a sixth of the peak rotation of the storm that made it, even on ground that keeps fifteen-sixteenths of every excursion, and with ordinary soils it was about a tenth.

By hand: what a tenth of a milliradian is #

The leans here are small. An 8 m strip tilted 0.117 milliradians has one end 0.94 mm lower than the other. A column 40 m tall standing on it is out of plumb at its top by $40 \times 0.117 = 4.7$ mm, a small fraction of the out-of-plumb the frame was allowed when it was erected and far below what anyone could see. The fifty-year lean of 0.544 milliradians puts the same column 22 mm out at its top, which is a measurable and permanent addition to the frame’s initial lean — the imperfection that the load that is really a lean turns into a notional horizontal force on every storey. Against Eurocode 3’s allowance of about a two-hundredth of a radian for that imperfection, 0.54 milliradians is a tenth.

So the footing in a prevailing wind does lean, and its lean is permanent and directional, but it is self-limiting and slow, and on ordinary ground it is an order of magnitude below the frame’s own fabrication tolerance. What matters more for design is the lopsided rocking stiffness, which changes how the frame shares its wind between bays, and the fact that the lean arrives in steps at the largest storms — so a survey that finds a footing leaning more than it did is reporting the largest storm since the last survey, not a failure in progress.

A lean that cracks nothing #

A permanent tilt sounds like damage, and settlement limits are often written as angles. But the crack is a strain, not a slope: what cracks a building is the tensile strain its distortion puts into walls and partitions, and a rigid tilt puts in none. The lean here is a rotation of one footing, and if every footing of a frame leaned by the same angle the frame would rotate as a body and nothing in it would know.

They do not lean by the same angle. The corner footing of a frame in a prevailing wind takes the largest rocking moment and the leeward footings of each bay take more than the windward, so the leans differ across the plan, and the difference between two neighbouring footings’ leans is a relative rotation of the beam that joins them. For the leans computed here — a few tenths of a milliradian at most, against the one to three milliradians of angular distortion at which finishes begin to crack — that relative rotation is an order of magnitude too small to matter. A footing on softer ground, or with storms that lift it further, would lean several times more; on that footing, the lopsided stiffness and the record-driven steps are worth computing, and the differences between its neighbours are worth a look before the finishes find them.

Springs that forget their neighbours #

Each spring forgets its neighbours. Real ground spreads load sideways, so pressing one end of the strip compacts ground under its middle as well; a Winkler bed concentrates the memory at the ends that were pressed. The bowl in real ground is smoother and its tilt slightly less.

Ground does not ratchet within a storm. Sands under many thousands of small cycles creep and densify even at loads below their previous maximum, which a three-branch spring cannot represent; the record steps here would then be joined by a slow drift, and the drift would have the prevailing wind’s sign.

The storms are quasi-static and one-dimensional. Real wind has a direction that wanders, so a corner footing sees a moment that sweeps round rather than reversing on one line, and the ground’s memory then has a plan shape. And under an earthquake the column load itself varies with the rocking, which the beam that sits on the ground never had to consider.

Still open: the footing that is pulled as well as pushed #

Every footing here carries a column load large enough that the strip, even lifted at one end, is always pressing down somewhere. A footing under a bracing bay’s tension column is different: its column load reverses with the wind, so it is pressed in one storm direction and lifted in the other, and the ground under it is pressed only in half the storms. Whether a footing that is alternately pressed and pulled leans the same way as one that is only rocked — or whether being lifted clear of its bowl in every reverse storm lets it come down somewhere new each time, and walk — is the question of what memory does to a footing that is sometimes not touching the ground at all.