The ends the ground was holding down

A beam that sits on the ground is held everywhere by something that pushes back in proportion to how far it is pushed, and that one change hands it a length nobody chose: $1/\beta = (4EI/k)^{1/4}$, the characteristic length, past two or three of which a column load has been forgotten. The same solution has the ground pulling. Beyond $3\pi/4\beta$ from the column the strip rises, and the springs of Winkler’s model, which are springs, pull it back down. Real ground under a real footing pulls on nothing. The question here is what changes when it is not allowed to.

The quick answer is: not much, and that answer is right about the numbers people usually look at. It is wrong about almost everything else.

Thirty-four kilonewtons a metre of glue #

The strip drawn is 16 m long, a metre wide and 600 mm deep, so its $EI$ is $5.4 \times 10^5\ \text{kN·m}^2$, resting on ground of subgrade modulus 50 × 10³ kN/m³ — 50 × 10³ kN/m² of stiffness for each metre of its length — with a column of 1,000 kN at its middle. The characteristic length is 2.56 m and the strip is 6.24 of them long, which Hetényi’s classification calls infinite.

The linear answer is the dashed curve, and it has two parts that matter. Under the column the bed pushes back with 197 kN/m, a little above the 195 that the infinite-beam formula $P\beta/2$ gives, because the strip is finite and its ends add a small reflection. Towards the ends the pressure crosses zero 5.7 m from the column and keeps going: at the free ends it is −34 kN/m. That is the ground holding the ends of the strip down with a force of 34 kN on every metre — glue, in effect, which the solution has used without saying so.

The pull is not a small fraction of anything a designer looks at under the column, but it is not small against the end pressure either; it is the end pressure. And the solution has used it to do work. The moment the strip carries under the column is set by the whole distribution of reaction along its length, and part of what keeps that moment down is ground on the far side of the zero crossing hauling the ends back.

The solid curve is the same strip with every spring that would have to pull released. It touches the ground from 3.97 m to 12.03 m, 8.05 m in all, and is clear of it outside that. Under the column it presses 213 kN/m, eight per cent more than the linear answer. The bed now does only what a bed can do, and the strip has had to find a different shape to make that enough.

Five passes to find the edge #

There is no closed-form way into this problem from the front. A spring that cannot pull makes the stiffness depend on the displacement, and the displacement on the stiffness, so the answer has to be found by asking the question several times.

The procedure is the obvious one and it works. Solve the linear problem; find every point where the strip has risen; take away the springs at those points; solve again. Each pass removes the springs that are pulling, which lets the ends rise further, which can release a few more springs inward of them. When a pass releases nothing new, the contact has been found. For the 16 m strip it takes five passes, and the contact edge settles at 4.03 m either side of the column.

The settlement shows what the released strip actually does. Inside the contact it is the familiar bowl, a little deeper than before — 4.25 mm under the column against 3.95. Outside it the strip is a straight line. It has no load on it and no ground under it, so it carries no shear and no moment and cannot curve; it simply continues at the slope it had at the edge of the contact, and by the ends it is 5.25 mm in the air. The linear strip’s ends rose too, but by 0.69 mm, because the springs were holding them. Released, the ends rise nearly eight times as far, and that is the clearest picture of how hard the linear answer had been leaning on them.

Two things follow from those straight lines. The first is that the outer four metres at each end of this strip do nothing. They carry no moment, so they need no reinforcement for bending; they touch no ground, so they spread no load. The second is that the strip inside the contact does not know they are there. A weightless cantilever with nothing on it applies nothing to the section it is attached to, so the contact zone is solving its own problem, as a strip with free ends where the ground lets go.

Eight metres, whatever is poured #

That second point has a startling consequence, and it is the one the quick answer misses.

Short strips stay down along their whole length: a 4 m strip, a 6 m strip, an 8 m strip all keep full contact, because the linear answer for each of them presses on the ground everywhere. At 8 m the pressure at the ends has fallen to almost exactly zero — 1.8 kN/m — and this is the strip that sits on the boundary. Every strip longer than that keeps exactly 8.05 m of itself on the ground: at 10 m, at 16 m, at 30 m. The line goes flat and stays flat.

The number is not arbitrary. With $\beta = 0.390$ per metre it is $\pi/\beta$ — two half-lengths of $\pi/2\beta$ — which is the result Weitsman obtained in 1970 for an infinite beam on a tensionless Winkler bed. It is a pleasingly clean answer to a problem whose linear version produces a tail of decaying waves, and the reason it is clean is the one the settlement figure showed: once the ends have lifted, the contact zone is a free strip of its own, and the only free strip that can carry a central load on this bed without pulling at its ends is the one of length $\pi/\beta$. Shorter, and the bed presses at the ends and there is nothing to release. Longer, and the ends would have to pull, so they lift until the strip in contact is that length again.

It is worth setting this beside the linear result. The first essay on this foundation made the point that the peak moment and the peak pressure under a column contain no length: past two or three characteristic lengths the strip is effectively infinite and the column cannot tell how much more of it there is. On a bed that cannot pull the same thing is true in a stronger form. The strip does not merely stop noticing its length; it stops using it. Concrete poured beyond 4 m either side of this column is not carrying the load less efficiently. It is not carrying it at all.

Nine per cent the linear answer borrowed #

Under the column, where the reinforcement is decided, the effect is modest, and the comparison is worth having exactly.

On short strips the two analyses agree to the last digit, because there is nothing to release: a strip that presses on the ground everywhere is not affected by forbidding it to pull. The gap opens once the strip is longer than $\pi/\beta$ and grows over the next few metres, to 9 per cent on the moment — 699 kN·m against 641 — and 9 per cent on the pressure, 213 kN/m against 195. Beyond about 14 m nothing changes, because the released strip has already stopped changing and the linear strip is already, to three figures, infinite.

The column’s settlement moves in exactly the proportion its pressure does, since one is the bed’s stiffness times the other. A column that the linear analysis says settles 3.90 mm settles 4.25.

Nine per cent on a moment is the kind of difference that disappears into load factors, and on that basis the quick answer is defensible. What it hides is on the other side of the zero crossing. The linear strip carries a reverse moment, putting its top face in tension, of 117 kN·m at 3.8 m from the column — the ground pulling the ends down bends the strip upward between the column and the ends, as the cut at that section would show. A strip designed from the linear answer is given top reinforcement for that moment. The released strip has none of it: the moment is of one sign throughout the contact and zero outside it. The linear answer is nine per cent light where the steel is heaviest and wrong in sign where it is lightest. Neither error is dangerous on its own. Both are the same mistake, and the second is the one that says the model was answering a different problem.

What holds the ends down instead #

A real strip is not weightless, and its weight is the thing in the real world that does what the pulling springs did in the model.

A uniform load on a uniform strip settles every point by the same amount, $q/k$, and produces no bending at all; on the linear model it simply adds $q$ to the contact pressure everywhere. So a self-weight of $q$ cancels exactly $q$ of the pull. The whole 16 m strip stays down once its weight passes 34.5 kN/m, the deepest pull the linear answer asked of the ground, and the figure confirms the arithmetic: the contact reaches the full length between 34 and 36 kN/m.

The strip drawn — 1 m wide and 600 mm deep — weighs 15 kN/m. That is not enough. Under its own weight and the column it keeps 11.5 m of its 16 m on the ground, and the outer 2.2 m at each end are still in the air. Backfill over the strip adds weight in exactly the same way, and a strip buried a metre deep in soil of 18 kN/m³ gains another 18 kN/m, which would hold this one down. Whether the ends of a footing lift is therefore a question about the ratio of the column load to the weight of everything on and in the strip, set against how hard the linear answer leans on its ends — not a question about the soil.

The lean is the surprising part, because it is not largest on the longest strips. On an infinite strip the deepest pull is $P\beta e^{-\pi}/2$, 8.4 kN/m, and 15 kN/m of self-weight cancels it with room to spare: a 20 m strip and a 24 m strip both keep full contact under their own weight. On a 12 m strip the free ends reflect the column’s disturbance back towards it, and the pull at the ends reaches 54 kN/m, more than six times as much. With the same 15 kN/m that strip keeps only 9.4 m of its 12 m down. A strip four to six characteristic lengths long is the one most likely to lift, because it is long enough to bend and short enough for its free ends to amplify the bending, and it is also the size of an ordinary combined footing under two or three columns.

The middle third belongs to a rigid footing #

The rule everyone uses for when a footing lifts is the middle third: keep the resultant of the load within the central third of the base, and the base stays in contact everywhere, because a rigid base under an eccentric load has a linearly varying pressure that only reaches zero at one edge when the eccentricity is a sixth of the length. The rule is exact, and it is exact for a rigid base.

The 8 m strip is the one that, under a central column alone, just keeps its ends on the ground. Its ends are barely pressed — 1.8 kN/m — so almost any tilt releases one of them. A moment of 27 kN·m, an eccentricity of 27 mm, lifts it; the middle third would allow 1,333, fifty times as much. At 1,000 kN·m the rigid footing is still in full contact and the strip keeps 6.56 m of its 8; at 2,000 kN·m the rigid footing’s formula gives 6.0 m and the strip keeps 4.56. The flexible strip is never in as much contact as the rigid rule promises, and over most of the range it is in much less.

The peak pressure is subtler, and does not err in one direction. At small eccentricity the flexible strip concentrates its pressure under the column, so it presses harder than the rigid footing’s straight-line distribution — 235 kN/m against 219 at 1,000 kN·m. At large eccentricity the rigid footing’s triangle piles everything onto the edge, and the flexible strip, which bends away from that edge, presses less: 275 kN/m against 333 at 2,000 kN·m. A rigid-footing check is unconservative for the first and conservative for the second, and nothing in the check says which regime it is in.

The 4 m strip is the counter-example that shows where the rule comes from. It is 1.56 characteristic lengths long, short enough to behave almost as a rigid body, and it keeps full contact until the moment is 609 kN·m — 91 per cent of the way to the kern’s 667. Past that its contact follows the rigid footing’s $3(L/2 - e)$ to within a tenth of a metre: 2.12 m against 2.16 at 1,280 kN·m. The middle third is not a property of footings. It is a property of footings short in characteristic lengths, which is most pad footings and very few strips, rafts or combined footings. Whether a footing is rigid is not settled by its thickness alone but by its length times $\beta$ — exactly the same number Hetényi used to decide whether a beam on springs is short or long.

The lifted edge is not a disaster in itself. A base that is allowed to lift — a wall rocking on its foundation is the extreme case — can be perfectly stable, and Eurocode 7 asks for special precautions only once the eccentricity passes a third of the width, well outside the middle third, so a footing is allowed to lose part of its contact under some combinations as long as the bearing check and the check of whether it tips or slides are met. What lifts is the assumption that the pressure diagram is known.

One strip, by hand #

The numbers that matter can be got without a computer, and the hand version shows where each one comes from.

The bed and the strip give $\beta = (k/4EI)^{1/4} = (50{,}000 / 2{,}160{,}000)^{1/4} = 0.390$ per metre, so $1/\beta = 2.56$ m. A linear infinite strip under 1,000 kN presses $P\beta/2 = 195$ kN/m under the column, carries $P/4\beta = 641$ kN·m there, and changes the sign of its pressure at $3\pi/4\beta = 6.04$ m; the deepest pull, at $\pi/\beta$, is $(P\beta/2)e^{-\pi} = 8.4$ kN/m. On a bed that cannot pull, the contact half-length is $\pi/2\beta = 4.03$ m, so the contact is 8.05 m long, and the average pressure over it is 1,000/8.05 = 124 kN/m. The peak of 213 kN/m is 1.71 times that average; a rigid footing of the same 8.05 m would press 124 everywhere, and the difference is the strip’s own flexibility, concentrating the load under the column.

For the eccentric footing the rigid rule is one line: $e = M/P$, full contact while $e \le L/6$, and contact $3(L/2 - e)$ beyond. An 8 m strip is $\beta L = 3.12$ long and a 4 m strip 1.56; Hetényi’s dividing lines put the first just under the long-beam boundary at $\pi$ and the second well inside the range where a beam is genuinely finite, and the two strips behave accordingly.

Springs that do not know their neighbours #

The calculation rests on choices that limit it.

The springs are independent. Winkler’s bed is a row of springs — the same idealisation that makes the ground a spring under a vibrating structure — that do not know about each other, and the gap under a lifted end is a gap with nothing in it. Real soil is a continuum, and a strip lifting from it leaves the soil beneath unloaded, rebounding a little, and the soil next to the contact edge dragged down by the contact. The edge of the contact on a real half-space is not where Winkler puts it, and the pressure there tends to a peak rather than to zero.

The bed is elastic and the strip returns. A strip that lifts and sets down repeatedly under a reversing moment — wind on a frame, a crane slewing — does not find the same contact each time. Soil compressed at the edge of one contact does not spring back fully when the other edge loads, and a footing that rocks can ratchet down on its edges in a way no single static analysis shows.

The column is a point and the strip prismatic. A column delivers through a base, and a strip thickened under its column has a different $EI$ there; both move the numbers by a few per cent and neither changes the shape of the answer.

The ground has no strength. The bed is a set of springs with no limit. Under 213 kN/m on a 1 m strip, the bearing pressure is 213 kPa, comfortably inside the capacity of most ground, but a strip on weak ground has a mechanism of its own, and the plastic redistribution under the column then spreads the contact rather than concentrating it.

Still open: the strip that lifts on alternate sides #

A footing under a frame carries a column load that is nearly steady and a moment from wind or earthquake that reverses. On the 8 m strip, a moment of 1,000 kN·m one way releases 1.44 m at one end, and the same moment the other way releases 1.44 m at the other: the contact rocks between two positions, and the soil under the outer metre and a half at each end is loaded only half the time. Whether the cycles of loading and unloading under those ends compact the soil into a shallow bowl that the strip then sits in — changing its contact, and therefore its moments, from one storm to the next — is a question about what the soil remembers. The springs here do not remember anything, and that is the thing a strip that lifts on alternate sides is most likely to be asking about.