The ground that never quite gives way

The ballast that helps it over put a block on a bed of springs that could not pull and could not push harder than the ground’s bearing limit, and found its tipping push below the rigid-ground answer, falling as the block got heavier past half the ground’s capacity. It tips inside its own hull had before it used springs with no limit at all. Between them they left the question the second one asked explicitly: real ground is not a spring that stops. It is stiffer the more it is compressed for a while, and then softer as it approaches failure — and “whether a body on nonlinear ground has a well-defined tipping push at all, or only a load at which its lean stops converging” is a question about what that curvature does.

The same ground, three ways #

The block is the one the ballast essay pushed over: 6 m square, 3,000 kN, pushed 8 m above its base, its weight 5 m up. The ground bears 300 kPa at most. To compare curved grounds fairly, each is matched to the capped spring at two points: the same limit, and the same settlement at which half of it is mobilised — 7.5 mm, which for the capped spring is a stiffness of 20 MN/m³ per metre of settlement.

Standing still, the block hardly distinguishes them. Its own weight puts 83 kPa on the ground, and at that pressure the three laws settle by 3 to 5 mm with nearly the same tangent stiffness — the hyperbola is a little stiffer there than the capped spring, the S-curve a little stiffer again. A plate test that loaded the ground to the block’s own pressure and a little beyond could not tell them apart.

They differ at the other end. The capped spring reaches the limit at 15 mm and stops; the S-curve reaches nine tenths of it at about 16 mm; the hyperbola reaches nine tenths only at 68 mm and never reaches the limit at all. What distinguishes these grounds is how they approach their strength, and that is exactly the part of the curve a block’s toe explores as it is pushed over.

Three grounds, three tipping pushes #

The capped spring gives the ballast essay’s answer, 776 kN. The S-curve gives almost the same, 770, at a slightly larger lean: its early softness and later stiffness roughly cancel, and near its limit it arrives as briskly as the capped spring does. The hyperbola gives 701 kN, 10 per cent less, at twice the lean.

The reason is the toe. As the block leans, the ground under its toe is pressed toward its limit and the ground under its heel unloads. The capped spring’s toe reaches the limit and then carries exactly the limit, a block of yielded ground whose width grows until the block tips — the rigid-plastic picture. The hyperbola’s toe never reaches the limit. Pressed harder, it settles more and gives a little more, so the pressure under the block stays rounded rather than flat-topped, and its centroid — the point the block’s weight turns about — stays further inside the base than the rigid-plastic block puts it. A ground that never quite reaches its strength is a ground whose strength is never quite used.

A tipping push made by the weight’s lean #

Why, then, does the hyperbolic block have a peak at all? The resisting moment of the ground under it rises as the block leans, toward the rigid-plastic limit and never beyond it, and on the hyperbola it never stops rising. What makes the push turn over is the other term: as the block leans, its weight moves toward the toe by the lean times the height of its centre of gravity, and that moment works against the ground’s.

Take the weight down to the base and the second term vanishes. The push then climbs toward the limit and never reaches it: at a lean of a fifth of a radian it is 801 kN against 813, and every further kilonewton costs a larger lean than the last. That block has no tipping push. It has a push it can never quite be given, and near it the lean does not converge — which is exactly the second of the two possibilities the question named. The answer to “is there a well-defined tipping push” is that the ground does not supply one. The block’s own weight does.

Lift the weight and a peak appears, at a push that falls and a lean that shrinks as the weight rises: 758 kN at 67 thousandths with the weight a metre up, 676 at 19 thousandths with it 8 m up. The tipping of a block on curved ground is a limit point of its equilibrium path, made by a second-order effect, the load that makes itself worse — the same thing that turns a column’s bending into buckling.

How far below the limit, and on which ground #

On all three grounds a higher weight tips at a smaller push, because a higher weight makes the lean more expensive. On the capped and S-shaped grounds the effect is modest — the block tips at a small lean, so its weight has not moved far. On the hyperbola the lean at tipping is two or three times larger and the weight’s moment grows with it: with the weight 10 m up the block tips at 82 per cent of the limit.

The ranking of the three grounds does not change with the height, and the lesson is in which two agree. The S-curve and the capped spring are different in shape everywhere except near the limit, and they give nearly the same answer. The hyperbola matches the capped spring at half the load and differs only near the limit, and it gives a different answer. A ground’s resistance to overturning is set by how it approaches its strength, and the stiffness it is tested at, at the pressures a structure normally applies, says almost nothing about it.

The heavier the block, the more it matters #

The ballast essay’s parabola — the tipping push rising with the block’s weight to a peak at half the ground’s capacity, and falling after it — survives on every ground, and on the curved one it is flattened and shifted. A light block tips on a narrow strip at its toe, pressed only part-way along the curve, and the three grounds agree to within a few per cent. A heavy block is already far along the curve under its own weight before it is pushed, and on the hyperbola that is where the curve is flattest: at four fifths of the capacity the hyperbolic ground allows 286 kN against the capped spring’s 573. The curvature costs little for a light block and half the resistance for a heavy one. The hyperbola’s peak moves down to 0.4 of the capacity, so the weight at which ballast stops helping comes sooner than the capped spring says.

Indistinguishable in service #

A designer who wanted to know which ground a block stands on could watch it lean under ordinary pushes. It would not help. Under 300 kN — two fifths of the tipping push on any of the three grounds — the block leans 1.12, 1.08 and 0.96 thousandths of a radian on the capped, hyperbolic and S-shaped grounds; under 500 kN, 2.02, 2.53 and 1.82. The hyperbolic block has begun to lean a little more, but by an amount no survey of a real structure would separate from the ground’s own scatter.

At 650 kN the three separate: 3.56, 7.93 and 3.46 thousandths. The hyperbolic block is leaning more than twice as far, because it is 93 per cent of the way to its tipping push and the others are 84. Service behaviour is decided by the ground’s middle, and the margin against tipping by its end, and the middle is where the three grounds were made to agree. A factor of safety of 1.55 on the capped spring is 1.40 on the hyperbola for the same block under the same 500 kN push — a difference the block would not show until the push was nearly enough to tip it.

The tower whose push falls to nothing #

The height of the weight lowered the tipping push on every ground, and the argument has a limit. A structure whose weight is carried high enough on ground soft enough has no push left at all: its own weight, leaning with it, overturns it. The condition is that the ground’s resistance to rotation, the moment it supplies per radian of lean, is no larger than the weight times the height of its centre of gravity — the column’s buckling criterion with the ground as the column’s stiffness and the structure’s weight as its load.

For the block drawn here that is nowhere near: under its own weight the hyperbolic ground resists rotation with 2.25 million kN·m per radian, against the 15,000 kN·m per radian its weight and height supply. A squat block cannot fall over by itself. A tall, heavy, narrow structure on a small foundation can, and the most famous one does not need to be named: a campanile of about 14,500 tonnes whose weight is some twenty metres up, on a foundation under twenty metres across, on soft clay, whose lean for most of its history was a slow approach toward exactly this limit. On curved ground the limit is closer than a straight spring suggests, because the ground’s rotational stiffness is the tangent of its curve at the pressure the structure applies, and it falls as the lean presses the toe further along the curve.

Why a smooth ground is the realistic one #

Every real ground is curved, and a ground loaded toward failure in a test is usually closer to the hyperbola near its limit than to the capped spring. A plate load test on sand or clay produces a curve that bends over gradually as it approaches failure; the load at which a footing “fails” in a test is usually read off by a convention — a settlement of a tenth of the width, or the intersection of two tangents — because the curve does not stop. The capped spring is the idealisation that makes the arithmetic of the ballast essay exact, and it is the optimistic one.

The S-curve is realistic too, for ground that has been loaded before and unloaded — an overconsolidated clay, a compacted fill — which is stiff at small strains, softer through the middle and stiffer again as it densifies. The finding that it behaves like the capped spring is a finding about its last part: an S-curve whose last part softened like the hyperbola would behave like the hyperbola, whatever it did first.

What a check can use instead #

The ordinary overturning check with a bearing limit is the rigid-plastic one: the effective base narrowed by the eccentricity, the toe carrying the ground’s full capacity over a block of it. That is the capped spring’s answer in the limit, and for the block drawn here it is 813 kN of push. The figures say by how much a curved ground falls short of it, and the shortfall has a simple structure: it grows with the block’s weight as a share of the ground’s capacity and with the height of its weight, and it is set by how slowly the ground approaches its limit.

A check can respect that without modelling the curve. The ground’s strength near failure is a number read off a curve that never stops, by a convention — a settlement of a tenth of the width, or a point where two tangents meet — and the convention chosen fixes how much of the curve’s last part is counted as strength. Reading the limit at a small settlement, as a hyperbola forces a test to, gives a lower limit and so a lower rigid-plastic push; reading it at a large one gives the optimistic number the capped spring uses. For overturning the conservative reading is the right one, because the toe of a tipping block is where the ground is asked to go furthest along its curve, and the block tips before the toe gets to the end of it.

Which free body produced the number #

The free body is the block, cut from the ground along its base. For each lean the settlement at the middle of the base is found by bisection so that the contact pressure, each strip at the pressure its own settlement gives on the ground’s law, carries the weight exactly. The pressure’s moment about the base’s centre is the ground’s resistance; the push is that resistance less the weight’s own moment — its weight times its height times the lean — divided by the height of the push. The tipping push is the largest push along the path, and each figure checks that no ground ever resists more than the rigid-plastic limit and that the capped spring reproduces the ballast essay’s numbers.

A plane block, a Winkler bed and a push that only rises #

The ground is a bed of independent springs, each square metre settling according to its own pressure. Real ground spreads load: the ground under the toe is loaded partly by the ground beside it, and its failure is a mechanism of the kind a bearing capacity is, whose load falls as the load’s eccentricity grows. The springs capture the pressure’s shape and the curve’s influence; they do not capture the mechanism.

The push only rises. Every path here is a single push increasing from nothing. Wind and waves push and release, and on curved ground each cycle leaves a little permanent settlement under the toe — the ground does not return along its loading curve — so the lean at which a block is found after years of weather is not the lean of any single push, and it ratchets.

The block’s weight is constant, as it was not in the ground that arrives after the weight. A curved ground that is also consolidating is two effects at once: the curve’s softening near its limit, and the limit itself rising with time.

What the pictures cannot show #

That the curve of a real ground is not known this well. A site investigation gives a strength and, at best, a modulus at small strain; the shape of the curve between them is inferred from correlations, and the difference the figures show between a hyperbola and a capped spring — 10 per cent for this block, half for a heavy one — is larger than the scatter most designs allow for. Which curve the ground follows is a question a plate test to failure answers and a routine investigation does not.

Nor can they show the dynamics of the last few thousandths. At the peak of the push–lean path the block is at a limit point, and a push held there is not an equilibrium it can stay at: it falls. How fast, and whether it falls clear or comes to rest leaning on its toe, depends on the ground’s curve beyond the peak, which none of these figures follows.

Still open: the block that is pushed and released #

A structure on curved ground is rarely pushed once. A wind turbine’s foundation is pushed by every gust in one direction and then the other, a tank by every storm, a crane by every lift, and on curved ground each push leaves the toe a little further down and the heel a little less supported than before. Whether that accumulated rotation converges to a lean the structure can live with, or grows without limit at a push well below the tipping push computed here, is the question of shakedown in the ground — and for a monopile or a gravity foundation it is the question that governs its design rather than the one the tipping push answers.