The ballast that helps it over

Weight is the only thing resisting it reduced overturning to one comparison: the push times its height against the weight times half the base. No strength enters it; a structure strong enough everywhere can still be blown over, and the defence is weight and width. It tips inside its own hull gave the ground stiffness, and found the body leaning before it tips, its effective base shrinking as the soft side goes down.

Both left the ground unbreakable, and both said so. The spring model’s final section named the omission exactly: real ground has a bearing capacity, at which its stiffness is zero, and a body tipping onto its toe presses the ground under the toe harder than anywhere else. The middle third said the same from the other side — that a material with no tension imposes a kern, and that a peak pressure runs into a bearing limit before the contact length reaches zero.

This essay gives the ground its limit. The result is not a correction to the overturning check. It reverses the one rule the check rests on, that weight always helps.

The toe runs out of ground #

The block drawn below is 6 m square in plan and weighs 3,000 kN. The ground under it is a bed of springs — each square metre settling a metre for every 20,000 kN pressing on it — that cannot pull and cannot carry more than 300 kPa. Over the whole base that is a bearing capacity of 10,800 kN; the block uses 28 per cent of it standing still.

Push the block sideways, 8 m up, and watch the pressure under it. At first it is a trapezium, more at the toe than at the heel. By a push of 629 kN the heel has lifted: the block touches the ground over 3.9 m of its 6, with a triangle of pressure peaking at the toe. By 770 kN the peak has reached the ground’s limit and flattened into a block of pressure 300 kPa high and 0.8 m wide. At 776 kN the block of yielded ground is 1.1 m wide, the contact 2.3 m, and the block tips.

On rigid ground the same block would tip only when its weight’s line reached the edge of its base: at a push of $W \cdot B/2 / h = 3{,}000 \times 3/8 = 1{,}125$ kN. The yielding ground has taken a third of that away, and the reason is visible in the last panel. The weight no longer turns about the toe. It turns about the centre of the yielded block, which is inside the base by half the block’s width, and the block’s width is the weight divided by the ground’s limit.

The restoring moment, with a strength in it #

That picture has a closed form in the limit where the ground is rigid up to its bearing pressure and then yields — a rigid-plastic bed. At the point of tipping the whole weight rests on a block of pressure $q_u$ at the toe, of width $b = W/(q_u L)$ for a base of length $L$. The block’s centre is $b/2$ in from the toe, so the weight’s lever arm about it is $B/2 - b/2$, and the restoring moment is

$$M_u = \frac{W B}{2}\left(1 - \frac{W}{W_u}\right), \qquad W_u = q_u B L.$$

The first factor is the rigid-ground answer. The second is new, and it contains a strength — the ground’s — where the overturning check was proud of having none. For a block weighing a tenth of the ground’s capacity it takes 10 per cent off; for one weighing half, 50.

And it makes the restoring moment a parabola in the weight. It rises from zero, peaks at $W = W_u/2$, and falls to zero again at $W = W_u$, where the ground can only just carry the block standing still and has nothing left to resist a push with.

Heavier, and easier to push over #

Push three blocks of the same size and different weight over, on the yielding bed, and the parabola appears in the results. The block weighing a fifth of the ground’s capacity tips at 618 kN. The block weighing half tips at 958 kN, the most of the three. The block weighing four fifths tips at 573 kN — less than the lightest, although it is four times as heavy and on rigid ground would have been four times as hard to push over.

It also tips at the smallest lean, 9.3 thousandths of a radian against 14.5 for the lightest, because it has less ground to spare. The lightest block rocks onto a narrow strip at its toe and has room to lean; the heaviest is already pressing a wide strip toward its limit before it is pushed at all, and the push only has to finish the job.

Across every weight the pattern is the parabola, with the elastic bed a little below it — the spring bed wastes some of the base in the elastic tail of its pressure diagram, which the rigid-plastic limit does not. The rigid-ground line and the two curves agree for light blocks and part company quickly: at a tenth of the capacity the yielding ground costs 10 per cent, at half it costs half, at four fifths it costs more than four fifths.

Ballast #

The practical form of the reversal is ballast. A body that fails an overturning check — a tower crane’s base, a temporary grandstand, a wind-loaded sign, a tank empty in a gale — is steadied by adding weight, and the check says every tonne helps in proportion.

Give the 3,000 kN block a design push of 675 kN. On rigid ground its factor of safety is 1.67; on the yielding ground, 1.15, because the ground has already taken a third of its resistance. Add ballast and the factor on the yielding ground rises — but slowly, and only to 1.42, reached with 2,254 kN added, when the block weighs 49 per cent of what the ground can carry. Past that, each tonne of ballast lowers the factor it was added to raise. With the most ballast drawn the factor is 0.34: the block would be pushed over by a third of the design load.

The rigid-ground check reports 5.52 for that last block. It is not merely unconservative; it is pointing the other way, recommending exactly the change that makes the structure worse. And it does so at weights that are not extreme — a block using three quarters of its ground’s bearing capacity under permanent load is a block with an ordinary factor of safety against bearing failure of about 1.3.

Where the toe block comes from, and where it goes #

The width of the yielded block at tipping is the mechanism in one number. On the rigid-plastic bed it is exactly $W/W_u$ of the base, because the block has to carry the whole weight at the ground’s limit. On the spring bed it lags a little, because part of the weight is still carried on the elastic slope of the pressure next to it. A light block tips on a sliver of its toe; a heavy one tips on most of its base, having never lifted its heel very far, and the difference between them is the entire content of the second factor in $M_u$.

That width is also where the two earlier essays’ pictures meet. The middle third found that a resultant outside the kern lifts the heel and concentrates the pressure; the essay on a body on springs found the effective base shrinking as the soft side goes down. Both described a body tipping about a point that moves inward. On ground with a bearing capacity the point stops being a point: it becomes the centre of a block whose width the weight decides, and the heavier the body the further inward the pivot.

Stiffer ground is not stronger ground #

It is tempting to read the gap between the rigid-ground answer and the yielding one as a matter of the ground being soft, and to expect a stiff ground to close it. It does not. Make the ground a thousand times stiffer, from a bed that sinks a metre under a thousand kilonewtons a square metre to one that sinks a millimetre, and the tipping push rises from 541 kN to 810 — and stops, at the rigid-plastic limit of 813.

The softest bed does worst for a reason of its own: it lets the block lean nearly a tenth of a radian before it tips, and at that lean the weight’s own line has moved 0.48 m toward the toe, taking an overturning moment of its own with it. A stiff bed removes that effect entirely. What it cannot remove is the toe block, because the toe block is not a deformation; it is the ground’s strength, and the rigid-ground check does not assume the ground is stiff — it assumes the ground cannot fail. The two are different assumptions, and only the second is wrong for any real ground.

The same parabola in a column #

The formula for the restoring moment has a familiar shape, and recognising it is the quickest way to trust it. A column of a material that cannot carry tension — masonry, unreinforced concrete, a stack of blocks — carrying a compression $N$ at an eccentricity has a moment capacity that is exactly the same expression: the compression concentrates at one face in a block at the material’s crushing stress, the block is $N/(f_c b)$ deep, and the moment about the column’s centre is $N (h/2)(1 - N/N_c)$ with $N_c$ the squash load.

The block on its ground is that column lying on its side. Its base is the column’s section, the ground’s bearing limit is the column’s crushing strength, and its weight is the column’s axial load. The interaction curve of a column peaks at half the squash load for the same reason the tipping push peaks at half the ground’s capacity: below it, more compression means a larger lever of weight to resist the moment with; above it, the compression block has eaten so much of the section that the lever is shorter than the extra weight is worth. A masonry designer knows the second half of that curve well — a wall already highly compressed has less moment capacity, not more — and the overturning check is the one place in the subject where the same curve is used as though it had only its first half.

Check the ratio, shorten the lever, widen the base #

Three things follow, and each is a single line of arithmetic.

Check the weight against the ground before counting it as resistance. The ratio $W/W_u$ — the body’s weight over what the ground under its whole base could bear — tells at once which half of the parabola the body is on. Below a quarter, the rigid-ground check is within a quarter of the truth; above a half, adding weight makes things worse; in between, it helps less than the check says.

Use the reduced lever arm. Replace $B/2$ in the restoring moment with $(B - W/(q_u L))/2$, which is the rigid-plastic toe block’s lever. It is the effective-width idea of geotechnical practice, turned the right way round for an overturning check, and it is exact in the limit the figures approach from below.

Widen before ballasting. The peak of the parabola is $W_u B/8 = q_u B^2 L/8$, which for a square base grows as the cube of its width, while ballast only moves the body along a parabola whose peak the ground has already fixed. A body short of overturning resistance on weak ground needs a larger base, not a heavier one, and the same tonnage spent on a wider footing buys resistance that ballast cannot.

The whole of it, once, at half the capacity #

The base is 6 m by 6 m and the ground bears 300 kPa, so its capacity under the whole base is $300 \times 36 = 10{,}800$ kN. At a weight of half that, 5,400 kN, the rigid-plastic toe block is $5{,}400/(300 \times 6) = 3$ m wide — half the base — and its centre is 1.5 m in from the toe, 1.5 m from the base’s centre. The restoring moment is $5{,}400 \times 1.5 = 8{,}100$ kN·m, which is $(WB/2)(1 - W/W_u) = 16{,}200 \times 0.5$. Pushed 8 m up, that resists 1,012 kN; the spring bed, with its elastic tail and the weight’s own lean, reaches 958.

At 8,694 kN, 80 per cent of the capacity, the toe block is 4.83 m wide, its centre 0.59 m from the base’s centre, and the restoring moment 5,090 kN·m — 636 kN of push, and 561 on the spring bed. The rigid-ground answer for that block is $8{,}694 \times 3/8 = 3{,}260$ kN of push. The block is five times heavier than it needs to be to resist 561, and the extra weight is the reason it resists so little.

The block cut from its ground #

The free body is the block, cut from the ground along its base. Crossing the cut is the contact pressure, which on the spring bed is the subgrade modulus times the local settlement, capped at the bearing limit and never tensile. For each lean of the block the settlement at its centre is found so that the pressure carries the weight; the moment of the pressure about the base’s centre is the restoring moment; and the push follows from moment equilibrium of the whole block, less the weight’s own moment once its line has moved toward the toe by the lean times the height of its centre of gravity. The tipping push is the largest push along that path. Each figure checks that the spring bed never resists more than the rigid-plastic limit, and the rigid-plastic limit never more than rigid ground.

A soil mechanism, repetition, dynamics and sliding #

Ground that is not a bed of springs. Real bearing failure is a mechanism in the soil, a wedge and two fans, and its capacity under an eccentric load is reduced by the eccentricity itself — the effective-width rule of geotechnical practice. The springs capture the pressure distribution; the soil mechanism decides what the limit really is, and it falls as the load moves toward the toe.

Repetition. Every push here is applied once. A tower rocked by wind thousands of times ratchets its toe into the ground, and the yielded block grows cycle by cycle — shakedown in the ground rather than in the steel.

Dynamics. A block that tips under a static push behaves very differently under a pulse, where the time it takes to rock matters more than the push that starts it.

Sliding. A heavy block on weak ground may slide before it tips; the essay on whether it tips or slides set that crossing by friction, and a yielding toe moves it.

One bearing limit under the whole base #

That the ground’s limit is the same under the toe as under the rest of the base, and does not change as the block tips. It does neither: bearing capacity rises with the depth the toe has been pushed to and falls with the inclination of the load, and a tipping block does both at once. The parabola is the shape the answer takes when the ground has one limit; a ground whose limit moves gives a curve of the same shape with its peak somewhere else, and the peak exists whatever the ground does, because a heavier block always leaves less of the ground’s capacity for the push.

Still open: the body lowered onto its ground #

Every block here arrives on its ground whole, with its full weight. A real one is built there: a tower rises course by course, a tank fills, a crane takes on its counterweight, and the ground under it consolidates as the weight arrives, gaining strength as it drains. The ground’s capacity at the moment of the design push is therefore not its capacity on the first day, and neither is the weight. Whether the parabola moves faster than the weight climbs up it — whether a structure built slowly on soft ground passes through its worst point during construction rather than after it — is a question about the order in which the weight and the strength arrive.