The ground that arrives after the weight

The ballast that helps it over gave the ground under a block a bearing capacity, and found that the resistance to tipping is no longer the weight times half the base but

$$M_u = \frac{WB}{2}\left(1 - \frac{W}{W_u}\right),$$

with $W_u$ the most the ground under the whole base can bear. The resistance is a parabola in the weight. It peaks when the block weighs half the ground’s capacity, and past that every tonne added to steady the block brings it nearer to going over.

Weight is the only thing resisting it had made the check a comparison of two moments with no strength in either, and the body on settling pads had shown the ground’s stiffness taking some of the width away. The ballast essay’s parabola takes more, and it puts the ground’s strength into the overturning check where the check had been proud of having none. Its blocks arrived whole and stood on ground whose capacity was fixed. It ended on the case that is neither: a structure built in place on soft clay, whose weight arrives course by course while the clay under it drains and gains strength. The capacity $W_u$ is then moving, and so is the parabola. This essay follows a tower and a silo up it.

Clay that is stronger for having been loaded #

A saturated clay loaded quickly carries the load in its pore water, which cannot escape in the time available, and its strength is the undrained strength it had before — a single number, $s_u$, and the bearing capacity of a square base on it is about $6.2\,s_u$ per square metre, the 2 + π of a strip raised by a fifth for the square. Leave the load there and the water drains, the clay consolidates, the load passes from the water to the grains, and the clay’s strength rises with it: in a normally consolidated clay, by about 0.22 of whatever effective stress it has gained.

The drainage is slow, and it is the same drainage that makes a building on clay go on settling for years after it is finished — the settlement and the strength gain are two readings of one process, the water leaving. For a layer drained top and bottom over 5 m, with a coefficient of consolidation of 3 m² a year, Terzaghi’s theory puts 90 per cent of the drainage at 7.1 years. A weight that arrives over a year has, on the day it is complete, had only a fraction of that time to act, and the earliest of it has had longest.

The bodies here stand on that clay, 10 m square in plan, with an undrained strength of 35 kPa before anything is built. Under the whole base it can bear 21,600 kN at first; under 150 kPa of fully drained load it will bear 42,000.

A tower that is weakest when it is finished #

The tower weighs 15,000 kN when complete, and the wind on the finished tower is a push of 1,250 kN at half its 40 m height. As it rises its weight grows in proportion to its height, and so does the area the wind acts on; the push’s lever grows too, so its overturning moment grows as the square of the height.

Every curve does the same thing. The factor of safety falls throughout construction, reaches its lowest on the day the last of the weight arrives, and climbs from then on for years. Built in a year, the tower is at 1.33 on its first day complete and 1.93 a decade later, and the difference is entirely the clay’s.

The fall during construction is the first result, and it holds whatever the ground does. The tower’s weight grows with its height and the wind’s moment with the square of its height, so the ratio of the two falls as it rises — on rigid ground in proportion to the height, which is why a finished tower is always checked and a half-built one rarely is. On yielding ground the second factor, $1 - W/W_u$, falls as well, because the weight climbs faster than the capacity can follow. Nothing about the clay can make the half-built tower worse off than the finished one, since both factors are moving the same way.

And nothing can make any later day worse than the day it is finished. From then on the weight is constant, the wind is what it will always be, and the capacity can only grow. The worst point of a tower’s life on consolidating ground is the day it is finished. The lead the ballast essay left asked whether a structure built slowly passes its worst point during construction rather than after; for a body whose push grows as it rises the answer is exactly at the boundary between the two, every time.

The parabola that moves #

The two parabolas are the ground on the first day and the ground a decade later, and the tower’s path runs between them. It leaves along the lower one — with little weight on it the clay has little to consolidate under — and as it climbs it moves up and away from it, gaining capacity while it gains weight, but never catching the upper parabola while construction goes on. When the last of the weight arrives the path turns vertical, and the whole of its remaining gain happens at constant weight.

The dashed curve is the demand, and it rises as the square of the weight. It is closest to the path where the path turns: the day the tower is finished. The finished tower weighs 15,000 kN, which is 69 per cent of the undrained capacity and 36 per cent of the consolidated one — past the peak of the first parabola and before the peak of the second. It is on the wrong side of the parabola it was built on and the right side of the one it will stand on.

What the construction time buys #

The finished tower’s factor rises smoothly with the time taken to build it and has two limits. Built faster than the clay can respond — a week, a month — it is the undrained tower; built much more slowly than the clay drains it is the drained one; and the transition between the two is spread over more than two decades of construction time, centred on the clay’s own consolidation time.

What matters is the ratio of the two times, not either alone. A year is a fast build on this clay and would be a slow one on a clay draining in a month. The same tower on the same foundation can be safe or not depending on nothing but how quickly it was put up, and the rigid-ground check, reporting 3.00 whatever the schedule, cannot see the difference.

The figure also shows what the ballast essay’s warning becomes on this ground. The ratio $W/W_u$ that decides which side of the parabola a body is on is not a property of the body and its ground; it is a property of the body, its ground and its programme. The tower built in a week is at 67 per cent of its ground’s capacity on the day it is finished and the tower built over twenty years at 38 per cent — the same weight on the same clay.

A pause is worth most near the top #

Construction on soft ground is often paused deliberately, to let the clay catch up. Where the pause falls turns out to matter as much as how long it is.

A year’s pause after a twentieth of the weight is worth almost nothing: the clay drains for a year under a weight too small to strengthen it much, and the other nineteen twentieths still arrive in eleven and a half months. The same year taken after nineteen twentieths is worth more than spreading the whole construction over two years, because the clay drains for that year under nearly all of the final weight. A pause is a period of consolidation under whatever load is present, and its value is proportional to the load it is taken under.

The rule has a limit, and the silo shows where it is.

The silo, which is filled rather than built #

A silo is a body whose weight arrives the other way. Its structure is built first, light, and the weight comes when it is filled — here 2,000 kN empty and 20,000 kN full on the same 10 m square of clay. The wind’s push on its shell is the same full or empty, and what fills it does not sit exactly on the centre of the base: grain loaded into one bin before another, or a load poured down one side, puts the weight 0.5 m off centre, and that moment grows with the fill.

So its demand is not the tower’s. It starts at a fixed wind moment on the empty shell and grows only slowly, while the resistance runs along the parabola: up while the silo is less than half as heavy as its ground can bear, down after.

Every filling rises from the empty silo’s 1.89 and peaks at about 3.3 when the silo is half to two thirds full. What happens after depends on the clock. Filled in a week, the silo goes over the top of its parabola and down the far side to 0.84: the full silo, on its first filling, cannot resist the moment of its own off-centre fill and the wind together. Filled over a year, the clay has drained under the rising weight, the peak has moved to the right while the silo climbed toward it, and it ends at 2.25.

The silo passes its worst point during construction in a sense the tower cannot: the first filling is a test that the silo either passes or does not, and the empty silo and the long-term full silo are both safe. It is the reason tanks and silos on soft ground are filled for the first time in stages, with settlement watched between them — a hydrotest is a load test of the ground as much as of the tank.

It is also the most famous way a structure has gone over. The grain elevator at Transcona, near Winnipeg, was filled for the first time over a few weeks in the autumn of 1913, and within a day of the filling reaching its height it tilted to nearly 27 degrees, intact, into the clay beside it. Peck and Bryant showed forty years later that the clay’s undrained bearing capacity accounted for it. In the terms of the parabola the elevator had been filled past $W/W_u = 1$, where the resistance to tipping is zero and the smallest eccentricity is enough: a bearing failure under a body that is not perfectly centred presents itself as an overturning.

Where the hold goes #

The silo’s filling can be paused, and the tower’s rule — the later the better — holds for the full silo’s factor. But the silo has a worst point that is not its end.

The full silo prefers its hold as late as possible: 1.66 with the hold at a twentieth, 2.38 with it at nineteen twentieths. But a hold at nineteen twentieths is reached in less than three months with almost all of the weight on clay that has had almost no time — which is a nearly undrained filling to 95 per cent, and its own worst point, 1.75, arrives on the way up to the hold rather than at the end. Between a third and nine tenths, the worst moment of the whole filling is neither: it is the empty silo in the wind, which no schedule can change.

The best hold is the latest one that does not become the worst moment itself, which for this silo is somewhere past four fifths of the fill. The tower does not have this problem, because it approaches its end with less than all of its weight and cannot be worse before it than at it.

The opposite clock #

The tower’s capacity rises with time and its worst day is the first day it carries everything. There is a family of structures whose clock runs the other way, and setting the two side by side shows what the tower’s result depends on.

A concrete column under sustained load creeps: its deflection grows at constant load, its second-order moment grows with the deflection, and its capacity falls with time, so a column that stands on its first day can fail years later without any change in what it carries. Its worst day is its last. A pile driven through fill that is still settling is the same shape of problem in the ground: the fill goes on consolidating, drags the pile down and adds load to it that was not there when it was installed, so the pile’s worst day, too, is late.

The difference is in what the time-dependent process does to the capacity. Creep and negative friction take capacity away as the years pass; consolidation under a structure’s own weight adds it. A check made on the day a structure is finished is the worst case for one family and the best case for the other, and the two are told apart by asking which way the slow process runs — which is a question about the material and the ground, not about the load. A tower on clay is checked at completion because that is its minimum, and a creeping column at the end of its life for the same reason read the other way.

Preloading, and why it is the pause taken to its limit #

The pause figure has a limit it approaches but does not reach: a pause taken after the whole of the weight is in place. Taken literally that is simply waiting, and it does nothing for the day the tower is finished. But a heavier weight, placed early and removed before the structure is built, does exactly what a late pause does, and more.

That is preloading: an embankment of earth heavier than the future structure is placed on the site, left while the clay consolidates under it, and removed before construction starts. The clay the structure is then built on has already gained the strength of a load larger than its own, and the structure starts its life on the upper parabola rather than climbing toward it. In the terms of the figures, preloading moves the whole tower path onto the consolidated parabola before the first course is laid, so the day it is finished is no longer a minimum at all, and the rigid-ground factor of 3.00 is the only one that has to be approached. The price is the time the preload has to stand, which is the clay’s consolidation time and not the structure’s — often shortened with vertical drains that halve the drainage path and so quarter the time.

A preload is also the one arrangement in which the ballast essay’s warning does not apply. There, weight added to steady a block moved it along a fixed parabola toward its descending side. Here weight added and then taken away moves the parabola itself, and leaves the structure on the rising side of a larger one.

One tower, by hand, on its first day #

On the first day the clay’s strength is 35 kPa and the square base’s capacity is $6.17 \times 35 \times 100 = 21{,}600$ kN. Placed in an instant, the finished tower’s 15,000 kN would use 69 per cent of it, and its resisting moment would be $15{,}000 \times 5 \times (1 - 0.69) = 22{,}900$ kN·m against the wind’s $1{,}250 \times 20 = 25{,}000$: a factor of 0.92.

Fully drained under 150 kPa, the clay gains $0.22 \times 150 = 33$ kPa, to 68, and the capacity is $6.17 \times 68 \times 100 = 42{,}000$ kN. The tower then uses 36 per cent of it, resists $15{,}000 \times 5 \times 0.64 = 48{,}200$ kN·m, and has a factor of 1.93.

Built over a year, the clay under the tower has drained about a quarter of the way on the day it is finished — the early weight more, the late weight hardly at all — and its strength is about 44 kPa. That gives a capacity of 27,000 kN, a ratio of 0.56, and a factor of 1.33.

One capacity, a uniform gain and a rigid-plastic toe #

The clay drains one-dimensionally and uniformly under the base, gaining the same strength under the toe as under the heel. A body that is leaning presses its toe harder, so the clay there has more to consolidate under and gains more — which helps — and is also nearer to failing in the meantime, which does not.

The capacity is the undrained bearing capacity of a square surface footing, with no allowance for the eccentricity of the load or for the depth the toe has been pushed to, both of which the ballast essay set aside for the same reason: they move the peak of the parabola without removing it.

Tipping is the only way out. A body on weak clay can also slide, and the choice between tipping and sliding is made by the ratio of base to height against a friction that, on undrained clay, is the same strength $s_u$ that sets the capacity; for a squat body the sliding check depends on the schedule exactly as the tipping check does, and the strength gain helps both.

The ground is rigid up to its limit — the rigid-plastic toe, which the spring bed of the ballast essay approaches from below. The off-centre fill is also assumed to stay inside the middle third of the base before the wind acts, so that the whole base is in contact while the clay drains. A real tower on soft clay also settles and tilts, and a tilt moves its weight’s line toward the toe as the height of its centre times the tilt; on the softest beds that essay drew, the lean alone cost a third of the resistance.

What the pictures cannot show #

That the clay is not known to three figures. Every curve here is drawn from one strength, one rate of gain and one coefficient of consolidation, and each of those is a site investigation’s estimate with a scatter of tens of per cent. The shape — a minimum on the day the weight is complete, a hold worth most when it is late — survives any such scatter; the numbers on it do not.

Nor can the pictures show how the clay is watched. A body built on soft ground is instrumented, with settlement markers and piezometers reading the pore pressure the weight has put into the clay, and the construction is slowed or stopped when the pore pressures are not falling as fast as expected. That practice is the measurement of the quantity every figure here computes, and it is why structures like these are built successfully far more often than the parabola suggests they should be.

Still open: the body that is also compressing its ground unevenly #

Every body here consolidates its clay uniformly and stays upright while it waits. A body whose weight is off centre consolidates its clay more under the heavier side, settles more there, and tilts — and a tilt moves the weight toward the side that is already settling most. Whether consolidation under an eccentric load is self-correcting, because the more heavily loaded clay also gains more strength, or self-aggravating, because it also settles more, is the question the silo’s half-metre of off-centre fill leaves after the filling is done.