The building held down by its pump

The basement that leaks before it floats followed a finished basement as its water table rose, and found that the slab’s cracks ran out of room long before anything else: each metre of rise added a sixth to the building’s uplift but more than a quarter to the net load on the slab, which carries the water less only its own weight. A slab with a quarter more steel than its crack limit needed at the design head tolerated about 1.6 m of rise before its cracks reached 0.2 mm.

That essay ended with the usual repair. A basement whose slab has started to leak is given back its design head by drains under the slab and a sump pump that lowers the water beneath it, so that the slab sees what it was designed for whatever the water table outside is doing. That is permanent drainage — the arrangement a basement is a boat found usually refused at design, because a pump’s life is a fraction of a building’s — arriving as a repair rather than a design. The question is what the basement is once it depends on a pump, and what happens in the hours after the pump stops.

The same basement, under a lighter building #

The basement is the earlier essays’: 20 by 30 m in plan, dug 9 m, its slab 0.9 m thick and held down at six columns, its slab’s steel a quarter more than its 0.2 mm crack limit needed. It was designed for a water table 3 m below the ground, a head of 6 m under the slab. The water table has since risen 3 m, to the ground, and a pump holds the head under the slab at 6 m.

The earlier essay put eight storeys over it. This one puts five, for a reason that the essay will reach: with eight storeys the building’s weight was so far ahead of its uplift that flotation was the last thing the rise troubled, and that is a property of a heavy building, not of a basement. Under five storeys, at the design head, the flotation check — the uplift at its full value against nine tenths of everything’s weight, the check EN 1997-1 calls uplift — stands at 0.86. The building was comfortably safe as designed and is safe again while its pump runs.

The water comes back on the ground’s clock #

When the pump stops, the drains under the slab fill and the head under it returns towards the water table outside. How fast depends on the ground round the basement: the water has to flow back through it into the volume the pump had kept drained, and the inflow is proportional to the head still to be made up. That makes the recovery first-order, as the water that comes back while the floors go up took it for a construction dewatering:

$$h(t) = h_0 + \Delta h\left(1 - e^{-t/T}\right),$$

with $h_0$ the design head, $\Delta h$ the rise the pump was holding back and $T$ the ground’s recovery time constant. In open sand and gravel $T$ is hours; in silt, days; in clay the basement would hardly have needed the pump.

The curves are the same curve on three clocks. In ground that recovers in 12 hours, the head is back half way, at 7.5 m, after 8.3 hours, and within a few centimetres of the full 9 m after a day and a half. In 3-hour ground the whole return takes most of a working shift; in 48-hour ground it takes most of a week.

Two levels are drawn across them. At a head of 6.97 m — 0.97 m of the rise back — the flotation check is used up. At 7.65 m the slab’s cracks reach 0.2 mm. Between them and above them nothing else happens: the slab’s strength, the check one would expect to be the dangerous one, is not on the figure, because it is never reached.

What the pump was holding #

Read across at any hour and the three checks are in the same order. Flotation starts highest, at 0.86, and grows fastest in proportion, because a building’s weight is fixed and its uplift is the whole head over the whole plan. The cracks start at 0.70 of their limit — the steel’s quarter margin, which buys more than a quarter because a crack’s width falls faster than the steel’s stress — and reach it later. The bending starts at 0.37 and ends at 0.61, a long way below its strength.

That last line is the one worth looking at twice. The slab is the element that carries the water, and an engineer asked what a rising water table threatens would name the slab first. It does carry a net load that grows by more than a quarter per metre of rise. But a slab sized for its crack width at the design head has steel well beyond what its strength needs — the basement that leaks before it floats found the crack limit governing its steel on both faces — so the slab’s strength has room for the whole rise and more. What the pump holds is not the slab’s strength. It is the slab’s watertightness, and, under a light enough building, the building’s flotation.

A building that fails its flotation check at the ultimate limit state does not necessarily rise out of the ground at that instant: the check counts nine tenths of the weight against the full uplift, and the real weight has that tenth in hand. But the check is the one the building was designed to pass, and passing it was the condition on which the basement was built. After 36 hours it stands at 1.27, past the point where even the full weight balances the water.

Under a light building the building goes first #

The cracks’ deadline is a horizontal line: 9.5 hours, whatever is built over the basement, because the slab carries the water against its own weight and the building does not enter its sum. Flotation’s deadline is set by the building, and it climbs steeply with the storeys: at once under two or three storeys, one hour under four, 4.7 under five, 10.1 under six, twenty hours under seven, and never under eight, whose weight carries the whole rise with a margin.

The two lines cross between five and six storeys. Above the crossing the pump is protecting the slab’s cracks, and its failure costs a damp basement and a repair. Below it the pump is protecting the building’s stability, and its failure costs the condition the building was designed on. The same pump under the same slab is a maintenance item under one building and a structural element under another, and nothing about the pump says which.

That is the reverse of the order the earlier essay found under eight storeys, where the slab’s cracks ran out first and flotation hardly moved. The order was never a property of basements. It is a property of how much building there is above one.

A deadline in time constants #

The first-order recovery makes every deadline a pure number of the ground’s time constants: $t_{lim} = -T\ln(1 - \Delta h_{lim}/\Delta h)$, where $\Delta h_{lim}$ is the part of the rise a check can take. For flotation under five storeys that is 0.39 of $T$; for the cracks, 0.80. In gravel that recovers in two hours the building has 48 minutes. In silty sand that takes four days it has a day and a half.

The ground that makes a pump necessary is the ground that gives it least time. A basement in tight ground leaks slowly and needs little pumping, and if its pump stops the water takes days to come back. A basement in open gravel needs a large pump to hold its head down at all, and gives that pump almost no grace when it stops. The volume the pump keeps drained is roughly the same in both — the ground’s storage over the depression it has made — and the time to refill it is that volume over the flow, which is the flow the pump was delivering. A pump that has to work hard is a pump whose failure is quick.

Steel that takes the cracks out of the pump’s keeping #

The cracks’ deadline depends on how much steel the slab has over what its crack limit needed, because that margin is the rise the slab can take on its own.

A slab with exactly the steel its crack limit needed at the design head has no deadline at all: the first centimetre of returning water takes its cracks past 0.2 mm. A tenth more steel buys 2.9 hours, a quarter more 9.5, and from 45 per cent more the cracks never reach the limit at any head the ground can supply. That last slab was, for watertightness, designed for the water table at the ground — which is the design a basement is a boat recommended for exactly this reason, that a water table’s design level is a prediction and the ground level is a bound.

Flotation has no equivalent. No amount of steel in the slab changes the building’s weight, and the ways of buying flotation back — ballast, a thicker slab, tension piles of the kind the piles that would lift the ground sized — are not repairs that can be made to a finished building cheaply. That asymmetry is why a pump is installed in the first place.

The pump’s reliability is now a load #

Once a check depends on a machine, the machine’s failure rate is part of the check. A sump pump stops for two reasons: it fails, or its power does. Suppose the two together give two outages a year, and that each lasts an exponentially distributed time before the pump is running again. The probability that an outage outlasts a deadline $t$ is then $e^{-t/\bar\tau}$, $\bar\tau$ being the mean time to restore it.

A pump that is restored in six hours on average — a call-out to a contractor, a replacement fitted — lets the building’s flotation check be used up 0.91 times a year. One restored in two hours on average does it 0.21 times a year, about once in five. Even the second is not a structural reliability anyone would accept for an ultimate limit state, which designs for events of the order of once in thousands of years. To bring the flotation check’s frequency down to that level the outage has to be made essentially impossible or almost instantaneous: a second pump that starts by itself, on a supply that does not share the first’s power, with an alarm that someone answers.

That is what the counterweight that knows the load faced with an active counterweight on a crane: a device that improves the structure while it works makes the structure depend on its working, and its failure becomes a design situation rather than an accident. In the language of the structure that survives losing a member, the pump has become a key element: one whose loss the structure is not designed to survive. The basement’s pump was installed as a repair to keep a slab dry. Under a light building it has quietly become what holds the building down.

What the pump is standing in for #

It is worth pricing the pump’s job in the currency the basement was designed in, which is weight. For the building under five storeys never to use up its flotation check, whatever the ground does, nine tenths of its weight would have to match the uplift of the full 9 m head: $9.81 \times 9 \times 600 = 53{,}000$ kN, so a weight of 58,900 kN against the 45,600 it has. The 13,300 kN between them is 0.92 m of concrete over the whole of the basement’s floor — or three more storeys of the building, which is the arithmetic the storey axis of the figures already showed.

Neither can be added to a finished building. A metre of ballast on the slab takes a metre of headroom out of a basement whose levels were set by its use, and adds its own weight to a slab that then carries less net upward load and more downward at its columns. Tension anchors drilled through the slab into the ground below are the usual passive repair, and they have to be designed for the slab’s bending between them as well as their own pull. The pump is doing the work of 13,300 kN of concrete, and it is doing it with an electric motor and a float switch.

That is the honest comparison when the repair is chosen. A pump is cheap to install and cheap to run, and its cost is not in either: it is in the obligation it creates, which is to keep it running for the life of the building, under owners who did not install it, through every power cut the building will see.

An alarm is a deadline spent in advance #

The usual answer to a pump that might fail is a high-level alarm: a second float, above the pump’s own switch, that raises an alarm when the head under the slab passes it. An alarm does not change the deadline; it changes when the clock is seen to have started.

Set the alarm 0.3 m above the design head, where the water table’s normal fluctuation will not trip it. In ground that recovers in 12 hours the head reaches it $-12\ln(1 - 0.3/3) = 1.3$ hours after the pump stops, and the flotation check is used up 4.7 hours after: the alarm leaves 3.4 hours for someone to answer it and get a pump running. In 2-hour ground the same alarm sounds after 13 minutes and leaves 35. A response time that is adequate in one ground is not adequate in another, and the ground’s time constant is something nobody measures once the basement is built.

The alarm has a second use that is easier to justify. A basement whose alarm sounds when the pump has not stopped is a basement whose drains are failing to deliver water to the pump — the slow failure the last section of this essay comes to.

The deadline by hand #

The building under five storeys weighs $W$ such that the flotation check at the design head is 0.86: $9.81 \times 6 \times 600 = 35{,}300$ kN of uplift against $0.9W = 35{,}300/0.86 = 41{,}100$ kN. The check is used up when the uplift reaches 41,100 kN, at a head of $41{,}100/(9.81 \times 600) = 6.98$ m: 0.98 m of the 3 m rise. With a 12-hour time constant,

$$t = -12\ln\left(1 - \frac{0.98}{3}\right) = -12\ln 0.674 = 4.7\ \text{hours}.$$

The cracks take 1.65 m of the rise, the margin the quarter of extra steel gives, and $-12\ln(1 - 1.65/3) = 9.5$ hours. Under eight storeys the building weighs $8 \times 8 \times 600 = 38{,}400$ kN more than under none and $3 \times 8 \times 600 = 14{,}400$ kN more than under five, and the flotation check at the full 9 m head is 0.98: never used up.

What the recovery leaves out #

The recovery is first-order. The ground round a pumped basement does not refill as one reservoir: the water nearest the basement comes back first, and the far field more slowly, so a real recovery is quick at first and then has a long tail. For the deadlines that matter, which fall in the first half of the recovery, the first-order curve is a reasonable fit; for a check reached near the end of the recovery it would be optimistic about the time.

The pump either runs or it does not. Sump pumps run on level switches and cycle, so the head under the slab is a sawtooth between two levels rather than a constant, and the design head is the top of the sawtooth. A pump that fails partly — a worn impeller, a drain half silted — holds a head above its design value rather than losing it, which is the slow failure the next question is about.

The slab’s cracks are taken as they would be in a new slab. A slab that leaked before its pump was installed has cracks that opened under the risen water and may not fully close under the lower head; they are already wider than a new slab’s at the same stress.

Flotation is a check on the whole building. It counts every storey’s weight as if it arrived at the basement uniformly. A building lighter at one end tips before it lifts, which a check of totals cannot see.

Still open: the drain that silts up #

Every failure here is sudden: the pump stops and the head returns. Drains under a slab fail slowly as well. Fines carried by the water they collect settle in the drainage blanket and the pipes, the drains’ capacity falls year by year, and the pump — still running, its level switch still cycling — holds a head that creeps upwards because the water reaches it more slowly than it reaches the slab. Nothing alarms. Whether a basement relieved by drainage loses its margins faster to the silt than to the pump, and whether the head under the slab can be watched cheaply enough — a standpipe through the slab, read once a year — to see the creep before the cracks do, is what the drain’s slow failure asks of the pump’s sudden one.