The drain that is mostly blocked

A flat roof ponds when its sag makes room for water and the water deepens the sag, and the water that will not run off is the same equation as a column that was never straight, with rain in place of the axial load. The depth of water the check starts from is the drainage designer’s, and the roof that drains towards its sag found that where the outlet sits matters as much as how high it is. A scupper through the parapet fixes the water’s surface, so every millimetre of sag fills. A drain at the middle of the bay goes down with the roof, so the surface follows the sag down, and the members carry less than a rigid roof’s load.

The codes ask for the check with the primary drains blocked, because blocked drains are what happen in the storms that matter: leaves, gravel, a dead bird, a membrane lifted over the outlet. But a drain is seldom blocked outright. It is choked to a fraction of its flow. The roof is then draining through both outlets at once — some of the storm leaving by the partly blocked drain at the middle of the bay, the rest, if the water rises far enough, over the scupper at the edge. The question here is how the roof behaves between the two roofs the earlier essay drew, and how much of the drain has to be working for the roof to behave like the drained one.

A drain that passes a share of its flow #

Take the bay from that essay: girders 12 m long carrying joists 9 m long at 1.5 m centres, each sized to deflect L/240 under 1.5 kN/m², with a capacity of 2.0 kN/m² of which the roof’s own weight takes 0.5. A storm of 150 mm an hour falls on its 108 m², which is 4.5 litres a second.

At the middle of the bay is a drain whose inlet is on the deck. Clear, it passes the storm with 30 mm of water over its inlet. Like a weir, its flow rises with the depth over it to the power one and a half, and a drain choked to a share of its clear flow passes that share at every depth. At the edge of the bay is a scupper through the parapet, 400 mm wide, its sill 50 mm above the roof’s supports. Water leaves over it once the level is above the sill, at $1.78\,b\,h^{1.5}$ for a head $h$ over the sill — 34 mm of head for this storm alone.

The two outlets read the water from different places. The scupper reads the level, measured from the supports, which do not move. The drain reads the depth over its inlet, at the bottom of the sag, which is the level plus however far the middle of the bay has gone down. So the roof’s sag, which is what makes ponding dangerous, is also water over the drain. For any level the bay is solved under a water surface fixed at that level, wet only where it has sagged below it if the level is under the supports; the level the storm settles at is the one at which the drain and the scupper together pass 4.5 litres a second.

The four states are very different roofs, and each is the girders’ and the joists’ sags added, the water filling both. With the drain clear the water stands 14 mm above the supports and the middle of the bay is down 16 mm, so the drain has 30 mm over it, which is exactly what it needs. With the drain blocked the water rises to the sill and over it by the scupper’s head, 84 mm, and the bay, now ponding under a fixed level, goes down 95 mm at the middle. A fifth open, the level is 41 mm, below the sill — the scupper is dry and the whole storm is going down the drain — and the middle is down 47 mm, so the drain has 88 mm over it. That depth, three times the clear drain’s, is what lets a fifth of a drain pass the whole storm.

The level, and the depth the roof makes for itself #

Follow the level and the depth over the drain as the drain’s open share runs from nothing to all.

From 15 per cent open upward the scupper never flows. The drain alone takes the storm, and the level settles wherever the depth over the drain is enough: for a drain passing a share $s$ of its clear flow, the depth must be $30\,s^{-2/3}$ mm — 88 mm at a fifth, 48 at half, 30 clear. The level is that depth less the bay’s sag at its middle, and the sag grows with the water, so the level climbs much more slowly than the depth over the drain does.

Below 15 per cent the required depth over the drain would put the level above the sill. The scupper starts to flow and the level is pinned near the sill — between 50 mm and the scupper’s 84 mm with the drain wholly blocked. The bay then ponds against a fixed surface, and its sag runs on toward the 95 mm of the blocked roof.

The difference between the two lines is the bay’s own deflection at its middle, and it is depth over the drain the roof has made for itself. On a rigid roof the two lines would coincide: the level would have to be the depth over the drain, and a fifth-open drain would need its 88 mm entirely from the level — above the sill, so the scupper would be running.

Whose drain carries the storm #

That last point is a comparison between roofs, so make it directly: the share of the storm that leaves by the drain, against how much of the drain is open, for the bay at L/240 and for a rigid roof of the same plan.

A rigid roof’s drain has only the level over it, and the level cannot rise past the sill without the scupper taking a share. So its drain carries the whole storm only while it can do so with at most 50 mm over it, which takes $(30/50)^{1.5}$, 46 per cent, of its clear flow. Below that the scupper shares the storm, and at a tenth open the drain takes only 39 per cent of it.

The bay at L/240 has its sag over its drain as well as the level. With the level at the sill the middle of the bay is down 56 mm, so its drain has 106 mm over it and needs only 15 per cent of its clear flow to pass the storm. At a tenth open it still takes 86 per cent. The flexible roof is better drained through a partly blocked outlet than the rigid one is, for the same reason that it ponds: it collects the water toward its middle, which is where the drain is.

This is the drained-roof essay’s finding — that a drain riding with the roof turns the sag from a liability into a help — carried down to drains that barely work. It depends on the drain being at the low point. A drain at a column, or a scupper, reads only the level, and gets no help from the sag at all.

What the members carry #

The level and the sag decide the members’ load. Read it as the more heavily loaded member’s load, with the roof’s own weight, over its capacity, against the drain’s open share, for members sized to L/180, L/240 and L/360 and for a rigid roof.

The curves are nothing like a straight line between the two ends. At L/240 the members carry 0.37 of their capacity with the drain clear and 0.99 with it blocked, and half open they carry 0.45 — a sixth of the way from one to the other. At a fifth open they carry 0.61. Almost all of the rise from the drained roof to the scuppered one happens in the last fifth of the blockage, and most of it in the last tenth.

Stiffness changes sides partway along. With the drain blocked the order is the one a ponding check expects: the softer the roof, the worse — 1.25 at L/180, over its capacity, against 0.66 for a rigid roof. With a fifth of the drain open the four are within four hundredths of each other. Beyond that the order reverses, slightly: half open, the L/180 roof carries 0.44 and the rigid roof 0.48, because the soft roof’s sag has put its water over its drain and the rigid roof’s water is spread evenly and standing higher. Stiffness is not strength, and here it is not even always stiffness: what it buys depends on which outlet is setting the water.

The same bay at a larger plan shows the reversal more sharply.

With the drain blocked, the 18 by 12 m bay at L/180 ponds to more than three times its capacity, and at L/240 to 1.62; deflection goes as the fourth power of span and the amplification grows with it. But a tenth of the drain is enough to bring every one of the four to under 0.8, and from about a fifth open the softest roof is the least loaded. The larger bay’s sag is so much deeper that a nearly blocked drain at its middle has a great deal of water over it. The stiffness that a blocked-drain check would demand for this bay — better than L/360, which still reaches 1.13 — buys almost nothing once a tenth of the drain is working.

How much of the drain must work #

The share at which the scupper starts — below which the roof is effectively scuppered, and above which it is effectively drained — is the number that summarises a roof’s tolerance of a blocked drain. It depends on the bay’s size and stiffness.

For a rigid roof the share is fixed at 46 per cent whatever the bay: the drain was sized for the bay’s storm, so a larger bay’s drain is larger in proportion, and the depth it needs is set by the sill. For a flexible roof the share falls as the bay grows, because the sag at the sill’s level grows faster than the storm. At 18 m an L/180 roof would need its drain less than a tenth of a per cent open — its sag alone gives a choked drain more depth than a clear one needs.

That is a tolerance, not a safety margin. The same 18 m bay at L/180 ponds without limit once the drain is wholly blocked. Its tolerance of a mostly blocked drain is real; its intolerance of a fully blocked one is fatal. A roof designed on the strength of its drains always working a little is a roof whose failure is decided by whether a drain can be blocked completely — which on a roof with a single drain per bay, under a membrane that can lift, it can.

Where the sill’s height matters #

The scupper’s sill sets the level a blocked roof ponds against, and the drained-roof essay found that every millimetre of sill comes off the head a bay can carry. With a partly working drain the sill matters less.

With the drain blocked the member load climbs steeply with the sill — from 0.77 at 25 mm to 1.43 at 100 mm — because the sill is the level the bay ponds against. A tenth open, it climbs much more slowly and levels off: past about 65 mm the drain is passing the whole storm before the water reaches the sill, and a higher sill changes nothing. Half open, the sill is never reached and the line is flat. A low sill is the remedy for the fully blocked drain and almost nothing else.

The bay, by hand #

The threshold can be found without the grid. With the level at the sill, $h_s$, the drain has $h_s + \delta$ over it, where $\delta$ is the bay’s sag at its middle under a level of $h_s$. The drain passes the storm alone if $s\,(h_s + \delta)^{1.5} \ge h_0^{1.5}$, so the threshold share is

$$s^* = \left(\frac{h_0}{h_s + \delta}\right)^{1.5}.$$

For the rigid roof $\delta = 0$ and $s^* = (30/50)^{1.5} = 0.46$. For the bay at L/240, $\delta$ at a 50 mm level is the two-way ponding answer — 56 mm — and $s^* = (30/106)^{1.5} = 0.15$. The sag is the whole difference, and the two-way ponding amplification that makes the sag large is what makes the threshold small. The sag at the drain is the only structural number in the formula, and any way of finding one deflection without solving everything will give it, provided the water load it is found under includes the water the sag itself collects.

Above the threshold the level is wherever $s\,(H + \delta(H))^{1.5} = h_0^{1.5}$, with $\delta$ growing with $H$ by the ponding amplification; below it the level is the sill plus the scupper’s head for the share of the storm the drain does not pass. Both are one-line iterations once $\delta(H)$ is known, and $\delta(H)$ is the earlier essay’s.

One bay, a weir and a steady storm #

The bay is one bay, with its own drain and its own scupper. A real roof has interior bays with no scupper of their own, whose water reaches the edge only by rising over the girder lines into the next bay; there the partly blocked drain is the interior bay’s only outlet below the level of its neighbours, and its tolerance is higher still, since the level can rise further before anything else takes over.

The drain’s flow law is a weir’s, the depth to the power one and a half. A drain that runs full — a siphonic system, or a gravity drain whose pipe rather than its inlet limits the flow — passes flow as the square root of the head, and gains less from a deeper sag: the threshold rises and the soft roof’s advantage shrinks, though it does not vanish. A blockage is taken as a share of flow at every depth; debris that floats onto a grate and is pressed down by deeper water would block more as the water deepens, and a roof relying on its sag to drive water through a choked drain would find that the sag also presses the debris harder.

And the storm is steady. A real storm peaks and falls away, and a ponded bay holds water — 14 cubic metres on this bay at the scuppered level — that a partly blocked drain takes time to empty. The peak level is then below the steady one, by an amount that depends on how long the peak lasts against how long the bay takes to fill.

The share that matters #

A partly blocked drain is not part-way between a working drain and a blocked one. At the middle of a flexible bay it sits at the bottom of the sag, with the bay’s own deflection over it as well as the level, and that depth lets it pass the whole storm down to a small fraction of its clear flow — a seventh at 12 m and L/240, much less on a larger or softer bay. Down to that fraction the roof behaves like the drained roof, and stiffness barely matters or slightly hurts. Only below it does the scupper set the level, the bay pond against a fixed surface, and stiffness decide the outcome.

So the useful question for a roof is not how stiff it must be with its drains blocked, but whether its drains can be blocked completely. If each bay has one outlet that can be sealed, the blocked-drain check is the governing one and stiffness is its remedy. If every bay has two drains, or a drain with a grate that cannot be sealed by one leaf, the roof is designed by the last fifth of the blockage curve, where it barely matters how stiff it is. It is the drainage’s version of the truss sized to lose a diagonal: the question is not how strong each path is, but whether a single event can close the only one.

Still open: the roof whose bays share a drain #

Every bay here has its own drain at its own low point. Many roofs are drained by fewer outlets than bays, with the deck laid to falls toward them, and a bay without a drain empties over its neighbours’ girders into a bay that has one. If that drain is partly blocked, the bays that feed it pond against a level set by the drained bay’s sag, which is deeper because it holds their water as well as its own. Whether a roof drained through a few low points tolerates a partly blocked drain as well as a roof drained at every bay, or whether the bays without drains concentrate their water onto the one bay whose sag is already deepest, is the question a falls-and-drains layout puts to the sag that helped here.