The roof that drains towards its sag

The roof that ponds on its girders coupled a flat roof’s joists to the girders they sit on and found the water fills the sum of both sags. A 12 by 9 m bay with both members sized to deflect $L/240$ under 1.5 kN/m² has its joists’ sag amplified 1.79 times by the water that collects in it, where each member alone would say 1.24 and 1.35; a square bay sized the same way ponds outright at 24.5 m. Every one of those numbers was the roof’s answer to a depth of water it was handed.

The depth is the drainage designer’s. A flat roof drains through primary outlets, and a design for rain assumes they are blocked — by leaves, by a plastic bag, by ice — so that the water rises until it reaches a secondary outlet set somewhat higher. Its height above the roof is the static head. To pass a design storm through it, the water must stand above its sill by a further hydraulic head, set by how much roof it drains and how long it is. The static head plus the hydraulic head is the head of water the roof carries, and it is usually on another drawing.

That drawing fixes more than the head. It fixes where the outlet is: through the parapet at the roof’s edge, at a column, on a girder line, or in the middle of a bay. Ponding as the earlier essays described it depends on that choice as much as on any stiffness.

What fixes the surface #

Ponding is a feedback loop: the roof sags, the sag makes room for water, the water sags the roof further. The loop needs one thing that is easy to miss. The water’s surface must stay where it was while the roof goes down. Only then is every millimetre of sag a millimetre more water.

A scupper through the parapet does exactly that. Its sill is in the wall, at the edge of the roof, on the column line, and the wall does not move. However far the bays sag, the water’s surface stands at the sill’s height plus the hydraulic head, measured from the roof’s supports. A primary drain at a column, with a dam ring around it to make it a secondary outlet, does the same: a column top does not deflect.

A drain in the middle of a bay does not. It is fixed to the deck, and the deck at the middle of the bay is the lowest point of the sag. When the roof deflects, the drain goes down with it, and the water’s surface — which stands a head above the drain’s sill — goes down too. The sag still makes room for water, but the surface it is measured from has fallen by exactly the sag at the drain.

A drain on a girder line, half-way between columns, is between the two: it goes down with the girder’s sag but not with the joists’.

To see what each does, the bay is the same two half-sines as before — the girder’s sag across the bay and the joists’ sag between girders — but the water is now integrated over a grid instead of in closed form, so that the depth at each point is the surface’s height above the deflected roof there, and nowhere less than nothing. For the fixed level the result is the closed form exactly, and the grid adds nothing; for an outlet that moves, it lets the shallow edges of the bay run dry if the sag outruns the head.

With the level fixed, the member load is the amplification the two-way analysis found: twice a rigid roof’s at 12 m, four times at 18 m, and without limit at 24.5 m. With a drain on the girder line, the girder’s sag no longer collects water, and only the joists’ sag relative to their supports does. The amplification is the joists’ one-way answer and a little more: 1.24 at 12 m, 1.41 at 18 m, still rising, but at 30 m still under two and with no instability in sight. With a drain at the middle of the bay the curve goes the other way. The members carry 0.82 of a rigid roof’s load at 12 m and 0.75 at 18 m, and the larger and softer the bay, the less they carry. A roof that drains towards its own sag carries less water the more it sags.

Why the middle drain sheds load #

At the middle drain the water stands at the head above the deck, $h$. Everywhere else the deck is higher than at the drain, because the drain is at the lowest point, so the water is shallower. The average depth over the bay is less than $h$, and the members, carrying the average, are loaded less than a rigid roof under $h$ would be.

How much less depends on how deep the sag is compared with the head. The sag is caused by the water, so it is proportional to the head, and the ratio between them is fixed by the bay’s stiffness. A stiff bay sags little and the water is nearly level, the load nearly $\gamma h$. A soft bay sags a lot, the water is deep at the middle and shallow at the edges, and the members carry a smaller share. The fixed-level bay is the same arithmetic with the opposite sign: a soft bay sags a lot and every bit of it is filled from a level that does not drop.

The three profiles are the same bay under the same 50 mm head. With the level fixed the water is 50 mm at the supports and 168 mm at the middle: the head plus the whole sag, amplified. With a drain on the girder line the surface has gone down with the girder, so the supports — which are at the joists’ ends, on the girders — have only 22 mm over them and the middle 75. With the drain at the middle of the bay the water is 50 mm where it is deepest and 15 mm at the supports.

None of this is ponding in reverse, and nothing about the bay is safer except its load. A drain at the middle of a bay is a drain at the bottom of a hollow. The hollow was there before any water was, from the roof’s own weight, and the water finds it.

The head each bay can carry #

The members of a roof have a capacity, and the roof’s own weight takes part of it. With 2.0 kN/m² of capacity and 0.5 of roof weight, there is 1.5 kN/m² for water, which on a rigid roof is $1.5/9.81 = 153$ mm. Divide that by each outlet’s amplification and the head each bay allows follows.

The curves fan out from the rigid roof’s 153 mm. With the level fixed, a 12 m square bay allows 76 mm, an 18 m bay 39 and a 24 m bay 3, half a metre short of ponding outright. With the drain on the girder line, 124, 109 and 93: the joists’ sag still costs something, but it grows slowly. With the drain at the middle of the bay, 188, 205 and 221. The middle-drained bay allows more head than a rigid roof, because a rigid roof under $h$ has $h$ everywhere and the sagging bay has $h$ only at the drain.

The 12 by 9 m bay of the earlier essay allows 85 mm of head with its level fixed, 134 with a girder-line drain, 182 with a drain at its middle. Those are the three numbers the drainage design has to fit inside, and they differ by more than a factor of two for one structure.

The head that pushes the storm out #

The allowed head has to hold two things: the height of the overflow’s sill and the head that drives the storm through it.

A scupper is a short weir. The flow over a weir grows as the head over its sill to the power one and a half, so the head needed grows as the flow to the two-thirds: double the roof a metre of scupper drains and the head rises by 59%. At a storm intensity of 150 mm an hour — a short, heavy storm of the kind overflows are sized for — a metre of scupper draining 400 m² needs 44 mm of water over its sill. A roof drained through two short scuppers at each end of a long building can need a hydraulic head larger than the static one.

What is left for the sill #

Subtracting the hydraulic head from the allowed head leaves the height the overflow’s sill can stand above the roof.

For the 18 by 12 m bay with scuppers fixing the level, 57 mm is all the water the members can carry, and at 400 m² of roof per metre of scupper the storm takes 44 mm of it. The sill can stand 13 mm above the roof — less than a typical membrane upstand — and once a metre of scupper drains 592 m², it would have to stand below the roof, which is to say there is no overflow height that works. The roof needs more metres of scupper, or a stiffer structure, or a different kind of outlet. The same bay with a drain at its middle can have its overflow sill at 148 mm and still be inside its members’ capacity.

This is the drawing the earlier essay said belongs to somebody else. Read with the structure, it is not a separate decision. The structural engineer’s ponding check and the drainage designer’s choice of overflow are one inequality, and both of its sides are set by where the water leaves.

Stiffness works in opposite directions #

The earlier essay found that a stiffer deflection limit buys span in proportion, because deflection goes as the fourth power of the span and a member sized to a limit keeps one of those powers as its ponding parameter. That is true under a fixed level and false under a drain that rides with the roof.

With the level fixed, stiffening the bay from $L/180$ to $L/720$ raises the allowed head from 63 mm to 130. With a drain on the girder line it raises it a little, from 127 to 147. With a drain at the middle of the bay, stiffening lowers it, from 191 mm to 163, because a stiffer bay sags less towards its drain and its water is more nearly level. All three approach the rigid roof’s 153 mm in the limit, which is the only roof on which an outlet’s position does not matter.

That reversal is the clearest statement of what ponding is. It is not a property of a flexible roof. It is a property of a flexible roof under a fixed level. Take the fixed level away and flexibility helps.

The catch in the middle drain #

The middle drain is not a free answer, and the reasons it is not are what decide where outlets actually go.

A drain at the middle of every bay is many drains. A warehouse of a hundred bays has a hundred secondary drains, each with its own pipe run through the roof void to a downpipe, each a place to leak and to block. Scuppers through a parapet need no pipework at all, which is why they are the commonest overflow and why the commonest overflow is the one that lets a roof pond.

Every bay must have one. The argument holds for a bay that drains to its own low point. A bay without its own drain, beside one with a drain, is a bay whose water level is set by its neighbour’s drain — a fixed level, as far as it is concerned, if its neighbour is stiff — and it ponds on that level. A roof with drains in some bays is as weak as its bays without them.

The drain must be where the sag is deepest. A bay that sags most somewhere other than at its drain has a low point the drain does not reach, and the water there stands on a level fixed by the drain’s sill, which no longer falls as fast as the low point does. Edge bays, bays with a heavy plant load and bays over a transfer girder — where a support that deflects adds its own sag — sag differently from their neighbours, and a drain placed on a regular grid will not be at all of their low points.

A blocked drain is the design case. The secondary outlet exists because the primary one may be blocked. A secondary drain beside the primary one, at the same low point, may be blocked by the same leaves. The overflow at the parapet is preferred partly because it is not at the low point: it is where nothing collects.

So the commonest arrangement is the worst one structurally for good reasons, and the arrangement that removes ponding is a set of obligations the drainage has to meet bay by bay. The roof’s structure and its drainage are one system either way. The question is only which side carries the burden.

The 12 by 9 m bay, by hand #

The members of the 12 by 9 m bay are sized to deflect $L/240$ under 1.5 kN/m², so the girders have $EI = 5 \times 1.5 \times 9 \times 12^3 \times 240/384 = 72\,900\ \text{kN}\cdot\text{m}^2$ and the joists, at 1.5 m centres, $5\,126\ \text{kN}\cdot\text{m}^2$. Their ponding parameters are $C_p = 0.258$ and $C_s = 0.193$, as before.

With the level fixed the joists’ amplification is $1/D = 1.79$. A head $h$ puts $1.79\,\gamma h$ into the joists. Setting $0.5 + 1.79 \times 9.81\,h = 2.0$ gives $h = 85$ mm. Under 50 mm of head the girder sags 28 mm and the joists a further 28 relative to it, and the water at the bay’s middle is 106 mm deep.

With the drain at the middle of the bay the surface is $h$ above the middle of the sagging deck, so the water at any point is $h$ less the difference between the middle’s sag and that point’s. Under 50 mm of head the girder sags 13 mm and the joists 13 relative to it — half the fixed-level sag, because half the water is gone — and the joists carry 0.82 of $\gamma h$: $0.5 + 0.82 \times 9.81\,h = 2.0$ gives 182 mm. The factor between the two outlets, 2.1, is the ratio of what a fixed level and a falling one leave in the bay.

The scupper’s hydraulic head at 150 mm/h for 400 m² a metre is $Q = 0.150/3600 \times 400 = 0.0167\ \text{m}^3\text{/s}$ per metre, so $h = (0.0167/1.78)^{2/3} = 0.044$ m.

Two half-sines and a level surface #

The model is the earlier essay’s with the water’s surface free to move, and it keeps that essay’s limits along with some of its own.

One shape each. The girder and the joists sag as half-sines whatever the water does. When a middle drain dries the corners of a large bay, the load concentrates towards the middle and the true sag is more peaked than a sine; the share of load the members see is then a little smaller still, and the comparison between outlets is unchanged.

The roof is cambered for its own weight. A camber is rarely exact, and a roof that sags under its own weight before any rain has a hollow the water fills from the start. Under a fixed level that hollow is filled and amplified; under a middle drain it does not matter, because the drain is at the bottom of the hollow either way. An uncambered roof makes the fixed-level outlet’s numbers worse and the middle drain’s unchanged.

The members do not creep. A timber or concrete roof under standing water loses stiffness as creep proceeds, which moves a fixed-level bay towards its limit and a middle-drained one, slightly, away from it.

The storm is steady. The hydraulic head is the head at the storm’s peak flow, and a roof’s water level lags the storm because the roof stores water. A short storm on a large roof may never reach its steady head; a long one will.

Every bay is the same. The bay is one of a field of identical bays, each with its own outlet or the same fixed level. A roof’s edge bays, and a bay next to a different one, break that symmetry, and the argument then has to be made bay by bay.

Capacity is a single number. Each member is checked against a uniform load of the same intensity that would give its sag the same size, which is the steel specification’s approach to ponding stress and ignores the change in the shape of the moment diagram under a peaked load.

What the outlet is part of #

The essays on ponding have treated it as the column that was never straight with rain in place of the axial load: an amplification $1/(1 - C)$ that runs away as the stiffness falls. That analogy holds exactly, and it carries an assumption the column carries too. The axial load on a column does not change direction as the column bends. Here the equivalent is the level of the water, and the outlet decides whether it moves.

A level that falls as the roof sags is a load that follows the structure, and a load that follows the structure can stabilise as well as destabilise. It also gives up a symmetry the fixed level had. Under a fixed level the water a girder’s sag admits loads the joists exactly as much as the water a joist’s sag admits loads the girder, Maxwell’s reciprocity applied to a load the roof makes for itself. Under a falling level each member’s sag also lowers the water over the whole bay by its sag at the drain, and a uniform lowering takes load off the girder’s broad shape and the joists’ narrower one by different amounts, so the two cross effects are no longer equal. The water that will not run off is water whose surface is held up by something the roof cannot move. A drain that goes down with the roof is a surface the roof does move. The difference between them is a few metres of pipe, and it is worth more than any change of section.

Stiffness is not strength, and on a flat roof the stiffness the deflection check asks for is a proxy for something else: a limit on how much water a sag can be made to hold. That limit is not only set by the sag. It is set by what the water’s surface does while the sag forms, which is a question for the drawing that shows where the water leaves.

Still open: the roof whose outlets are not all clear #

Every bay here drains through one kind of outlet, and that outlet works. A real roof has primary drains at its low points and overflows at its edge, and a design storm with the primary drains blocked meets a roof on which some bays are draining to a falling surface and the rest to a fixed one. If the primaries are partly blocked — passing some of the storm, not all — the water’s level in each bay is set by a balance between what the partly blocked drain at its middle can pass and what reaches the scupper at the edge, and that balance shifts as the bays sag. Whether a roof with partly working middle drains behaves like the middle-drained roof or the scuppered one, and at what share of blockage it changes from one to the other, is the question of whether the structure can be designed for drainage that works some of the time.