The water that will not run off

Deflection limits are usually presented as a comfort matter. A floor that bounces is unpleasant, a beam that sags cracks the plaster below it, a lintel that drops jams the door. All true, and all of it makes deflection sound like the limit state that does not kill anybody.

There is one deflection problem that does. A flat roof that deflects under rainwater has made room for more rainwater, and the extra water deflects it further. If the roof is stiff enough the process converges to a puddle and stops. If it is not, it does not stop, and there is no equilibrium at any depth — the roof fills until it fails.

The reader who has met the imperfect column will recognise that curve, and the recognition is not a resemblance. It is the same equation — which is worth demonstrating rather than asserting, because the same generator draws both and only the axis labels change.

Everything a designer knows about columns is therefore available here, and the transfer is exact rather than analogical: the imperfection amplifies rather than initiates, the amplification is unbounded at the ratio of one, and the structure never reaches the critical value because it fails on its way there.

The loop, and the number that closes it #

Let the roof bay have a stiffness such that a uniform load $w$ produces a sag $\delta = w/k$. Water standing to a depth $\delta$ over the bay weighs $\gamma \delta$ per unit area, where $\gamma$ is the unit weight of water. So the sag caused by the ponded water is

$$\delta = \frac{\gamma \delta}{k}$$

which has the solution $\delta = 0$ unless $\gamma = k$, at which point it has infinitely many. That is a bifurcation, and it is Euler’s problem written with different letters: a system that is indifferent at one particular value of a parameter and stable below it.

Now add rain that arrives independently of the sag — the depth $\delta_0$ that would stand there anyway, from a slight camber, a construction tolerance, or the depth at the drain. The sag becomes

$$\delta = \delta_0 + \frac{\gamma \delta}{k} \quad\Longrightarrow\quad \delta = \frac{\delta_0}{1 - \gamma/k}$$

which is the amplification factor exactly, with $\gamma/k$ in place of $P/P_{cr}$. Everything that is true of the buckling column is true here, term for term: the deflection is amplified rather than initiated, the amplification is unbounded as the ratio approaches one, and a small initial imperfection is what turns a bifurcation into a response.

What makes a roof stable #

The condition for stability is $\gamma < k$: the stiffness of the bay, expressed as load per unit area per unit deflection, must exceed the unit weight of water.

Writing $k$ out for a simply supported bay of span $L$ under uniform load, where $\delta = 5wL^4/384EI$, gives $k = 384EI/5L^4$, so the criterion is

$$\frac{384 EI}{5 L^4} > \gamma \quad\Longrightarrow\quad \frac{\gamma L^4}{EI} < 76.8$$

The $L^4$ is the whole story. Ponding sensitivity goes as the fourth power of the span, which is the fastest-growing quantity in the subject: a bay 20% longer is twice as susceptible, and one twice as long is sixteen times. A roof that is perfectly safe at 6 metres is a genuine hazard at 12 with the same section.

Which free body produced the number #

The free body is one bay of roof, one metre square, cut out of the deck with the water on it.

Down on it: the water it carries, of depth $\delta$ and weight $\gamma\delta$ per unit area, taking $\gamma = 9.81$ kN/m³. Up on it: the elastic restoring force of the deck, $k\delta$ per unit area. Equilibrium requires $k\delta = \gamma\delta + w_0$, where $w_0$ is any load not caused by the sag.

The instability is visible directly in that equation. Both the driving term and the restoring term are proportional to $\delta$, so when $\gamma$ exceeds $k$ there is no positive $\delta$ satisfying it — the equation has no solution, which is the mathematical form of “the roof fills up”.

Put numbers to it. A steel deck spanning 8 metres with $EI = 4000$ kNm² per metre width has $k = 384 \times 4000/(5 \times 8^4) = 75.0$ kN/m² per metre of deflection. Against $\gamma = 9.81$, the ratio is 0.13 and the amplification is 1.15 — safe, with the sag 15% larger than a static calculation would give. Take the same deck to 14 metres and $k$ falls to 8.0, the ratio is 1.23, and there is no equilibrium at all.

The assumption every figure here rests on is that the water is free to accumulate: that the drainage is either blocked or overwhelmed. A working drain removes the feedback entirely by fixing the depth, and every ponding failure in the record is a drainage failure first. The calculation is a statement about what happens after the drain has failed, and the reason it is worth doing is that a blocked drain is a maintenance event rather than a structural one.

The critical span, computed #

The criterion $\gamma L^4/EI < 76.8$ can be turned round to give the span at which a given deck becomes unstable, and doing so makes the sensitivity concrete.

$$L_{\text{crit}} = \left(\frac{76.8\,EI}{\gamma}\right)^{1/4}$$

For the 4000 kNm² deck above, that is $(76.8 \times 4000/9.81)^{1/4} = 13.4$ metres. Below it the roof converges on a puddle; above it, it does not. The fourth root is what makes this awkward to reason about intuitively: quadrupling the deck’s stiffness moves the critical span by only 41%, so stiffening a roof out of trouble is expensive in a way that stiffening it out of a deflection limit is not.

The same fourth root has a reassuring reading. A roof comfortably inside the criterion is comfortably inside it — the ratio is not sensitive to small changes in $EI$, so ordinary variation in material properties or in fixity does not move a safe roof into danger. What moves it is a change in span, and spans are not variables that drift.

The other direction is the one that matters for a roof somebody is standing on, because it is the case that gives no warning at all.

Which inverts what an inspection is worth. A visibly dished roof is a roof whose imperfection is large, and a large imperfection is a large reading rather than a large danger — the three curves above all become unstable at the same ratio. The frightening bay is the flat-looking one, because the same physics is running in it with the instrument switched off.

Reading the critical depth off a roof nowhere near it #

That last observation looks like a dead end — the dangerous case is the one that shows nothing — and it is not, because the small readings contain the answer. The amplification $\delta = \delta_0/(1-\lambda)$ rearranges into a straight line, and fitting that line finds the critical value from measurements taken a long way below it.

Divide the growth in sag by the ratio that caused it and plot the result against the growth. The relation is linear, its slope is the reciprocal of the critical value, and its intercept is the imperfection the bay started with. Neither quantity had to be approached to be measured.

Seventy-five per cent is comfortable and it is still further than anybody would want a real roof taken. The test worth doing is the one at a depth that could be reached with a hose.

The practical version is a hose, a level and an afternoon: flood a bay to a measured depth, record the sag, repeat at three or four depths, and fit. What comes back is the bay’s own critical depth, including the continuity, the partial fixity, the composite action with the finishes and every other stiffness nobody could have estimated — and including the softening of a deck that has already crept. It is the same reasoning a measured stiffness beats an assumed one rests on, applied to a quantity that would otherwise only be known by reaching it.

Why this is a collapse and not a nuisance #

Three features make ponding the deflection problem that is genuinely dangerous, and each of them separates it from the rest of serviceability.

It has no equilibrium to settle into. An overloaded floor deflects further and stops. A ponding roof past its critical stiffness does not stop, because every increment of deflection recruits its own load.

The load is not in the load schedule. The design rainfall gives a depth on a flat roof; the ponded depth is that depth amplified by a factor nobody looked up, and the amplification is invisible in every drawing and every schedule.

It is a strength failure caused by a stiffness deficiency. The roof does not fail because it was not strong enough for the design load; it fails because it was not stiff enough to stop collecting one. That is the same inversion stiffness is not strength is about, taken to its limit — here the stiffness deficiency does not merely make the structure unserviceable, it manufactures the load that breaks it.

It attacks the weakest bay first. Roofs are not uniform. The bay with the longest span, the shallowest section, or the most creep-affected concrete is the one with the lowest $k$, and it is the one where water collects — so the geometry actively concentrates the hazard where the resistance is least. Nothing else in the subject does that.

The size of the effect is worth one line of arithmetic, because “amplified” sounds like a correction and is not. A roof sitting at an ordinary serviceability limit has sagged by about a three-hundredth of its span, which on a 12 metre bay is 40 mm. Forty millimetres of water lying over the middle of that bay weighs $0.040 \times 9.81 = 0.39$ kN/m². Against a roof designed for perhaps 0.6 kN/m² of imposed load, the deflection has manufactured two thirds of the design load, before any amplification and before anything went wrong.

Three stiffnesses in series #

A real roof is not one member, and the single-bay criterion flatters it badly.

Water at a point sags the deck spanning between secondary beams; it also sags the secondary beam carrying that deck; and it sags the primary beam carrying the secondary. All three deflections add at the point where the water is, so the stiffness the water feels is the three in series:

$$\frac{1}{k_{\text{total}}} = \frac{1}{k_{\text{deck}}} + \frac{1}{k_{\text{sec}}} + \frac{1}{k_{\text{pri}}}$$

Series stiffness is dominated by the softest member, exactly as a bracing chain is, so a stiff deck on a flexible primary is a flexible roof. Three members each individually satisfying the criterion by a comfortable margin can combine to fail it — three components each at three times the required stiffness give a total of exactly the required stiffness, with no margin at all.

This is why ponding checks in practice are framed as a criterion on the system rather than on any member, and why the check is easy to omit: no single element’s design calculation contains it, and the person checking the deck and the person checking the primary beam may not be the same person.

The two defences that do not need anybody to do anything #

The calculation above is a statement about what happens once the drain has stopped working, and every ponding collapse in the record began there. So the useful design question is not how stiff to make the roof but what protects it when the maintenance has not happened, and there are two answers that require nothing of anybody after the building is finished.

A fall, which has to survive the deflection. A roof laid to a slope drains by gravity whatever the outlet is doing, provided the water has somewhere to go. The catch is that the fall is specified on the drawing and the roof is not on the drawing after it deflects. A nominal fall of 1:80 across a 12 metre bay is 150 mm of drop; a sag of $L/300$ at the middle of that bay is 40 mm. The fall survives. Halve the fall to 1:160 and it is 75 mm against a sag that grows as the fourth power of the span — and somewhere in between the deflected roof has a low point in the middle of a bay that the drawing shows sloping continuously.

Which is the reason minimum falls are specified at roughly twice what drainage alone would need. The extra is not for the water; it is for the deflection and the construction tolerance, and the criterion is that the deflected surface still falls everywhere, which is a structural check on a geometric quantity nobody thinks of as structural.

An overflow, which caps the depth. A weir cut through the parapet at a known level cannot block in the way an outlet can, because it has no pipe. What it does structurally is convert an unbounded problem into a bounded one: the depth of water on the roof can never exceed the overflow’s height, so $\delta_0$ has a ceiling, the amplification $1/(1-\gamma/k)$ still applies to it, and the product is a finite load that can be designed for.

That reframes the whole subject into a load case that could be written down. The roof shall carry water to the level of the overflow, amplified for ponding, with the primary drainage assumed blocked. It is not in any load schedule, it is not in any of the drawings the structural engineer produces, and the height of the overflow — the one number in it — is usually set by somebody choosing where a slot looks acceptable in a parapet.

Between the two, the overflow is the more reliable defence and the fall is the more forgiving one. A roof with both is safe against the mechanism this essay describes without depending on a single act of maintenance, which is the only kind of safety worth having against a failure whose trigger is a leaf.

Where the model stops #

Simply supported bays. The stiffness used is that of a simply supported member. Continuity over a support stiffens the bay substantially — the same load produces a fifth of the deflection for a fully fixed member — so a continuous roof deck is considerably more resistant than this calculation says, and an end bay considerably less so than the interior ones beside it.

One bay, one direction. A real roof is a grid: a deck spanning onto secondary beams spanning onto primaries, and all three deflect. The total sag at a point is the sum of three contributions, so the effective stiffness is the three in series and the criterion is considerably more severe than any single member’s. The classic treatment handles the two-way case with a pair of coupled stiffness parameters.

Uniform water, static analysis. The water is treated as a uniform depth over the bay. It is not: it forms a lens deepest at the point of maximum sag, and the true problem is a coupled one in which the load distribution depends on the deflected shape. The uniform treatment is conservative in some geometries and not in others.

Elastic. Once the deck yields — and the first yield is not the end of a steel deck’s usefulness in any other context — its stiffness falls, which lowers $k$, which raises the ratio, which accelerates everything. Ponding failures are rapid at the end for this reason.

The water is assumed to be water. Snow behaves similarly and worse in one respect: it does not flow to the low point, so a drifted load is placed by wind rather than by the roof’s own shape, and the feedback is weaker. But snow that thaws and refreezes at a blocked outlet produces a dam, and the depth behind it is set by the dam rather than by the roof — which converts an unstable feedback into a large static load, and is a different calculation with a similar outcome.

Camber is assumed helpful. A cambered beam has less initial water and is better off — but camber is lost to creep in concrete and to relaxation in prestressed members, and a roof that was safe when built can become unsafe without anything happening to it. This is the one failure mode in this collection that arrives purely with time.

The figures cannot show the thing that makes ponding dangerous, which is that the curve has no end. Every plot here stops at some deflection because the canvas stops, and the impression it leaves is of a quantity growing large. The actual behaviour past the critical ratio is that there is no curve: no depth is an equilibrium, and what the graph should show is an absence rather than a steep line. No drawing conveys the difference between “very large” and “no solution exists”.

The generalisation #

This is the third appearance in this collection of the same feedback loop, and putting the three side by side is the point of writing about the last one.

A column deflects, the axial load acts through the deflection, the moment deflects it further. A frame sways, the gravity load acts through the sway, the moment sways it further. A roof sags, the water fills the sag, the weight sags it further. In all three the loop gain is a ratio of a destabilising effect to a restoring stiffness, and the response is $1/(1 - \text{gain})$.

What distinguishes them is where the destabilising effect comes from. In the column it is an external load already present, amplified by the imperfection that was always there. In the frame it is gravity, which is unavoidable. In the roof it is a load that the deflection itself creates — the water was not there before the roof sagged, and the structure has manufactured its own action. That is the strongest form of the phenomenon and it is worth having a name for: a structure that generates load in proportion to its own response has a stability problem regardless of how strong it is.

Other members of the family, once the shape is recognisable: snow drifting into the sag of a roof; a silo whose wall bulges and admits more material; a retaining wall that rotates and mobilises more active pressure behind it; a soft-storey building whose drift concentrates in the storey that is already drifting, which is the sway amplification with a weak link in it. In every case the diagnostic question is the same — does the response create the action? — and if the answer is yes, no amount of strength closes the loop, because strength is not in the denominator.

The mathematics of ponding was worked out by Marino in 1966 and by Chinn shortly before, prompted by a run of flat-roof collapses in the United States in the early 1960s that had been attributed to snow and turned out mostly to be water. The failures were not in unusual buildings; they were in ordinary long-span roofs whose drains had blocked, and the analysis that explained them had been available in another field, on another structure, for two hundred years.

The ladder from here #

Later rungs on this anchor: the two-way ponding problem and its coupled stiffness parameters. Snow drift as a load created by geometry. Camber, creep and the loss of both. The serviceability limit as a set of distinct requirements — appearance, function, damage to finishes, vibration — which the single number $L/250$ compresses. Vibration and the frequency limits that govern long-span floors, where the criterion is not a deflection at all. And the drainage design itself, which is the actual defence and is usually somebody else’s drawing.