The roof that ponds on its girders

The water that will not run off treated a flat roof as one member. Its sag makes room for water, the water deepens its sag, and the sag grows as $1/(1 - C)$, where $C$ is the weight of water a unit of sag admits over the stiffness that resists it — the same curve as a column with a bow in it, with rain in place of the axial load. At $C = 1$ the roof fills faster than it stiffens.

That essay ended by noting that a real roof is three members in series — the deck, the joists that carry it, the girders that carry them — and that three members each comfortably inside the criterion can fail it together. A roof designer is handed two of those members by two different calculations, and the questions it left are how much worse two members are than one, how the specification’s simplified check accounts for it, and what it means for a roof whose members were all sized to a deflection limit and checked no further.

Joists on girders #

The roof is a grid of bays. Each bay has joists spanning $L_s$ between two girders, at spacing $S$, and the girders span $L_p$ between columns. Water standing on the roof fills whatever shape the two together make: the girder’s sag carries every joist down with it, and each joist sags further between its two girders.

Marino’s 1966 analysis of exactly this system measures each member by its own ponding parameter — the weight of water a unit of its sag admits over its own bending stiffness:

$$C_p = \frac{\gamma\,L_s\,L_p^4}{\pi^4 E I_p}, \qquad C_s = \frac{\gamma\,S\,L_s^4}{\pi^4 E I_s}.$$

Each on its own is a one-way roof: stable while its parameter is below 1. Together they share the water, because the water above a joist is also above the girder carrying it.

The stable region is not the square that the two one-way checks draw. It is bounded by a curve that runs from 1 on each axis inward, through 0.526 on the diagonal. Two members, each about half-way to ponding on its own, are critical together.

Where the curve comes from #

Take each member’s sag as a half-sine: the girder by $a\sin(\pi x/L_p)$ and each joist, relative to its supports, by $b\sin(\pi x/L_p)\sin(\pi y/L_s)$, the factor in $x$ because a joist near the middle of the girder carries water deepened by the girder’s sag. The water fills the sum, and its weight does work through both.

Each member’s own stiffness, scaled by itself, is $1 - C$ — the one-way answer. The coupling comes from the water above both: the water a girder’s sag admits also loads the joists, and the water a joist’s sag admits also loads the girder. Scaled the same way, the cross term is $m = \sqrt{8 C_p C_s}/\pi$, and the roof’s stiffness matrix is

$$\begin{bmatrix} 1 - C_p & -m \\ -m & 1 - C_s \end{bmatrix}, \qquad D = (1 - C_p)(1 - C_s) - \frac{8}{\pi^2}\,C_p C_s.$$

The matrix is symmetric, and that is not a convenience of the algebra: the water a girder’s sag admits loads the joists by exactly as much as the water a joist’s sag admits loads the girder, which is Maxwell’s reciprocal theorem applied to a load that the structure makes for itself. The roof is stable while $D > 0$. On the diagonal $1 - C = (\sqrt 8/\pi)\,C$, so $C = 1/(1 + \sqrt 8/\pi) = 0.526$.

The essay on a single bay gave the simpler estimate of two springs in series, which puts two equal members at 0.5. The half-sines are a little kinder, because the joist’s sag is concentrated mid-bay and the girder’s spread across it, so the two shapes do not overlap fully: $8/\pi^2 = 0.81$ rather than 1. The difference is five per cent of the critical parameter, and in the right direction for the series estimate to stand as a safe shortcut.

Each member’s own answer misses most of the growth #

Stability is the end. What a roof designer meets first is growth: rain that collects to some depth on a roof that does not fail, but sags further under the water than the members’ own calculations said.

Solving the two-by-two under a uniform head of water gives the amplification of each member’s sag in closed form:

$$\text{joist: } \frac{1}{D}, \qquad \text{girder: } \frac{1 - C_s + 8C_s/\pi^2}{D} = \frac{1 - 0.19\,C_s}{D}.$$

Both share the denominator, so both grow together and both go to infinity at the same point. The joist’s is the larger: it is carried down by the girder’s sag as well as its own, so it holds the deepest water. With rigid joists, $C_s = 0$, the girder’s reduces to $1/(1 - C_p)$, the one-way answer; with rigid girders the joist’s does.

On roofs with equal parameters the coupled amplification runs away from the one-way answer early. At 0.2, a member checked alone expects its sag to grow by a quarter, and the joist’s grows by two-thirds. At 0.3 the one-way answer is 1.43 and the coupled joist 2.40. At 0.4, 1.67 and 4.34. A designer who checks each member against the one-way criterion, finds each at 0.4 and comfortably below 1, and so expects a sag two-thirds larger than the dry one, has a roof whose joists sag more than four times their dry sag.

A roof sized by its deflection limit #

The parameters can be read straight off the deflection limit a roof was designed to. A member of span $L$ at spacing $S$ sized so that a uniform load $q$ deflects it to $L/n$ has $EI = 5qSL^3n/384$, and substituting it into its ponding parameter leaves

$$C = \frac{384\,\gamma\,L}{5\pi^4\,n\,q}.$$

A deflection limit is a ponding parameter that grows with the span. Sized to $L/240$ under 1.5 kN/m², every member has $C = 0.0215$ per metre of its span. The 12 m girders of the marked roof have $C_p = 0.26$; its 9 m joists $C_s = 0.19$.

Each, checked alone, is far from trouble: amplifications of 1.35 and 1.24. Together, $D = 0.74 \times 0.81 - 0.81 \times 0.26 \times 0.19 = 0.56$, and the joists’ sag is amplified 1.79 times and the girders’ 1.73. Rain that would sag the joists 50 mm if none of it could collect in the sag sags them 90 mm once it does.

That is what stiffness is not strength means on a roof. Both members meet their deflection limit by construction; the deflection limit is a statement about one load, and ponding is the load the deflection invites.

What the growth does to a stress #

An amplification of a sag is an amplification of the load that made it, because the extra sag is filled with water and water is load. So the joists of the marked roof carry 1.79 times the moment the rain alone would put in them, and the girders 1.73 times theirs, on top of the roof’s own weight, which does not grow.

How much that matters depends on how large the rain load is beside everything else. A roof whose overflow is set 50 mm above its drain carries about 0.5 kN/m² of standing water before any of it collects in a sag; amplified 1.79 times it is 0.9 kN/m², which on a roof sized for 1.5 kN/m² is more than half of the load it was sized for, spent on water. A roof with its overflow at 100 mm has nowhere to put the amplification. The specification’s exact method is this sum done member by member: the rain stress multiplied by the coupled amplification, added to the stress from everything else, and checked against a limit.

And the stresses are largest where the water is deepest. The joist’s amplification is the larger of the two, and the joist at mid-bay, on the middle of the girder, carries the deepest water on the roof: the girder’s sag beneath its supports and its own between them. It is the joist that a two-way roof fails first, and the girder that decides when.

The size at which a bay ponds #

Make the bay square and grow it, keeping every member sized to the same limit, and the parameter of both members grows with the side.

Without ponding, the depth at mid-bay already exceeds the rain depth, because the dry sag of both members is part of the hollow; it grows linearly with the bay. With ponding counted member by member, the hollow grows faster. With the two coupled, the depth at mid-bay is 2.47 times the rain depth at 12 m and 5.25 times at 18 m, and it goes to infinity at 24.5 m — a bay size at which each member, checked alone, has a ponding parameter of 0.53 and would last to 46.5 m.

The linear growth of $C$ with span is the whole difficulty. Deflection goes as the fourth power of the span, and a member sized to a deflection limit buys back three of those four powers with its second moment; the one left over is the ponding parameter.

The simplified rule is a line of constant growth #

Codes do not ask a designer to compute the two-by-two. AISC’s specification gives a simplified rule — a roof is acceptable if $C_p + 0.9\,C_s \le 0.25$ — and an exact method for roofs that fail it. The rule is the dashed line on the first figure, a long way inside the stable region, and its numbers look arbitrary until they are traced along the line.

Along the line the amplification hardly moves: 1.33 at one end, where the joists are rigid, 1.38 at the other, where the girders are, and 1.31 to 1.36 between. The 0.25 is the growth a designer accepts without further work — a third, the one-way amplification at $C = 0.25$ — and the 0.9 is close to the slope that keeps the coupled growth constant as the sagging moves from girders to joists. The simplified rule is not a margin on instability; it is a limit on growth, and it holds that growth to a third whatever the split between the two members.

That is also why the rule is so much stricter than the stability boundary, and why a roof that fails it is not unsafe. The square bay of the figure above meets the rule only up to 6.1 m — a bay most warehouses would call small — and is stable to 24.5. Between the two lies the exact method, which computes the amplified stresses and asks whether they are acceptable: the coupled amplifications above, multiplied into each member’s stress under the rain load.

A larger bay #

The marked roof is a modest one. A larger bay, with its members sized the same way, sits further out.

With 18 m girders and 12 m joists the parameters are 0.39 and 0.26. Each member alone expects its sag to grow by a third to two-thirds; the joists’ sag grows by a factor of 2.67, and the water at mid-bay stands 3.37 times as deep as the rain, against 2.37 by adding the two members’ own answers. It is two and a half times the simplified rule’s limit and still well inside the stability boundary, and the water at its middle already stands more than three times as deep as the rain.

Stiffer limits buy span linearly #

The deflection limit enters the ponding parameter in the denominator, so a stiffer limit moves everything outward in proportion.

At $L/360$ the bay ponds at 36.7 m and meets the simplified rule to 9.2 m — exactly half as far again as at $L/240$, because the parameter is inversely proportional to $n$. An 18 m bay at $L/360$ behaves exactly as a 12 m bay at $L/240$: the curves are the same curve with the span axis stretched. The same is true of the load: a roof sized for a heavier load has stiffer members and a smaller ponding parameter, which is why a roof sized for snow is rarely the one that ponds, and a light roof in a climate with no snow — sized for a nominal maintenance load and its own weight — is.

Which member to stiffen #

A roof that grows too much can be stiffened in either member, and the two are not equally effective.

Double the girders’ second moment on the marked roof and $C_p$ halves to 0.13: the joists’ amplification falls from 1.79 to 1.47, and the depth at mid-bay from 2.13 times the rain depth to 1.69. Double the joists’ instead and $C_s$ halves to 0.10: the amplification falls to 1.54 and the depth to 1.74. Stiffen both by half as much again and the parameters fall to 0.17 and 0.13: the amplification is 1.42 and the depth 1.60.

The girder is the better buy here because its parameter is the larger, and the sensitivity of $D$ to each parameter is nearly the same — $\partial D/\partial C_p = -(1 - 0.19\,C_s)$ and $\partial D/\partial C_s = -(1 - 0.19\,C_p)$, both close to $-1$. So the rule is simple: reduce the larger parameter. That is usually the girder, because its span enters the parameter to the fourth power and its tributary width is the whole joist span, and it is usually the member nobody thought about ponding for, because the water does not stand on it.

The rule is the same one a column with a bow in it obeys: the amplification is set by how close the system is to its critical point, and the cheapest way to move it away is to stiffen whichever part is doing most of the moving.

The marked roof, by hand #

For the 12 by 9 m bay, with $\gamma = 9.81$ kN/m³ and $q = 1.5$ kN/m²,

$$C = \frac{384 \times 9.81 \times L}{5 \times 97.4 \times 240 \times 1.5} = 0.0215\,L,$$

so $C_p = 0.0215 \times 12 = 0.258$ and $C_s = 0.0215 \times 9 = 0.193$. Then

$$D = (1 - 0.258)(1 - 0.193) - 0.811 \times 0.258 \times 0.193 = 0.599 - 0.040 = 0.558,$$

and the joists’ amplification is $1/0.558 = 1.79$, the girders’ $(1 - 0.19 \times 0.193)/0.558 = 1.73$. The cross term, 0.040, is only seven per cent of the product of the two one-way terms, and it costs the roof a third more sag than the one-way answers: the denominator is a difference, and a difference near its zero is sensitive to everything subtracted from it.

What the two half-sines leave out #

The model is the simplest that couples the two members, and its choices limit it.

One shape each. The half-sines are close to the true shape of a uniformly loaded simple span, and the ponding load sharpens the shape towards mid-span; a higher-order analysis gives slightly larger amplifications near the boundary.

No deck. The deck spanning between joists is a third member in series, and on a flexible metal deck it moves the boundary further in. The specification bounds it separately, with a minimum second moment for the deck.

Simple spans. Continuous joists or girders are stiffer against a sag confined to one bay, though a uniformly ponding roof loads every bay at once and gains less from continuity than a deflection check would suggest.

No camber, no creep. A camber changes how much water arrives before the roof is flat, not how fast it grows once it is there; and on a timber or concrete roof creep lowers the stiffness under sustained water and moves every number here towards the boundary.

Supports that do not move. The columns are rigid. A roof supported on a transfer girder or a long-span truss has a third and fourth member in the series, and a support that deflects is in the ponding calculation whether or not it was in the deflection one.

Still open: drainage as the criterion #

Every number above is the roof’s answer to a depth of water it was given. The depth is set by the drains: a primary drain that is blocked and an overflow set at some height above it, so the rain head is the overflow’s height plus whatever head drives the design storm through it. That height is a drainage designer’s choice, usually on another drawing, and the ponding check takes it as data. Whether a roof’s overflow could be set from its own coupled amplification — so that a bay at 18 m with its joists amplifying 2.67 times is given a lower overflow than a bay at 12 m — is the question of whether the drainage and the structure are one design or two, and of which of them should be made to answer for the other.