Structural Engineering & Statics
where the load goes
Series

Serviceability — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Which limit arrives first. Utilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry.

    Stiffness is not strength, and usually it is the one that governs

    A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all.

    part 1 · deflection
  2. The water that will not run off. Water depth against sag at the middle of the bay, for a roof bay starting with a reversed construction camber of 0.01. There is no critical value to reach: the deflection grows from the first increment, slowly at first and then without bound as the ratio approaches 1.00 — which is the depth at which the bay fills faster than it stiffens, and which the roof bay therefore never attains. The perfect roof bay, drawn for comparison, sits on the vertical axis until it arrives there and then has no answer at all.

    The water that will not run off

    A flat roof deflects, the deflection makes room for water, the water deepens the deflection. It is the same equation as a buckling column, with rain in place of the axial load.

    part 2 · deflection
  3. Two members each half-way to ponding are unstable together. The girder's ponding parameter Cp against the joist's Cs. Either member alone is stable up to 1; together the roof is stable only below the solid curve, (1 − Cp)(1 − Cs) = (8/π²)·Cp·Cs, which meets the diagonal at 0.526 — two members each about half-way to ponding on their own are critical together. The dashed line is the simplified rule Cp + 0.9Cs ≤ 0.25. The marked roof is a 12 by 9 m bay whose girders and joists are each sized to deflect L/240 under 1.5 kN/m²: Cp = 0.26 and Cs = 0.19, stable, with its joists' sag amplified 1.79 times where the joist alone would say 1.24 and the girder alone 1.35.

    The roof that ponds on its girders

    A flat roof's joists sit on girders, and the water fills the sum of both their sags. Two members each half-way to ponding on their own are unstable together, and long before that the growth of the water is far larger than either member's own amplification says. AISC's simplified rule turns out to be a line of constant growth, and a member sized to a deflection limit has a ponding parameter proportional to its span — so a bay sized to L/240 under 1.5 kN/m² meets the rule up to 6 m, doubles its water at 12 and ponds at 24.5.

    part 3 · deflection
  4. Where the outlet sits decides whether the roof ponds at all. The load the water puts in the more heavily loaded member of a square bay, over what the same head of water would put in it on a rigid roof, against the bay's side, with girders and joists each sized to deflect L/240 under 1.5 kN/m². With the level fixed by a scupper it is 2.01 at 12 m and 3.92 at 18 m, and the bay ponds at 24.5 m. With a drain on the girder line it is 1.24 and 1.41; with a drain at the bay's middle, 0.82 and 0.75, falling as the bay grows, because the water's surface goes down with the roof.

    The roof that drains towards its sag

    The depth of water a ponding check takes as given is set by the drainage: an overflow's height above the roof, plus the head a storm needs to leave through it. Which head a roof can carry depends less on how stiff it is than on where the overflow sits. With the level fixed by a scupper in the parapet, a 12 by 9 m bay allows 85 mm of water and an 18 m square bay 39. With a drain at the middle of the bay, whose inlet goes down as the roof sags, the same bays allow 182 and 205 mm — more than a rigid roof would — because the water's surface follows the sag down instead of filling it.

    part 4 · deflection
  5. How much of the drain has to work. The load in the more heavily loaded member, with the roof's own 0.5 kN/m², over its 2.0 kN/m² capacity, for a 12 by 9 m bay in a 150 mm/h storm, with a drain at its middle that passes the storm at 30 mm when clear and a 400 mm scupper at its edge whose sill is 50 mm above the supports, against the share of a clear drain's flow the drain still passes. Members at L/180, L/240, L/360 and rigid: with the drain fully blocked, 1.25, 0.99, 0.84, 0.66; a tenth open, 0.80, 0.77, 0.71, 0.62; half open, 0.44, 0.45, 0.46, 0.48; clear, 0.37, 0.37, 0.38, 0.40. A softer roof is the worse with the drain blocked and very slightly the better once about a fifth of it works.

    The drain that is mostly blocked

    A roof's ponding check is made with its primary drains blocked, and a drain is seldom blocked outright: leaves and gravel choke it to a fraction of its flow. A fraction turns out to be nearly enough. On a 12 by 9 m bay sized to L/240, a drain passing a seventh of its clear flow still takes the whole of a 150 mm/h storm, because the water standing over it at the bottom of the sag is the bay's own deflection plus the level. A softer roof's drain works harder still. Only near complete blockage does the scupper take over, and only there does stiffness help.

    part 5 · deflection

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