Structural Engineering & Statics
where the load goes
Series

Camber — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four camber rules, and what each leaves on the finished beam. The same 12 m composite beam, cambered against four different things, followed through its own load history. Positive is a sag and negative a hog, and the point at the left of each line is the shape it was fabricated to. Cambering against the wet concrete leaves 12.7 mm of sag at the end and a flat beam on the day the slab is poured; cambering against the total load leaves the beam dead flat when fully loaded and hogged 37.9 mm — one part in 316 of the span — before anything is on it at all.

    Built to the wrong shape on purpose

    A cambered beam is fabricated curved upward so that load bends it down to something like straight. Nothing in the analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads count.

    part 1 · deflection
  2. What a millimetre on each member does at mid-span. A Pratt truss of eight panels, 24.0 m long and 2.4 m deep, carrying 60 kN of dead load at each top node, drawn with its depth exaggerated. Each member is drawn as thick as, and labelled with, the distance mid-span moves for a millimetre's change in that member's length: its force under a unit load at mid-span. The middle top-chord members move it 2.50 mm per mm — the span over four depths — and the chord members fall off towards the supports: 1.25, 1.88, 2.50, 2.50, 1.88, 1.25 along the top chord. The diagonals move it 0.80 per mm and most verticals 0.50; three verticals — the one at mid-span and the two beside the supports — carry nothing under a load at mid-span and move it not at all (dashed). A truss's shape is its members' lengths, weighted like this.

    The camber that lives in the member lengths

    A beam is cambered by bending it; a truss cannot be, because it has no curvature to bend — its shape is nothing but the lengths of its members. So a truss's camber is a cutting list, and each millimetre on that list arrives at mid-span multiplied by the member's force under a unit load there: two and a half for the middle chords of a truss ten times as long as it is deep, nothing at all for some verticals. Build only the chords to their dead-load strain and a quarter of the sag stays, the webs' share. Cut every member to a millimetre of scatter and the mid-span misses by six, an eighth of the camber.

    part 2 · deflection
  3. Fit the counters last, and the tolerance locks in nothing. The largest force random length errors lock into an eight-panel Pratt truss with a counter-diagonal in each of its six interior panels, 24.0 m long and 3.0 m deep, carrying 60 kN of dead load at each interior bottom joint, against the errors' standard deviation: the median draw (solid) and the 95th percentile (dashed) with every member fitted in the shop, and the same with the counters left loose until the truss carries its dead load and then cut to the gap (on the axis). With everything fitted, 174 and 288 kN at 1 mm, 348 and 577 at 2 — straight lines, because the force is proportional to the misfit. With the counters fitted last, nothing at any tolerance: the Pratt truss without its counters is determinate, and a determinate truss takes any length error by moving.

    The counter fitted after the load

    A truss with a counter-diagonal in every panel cannot take a change of length for free, so a camber cut into its members might seem to lock force into it. One cutting list locks in nothing — every member shortened by its own dead-load stretch — and the simpler lists lock in little. What locks in force is the shop: a millimetre of error per member puts 174 kN into some member of a truss whose largest dead-load force is 471. Leave the counters loose until the dead load is on and cut them to the gap, and the same errors lock in nothing at all.

    part 3 · deflection

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