Structural Engineering & Statics
where the load goes
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Noding eccentricity — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The moment a joint makes out of geometry. Bending stress in the chord divided by the axial stress already in it, against how far the diagonal's working line misses the node. A truss analysed as pin-jointed is drawn with its members as lines through their own centroids meeting at a point, because that is what makes the joint a pin with nothing but forces in it. A fabricated joint is not obliged to oblige, and the resultant then has a moment about the node — 16 kNm at the 60 mm drawn. It divides between the members meeting there in proportion to their rotational stiffnesses, 4EI/L or 3EI/L, which is the moment-distribution rule applied to an unbalanced moment that came from geometry rather than from load. The chord takes 48 per cent of it and carries 25 per cent of its axial stress again in bending. Ten per cent arrives at 28 mm, which on a chord of any depth is a detailing decision rather than a mistake.

    The joint that is not where it was drawn

    A truss is drawn as lines through the centroids of its members meeting at points. A fabricated joint is under no obligation to oblige, and when the working lines miss each other the resultant has a moment about the node that has to go somewhere.

    part 1 · connections
  2. A tubular K joint with a 12.6 mm gap. A 168.3 × 8.0 mm chord with two 114.3 × 6.3 mm braces at 45°, their toes 12.6 mm apart on the chord's crown. The braces' working lines, carried down through the chord, meet 3.0 mm below the chord's axis — an eccentricity of 0.02 chord diameters, inside the band of −93 to +42 mm in which the joint rules let its moment be neglected. Nobody chose that number. It follows from the gap, the brace diameter and the angle, and the gap was chosen so both toes could be welded. The working lines would meet on the chord's axis at a gap of 6.7 mm, which is a joint no rule permits.

    The joint whose lines may not meet

    In a welded tubular truss the noding eccentricity is not a fabrication error. It is fixed by the brace diameter, the brace angle and the gap between the toes, and the gap is chosen for the welder. For a third of ordinary proportions the one gap at which the working lines would meet lies in a band no fabrication rule permits — so the concurrent joint the truss was analysed with is the one joint that cannot be built.

    part 2 · connections
  3. The room left between two planes of braces. The surface of a 168.3 mm chord unrolled flat — along the chord across, round it up — with the footprints where 114.3 mm braces at 45 degrees are welded to it: two from a K joint in one plane (solid), two from a K joint in a plane 90 degrees round the chord (dashed), both nodes at the same section, each with a 12.6 mm gap between its own toes. The narrowest gap between a brace of one plane and a brace of the other is 6.5 mm, against the 12.6 mm two welds need: too little to weld. Each footprint spans 85.6 degrees of the chord's circumference.

    Two planes of braces and one circumference

    A tower leg or a triangular girder's chord receives K joints from two planes at once, and each can be checked as a planar joint: its gap, its eccentricity, its band. What the planar checks cannot see is that the two planes' braces are welded to the same tube. A 114 mm brace takes up 86 degrees of a 168 mm chord's circumference whatever its gap, so a square tower's two planes leave 6.5 mm between them where the welds need 12.6, and a triangular girder's braces land on each other. And two eccentricities at the edge of their band give the chord √2 or √3 times the moment the band was written for.

    part 3 · connections

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