Structural Engineering & Statics
where the load goes
Series

Corner moment — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The moment does not stop at the end of the beam. A portal frame of 8 m by 4 m with fixed bases, carrying 20 kN/m on the beam. The bending moment is drawn on the tension side of every member, and it runs round the corner without a break: 65.2 kNm arrives at the end of the beam and 65.2 kNm leaves down the column, which is the same number, since joint rotational equilibrium is one of the equations the frame solve satisfied. Midspan carries 94.8 kNm, and the two add to 160.0 — the 160.0 kNm of a simply supported span, to 0.0e+0 kNm. The corner takes 61% of the wL²/12 a fully built-in beam would have carried, because the columns are springs rather than walls: the beam-to-column stiffness ratio is 1.27. The beam's moment crosses zero 0.92 m from the corner and the column's 1.33 m above its base.

    The moment that goes round the corner

    At a rigid knee the bending moment does not stop at the end of the beam. It turns and runs down the column, and in the same instant the beam's shear becomes the column's axial force — while the block of steel that has to carry the turn appears on no member diagram anywhere.

    part 1 · internal-forces
  2. The interior panel is designed by the case that designs nothing else. The shear across the column-web panel at each kind of joint of a frame of two 8 m bays and two 4 m storeys with fixed bases, 457 mm beams and 254 mm columns with 8.6 mm webs, under five load cases, the worse direction of sway taken where there is one. Full gravity: roof corner 106, roof interior 0, floor exterior 143, floor interior 0 kN; One bay loaded: roof corner 115, roof interior 117, floor exterior 150, floor interior 154 kN; Sway: roof corner 25, roof interior 40, floor exterior 53, floor interior 89 kN; Gravity + sway: roof corner 131, roof interior 40, floor exterior 196, floor interior 89 kN; Pattern + sway: roof corner 141, roof interior 157, floor exterior 203, floor interior 242 kN. The floor interior panel is empty under full gravity and carries 89 kN under gravity with sway, the case that governs the members; under one bay loaded with sway it carries 242 kN — 2.7 times as much, more than any exterior panel's 203 kN, and 70 per cent of the web's shear resistance.

    The panel that gravity leaves empty

    A corner joint's panel carries the difference between a beam's flange couple and half a column's shear, and full gravity load is what makes that large. An interior joint has a beam on each side, and under full gravity their two couples cancel: the panel between them carries nothing at all, in the load case that sizes every member around it. It fills under the two loads that are not symmetric. A pattern load hands it one beam's moment, and sway hands it two beams' moments turning the same way. Together they put more shear through it than through any exterior joint, in a combination that governs no member.

    part 2 · internal-forces

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