Structural Engineering & Statics
where the load goes
Series

Single lap — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The load path has a kink in it, and the kink is a plate thick. Two 10 mm plates lapped over 60 mm and pulled with 60 kN. The two load paths are offset by the thickness of a plate, so the joint carries a moment nobody applied: P × 10.0 mm / 2. Taken at face value that gives a peak stress 4.00 times the mean. The joint rotates under load and the moment falls to 86 per cent of it, leaving 3.57 times — a saving of 11 per cent and not, on a plate this thick, a rescue. The bolt is bent as well as sheared: 382 N/mm² of bending against 191 of shear.

    The joint that is crooked by construction

    Lap two plates and fasten them and the two load paths are offset by the thickness of a plate. The joint carries a moment nobody applied, the peak stress is four times the mean, and the rotation that is supposed to straighten it out saves eleven per cent — because the rescue works for thin sheet with a long lap and a bolted structural joint is neither.

    part 1 · connections
  2. The thin plate straightens onto the load line. The centre lines of a 10 mm plate lapped on a 20 mm plate, 100 mm wide, over 60 mm, carrying 60 kN with the grips 1200 mm from the overlap, over 480 mm either side of the overlap, with every vertical distance drawn 14 times the lengthwise scale; dotted, where the centres would be if the joint could not rotate. The load pulls along the straight line between the grips, each of which holds its plate on its own centre, and the moment where a plate enters the overlap is the load times the gap between that line and the plate's centre. The thin arm, flexible under tension, has swung toward the line: its gap is 0.49 of the rigid one, e/2. The overlap has turned with it, and the thick arm, too stiff to follow, is left 1.34 of e/2 from the line. Peak stresses: 193 N/mm² in the thin plate and 121 in the thick one, against 330 and 98 if the joint could not turn.

    The thicker plate takes the bend

    A lap joint's eccentricity puts a moment into both plates, and the rule is that each takes half of the load times the offset. That is true only while the joint cannot turn. Let it turn, and a thin plate lapped on a thick one straightens under its own tension and hands its share across: the thick plate ends up bending a third more than the rule says, the thin one half as much, and in the limit the thick plate would take the whole of it.

    part 2 · connections
  3. The area rule's even stress, and the stress the cut finds. The tensile stress through the depth of the two covers where they cross the gap of a 20 mm main plate 200 mm wide, spliced with a 15 mm cover on one face and an 8 mm cover on the other, carrying 900 kN; the main plate's centre line, on which the load acts, is at zero. Shared by area every fibre would work at 196 N/mm² (dashed). The free body puts the covers' resultant on the load line instead, and the stress varies across them: 107 to 179 N/mm² in the 15 mm cover and 275 to 313 in the 8 mm one — the thin cover's outer face at 1.60 times the area rule's stress.

    The thinner cover carries more

    A plate spliced with a cover on each face is the standard cure for a lap joint's eccentricity, and the covers are usually sized as if they shared the load in proportion to their areas. A free body cut across the splice says otherwise. With nothing but the two covers crossing the cut, their forces must balance about the line the load acts on, so they share by lever arm and not by area — and the thinner cover, which sits nearer that line, carries more than half the load on less than half the steel.

    part 3 · connections
  4. Across the gap the splice plates bend as one section. The top half of the splice of a 612 mm deep I-beam with 229 by 19.6 mm flanges, spliced at 700 kN·m by a 15 mm outer cover on each flange, two 8 mm inner covers beside the web under it and two 8 mm web plates 480 mm deep, to scale, cut at the gap where only the plates cross (the beam's own flange and web dashed beyond), and the stress through it. Bent as one section, the outer cover works at 206 N/mm² at its centre and the inner covers at 186, in proportion to their distances from the neutral axis; the hand design's area rule gives both 193; the plate splice's lever rule gives the outer 123 and the inner 350.

    The splice that bends as one section

    A plate spliced with two unequal covers shares its load by lever arm about the line the load is held on, and the thinner cover carries more. A beam's flange splice looks like the same joint, and the rule that worked for the plate puts the thin inner covers at the edge of yield. They are working at half that. A beam's halves are not held on any line; they turn, and every splice plate across the gap turns with them, so the plates bend as one section and share the beam's moment by their distances from its neutral axis. The outer cover works hardest, the inner covers less, and the web plates less than the hand design hands them.

    part 4 · connections
  5. Two neutral axes in one cut. Stress against height through a girder with a 300 by 30.0 mm top flange, a 229 by 19.6 mm bottom flange and a 573 by 11.9 mm web, spliced at 700 kN·m. The girder's own neutral axis is 374 mm above its underside, and its flanges work at 133 and 201 N/mm². Its splice, each flange's covers sized to that flange's force at 250 N/mm², has its own centroid at 326 mm — 48 mm lower — and across the gap the plates bend about it: the top outer cover at 244 N/mm², the bottom outer at 269, the web plates from 179 at their top to 212 at their foot. Shaded: the covers' heights (darker) and the web plates'.

    The splice that finds its own axis

    Splice a girder whose top flange is heavier than its bottom one and the splice plates, sized flange by flange, have a centroid 48 mm below the girder's. It looks as if the splice must then carry an axial force the girder does not, or be proportioned to keep the two axes together. It carries none: a cut through the gap carries what the girder carries, and the girder carries no force. Held at both ends, the mismatch makes less than a kilonewton. What the offset does instead is move the web's own force into plates that were not sized for it.

    part 5 · connections

All series