Structural Engineering & Statics
where the load goes
Series

Bolt group — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A bolt group under an eccentric load. A 3 by 2 bolt group carrying 100 kN at 150 mm from its centroid, with the resultant force on each bolt drawn to scale, by the elastic vector method. The load is shared equally and the torque is not, so the worst bolt carries 50.37 kN against 16.67 kN of direct shear alone — 3.02 times as much.

    The bolt that carries more than its share

    Six bolts, one hundred kilonewtons, and a worst bolt carrying fifty. The load is shared equally and the torque is not, and the second one is invisible on any drawing where the connection is a point.

    part 1 · connections
  2. One bracket, three neutral axes, three sets of bolt forces. A bracket 300.0 mm deep with three rows of two bolts at 75.0 mm pitch, carrying 15.0 kN·m about an axis in the plane of the bolts. Taking the axis at the group's centroid puts the outer rows at 50.0 kN of tension and 50.0 of compression, with the middle row idle. Taking it at the plate's compression edge puts every row in tension — 7.1, 14.3, 21.4 kN from the bottom up — with the top row at 21.4. Solving for it instead, with the plate bearing over 200.0 mm of width and the bolts as areas, puts it 40.2 mm above the edge and the top row at 24.6 kN. All three make the applied moment exactly. The top row differs between them by a factor of 2.33.

    The bolt group has no neutral axis

    A bolt group carrying a moment about an axis in its own plane has some bolts in tension and something in compression somewhere else. Where that somewhere else is decides the answer, nothing in the group decides it, and the two defensible choices give the worst bolt 50 kilonewtons and 21.

    part 2 · connections
  3. Capacity against load direction, layout by layout. The same capacities as a function of direction on straight axes, for a load through (150, 150) mm. Three rows of two is weakest at 138°, 135.8 kN. Two rows of three is weakest at 312°, 135.8 kN. A ring of six, r = 70.4 mm is weakest at 132°, 150.3 kN. The curves cross: no layout is strongest in every direction, and the one to choose is the one whose lowest point is highest.

    The bracket pushed from the wrong side

    A six-bolt bracket checked for a load straight down carries 198.5 kN. Push the same load through the same point at 138 degrees and it carries 135.8 — the direction changes the torque as well as the shear, and the worst direction is one no drawing shows. The capacity of a group whose load can turn is a closed curve, the bolt that governs it changes as the load turns, and what the curve rewards is not a larger polar moment but a smaller distance to the furthest bolt.

    part 3 · connections
  4. The two methods, direction by direction. Capacity against the direction of the load for three rows of two bolts of 100 kN each, loaded through (150, 150) mm. The elastic method bottoms out at 135.8 kN at 138°; the instantaneous centre at 168.3 kN at 136°, 23.9 per cent higher. Both spike to 600.0 kN where the load aims at the centroid and every bolt carries an equal share. The gap between the curves is not constant: it is widest where the elastic distribution is most uneven, which is the same place the group is weakest.

    The better method flatters the worse layout

    The instantaneous centre method finds capacity the elastic method cannot see, and how much it finds depends on the layout it is applied to. On a six-bolt rectangle under a load of unknown direction it is worth 24 per cent; on a ring of six with a smaller polar moment it is worth 14. The two methods agree on which layout to choose and disagree by a factor of five about the margin — and the reason is that a layout the elastic method already likes is one with nothing left to redistribute.

    part 4 · connections
  5. What each missing bolt costs. The weakest-direction capacity of six bolts at 75 by 75 mm, loaded through a point (150, 150) mm from the centroid, with the whole group and with each bolt in turn left out. The full group carries 135.8 kN. Leaving out bolt 3 leaves 90.2 kN, a loss of 34 per cent; leaving out bolt 2 leaves 131.1 kN, a loss of 4. One sixth of the bolts is not one sixth of the capacity, and which sixth it is matters by a factor of 10. The dashed line is the capacity a group that lost a proportional share would have, 113.2 kN.

    The bolt that was never fitted

    A six-bolt bracket found with five bolts in it has lost a sixth of its fasteners and between four and thirty-four per cent of its capacity, depending which one is missing. The share is the smallest of the three things that changed: the centroid moves away from the gap, which lengthens the load's own lever arm, and the polar moment falls by more than the count does. The bolt whose absence costs most is not the bolt that governed the check.

    part 5 · connections
  6. Four ways to put six bolts in one plate. Six bolts inside a 150 × 150 mm field of bolt centres, no two closer than 60 mm, loaded through a point (200, 0) mm from the field's centre. The two-column layout carries 193.5 kN complete and 138.8 kN with its worst bolt missing. The ring carries 167.2 kN complete and 117.8 kN with its worst bolt missing. The strongest found carries 234.3 kN complete and 149.7 kN with its worst bolt missing. The most robust found carries 231.6 kN complete and 174.7 kN with its worst bolt missing. The layout found by maximising the complete capacity and the layout found by maximising the worst omission are different layouts, 1 per cent apart when complete and 17 per cent apart with a bolt missing, and both beat the ring — the most evenly spread of the four — on both counts. The dashed line runs from each group's centroid to the load: 200 mm, 200 mm, 180 mm, 188 mm. The ringed bolt is the one each group can least afford to lose.

    The strongest layout leans on one bolt

    Search a plate for the six bolt positions that carry most and the answer carries 234 kN — and loses 36 per cent of it if one particular bolt is missing. Move that one bolt fifty millimetres, into the corner the optimum had just left, and the group carries 232 kN and loses 25 per cent whichever bolt goes. Robustness here costs one per cent of strength, and a search for strength alone will never find it.

    part 6 · connections
  7. The bolt that governs is worst at only one thing. The plate of a bracket of six M20 grade 8.8 bolts in three rows of two at 75 mm, loaded 150 mm to the side of the group and 150 mm in front of the column face, with the load at 60° from straight down, drawn at the group's capacity on the elliptical interaction, 227 kN. Each bolt's arrow is its shear in the plane of the plate and its disc grows with its tension, which comes from the load's distance in front of the column: bottom left 9 kN and 67 kN; middle left 33 kN and 76 kN; top left 74 kN and 84 kN; bottom right 40 kN and 35 kN; middle right 51 kN and 43 kN; top right 84 kN and 51 kN. The most shear is on the top right bolt and the most tension on the top left; the one that reaches its interaction first is the top left, and the next, the top right, is at 0.97 of its own.

    The bracket with no worst bolt

    A bracket that stands out from its column is pulled off the column by the same load that twists it in its plane, so every bolt carries shear and tension at once. Straight down, one corner bolt is worst at both and the check is that bolt's. Tilt the load and the most sheared bolt and the most pulled bolt come apart; the one that fails is whichever of them its interaction reaches first, and at the angle where they swap the group has two worst bolts and no single one.

    part 7 · connections

All series