Structural Engineering & Statics
where the load goes
Concept

Elastic method — where it appears

The analysis of an eccentrically loaded bolt group that treats each bolt as a linear spring about the group's centroid. Each bolt takes an equal share of the load plus a share of the torque in proportion to its distance, which is conservative against the instantaneous centre method.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Throat stress round a fillet weld group. A l shape weld group carrying 150 kN at 200 mm from its centroid. The peak throat stress is 2.17 kN per mm of throat, at (139.53, 100); the worst point at maximum radius from the centroid carries 1.99. Checking by radius is wrong here by 9.17%, and points at identical radius differ by a factor of 1.

The radius rule, and where it fails

A weld group under an eccentric load is checked at the point furthest from its centroid, on the reasoning that the stress from the twist grows with the radius. That reasoning ignores the direction the two stresses point in, and for one common shape it misses the peak by nine per cent.

connections · Weld group 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.

connections · Bolt group 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.

connections · Bolt group 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.

connections · Bolt group 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.

connections · Bolt group 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.

connections · Bolt group The worst direction is the long axis of an ellipse. For 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: the demand on the governing bolt, top right, per kilonewton of load, drawn as a radius in the direction the load acts — straight down at the bottom — under the elliptical interaction (solid), which in each quadrant of directions is an arc of one ellipse, and under the straight line (dashed), which is not. The ellipse's longest radius, at 16.0°, is its eigenvector: 0.0058 of the bolt's interaction used for every kilonewton, a capacity of 172.5 kN, found without trying a single direction.

The worst direction is an eigenvector

A bracket whose load can come from any direction in the plane of its bolts has a capacity that is a closed curve, and once the load also pulls the bracket off its column the curve becomes a surface over direction and distance. Finding its lowest point looks like a search. Under the elliptical interaction of shear and tension it is not: inside each quarter-turn of directions every bolt's demand squared is a quadratic form in the load's direction, so the worst direction is that form's eigenvector, and the bracket's least capacity is a 2 × 2 eigenvalue per bolt. Under the straight line EN 1993-1-8 prints it is a search, because a straight line has no curvature to make an ellipse of — and the hand check that sets the worst shear anywhere against the worst tension anywhere is a tenth too low in the plane and a fifth too low once the bracket is pulled.

connections · Bolt group

Named alongside it

The objects these essays reach for when they reach for this one.

Bolt groupEccentricityPolar momentLoad directionBracketInstantaneous centreBolt tensionInteractionPlastic redistributionRedundancyRobustnessTorsion

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