Structural Engineering & Statics
where the load goes
Concept

Plastic redistribution — where it appears

The transfer of moment from a yielded region to one still elastic, which continues until enough hinges have formed to make a mechanism. It is what allows a plastic analysis to reach a collapse load above first yield, and it requires rotation capacity the section has to be detailed for.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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.

connections · Bolt group 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 Three fasteners, three completely different clocks. What each of three shear fasteners carries against how far the joint has moved. A 400 kN fillet weld is linear to 0.4 mm and then gone — it is stiff and it is not ductile. A 380 kN bolt in a hole 2 mm larger than itself carries nothing until the hole closes and then rises over several millimetres of hole elongation. A 250 kN preloaded bolt is at its slip resistance in under a tenth of a millimetre and holds it until it slips. Two of these in one joint are at the same displacement, so the one that gets there first carries the load — and the weld gets there 23 times sooner than the bearing bolt does.

Two fasteners that never arrive together

Every steel code forbids adding a weld's capacity to a bolt's in one shear joint, and states it as a rule rather than deriving it. It is derivable. Two fasteners in parallel are at the same displacement rather than the same force, and a weld has ruptured at four tenths of a millimetre while a bolt in a standard hole has not yet touched the side of it.

connections · Weld group Two collapses for the same plate, and only one of them fits. The same three bolt rows at 90 mm pitch, folding two ways. On the left each row folds on its own pattern — 249 mm of hinge line round each bolt — and the patterns overlap, because 249 mm of fold cannot fit in a 90 mm pitch. On the right the plate does what it can actually do: the hinges run straight from one row to the next, and the whole group folds on 429 mm rather than 746. The arithmetic follows the drawing — 377 kN for the group against 657 for the rows added up.

The rows have to share one fold

Each bolt row of an end plate is checked on its own, and the answers are added up. Two rows ninety millimetres apart cannot each fold the plate on a pattern two hundred and fifty millimetres long, because there is only one plate — so the group folds on one shorter pattern, and the sum was never available.

connections · Prying 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 A plate pulled off an unstiffened flange. Looking down the column at a 250 × 20 mm plate on a 300 × 19 mm flange with a 11 mm web and 27 mm root radii, the plate pulled away from the flange. Shaded on the plate's welded edge: the force per millimetre of weld while everything is elastic, 47% of it entering over the web and its root radii and peaking at 2.67 times the uniform value at the toe of the radius. Dotted: the uniform force P/bp a rigid flange would give. Dashed: the same force spread over the effective width EN 1993-1-8 gives the joint, 191 mm — a plastic width, 2.8 times the 69 mm the elastic share fits at the web's average.

A quarter of the weld takes half the pull

A beam flange welded straight on to an unstiffened column flange is checked with an effective width of 191 mm out of 250. While the joint is elastic the plate's force does not use anything like that width: 47 per cent of it goes through the 65 mm over the web, and the flange has already yielded at 30 per cent of the force the rule allows. The rule's width is a plastic width, and the full-strength weld it insists on is what lets the joint get there.

connections · Weld group

Named alongside it

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

Bolt groupDuctilityEccentricityFillet weldInstantaneous centreWeld groupConnectionEffective widthElastic methodFree bodyPolar second momentTorsion

All concepts