Structural Engineering & Statics
where the load goes
Concept

Plate bending — where it appears

Bending of a plate out of its own plane, resisted by a stiffness proportional to the cube of the thickness. It is a very weak action compared with anything in-plane, which is why a thin web bends away from a brace and why a base plate can only reach so far from its column.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

An eccentric load is three load cases, and only two of them are checked. A line load of 40 N/mm at 1.5 m from the axis of a 3.0 by 2.0 m box, replaced by the three cases it is equivalent to. Bending is the load on the axis. The torque 60 kNm per metre then splits into a set of edge forces that drives Bredt's shear flow and distorts nothing, and a set with the flange forces reversed — 10.0 kN/m up one web and down the other, 15.0 kN/m across the flanges — which carries no torque at all and squashes the rectangle into the rhombus drawn behind it. Its generalised load is exactly half the torque, so a box girder spends half of an eccentric load's torsion on changing its own shape, and no torsion calculation contains that half.

The section that will not keep its shape

A box girder is closed, so torsion costs it almost nothing. What an eccentric load actually does to it is something a torsion calculation contains no term for — the rectangle becomes a parallelogram, in its own plane, along the whole length of the span.

sections · Box distortion The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything.

The plate that is only as big as it is thick

A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all.

connections · Base plate The compression is under the flange, not at the edge of the plate. A 500 × 400 mm base plate, 20 mm thick, under a 260 mm column carrying 300 kN and 120 kN·m, with holding-down bolts 60 mm from the tension edge. A plate this thick can deliver bearing only 43 mm past the compressed flange, so the compression sits under that flange, 124 mm from the column's centre, and the lever arm to the bolts is 314 mm: the bolts carry 264 kN and the flange zone 564 kN. The dashed block is the rigid-plate answer, 54 mm deep at the plate's far edge with its resultant 223 mm out, which needs only 128 kN in the bolts. Both satisfy equilibrium; only one is a plate that can carry the pressure where it is drawn.

The compression that stays under the flange

Put a moment on a base plate and a pressure-block calculation pushes the compression out to the plate's edge, where the lever arm to the holding-down bolts is longest. A plate that is not rigid cannot put it there. The compression stays within a few tens of millimetres of the compressed flange, the lever arm shrinks, and the bolts carry twice what the pressure block said.

connections · Base plate 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 The flange is bent hardest where it is thinnest, not where it curls most. The transverse bending stress at the root of the inner flange, where it meets the web, along a knee curved to 1,000 mm through a quarter turn, its flanges 600 mm apart on a 10 mm web, carrying 500 kN·m all round the curve; the dotted line is a yield stress of 355 N/mm². Uniform, 269 all round. Narrowing, 283 at the rafter end and 207 at the column end, largest where λ is; thinning, 191 at the rafter end and 397 at the column end, largest where λ is too. Tapered both ways, 205 at the rafter end and 325 at the column end: largest at the column end, where λ is 0.94 against 1.14 at the rafter end, because there the flange's own stress has nearly doubled.

The flange bent hardest where it curls least

A curved I-beam's flange curls towards its web and is bent across its own width at the root, by a stress that rises with the curling parameter λ. A portal knee's flange is cut from plate and usually tapers, and a taper changes λ along the curve. So the obvious guess is that the transverse stress peaks where λ is largest. For a flange that only narrows, or only thins, it does. Taper it both ways and the guess fails: λ is smallest at the narrow, thin end, and that is where the flange is bent hardest, because the same force is carried there on half the steel. The force the changing width hands to the web, meanwhile, is a few newtons a square millimetre.

sections · Curved beam

Named alongside it

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

Base plateBearing stressColumnEccentricityEffective areaEffective widthFoundationStress concentrationT-stubBeam on elastic foundationBox girderCantilever

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