Coped beam — the series · Structural Engineering & Statics

The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 120 mm to frame into a girder. What is left is a tee with a section modulus of 3.836e+5 mm³ against the whole section's 1.438e+6 — 27 per cent. The moment at the end of the cope is the reaction on a lever arm of 130 mm: 23.4 kNm, giving 61 N/mm² and a flexural utilisation of 0.17. The web now has a free edge along the cope, so its buckling coefficient collapses from 4 to 0.425 — a factor of 9.4 — and the re-entrant corner has a stress concentration of 5.5 on a 10 mm radius.

The section that is checked is not the one chosen

A beam framing into a girder has its top flange cut away so the two can sit at the same level. What is left is a tee with a quarter of the section modulus, a web with a free edge, and a re-entrant corner — and the beam was selected on a table entry that describes none of it.

part 1 · connections

Block shear: the metal between the holes. Three bolts in a 9 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 55 mm long. Shear ruptures first, and the capacity is the sum of two different strengths on two different planes: 524.79 kN, of which the shear plane carries 63.03%.

The end that is only a plate

Cut one flange off a beam's end and what is left is a tee. Cut both and what is left is a plate with holes in it — no flanges, no section modulus worth the name, and none of the checks the beam was selected by. Three plate checks replace them, and the one that governs depends on dimensions that appear in no section table.

part 2 · connections

Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 200 mm, carrying a reaction of 300 kN through three bolts, as utilisations, for the end as coped and with three repairs: an 8 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.46, shear 0.41, tear-out 0.57 and local buckling 0.46. The doubler thickens the web, which is most of what a coped tee is, and lowers all four; the edge plate gives the tee back a flange and lowers only flexure and buckling, leaving shear and tear-out exactly where they were. What governs: as coped, tear-out at 0.57; doubler, tear-out at 0.30; edge stiffener, tear-out at 0.57; both, tear-out at 0.30.

The repair that fixes the wrong check

A coped beam end that fails its checks is usually repaired by welding a plate along the edge the cope left, because the cope removed a flange and the plate puts one back. On ordinary proportions that repair restores the check that was not failing. The check that governs is carried by the web, and only a plate on the web reaches it.

part 3 · connections

The plate that arrives after the load

A repair plate welded to a beam already in use carries only the load that arrives after its weld has cooled. The elastic checks remember that order — a doubler welded under 60 per cent of the reaction removes 40 per cent of the stress it would remove from an unloaded beam — while tear-out, a plastic mechanism, gets the whole of the repair. Propping the beam is what hands the plate the load already there.

part 4 · connections

A crack at the cope's corner, and the hole its tip needs. A 457 × 190 beam end coped 150 mm long and 80 mm deep, reacting 350 kN of which 70 per cent comes and goes with traffic, to scale. A fatigue crack has grown 30 mm down the web from the corner where the cope's two cuts meet, the most stressed point of the coped section. At that depth the live load's range of stress intensity at its tip is 15.6 MPa√m, and a hole drilled with the tip inside it stops the crack only if its radius is at least (ΔK/10.5√fy)², 6.2 mm: a 12 mm hole. The crack and the hole together leave 341 mm of the 377 mm coped section below them.

The hole at the tip of the crack

The commonest reason to repair a coped beam end is a fatigue crack at the corner of the cope, and the commonest repair needs no plate at all: a hole drilled so that the crack's tip is inside it, turning a crack into a smooth notch that the traffic can no longer drive. The hole has to be big enough, and how big is set by the crack — its radius grows in step with the crack's length. A 30 mm crack at a stringer's cope needs a 12 mm hole, a 60 mm crack a 24 mm one. The repair stops working at about 100 mm, where the hole outgrows a site drill and the section left beside it runs out of capacity at nearly the same depth. Most of the crack's life is spent far shorter than that, which makes the repair a matter of how early the crack is found.

part 5 · connections

A doubler is put to work through its welds, and most of the work is at its ends. A 457 mm beam coped 200 mm long and 50 mm deep, carrying 300 kN, with a 8 mm doubler 393 mm deep on its web, the plate welded along its two long edges and running 200 mm past the end of the cope. The shaded bands along the two welds are the shear flow each passes into the plate, averaged over 50 mm: the top weld passes up to 1,056 N/mm and the bottom 821, both largest where the plate ends. At the cut through the cope's end the plate gives 92 per cent of the relief a plate joined at every point would give: the free edge is at 94 N/mm², against 152 as coped and 88 fully joined.

How far a doubler has to run

A doubler on a coped web is joined to the beam only along its welds, and the force reaches it through those welds from wherever it is welded. Stopped at the end of the cope, a plate welded along its two long edges gives about two fifths of the relief a fully joined plate would give at the very section it was put there for. Run on past the cope by four tenths of its own depth, it gives nine tenths. Welded all round, it is fully joined at any length.

part 6 · connections