The end that is only a plate

A beam framing into a girder at the same top level has its top flange cut away, and what is left is a tee whose section modulus is a quarter of the beam’s. A beam framing between the flanges of a column, or into a girder whose own depth leaves no room below, has both flanges cut away.

What is left then is not a reduced beam. It is a plate, 357 mm deep and 9 mm thick, with three holes in it.

The hero shows the check that governs it: a block of that plate tearing out around the bolt group, at 524.79 kN, along a shear plane 180 mm long and a tension plane 55 mm long. Nothing in that number came from the beam.

Where a double cope comes from #

Nobody draws a double cope for its own sake. It arrives from one of three geometric situations, and it is worth naming them, because each one puts a different constraint on the dimension the capacity turns out to depend on.

A beam framing into a column web. The beam has to pass between the column’s flanges, so both of its own flanges are cut back to clear them. The depth left is set by the column’s internal dimension, and the edge distances by whatever the fin plate’s own bolt line leaves.

A beam framing into a girder of similar depth. The top flange is cut so the two sit flush and the bottom flange is cut because the girder’s bottom flange is in the way. This is the common case in a floor where the secondary beams were sized by deflection and came out nearly as deep as the primaries.

A beam framing into a stiffened panel. The girder carries transverse stiffeners at the connection, and the beam’s flanges cannot pass them. This one is the least expected, because the stiffener was added for a reason unrelated to the beam and its presence changes the beam’s end condition.

In all three the second cut is forced by clearance rather than chosen, and in all three the person who decides the cope depth is not the person who checks the plate that is left. That is the same division of labour the rung below records, one cut deeper.

There is a fourth case that looks like a double cope and is not: a beam with both flanges cut back only over the width of the supporting member’s flange, with the web left full depth. That is a notch rather than a cope, and it fails differently.

What the second cut removes #

The first cope removes the top flange. The remainder is a tee, and a tee is still a bending member: it has a section modulus, a neutral axis somewhere near the surviving flange, and a compression zone that the flange stabilises.

The second cope removes that flange too, and three properties go with it.

The section modulus collapses to a rectangle’s. A 357 × 9 plate has $Z = t d^2/6 = 1.91 \times 10^5$ mm³ against the uncoped 457 mm beam’s 1.44 × 10⁶ — 13 per cent, against the single cope’s 27.

Both edges are now free. The single-coped web had one free edge along the cut and one held by the surviving flange; the double-coped web is a plate supported on one end only, and its buckling coefficient is the outstand value on both sides.

And the tear-out block can go either way. In a single-coped end the block is bounded above by the cut and the tension plane runs up to it. In a double-coped end there is a free edge above and below, so there are two blocks, and the governing one is the one with the smaller edge distance.

That third change is the one that decides the capacity, and it is why this essay is carried by a picture of a block rather than a picture of a section.

The edge distance, which is now the whole calculation #

In a plate 357 mm deep with a three-bolt group 140 mm long, there is plenty of depth to put the group where it likes. The edge distance is a choice, and it is the strongest term in the capacity.

It is also the dimension most likely to have been set by a minimum. A detailer placing a bolt group in a web plate reaches for the code’s minimum edge distance — around 1.2 times the hole diameter for a rolled edge — and centres the group on the plate only if nothing else has claimed the space. Two things usually have: the girder’s own flange above, and the fillet weld or bolt line of whatever the plate connects to below.

So the capacity of a double-coped end is, in practice, set by whatever depth was left over after the geometric constraints were satisfied — which is exactly the shape the cope length has one rung down, and for the same reason. A dimension that is the residue of four clearances is not a dimension anybody chose.

Three bolts that carry less than two #

That comparison is the clearest demonstration on this site of why block shear is not a check on any single plane.

Both groups tear out the same shape. The difference is what is inside the shear plane: three holes at 70 mm pitch remove 44 mm of a 180 mm plane, and two at 140 mm remove 22 mm of the same 180. The rupture capacity, which acts on the net area, therefore differs — 330.75 kN against 388.96 — while the yield capacity, which acts on the gross area, is 345.06 for both.

The check takes the smaller of the two. For the tight group rupture is smaller and governs; for the wide pair yield is smaller and governs. A bolt was removed and the governing mechanism changed, and the capacity went up by 2.7 per cent.

The gain is small and the lesson is not. On a connection whose capacity comes from areas rather than from fasteners, a bolt is a hole before it is a bolt. Adding one adds shear capacity through the fastener check and subtracts area from the tear-out check, and there is no reason the first has to win.

And a joint whose capacity is a shear-plane quantity is a joint that does not benefit from a better steel in the way a tension-governed one does, because the yield branch is competing with the rupture branch on the same plane and raising both moves the crossover rather than the answer.

Which free body produced the number #

The block is a closed shape and the calculation is the equilibrium of that shape, which is worth doing once, because it is the reason the answer is a sum rather than a minimum.

Cut the plate along three lines: down the bolt line on each side of the group to the end of the plate, and across the group at the far end. Lift the enclosed block out. Crossing the two vertical cuts is shear, distributed over the gross length of each; crossing the horizontal cut is tension, distributed over the edge distance less the half-hole the cut passes through.

The reaction the block is carrying has to be equilibrated by the sum of what crosses those cuts. That is the whole derivation, and it is why the capacity is $0.6 f \cdot A_{\text{shear}} + f_u A_{\text{tension}}$ rather than the larger or smaller of the two: the planes are in series along the boundary of one body, so they act together.

Two consequences follow immediately and neither is obvious from the formula.

The two planes do not fail at the same displacement. The shear planes must slide several millimetres before the shear stress on them peaks; the tension plane ruptures at a fraction of a millimetre of opening. Summing their peak strengths assumes the plate is ductile enough for both to be at capacity at the same moment, which is an assumption about the steel and not about the geometry.

And the shear planes carry a different strength depending on where the holes are. Yield acts on the gross length because the material between the holes yields; rupture acts on the net length because a crack must cross the metal that is actually there. That is the whole of the difference the previous section’s two-bolt group exploited.

The two free edges #

The curve is drawn for an outstand — one edge free, one supported — because that is what each half of the double-coped web is once the flanges are gone. It is the same collapse of the buckling coefficient from 4 to 0.425 that the single cope produces on one side, applied on both.

Two qualifications, and they pull in opposite directions.

It is worse than the single cope, because the plate has no anchor. In a single-coped end the surviving flange holds one edge straight; a plate free on both edges has nothing to hold the line of its compression zone.

And it matters less than it looks, because the compression is small. The moment on this section is the reaction times the lever arm to the bolt group, and at a simply supported end that moment is a fraction of what the beam carries at mid-span. A stress of 60 or 100 N/mm² over the top third of a 357 mm plate is not what buckles it.

Where the two qualifications meet badly is a connection that turns out not to be pinned. A joint detailed as simple still delivers a real hogging moment, that moment puts the top edge of this plate in compression over the whole of its unsupported depth, and the plate has no flange to hold it. The single-coped end has the same problem and half the exposure.

What is left of the beam checks #

Set the two families side by side and the point of this essay is a list.

Bending resistance. Was the beam’s section modulus; is now $t d^2/6$ over the remaining plate, at 13 per cent of the original.

Shear resistance. Was the web area of the whole beam; is now the web area of a plate 357 mm deep rather than 428, which is a loss of 17 per cent — much the smallest of the changes, and the reason a well-proportioned double cope usually still passes on shear.

Lateral-torsional buckling. Was a check on a section with two flanges; has no meaning at all on a plate, because there is nothing for the compression flange to be, and it is replaced by local buckling of an outstand.

Bearing, net section and block shear. Were checks on the connection and are now checks on the member, because the member at this location is the connection.

The list also says which checks arrive rather than which go, and one of them is easy to miss. The bolts now carry the reaction with an eccentricity. A fin plate connection puts the bolt line some 60 to 80 mm from the face of the support, so the group carries the shear plus a moment equal to the shear times that distance — and that moment is resisted by the same bolts, in the same plate, on the same block. Nothing in the block shear capacity contains it, and it is the reason a real check on this connection is a group check before it is a plate check.

That last line is the whole of the double cope. The distinction between a beam and its end connection has stopped existing over the length of the cut, and every check that arrives is a plate check.

The hole is a dimension, not a fastener #

This is the same detail that costs a preloaded joint fifteen per cent of its slip resistance, arriving through an entirely different mechanism. There the loss was a coefficient from a table standing in for statistical scatter; here it is arithmetic on an area, and the two are unrelated except in being invoked by the same decision on the same drawing.

The 6.7 per cent is smaller than the 15, which is worth noticing. The mode that looks most sensitive to the hole is the least sensitive to it, because block shear’s areas are long and a hole diameter is short, while slip resistance’s dependence on the hole is not a dependence on its area at all.

The thickness is linear in the answer and it is the one dimension nobody in the connection design chose. It arrived with the section, and it arrived because somebody upstream picked a beam by depth and deflection — which is the same section property talking whose two areas do different jobs.

There is one more consequence of the plate having replaced the beam, and it is about how the connection behaves rather than what it carries. A tee end and a plate end have very different rotational stiffness: the tee still has a flange to resist the beam’s end rotation, and the plate has almost nothing. So a double-coped end is, structurally, closer to the pin the analysis assumed than a single-coped one is — the second cut makes the connection weaker and simultaneously makes the assumption behind the analysis more nearly true.

That is a rare shape and it is worth stating plainly, because it cuts against the instinct that a weaker detail is a worse one. The double cope’s hogging moment is smaller than the single cope’s for the same frame, because the joint that would deliver it has lost the stiffness to do so. Whether the net effect is favourable depends on which of the two quantities was governing, and there is no general answer — only the observation that removing material from a connection changes both sides of its own check.

What to carry away #

A double cope changes the family of checks, not their values. The remainder is a plate, and a plate has no section modulus worth checking, no lateral-torsional buckling, and two free edges rather than one.

The edge distance is the strong variable and it is usually a leftover. Doubling it buys 46 per cent, and the depth to do it with is nearly always there.

A bolt is a hole first. Three bolts at a tight pitch carry less than two at a wide one, because the mode that governs is a tear through areas the holes come out of.

And the web thickness beats everything. Linear in every area, chosen by nobody in the connection, and two thirds of the check on a light section.

Where the model stops #

One block is drawn and there are several. A group in a plate with two free edges offers a block tearing upward, one tearing downward, and one taking the whole group out sideways; the check is the least of them, and the figures here show the governing one rather than the family.

The tension term is taken at full value. For a block loaded eccentrically — which a single line of bolts in a plate is — the tension plane unzips from one end rather than separating at once, and codes halve the term for that configuration. Every capacity here would be smaller.

The bolt group is assumed to share. Three bolts in a 140 mm line do; a long line does not, and the block’s arithmetic assumes an even pull along the shear plane that a long group cannot deliver.

And the connected part is not modelled. A double-coped web bolts to a fin plate, a cleat or a column web, and every check here has a twin on the other side of the bolts — including the bearing that ovalises the hole, which is checked on both plates and governed by the thinner.

The ladder from here #

Later rungs on this anchor: reinforcing a coped end — a doubler plate, a longitudinal stiffener welded along the free edge, or a horizontal plate at the cut — and which of the checks each of them actually repairs. The re-entrant corner at the second cut, where two stress concentrations sit on the same plate. Extended shear tabs, where the eccentricity is designed in rather than ignored. And the fabrication question underneath all of it: a cope is cut with a flame or a saw, its radius is a shop decision, and its surface finish is what a fatigue category is assigned on.

The double cope is old and its checks are new. Beams have framed between column flanges since the first riveted frames, and the block shear mode that governs them was not named until the 1970s, when tests on coped beams at Lehigh produced capacities that no combination of the existing checks predicted. What had been happening in the meantime is that connections were detailed generously, by rules of thumb about edge distance and bolt pitch that were not derived from anything and were, on this mode, roughly right.