Structural Engineering & Statics
where the load goes
Connections

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.

Assumes The section that is checked is not the one chosen, The metal between the holes, which comes out as a block and The plate that ripples, and the width that is left.

16 min read 7 figures Which failure arrives firstGeometry beats materialThe load must go somewhere

A beam framing into a girder at the same level has its top flange cut away, and the tee that is left has a quarter of the section modulus the beam was selected on, a web with a free edge, and a re-entrant corner. When that end fails its checks the repair that most detailers reach for is obvious from the drawing: the cope took a flange away, so weld a plate along the edge it left and put one back.

It is the right repair in one case and the wrong one in the case that is usual, and the difference is not a matter of the plate’s size. It is a matter of which property of the end each check depends on.

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.
Fig. 1 The four checks on a 457 mm beam coped 50 mm deep over 200 mm and carrying 300 kN through three bolts, for the end as coped, with an 8 mm doubler on the web, with a 100 × 10 mm plate along the free edge, and with both. The dot marks the check that governs each end. As coped it is tear-out at 0.57; the edge plate leaves it at 0.57, the doubler takes it to 0.30.

Four checks, four properties

A coped end is checked four ways, and the four do not ask about the same thing.

Flexure asks whether the tee can carry the moment the reaction produces at the end of the cope, and depends on the section modulus of what is left. Shear asks whether the reduced web can carry the reaction itself, and depends on the web’s area, its thickness times its remaining depth. Tear-out asks whether the bolt group can pull a block of material out of the web along a shear plane and a tension plane, and depends on the web’s thickness through every area of the block — the check a double cope turns into the whole calculation. And local buckling asks whether the web’s compressed free edge buckles sideways, and depends on the thickness squared and on whether the top edge is free.

The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 200 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 210 mm: 63.0 kNm, giving 164 N/mm² and a flexural utilisation of 0.46. 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.
Fig. 2 The unreinforced end: a 457 mm beam coped 50 mm deep over 200 mm. The tee has 27 per cent of the whole section’s modulus, the 300 kN reaction on a 210 mm lever arm gives a moment of 63.0 kNm and a flexural utilisation of 0.46, and the web now has a free edge along the cut.

A repair adds a plate in one place. It changes the properties that plate contributes to, and it leaves the rest exactly as they were. So the question to ask of any repair is not how much steel it adds but which of the four properties it touches — and the answer is fixed by where the plate goes, before any number is computed.

The edge plate reaches two of the four

A plate welded along the free edge the cope left is, structurally, a new top flange. It does two things and they are both real.

It moves the section’s extreme fibre out to a plate that is wide, so the section modulus climbs back toward the beam’s: on this end the flexural utilisation falls from 0.46 to 0.25. And it holds the free edge, so the web is no longer an outstand that can buckle sideways under compression — the coefficient that collapses from four to 0.425 when an edge is released is restored with the edge. Local buckling falls from 0.46 to 0.25 along with flexure.

It does nothing else. The plate is not part of the web, so it adds nothing to the web’s area and shear does not move. It is not in the block the bolts would tear out — the block lies in the web around the bolt group, below the plate — so tear-out does not move either.

The coped end with a plate along the free edge. A 457 mm beam coped 50 mm deep over 200 mm, carrying 300 kN, with a 100 × 10 mm plate welded along the free edge the cope left. Against the end as coped, the four utilisations move as follows: flexure 0.46 → 0.25; shear 0.41 → 0.41; tear-out 0.57 → 0.57; local buckling 0.46 → 0.25. The check that governs is tear-out, at 0.57, the same check as before the repair.
Fig. 3 The same end with a 100 × 10 mm plate welded along the free edge. Flexure falls from 0.46 to 0.25 and local buckling from 0.46 to 0.25. Shear stays at 0.41 and tear-out stays at 0.57, and tear-out was the check that governed: the repaired end has the same governing utilisation as the end before it was repaired.

That is the case the essay is named for. On ordinary proportions the check that governs a coped end is tear-out, because a reaction of a few hundred kilonewtons through three bolts in a 9 mm web asks a great deal of the block around them and very little of a moment on a 210 mm lever arm. The repair that looks most like undoing the cope restores the two checks that were comfortably passing and leaves the failing one untouched.

The doubler reaches all four

A doubler — a plate welded to the face of the web over the coped length — looks like a shear repair, and it is described as one in most connection guides. The calculation says it is more than that.

It thickens the web, so shear rises in proportion: an 8 mm plate on a 9 mm web divides the shear utilisation by 17/9. It thickens every area the bolt group would tear out, so tear-out falls in the ratio of the two blocks. It raises the web’s buckling resistance with the square of the thickness.

And it raises the section modulus substantially, which is the part that runs against intuition. The instinct is that a plate near the middle of a section adds little to its bending strength, and for a whole I-beam that is true. A coped tee is not a whole I-beam. Its surviving flange is at the bottom, far from the critical fibre at the top of the web, and between them is 390 mm of web that is most of what the section is. Thicken it and the modulus rises by seventy per cent.

The coped end with a web doubler. A 457 mm beam coped 50 mm deep over 200 mm, carrying 300 kN, with an 8 mm plate welded to its web over 300 mm. Against the end as coped, the four utilisations move as follows: flexure 0.46 → 0.27; shear 0.41 → 0.22; tear-out 0.57 → 0.30; local buckling 0.46 → 0.27. The check that governs is tear-out, at 0.30, the same check as before the repair.
Fig. 4 The same end with an 8 mm doubler welded to the web over 300 mm. Every check falls: flexure from 0.46 to 0.27, shear from 0.41 to 0.22, tear-out from 0.57 to 0.30 and local buckling from 0.46 to 0.27. Tear-out still governs, at 0.30 — the governing check has not moved, but its utilisation has nearly halved.

The two repairs are therefore not two versions of the same intervention at different weights. The edge plate is a flexural repair that happens to cure buckling. The doubler is a web repair that happens to help flexure, and because the web carries three of the four checks on a coped end, it reaches the governing one in almost every case.

The cope length where tear-out hands over to flexure

“Ordinary proportions” can be made exact, and the result has a property that makes it easy to use: the reaction is not in it.

Both checks divide the same reaction by a resistance. Tear-out divides it by the block’s capacity; flexure multiplies it by the lever arm from the bolt group to the end of the cope and divides by the tee’s section modulus times the yield stress. They are equal when

c+s=Ztee fyVblockc + s = \frac{Z_{tee}\,f_y}{V_{block}}

with cc the cope length and ss the setback of the bolt line from the support. For this 457 mm beam the tee’s modulus is 383,600 mm³ and three bolts give a block of 524.8 kN, so the two checks cross at a lever arm of 260 mm — a cope of about 250 mm. Shorter, and tear-out governs whatever the load; longer, and flexure does.

The boundary moves the way the argument says it should. Five bolts make a block of 778.8 kN, the crossing falls to a lever arm of 175 mm, and a cope of only 165 mm is already flexure-governed — which is why the long-cope end above, with its five-bolt group, needed the edge plate and the short-cope end with three bolts did not. A deeper beam, with a larger modulus left after the cope, pushes the crossing out again. So before anybody sketches a repair, the drawing already says which family of plate is aimed at the right check: compare the cope length with the crossing, and the crossing is two numbers from a section table and one from the bolt layout.

The same comparison explains why the shear on a coped web so rarely governs in its own right. Its resistance is the web’s shear area, which is far larger than the block’s shear plane, so shear sits below tear-out on the short cope and below flexure on the long one, and becomes the governing check only after a repair has lowered both of those — which is what the edge plate did to the long cope.

Fixing one check hands the end to the next

The edge plate is not always irrelevant. Lengthen the cope and put a longer bolt group in the web, and the balance of the four changes.

Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 300 mm, carrying a reaction of 250 kN through five 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 0.57, shear 0.35, tear-out 0.32 and local buckling 0.57. 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, flexure at 0.57; doubler, flexure at 0.34; edge stiffener, shear at 0.35; both, flexure at 0.23.
Fig. 5 The same beam coped 300 mm long, carrying 250 kN through five bolts. As coped, flexure governs at 0.57 and tear-out has fallen to 0.32 with the longer group. The edge plate takes flexure to 0.31 and the end is now governed by shear at 0.35, which the plate did not touch; the doubler takes flexure to 0.34 and it still governs.

With a 300 mm cope the moment on the tee is half as large again, and five bolts give the block a long shear plane, so flexure governs. The edge plate is aimed at exactly that check, and it succeeds: flexure falls from 0.57 to 0.31. But the repaired end is not governed by 0.31. It is governed by shear at 0.35 — the second-worst check before the repair, which the repair could not reach and so did not move.

A repair’s effect on an end is limited by the worst check it does not touch. The edge plate brought flexure below shear, and from then on every further millimetre of edge plate buys nothing. The doubler, reaching both flexure and shear, got the end to 0.34 and would keep improving with thickness. On this end the two repairs finish within a few per cent of each other, and the lighter plate is the one that reaches more checks.

The coped end with a doubler and an edge plate. A 457 mm beam coped 50 mm deep over 300 mm, carrying 250 kN, with an 8 mm plate welded to its web over 400 mm and with a 100 × 10 mm plate welded along the free edge the cope left. Against the end as coped, the four utilisations move as follows: flexure 0.57 → 0.23; shear 0.35 → 0.18; tear-out 0.32 → 0.17; local buckling 0.57 → 0.23. The check that governs is flexure, at 0.23, the same check as before the repair.
Fig. 6 The long cope with both repairs. Flexure falls from 0.57 to 0.23, shear from 0.35 to 0.18, tear-out from 0.32 to 0.17 and local buckling from 0.57 to 0.23. Flexure governs again, at 0.23 — the combination gives each check the plate that reaches it.

That sequence — fix the governing check, find the next one — is familiar from elsewhere, and it is the reason a block of metal between bolt holes can be stronger with a bolt taken out: when a capacity is the least of several mechanisms, changing one of them moves the answer only until another becomes the least. The practical rule it gives is short. Before choosing a repair, rank all four utilisations, and pick the plate that reaches the top two.

The repair that needs no plate

A tear-out governed end has a repair that is not a plate at all, and on a new connection it is the one to try first.

The block the bolts would tear out is bounded above by the cope, and its tension plane runs from the top bolt to that cut. Lengthening the tension plane — moving the bolt group down the web, away from the cope — raises the block’s capacity directly. On the same web and the same three bolts, doubling that edge distance from 55 mm to 110 mm takes the block from 524.8 kN to 767.3 kN, and the governing utilisation of the 300 kN end falls from 0.57 to 0.39 without a single weld.

That is not as large a reduction as the doubler’s, which reached 0.30, and it has a cost of its own: the bolts move down the web, the lever arm from the support to the cope is unchanged, and the connection’s other part — the fin plate or cleat on the girder — has to move with them. But it uses depth the web already has, it adds no heat to a thin plate and no new fatigue detail, and it can be read straight off the drawing.

The general form is worth stating because it applies to every check here. A check that depends on a dimension can be repaired by changing the dimension, and the four checks each have one: the edge distance for tear-out, the cope length for flexure and buckling, the web depth left for shear. A plate is the repair for a dimension that cannot change — an existing beam, a girder whose depth is fixed, a bolt group already drilled — and in those cases the four-properties ranking is what picks the plate.

The one case the edge plate is for

There is an end where the edge plate is the right answer outright, and it is worth identifying precisely, because it is the case the repair’s reputation was earned on.

Local buckling of a coped web depends on the square of its thickness and on how long the free edge is. On a 9 mm web with a 200 mm cope the calibrated buckling stress is above the steel’s yield, so buckling never governs by itself — it coincides with flexure. Thin the web and lengthen the cope and it separates.

Each repair reaches the checks its plate touches, and no others. The four checks on a 402.3 mm beam coped 40 mm deep over 400 mm, carrying a reaction of 100 kN through three bolts, as utilisations, for the end as coped and with three repairs: a 6 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.67, shear 0.28, tear-out 0.34 and local buckling 0.92. 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, local buckling at 0.92; doubler, flexure at 0.34; edge stiffener, tear-out at 0.34; both, flexure at 0.19.
Fig. 7 A 402 mm section with a 5 mm web, coped 40 mm deep over 400 mm and carrying 100 kN. As coped, local buckling governs at 0.92 — the calibrated buckling stress is 256 N/mm², well below yield — with flexure at 0.67. The edge plate takes buckling to 0.24 and the end is then governed by tear-out at 0.34; a 6 mm doubler takes it to flexure at 0.34.

Here the edge plate is decisive. It holds the edge that was buckling, the buckling check stops being an outstand problem at all, and the utilisation that governed the end falls from 0.92 to 0.24. The doubler helps too, and reaches the same governing figure by a different route, but the edge plate is the repair aimed directly at what was wrong.

The conditions are specific: a thin web — five millimetres, which is lighter than most rolled sections and typical of a plated or cold-formed one — and a cope long enough that the free edge has room to buckle. A 6.8 mm web on the same section and cope has a calibrated buckling stress of 473 N/mm², above yield, and buckling coincides with flexure again. The edge plate is the buckling repair, and buckling governs a coped end far less often than the edge plate is specified.

What each plate costs, against what it buys

The comparison is incomplete without the steel, because the edge plate’s appeal is partly that it is small.

As drawn on the 457 mm beam, the doubler covers the web from the cope down to the bottom flange over 300 mm: an 8 mm plate of about 300 by 390 mm, some 7.4 kg of steel with 1.4 m of weld round its edge. The edge plate is 100 by 10 mm over 250 mm: about 2.0 kg and 0.5 m of weld on its two sides. The edge plate is a quarter of the steel and a third of the welding.

On the short cope it buys nothing for the check that governs, so its cost per unit of governing utilisation removed is infinite. The doubler removes 0.27 of utilisation from tear-out for its 7.4 kg. On the long cope the edge plate removes 0.22 from the governing check before shear stops it, for 2.0 kg, and the doubler removes 0.23 for 7.4 — there the edge plate is the better buy by more than three to one, exactly because it is aimed at the check that governs.

That is the whole of the choice, stated as money: a light plate on the wrong check is not cheaper than a heavy plate on the right one. The ranking of the four utilisations decides which plate is light and which is wasted, and on a thin web the answer reverses once more, because a free edge’s buckling coefficient is the one quantity the small plate changes by a factor of nearly ten.

Where the plate is attached, and what that decides

The four-properties argument has a practical corollary about welding, and it is why some repairs do less than their calculation says.

A doubler acts with the web only if it is attached to it over the whole area it covers: welded round its perimeter, and on a large plate plug-welded through its middle as well, so that the two plates share their shear rather than the doubler sliding over the web. A doubler welded only along its edges is a separate plate for most of its area, and its contribution to the modulus and to the tear-out block is whatever the welds can transmit.

An edge plate acts as a flange only if it continues past the end of the cope far enough to develop its force in the uncoped beam. Stopped at the re-entrant corner, it ends exactly where the stress concentration of the corner already is, and it adds a second discontinuity there — a welded attachment’s end is a fatigue detail of its own, graded by its geometry, and a plate ending in a region of high stress range is one of the lower categories.

Both are detailing questions rather than design ones, and both are decided by somebody who may not have seen the ranking of the four checks. That is the same division of labour that produced the cope in the first place: the depth and length of the cut are set by clearances, and the repair is set by whoever is fixing the problem on the drawing in front of them.

What the four checks leave out

The doubler is taken as fully composite with the web. It acts as though it were one thicker web, which is true of a plate welded all round and plug-welded, and increasingly false of one attached only at its edges.

A stiffened free edge is taken as no longer buckling on its own. Once a plate holds the edge, local buckling of the coped web is taken to be covered by flexure of the stiffened section. A thin edge plate that itself buckles, or one that stops short of the cope’s end, does not earn that.

The block is the one-row block. A bolt group of two lines, or one with a large gauge, tears out along a different boundary, and a doubler that covers only part of that boundary reinforces only part of it.

The shear on the web is taken as uniform. Near the re-entrant corner it is not, and neither is anything else — the corner’s own concentration is the reason a crack in a coped end almost always starts there, whatever the utilisations say.

And the end moment is taken as the reaction on its lever arm. A connection detailed as simple still delivers some hogging moment, and a hogging moment puts the top of the tee in tension — where the edge plate is — which changes both the flexure check and whether the free edge is in compression at all.

Still open: whether a repair welded to a loaded beam helps with the load already on it

Every utilisation above assumes the repair was there before the beam carried anything. A repair is rarely added that way. It is added to a beam already in the building, already carrying its dead load, often with the floor it supports still in use — and a plate welded to a loaded member carries only the load that arrives after the weld has cooled. The stress already in the web stays in the web; the doubler joins in for the next increment.

That is a section that changed while it was being loaded, with the added plate playing the part of the slab poured onto a steel beam, and the same argument decides how much of its apparent benefit is real. Whether a repair can relieve a check that is failing under load already present — or can only stop the next load from making it worse — depends on how much of the reaction is dead load, whether the member can be propped while the plate is welded, and how much the heat of welding softens a thin web that is carrying stress at the time.

Where this goes next

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. 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. Block shear: the metal between the holes. Three bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 421.7 kN, of which the shear plane carries 70.43%. The metal between the holes, which comes out as a block A bolted end connection can fail without a single bolt breaking and without the plate reaching its tensile strength anywhere. A block of metal simply comes out, bounded by two surfaces with two different strengths on them. The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 6 kN/m of wet concrete and 9 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 146 MPa; propped, the finished composite section takes everything and reaches 93 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two. The section that changed while it was being loaded A stress is computed from a moment and a section modulus. When part of the moment arrived while the section was a different shape, there is no single section modulus to divide by — the stresses add and the properties do not, and two identical finished beams can differ by half again in stress with nothing on the drawing to say which is which.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Block shearConnectionConstruction sequenceDetailingLimit stateLoad pathLocal bucklingNet sectionPlate bucklingSection modulusShear areaStiffener