How far a doubler has to run

Every repair a coped beam end has been given so far was taken to act with the web the moment it was there. The repair that fixes the wrong check thickened the web by the doubler’s thickness at the end of the cope and read the four checks on the thicker section. The plate that arrives after the load added an order of loading — the doubler carries only what arrives after its weld has cooled — but kept the same section once it did. Both treat the plate as fully composite at the critical cut, which is to say joined to the web at every point of its face.

It is joined along lines. A doubler is a rectangle of plate laid against the web and fillet-welded round some part of its perimeter: along its top and bottom edges as a minimum, across its ends if the fabricator is asked to. Between the welds it simply lies against the web, and nothing transmits a longitudinal force across that contact except friction, which no design counts. Whatever stress the plate carries, it was given through a weld, and the force a weld gives has to spread into the plate’s interior through the plate itself.

So the question the staged calculation left open is not whether the plate is attached, but how far from its welds the plate is attached in any useful sense — and in particular, how far past the end of the cope a doubler has to run before it is doing at the cope what the section calculation said it would.

Two layers, joined only where the welds are #

The beam is the one every repair essay has used, and the one whose coped end was checked on a section nobody chose: a 457 × 190 mm section with a 9 mm web, coped 200 mm long and 50 mm deep, carrying a reaction of 300 kN. The doubler is 8 mm thick and covers the whole depth the cope leaves — 392 mm, from the top of the bottom flange to the cut edge.

A plate lying against a web is two plates in the same plane of bending, and the honest description of them is two layers of plane stress, one 9 mm thick and one 8, each with its own displacement field. Where they are welded, the two fields are the same: a fillet weld is very stiff in shear, and it is taken here as rigid. Everywhere else each layer moves as its own stresses tell it to. The flanges are bars along the web’s top and bottom lines, the top one only beyond the cope, and the reaction is applied at the girder face so that every section along the beam carries exactly the reaction times its distance from that face. That last fact is a check on the solution rather than an assumption of it: whatever the plate does, the two layers and the flanges between them must together carry R·x at every cut, and they do, to a few parts in a thousand.

What the plate does is then a matter of reading. At any cross-section, the plate’s own stresses integrate to its axial force and its own bending moment; the original steel’s integrate to the rest. The check a designer makes at the end of the cope reads the free edge’s stress off the tee as a linear distribution, so the same is done here, from the original steel’s own resultants. The one compromise is where: the re-entrant corner of a cope is a singular point in an elastic plane model, its stress growing without limit as the mesh is refined, so the section is read 15 mm inside the cut rather than at it. There the tee as coped carries 152 N/mm² at its free edge.

What a plate joined everywhere would give #

The reference is the plate the earlier essays assumed: one joined to the web at every point of its face. In the plane model that is the two layers sharing every node, which is a 17 mm web over the doubler’s area. Its free edge, at the same section, carries 88 N/mm².

That is the whole of what a doubler can deliver at this end, and every other arrangement is measured as a share of the step from 152 to 88. A share of one means the plate is doing exactly what the section calculation assumed; a share of a half means the repair is half the repair it was designed to be.

The plate that stops at the cope #

The most economical doubler is the one that covers the cope and no more — the coped length, 200 mm, from the beam end to the cut, which is also the length over which the bolts tear a block out of the web and the doubler’s thickness helps. Welded along its two long edges, it gives about two fifths of the relief at the section the cut reads.

That number is not well determined, and the reason is the finding rather than a defect in the arithmetic. As the mesh is refined, the share at a section 15 mm inside the cut drifts from 0.44 toward 0.38: the plate’s own moment is falling to nothing at its unwelded end, and the section being read is close enough to that end that a finer description of the plate shows it falling more steeply. Pushed to its limit, a doubler that stops at the cut gives nothing at the cut. It gives something at the sections behind it, where it has had room to take up its share, and nothing at the one section it was put there for.

The third figure follows the plate’s own bending moment along its length and shows why. Behind the cope, where both welds have been running for a hundred millimetres or more, the plate carries nearly as much moment as a fully joined plate would. Toward its unwelded end, over the last 150 mm or so, it hands that moment back to the web, and at the end it carries none.

Nine tenths at four tenths of its depth #

Run the same plate past the cope and its unwelded end moves away from the section that matters.

The share rises steeply and then stops rising. With 20 mm of extension the edges-only plate gives 0.60 of its relief; with 100 mm, 0.83; with 156 mm, nine tenths; with 300 mm, 0.94, and no further extension adds anything. The plate needs to pass the cope by about four tenths of its own depth before the section calculation is nine-tenths right.

The reason the share stops short of one is at the plate’s other end. A doubler welded along its edges and not across either end has two unwelded ends, and the one at the beam end — inside the coped length, close to the reaction — is in the same position as the far end of the plate that stopped at the cope. Its moment there is small because the beam’s moment there is small, but the plate’s share of the section’s work is built up from both ends at once, and a plate without a weld at its near end never quite reaches the fully joined state anywhere along it.

The depth sets the length, and the thickness hardly does. An 8 mm plate reaches nine tenths at 156 mm of extension and a 12 mm plate at 144. A plate of the same thickness but 300 mm deep, stopping short of the free edge, reaches nine tenths at about 44 mm. That is what a shear-lag length should do, and it is the same length that leaves an angle connected by one leg using half of itself. The force a weld gives the plate spreads into its interior by in-plane shear, and the distance it has to spread is half the plate’s depth; how fast it spreads is set by the ratio of shear stiffness to axial stiffness, G to E, which no choice of plate changes. A thicker plate carries more force and passes more shear in the same proportion, so its development length barely moves. A deeper one carries its force further from the welds that feed it, and needs longer.

A plate welded all round is a plate joined everywhere #

A weld across the plate’s far end changes the picture entirely. With its end welded and nothing past the cope, the doubler already gives 0.87 of its relief; with 31 mm of extension, nine tenths. And with a weld across the near end as well — a plate welded all round its perimeter — the share is one, exactly, at any length at all.

That last result is not a numerical coincidence. Inside the welded perimeter, neither layer carries any load except what its neighbours pass it. Each satisfies the same equations of plane elasticity, in the same material, over the same region, with the same displacements prescribed all round its boundary by the weld that ties it to the other. Such a problem has one solution. The plate’s field and the web’s field are therefore the same field, everywhere inside the weld, as though the two had been joined at every point. A doubler welded all round its perimeter is fully composite, in elastic plane stress, at any length — and the only thing that could make it otherwise is a part of its interior that is loaded independently of the web, which a doubler does not have.

It is the same argument that makes Saint-Venant’s principle work, read the other way round: a region’s interior is decided by what is prescribed on its boundary, so two regions with the same boundary have the same interior. What a perimeter weld prescribes is the web’s own displacement, and the plate inside it has no choice but to follow.

Where the moment reaches the plate #

The plate’s work at the cut is almost all bending. Its centroid sits at mid-depth of the coped web, and the coped tee’s own centroid is not far below it, so the axial force a fully joined plate would carry is small; most of what it contributes is its own second moment, 8 × 392³/12, joined to the web’s. That is why a one-dimensional reading of the doubler — two bars side by side, slipping along a line, as a steel beam and its slab are read — gets this repair wrong. Such a reading hands the plate its bending stiffness for nothing and lets only its axial force wait on the welds, so it credits a doubler stopped at the cope with most of its relief. The plane model is needed because what has to be put into the plate through its welds is a moment as well as a force: the top weld pulling one way and the bottom weld the other, a couple that builds up the plate’s own bending over the same shear-lag length as its force.

The figure shows that couple’s work. Behind the cut, the edges-only plate climbs to within a tenth of the fully joined plate’s moment. Past the cope it hands its moment back over its last 200 mm, steeply at the end. The plate with its end welded holds its moment almost to the end of the plate, because the end weld can take the moment out across the plate’s depth at once rather than through a long run of edge weld.

The free edge, along the coped length #

Read along the cope, the stopped plate’s failure has a shape. Near the beam end the four lines are close together because the moment is small. Through the middle of the coped length the stopped plate and the extended one are nearly the same, both well below the as-coped line. Over the last 60 mm before the cut, the stopped plate’s line turns upward toward the as-coped value, because that is where its own unwelded end is draining it. The extended plate’s line runs straight on to the cut a few newtons above the fully joined line.

That upturn is the whole difference between the two repairs, and it is exactly at the section the design check reads. A doubler that stops at the cope is a good repair everywhere except where it was needed.

The welds work hardest where the plate stops #

The flows along the two welds show where the plate’s share is actually being transferred. Along most of the plate they are small — a few hundred newtons per millimetre on the top weld, less on the bottom — because a plate already carrying its share needs very little more to keep carrying it as the moment grows. The welds’ work is concentrated at the plate’s ends, where its share starts and stops.

Over the plate’s last 25 mm the top weld passes about 1,050 N/mm, and averaged over its last 50 mm still about 710. A 6 mm fillet resists about 1,065 by the simplified method in EN 1993-1-8, so the weld at the plate’s end is at its limit while the weld everywhere else is lightly used. Two conclusions follow. A weld sized on the average shear flow — the plate’s force at the cut divided by the length of weld behind it — would be undersized where it matters, because the demand is a concentration at the ends and not a mean. And the plate’s end is where an unsound weld will first show, by shearing over a short length and returning the plate’s share to the web exactly where that share was being handed over.

There is a peak value at the very corner of the plate that is not in the figure, because the plane model’s stress at a corner of a welded plate grows without limit as the mesh is refined, as the re-entrant corner of the cope does. A real fillet yields a little at that corner and spreads the demand over a few millimetres; the averaged flow is the quantity a weld of finite size actually sees.

What the terminal lengths for cover plates say #

The design rule nearest to this question is the terminal development length of a partial-length cover plate in AISC 360. A cover plate is counted in a beam’s section only past a point some distance beyond its theoretical cut-off: one plate width with a continuous end weld at least three quarters of the plate’s thickness, one and a half widths with a smaller end weld, two widths with none. The doubler here is a cover plate whose width, in that rule’s sense, is its depth across the web — 392 mm — so the rule would ask it to run 392 mm past the cope with an end weld and 785 mm without.

The plane model asks 31 and 156. It is worth being clear why they differ, because the rule is not wrong for what it was written for. A cover plate on a flange carries its share as axial force, delivered by two edge welds into a strip of plate whose force spreads by in-plane shear from both edges at once. For such a strip the same shear-lag argument gives a decay of the unloaded part as $e^{-\pi\sqrt{G/E}\,x/w}$, which reaches nine tenths at about 1.2 widths — between the rule’s numbers for the two cases, and closer to the generous one. The doubler on a coped end differs in two ways. Its work at the cut is mostly bending, reached through a couple at its two edges. And it has already been welded along the whole coped length behind the cut, so much of its development length lies behind the section that needs it rather than beyond it. A cover plate on a flange has neither of those.

The terminal lengths are an upper bound for this repair, by a factor of two to ten, and a doubler detailed to them is not wrong — only longer than it needs to be. The cost of that extra length is not just steel.

The plate’s own end has a fatigue limit #

A doubler is often there because of fatigue, which is decided by a load that never comes near failing anything. The hole at the tip of the crack was about a coped end on a stringer, cracked at its re-entrant corner by the cycling of the live load. And a plate welded to a web is itself a fatigue detail. Its end, where a fillet weld stops on the face of the web, is a longitudinal attachment — the kind of place where the detail decides and the steel does not — and a longitudinal attachment more than 100 mm long sits in category 56 of EN 1993-1-9 — among the poorest details a girder carries. Its constant-amplitude limit is 41 N/mm².

The stress range at the plate’s end is the live load’s moment there times the web’s distance from the full section’s neutral axis, and the moment grows with the distance from the support. So the further the plate runs past the cope, the larger the range its own end sees. With 40 per cent of the reaction cycling, the range at the end of the cope is 14 N/mm²; at 200 mm of extension, 27; at 420 mm it passes 41.

That gives the doubler a window rather than a minimum. Below about 160 mm of extension the plate is not yet doing what the section calculation credits it with at the cope. Above about 420 mm its own end, on this live-load range, is a category-56 detail working above its endurance limit — a repair that has moved the place a crack will start from the cope’s corner to the end of the plate. For this beam the window is wide: a 250 mm extension sits comfortably inside it. For a beam with a larger live share, or a doubler on a beam nearer mid-span, the two ends of the window close toward each other. The terminal lengths for cover plates, applied to this doubler, would put its end at 785 mm, outside the window altogether.

The cover-plate rule has a fatigue history of its own, and it runs the same way. Partial-length cover plates on bridge girders were among the first details found cracking in service in the 1960s and 70s, at their ends, and the response then was a lower detail category for the end and a recommendation to stop the plate where the moment is small. The doubler’s end obeys the same logic at a smaller scale.

The plate that is welded under load #

The plate that arrives after the load found that a plate welded to a beam already carrying 60 per cent of its reaction carries only the 40 per cent that comes afterwards. That is a section that changed while it was being loaded, and nothing here changes it. The plane model is linear, so the plate’s share of the later load is the same fraction of the fully joined share whatever share was present when it was welded. The development length is the same, and the share of the relief at a given extension is the same; what the staged calculation adds is that the relief applies only to the later load.

The welds’ demand scales down in the same proportion: a plate welded under 60 per cent of the reaction never sees more than 40 per cent of the flows in the figure above, and its end weld is comfortably inside a 6 mm fillet’s resistance. That is a small consolation for a repair whose benefit is also only 40 per cent of what it would be on an empty beam. But it means the weld that is at its limit in the unloaded case is not the governing difficulty of a repair made in service. The dead load locked into the tee is.

The thickened tee and a strip in shear, by hand #

The fully joined reference can be checked without a mesh. The coped tee with an 8 mm plate over its web is a section of web thickness 17 mm and the same flange, and 195 mm from the support, carrying 58.5 kN·m, its free edge stress by beam theory is 89 N/mm². The plane model’s reading at the same section, 88, is a little lower, because close to the cope’s corner the stress across the section is not quite the linear distribution beam theory assumes. The tee as coped, by beam theory, carries 152 there, and the plane model with no plate reads the same 152 — a check that the section is far enough from the corner to be read as a section.

The development length can be roughed out the same way. A plate held along its two edges has a lowest shear-lag mode whose share decays as $e^{-\pi\sqrt{G/E}\,x/h}$, with $\sqrt{G/E} = 0.62$ for steel. With h = 392 mm the decay length is 392/(π × 0.62), about 200 mm, and nine tenths of a decaying share is reached at 2.3 times that, or 460 mm, for a plate developing from one end in a constant moment. The doubler does better than that by a factor of three because it is developing from behind the cut as well as ahead of it, and because the moment is rising toward the cut, so the share it needs at any point is less than the share it will need further on.

Where the plane model stops #

The welds are rigid and the plate is elastic. Neither is exactly true. A fillet weld has a stiffness, which softens the connection a little and spreads the concentrations at the plate’s ends; the steel at the end of a highly stressed weld yields a little, which does the same. Both make the plate’s ends less abrupt than the figures show, and neither changes the development length by much, because the plate’s own shear stiffness is far softer than either.

The doubler is in the plane of the web. A real doubler is on one face of the web, so its force acts 8.5 mm off the web’s mid-plane, and the pair bends out of plane under it. On a coped end that out-of-plane bending is resisted only by the web, whose free edge at the cope is the part least able to resist it — the same edge whose buckling was one of the four checks. A plane model cannot see that.

And nothing passes through the contact. The plate and the web touch over their whole area, and friction between them, clamped by the weld’s shrinkage, would carry some shear between the welds. It is left out, as design leaves it out, which makes the development lengths here a little long rather than short.

The weld the figures cannot see #

They cannot show the weld itself: its size, its root, its start and stop, and the defects a fillet weld tends to have at exactly the place where these figures say it works hardest. They cannot show the plate’s out-of-plane bending, nor the friction between the faces. And the fatigue figure uses the nominal stress at the plate’s end and the detail category’s allowance for the concentration there; it says where the end of the plate sits against a published category, not what the stress at the toe of that weld actually is.

Still open: the plate on one face of the web #

A doubler on one face of the web is an eccentric plate, and its share of the bending reaches it through a weld whose throat is pulled out of the web’s plane at every point. Whether that eccentricity bends the coped web’s free edge sideways enough to bring forward the local buckling a doubler was meant to delay — and whether a pair of thinner plates, one on each face, buys back more than their extra welding costs — is the question a plate on one side leaves at an end whose free edge is already its weakest part.