The plate that arrives after the load

The repair that fixes the wrong check compared three ways of reinforcing a coped beam end — a doubler on the web, a plate along the free edge the cope left, and both — and found that each reaches a different subset of the four checks a coped end is judged by. Every one of those comparisons assumed the plate was there before the beam carried anything. A repair is rarely added that way. It is added to a beam that is already in a building, already carrying its own weight and the floor’s, often with the floor above still in use, and a plate welded to a loaded member carries only the load that arrives after the weld has cooled.

Two stages, two sections #

The beam is the one the essay on repairs used, and the one whose coped end was checked on a section nobody chose: a 457 × 190 mm section, coped 200 mm long and 50 mm deep, carrying a reaction of 300 kN. At the end of the cope that reaction produces a moment of 300 kN on a lever of 210 mm, carried by the tee the cope left: a flange at the bottom and a web whose top edge is free.

Suppose 60 per cent of the reaction is dead load, and the repair is an 8 mm doubler welded to the web while that dead load is present. Until the weld cools, the doubler is a loose plate lying against a stressed web. It carries nothing, and it has no strain. The 180 kN already on the beam is carried by the coped tee exactly as before, with the tee’s own neutral axis and second moment.

From the moment the weld has cooled, the doubler and the web are one piece of steel, strained together. The remaining 120 kN, when it arrives, is carried by the repaired section — a thicker web, a neutral axis moved upward, a larger second moment — and each fibre’s stress increases by that section’s share. The original steel carries two stages on two sections; the plate carries one. The stress at any height in the web is the dead-load stress on the coped section plus the later stress on the repaired one.

That is a section that changed while it was being loaded, the arrangement familiar from a steel beam that carries its wet concrete alone and the finished floor compositely. The repair plate plays the part of the slab, and the same accounting decides how much of its apparent benefit is real.

What the stress across the tee shows #

The first figure draws that accounting across the depth of the tee.

As coped, with all 300 kN on the tee, the free edge of the web is at 164 N/mm². Had the doubler been there before the beam was loaded, the whole reaction would sit on the repaired section and the edge would be at 97. Welded under 60 per cent of the load, the edge is at the sum of 60 per cent of the coped value and 40 per cent of the repaired one: 137 N/mm². The repair has taken 27 N/mm² off a stress it could have reduced by 67 — 40 per cent of its possible effect, which is the share of the load that arrived after it.

The strains say the same thing in a different unit. When the weld cools, the web’s free edge is already strained by its dead-load stress, 99 N/mm² divided by the modulus, about half a millistrain, and the doubler beside it is strained by nothing. From then on they strain together, so the web is permanently half a millistrain ahead of the plate, and no later load can close that gap.

The doubler, meanwhile, is loafing. Its outer edge is at 39 N/mm², because the only moment it ever sees is the 40 per cent that came later. A plate that could share the web’s stress equally, had it been there from the start, is carrying less than a third of the stress the web beside it carries. Nothing is wrong with the plate; it simply joined late.

Which free body produced the number #

The free body is the end of the beam, cut at the end of the cope, and the only thing that changes between the stages is the section the cut passes through.

The moment on the cut is the reaction times the lever arm from the bolts to the cut, and that is fixed by statics however the section is made. The stress it produces at a height $y$ is $M(y - \bar{y})/I$, with the neutral axis $\bar{y}$ and the second moment $I$ of whatever section exists when the moment is applied. Two increments of moment applied to two different sections give

$$\sigma(y) = \frac{M_d\,(y - \bar{y}_0)}{I_0} + \frac{M_l\,(y - \bar{y}_1)}{I_1}$$

in the original steel and only the second term in the plate. The two sections disagree about where the neutral axis is — the doubler adds material over the whole web depth, above the flange, so the repaired section’s neutral axis sits higher — and the original steel keeps, locked into it, the distribution of the first stage.

There is no approximation in that sum as long as everything stays elastic, and every check that reads an elastic stress inherits it.

The checks that remember the order #

A coped end is judged four ways, and they read four different properties of the end. Following all four as the share of the reaction present at welding goes from nothing to all of it separates them cleanly.

Flexure reads the bending stress at the tee’s extreme fibre, and it moves in a straight line from the repaired value to the coped one as the share rises. Shear reads the shear stress in the original web, which carries the dead reaction alone and then its share — 9 mm of the 17 — of everything after; it moves the same way. Local buckling of the web’s free edge reads the compressive stress there against the critical stress of the plate that exists when it would buckle, which is the thickened one, and it moves too.

Tear-out — the bolt group tearing a block out of a web that is only a plate — does not move at all. Its utilisation is 0.30 with the doubler welded before any load and 0.30 with it welded under the whole reaction.

The consequence for this end is that the governing check changes. Welded to an unloaded beam the doubler brings every check below 0.30, and tear-out, at 0.30, governs. Welded under 60 per cent of the reaction, tear-out is still 0.30 but flexure has risen to 0.39 and governs instead. The repair is still worth having — the end was at 0.57 as coped — but it is worth less than the calculation that ignored the order of loading said, and the check it is worth least for is not the one it was designed around.

Why tear-out forgets #

The difference between the checks that move and the one that does not is the difference between a stress and a mechanism.

Tear-out is a block of web, bounded by the bolt holes, pulling out of the beam end. It fails when the steel along the block’s boundary has yielded in shear on one side and torn in tension on the other — a plastic mechanism, whose capacity is the area of the boundaries times the steel’s yield and ultimate strengths. The stress that was in the web before the doubler was welded is, from the mechanism’s point of view, a self-equilibrating stress already in the steel: it adds to the stress in some places and subtracts in others, and once the steel along the boundary yields, it flows until the whole boundary is at its strength regardless of where it started.

That is the reason the collapse load of a ductile structure is found from its mechanisms rather than by tracking its stresses: locked-in stresses change when yielding begins and not what the mechanism finally carries. It is the same reason a steel column’s residual stresses from rolling — yield that happened before any load was applied — lower its buckling load and leave its squash load alone. A check that is a mechanism gets the whole of a repair made under load; a check that is a stress gets the share of the load that came afterwards.

The coped end’s flexure is checked elastically, on its section modulus, and that is a choice with a reason: the web’s free edge buckles, and a section with a free compression edge is not trusted to develop a plastic moment. If the tee were compact enough to be checked plastically, its flexural capacity would forget the order of loading too. On a real coped end it usually is not.

The edge plate, under load #

The same repairs analysed on an unloaded beam gave a case in which the plate along the free edge was the right repair: a longer cope, where flexure rather than tear-out governed. Welded under load, that repair loses part of the only check it was chosen for.

As coped, this end governed in flexure at 0.57. The edge plate, welded to an unloaded beam, took flexure to 0.31 and handed the governing check to shear at 0.35 — a check the plate cannot reach. Welded under 60 per cent of the reaction, flexure only falls to 0.46, and it governs again. A repair that fixed exactly one check has been left fixing part of it, and in this case the loss decides the end’s utilisation rather than merely reducing its margin.

What propping buys #

The order of loading is not fixed by the building. It can be changed, for the price of a prop.

A prop under the beam near its end takes the reaction off the connection while the plate is welded. When the weld has cooled and the prop is released, the reaction returns — all of it — to a section that already includes the plate, and the end behaves as though it had been built repaired. The difference, on this end, is between 0.39 and 0.30.

The prop matters most where it is least convenient. If a check is already failing under the dead load alone, no plate welded under that load can bring it back: the dead-load stress is locked into the original steel on the original section, and the plate can only limit what the next load adds. Only removing the load while the plate is fitted — propping, jacking, or unloading the floor above — gives the plate a share of the stress that is already there. For a plastic check such as tear-out, the prop buys nothing; for an elastic one it buys the whole of the dead-load share.

The same end under a heavier reaction shows where that becomes the whole question. Every utilisation here is proportional to the reaction, so at 700 kN the end as coped would be at 1.07 in flexure and, repaired before loading, at 0.63. Welded under a dead-load share $s$, its flexure is $0.63 + 0.44s$, which stays below one only while $s$ is below 0.84. With more than 84 per cent of the reaction already present as dead load, no doubler welded to that beam can bring its flexure below one, however thick the plate: the locked-in stress on the coped section is already most of the limit. Propped, the same doubler gives 0.63. Tear-out on the same end is 1.33 as coped and 0.70 with the doubler whatever the share, because it is a mechanism.

How much of a reaction is dead load depends on what the floor is for. A plant room or a roof carrying heavy finishes is mostly dead load; an office floor carries a large share of imposed load that is not present when the building is empty. Before a repair is designed, that share has to be established for the day the plate will be welded, not the day the building was designed — the furniture and the people are not dead load, and they can be moved out.

The heat of the weld #

Welding a plate to a loaded web has a second cost, and it arrives during the work rather than after it.

A weld heats a strip of the parent steel beside it far beyond any of those temperatures for a few seconds. Steel’s strength falls with temperature, and a strip of web that is carrying 99 N/mm² of dead-load stress yields under it once the weld’s heat takes it past about 680 °C. The strip is narrow and the cooler steel around it takes up the load it sheds, so the beam does not fail. But the strip yields while it is being welded, the web distorts, and a thin web with a free edge — which is what a coped web is — can buckle locally under a stress it was carrying comfortably when cold.

The more load is on the beam, the lower the temperature at which that happens, and the wider the strip that passes it. The practical answers are the ones fabricators use: short intermittent runs of weld, so that no long length of web is hot at once; welding in a sequence that lets each run cool; and, again, propping, which takes the stress off the web before any heat is applied.

What the repair is still worth #

None of this makes a repair under load useless, and there is one purpose for which it is worth its full value.

A coped end with a fatigue crack at its re-entrant corner is repaired because of the stress range, which is what a fatigue check is made on, and the stress range comes from the load that varies — the live load, which arrives and leaves after the plate is in place. Every cycle of live load is carried by the repaired section, so for fatigue a doubler welded under the full dead load is as effective as one welded to an empty beam. It is only the static checks, which read the total stress, that remember the dead load was there first.

Where the model stops #

Both stages are elastic. The superposition holds until something yields. A web already close to yield under its dead load will yield locally when the later load arrives, and redistribute toward the plate more than the elastic sum says.

The plate is fully connected. The staged sum assumes the doubler’s welds transfer the shear flow that makes it act with the web. A plate welded only round its perimeter, or intermittently, joins in over a length rather than at once, and its share at the cope is less.

The repair weld’s own residual stresses are left out. The weld that attaches the plate shrinks as it cools and leaves tension in the weld and compression around it — a stress nobody applied and nothing restrains the steel from carrying, self-equilibrating, which, like the dead load, affects the elastic checks and not the plastic ones.

And the heat model is borrowed. The strength reduction factors were measured on steel heated slowly in fire. A welding thermal cycle is fast, and the metallurgy of the heat-affected zone is not that of steel in a fire, so the temperatures in the last figure are an indication of where the loaded web starts to give way rather than a calculation of it.

Still open: the plate’s own welds #

The staged calculation assumed the plate acts with the web. Whether it does is a question about the welds that attach it, and that question is still open. The shear flow a doubler’s welds must carry to bring the plate into the section, which under load is the live share only, and which fixes how long a doubler has to extend beyond the cope before it is doing anything. The re-entrant corner itself, where a fatigue crack is usually the reason for the repair, and where a drilled hole at the crack tip is the repair that needs no plate. And the heat-straightened coped end, where a distorted web is brought back by heating it under restraint, and the same temperatures appear from the other direction.