The hole at the tip of the crack

The repairs to a coped beam end considered so far all add steel. A doubler, a stiffener along the free edge or both each fixes some of the checks a cope fails and leaves the others, and a plate welded to a beam already carrying its load helps only with the load that arrives after the weld has cooled. Both essays took for granted that the reason for the repair was a static shortfall — a beam that could not carry its reaction through the section the cope left.

In practice the commonest reason is different. Coped ends on bridges and crane runways crack, at the re-entrant corner where the cope’s horizontal cut meets its vertical one, and they crack in fatigue under a load they carry comfortably. The commonest repair is not a plate. It is a hole.

A crack where the two cuts meet #

The beam drawn is a stringer of an old bridge, framing into a floor beam: 457 mm deep with a 190 mm flange, coped 150 mm long and 80 mm deep so that its top sits level with the floor beam’s, bolted to it through the web. Its end reaction is 350 kN, and 70 per cent of that comes and goes with the traffic.

The corner is the worst place on the beam. The coped section is a tee with its flange at the bottom and the free edge of its web at the top, so the reaction’s moment puts the highest bending stress exactly at the top of that web, at the end of the cope. The corner where the two flame-cut edges meet adds a stress concentration of several times, and the flame-cut surface adds roughness. Every passing lorry cycles that corner through a stress range of 48 N/mm² nominally, several times that locally, and cracks start there and run down the web.

A crack in a steel web is not an immediate danger. The section left beside it is most of the section, the steel is ductile, and the static checks of the coped tee barely notice a crack of a few tens of millimetres. What makes it dangerous is that it grows, and grows faster the longer it is. The repair has to stop it growing.

A hole that turns a crack into a notch #

A crack grows because its tip is infinitely sharp: the stress there is unbounded for any load at all, and fracture mechanics measures its severity instead by the stress intensity $K$, which grows with the applied stress and the square root of the crack’s length. Each cycle of load extends the crack by an amount set by the range of $K$.

Drill a hole centred on the tip and the tip is gone. What is left is a smooth notch whose radius is the hole’s, and a notch with a radius has a finite stress at its root — a hole multiplies the stress around it, but only by a finite factor. If that local stress range is low enough, the notch does not start a new crack, and the old one has stopped.

How low is low enough was measured rather than derived. Fisher and his colleagues at Lehigh tested drilled holes at the tips of fatigue cracks in bridge details around 1980, and found a simple boundary: a hole stops the crack if

$$\frac{\Delta K}{\sqrt{\rho}} < 10.5\sqrt{f_y}$$

with $\Delta K$ the range of stress intensity the crack had at its tip, in $\text{MPa}\sqrt{\text{m}}$, $\rho$ the hole’s radius in metres and $f_y$ the steel’s yield strength in MPa. The left-hand side is, within a constant, the stress range at the root of a notch of radius $\rho$ that had the crack’s $\Delta K$; the right-hand side is a threshold that rises with the steel’s strength. Solved for the hole,

$$\rho = \left(\frac{\Delta K}{10.5\sqrt{f_y}}\right)^2.$$

The radius goes as the square of the stress intensity. The stress intensity goes as the square root of the crack’s length. So the hole’s radius goes in proportion to the crack’s length, and a crack twice as long needs a hole twice as big.

The hole grows with the crack #

At the stringer’s cope that proportion is about two-fifths: the hole’s diameter is roughly 0.4 times the crack’s depth.

A 10 mm crack needs a 4.4 mm hole, which is smaller than anybody would drill. A 30 mm crack needs 12.4 mm; a 60 mm crack, 24.3 mm. The curve bends slightly upward, because the free edge’s correction to the stress intensity grows as the crack takes a larger share of the web, but over the range that matters it is nearly a straight line.

That straight line is the reason a “standard” arrest hole is the wrong idea. Practice often reaches for a 20 or 25 mm hole whatever the crack, on the reasoning that a hole is a hole. At this cope a 25 mm hole stops a crack up to about 64 mm deep, and a longer crack drilled with it starts again from the edge of the hole. The magnetic-base drills used on site cut holes up to about 50 mm in a web, which serves a crack of about 110 mm. Beyond that, the hole that would stop the crack cannot be made where the crack is.

The grade of steel matters through the threshold. A stronger steel tolerates a higher notch stress before it cracks again, so the radius goes as one over the yield strength: an S460 web needs a hole 23 per cent smaller than S355 for the same crack, and an S275 web one 29 per cent larger. Old riveted bridges, which are where cracked copes are found, were built of steels nearer the lower grade.

The section beside the hole #

The hole takes steel away from a section that was already the weak point of the beam, and that has to be checked too.

The coped tee is surprisingly tolerant. Uncracked, its static utilisation — the larger of its flexure and its shear under the whole reaction — is 0.52. A 30 mm crack with its 12 mm hole raises that only to 0.58, and a 60 mm crack with its 24 mm hole to 0.71. A crack starts at the top of the web, where the tee is least useful in bending — the flange at the bottom carries most of its moment — and the hole beneath it takes off a strip of web whose share of the shear is in proportion to its depth.

It runs out at 100 mm, where the crack and its 45 mm hole have taken the web down by a third and the tee reaches its capacity. The hole outgrows a 50 mm drill at 109 mm. Two independent limits — what the drill can make and what the steel beside it can carry — arrive within ten millimetres of each other, and either way the drilled repair has a ceiling at about a quarter of the way down the web. Past it the beam needs a plate, with all the questions about which check that plate repairs.

The coincidence is not a law. The static limit reads the whole reaction, and the drill limit reads only the cycling part of it, through the hole’s size. On a beam with a different share of live load they separate, and which one arrives first changes.

Most of a crack’s life is spent short #

A ceiling at 100 mm sounds generous for a crack that was found at 30. The growth curve says otherwise.

Each cycle extends the crack by an amount that rises as the cube of its stress intensity, and the stress intensity rises with the crack. So a crack grows slowly while it is short and very fast once it is long, and the loops it grows on are all the same size; only the crack changes. From a 2 mm flaw at the corner it takes 1.77 million cycles to reach 10 mm — 62 per cent of the whole of its growth to the 100 mm ceiling. It takes another 0.66 million to reach 30 mm, and from 60 mm to 100 mm it takes 0.15 million, five per cent of the whole.

A crack spends most of its life too small to see, and very little of it long enough to worry anybody. That is the general shape of fatigue crack growth and it is why a structure has to be designed to be found in time: the useful part of a crack’s life, for an inspector, is the part when it is long enough to see and still short enough to deal with, and that part is short.

How long there is to drill #

The practical question is not how long the crack takes to grow, but how long there is between finding it and losing the cheap repair.

A crack found at 10 mm leaves 1.09 million cycles before the drilled repair stops working. At a thousand heavy vehicles a day — a busy road, a modest railway — a million cycles is about three years, which is a comfortable margin for a repair that takes an afternoon. A crack found at 30 mm leaves 0.43 million cycles, about fourteen months. One found at 60 mm leaves 0.16 million, about five.

A visual inspection under paint reliably finds a crack at a cope only when it is some tens of millimetres long; with the paint removed and dye penetrant applied, a few millimetres. The difference between those two is the difference between three years and fourteen months of time to act, on the same beam, under the same traffic. The interval between inspections is set by the depth at which a crack is reliably found, not by the depth at which it becomes dangerous. On this detail a painted crack is seen at about three times the depth a cleaned one is, and the time left to act is two and a half times shorter for it.

More traffic, smaller holes allowed #

The share of the reaction that cycles enters the drill’s limit and the steel’s limit differently, and the figure shows how far apart they can move.

The static limit reads the whole reaction and feels the live share only through the hole’s size, so it moves slowly: from 113 mm at 40 per cent live to 85 mm at all of it. The drill’s limit reads the live range directly, and the hole goes as its square: 187 mm at 40 per cent, 109 at 70, 61 at all of it. They cross at about three-quarters live. Below that the steel beside the hole gives out first; above it the drill does.

A crane runway girder, whose end reaction is nearly all crane, is at the right-hand end: its arrest holes have to be big early, and a drilled repair runs out at a crack of about 60 mm. A floor beam in a building, where the reaction is mostly dead load, is at the left, and a drilled repair would serve far down the web — except that it would rarely crack in the first place.

The hole as a measurement #

A drilled hole is unusual among repairs in that it reports on itself. If it was big enough, the crack stops, and the hole sits in the web for the rest of the beam’s life with nothing at its edge. If it was too small — because the crack was longer than it looked, because the traffic is heavier than assumed, because the hole missed the tip — a new crack starts from the hole’s edge, usually within a few hundred thousand cycles, and the next inspection finds it.

That makes the first inspection after drilling part of the repair. The hole is cleaned, its edge checked with dye penetrant for any crack it failed to contain, and the result recorded. A crack that has restarted from an arrest hole is evidence that the stress range at the corner is higher than the calculation assumed, and it is a better measurement of that range than the calculation is. The response is not a bigger hole on the same reasoning, but a reason to look at the connection’s real behaviour: a coped end bolted through its web is supposed to be a simple support, and many are stiffer than that, so the cope carries a moment from the floor beam’s rotation that no simple calculation gives it. Cracks at copes on old riveted bridges are most often of that kind — a cope working harder than its design said, because the connection was more rigid than a pin.

Seen that way, the drilled hole and the plate that arrives after the load answer different questions. The plate is a response to a static shortfall: the section cannot carry what it must. The hole is a response to a fatigue crack in a section that can: it removes the crack’s tip and asks whether the stress range was what everyone thought. When the answer is yes, the hole is the whole repair; when it is no, it has bought the time and the evidence to design the plate properly.

One crack, by hand #

The live moment at the cope is the cycling part of the reaction times the distance to the end of the cope: $0.7 \times 350 \times 0.160$ = 39.2 kN·m. On the coped tee’s section modulus at its free edge, $8.1 \times 10^5$ mm³, that is a nominal stress range of 48.3 N/mm² at the top of the web.

A 30 mm edge crack in a strip in bending, 8 per cent of the 377 mm strip’s depth, has Tada’s geometry factor $1.122 - 1.40(0.08) + 7.33(0.08)^2 - \dots = 1.05$, so

$$\Delta K = 1.05 \times 48.3 \times \sqrt{\pi \times 30} = 493\ \text{N/mm}^{1.5} = 15.6\ \text{MPa}\sqrt{\text{m}}.$$

Fisher’s criterion with $f_y$ = 355 MPa sets $10.5\sqrt{355}$ = 198 MPa, so

$$\rho = (15.6 / 198)^2\ \text{m} = 6.2\ \text{mm}$$

and the hole is 12.4 mm across.

A strip in bending, a crack in air #

The calculation rests on choices that limit it.

The tee is treated as a strip for the stress intensity. The crack runs down a web with a flange at its far end, and the flange stiffens the section against the crack’s opening, so the stress intensity is lower than a rectangular strip of the same depth gives. The holes here are on the large side, which is the safe side.

The corner’s stress concentration is left out of the crack. A short crack at a sharp re-entrant corner feels the corner’s concentration as well as the nominal stress, which is why cracks start there. For cracks of tens of millimetres the concentration has decayed and the nominal stress governs; for the first few millimetres the growth is faster than drawn.

The growth is in air, at constant amplitude. Traffic is a spectrum of loads, not one range repeated, and the equivalent constant range is the cube root of the mean cube. A bridge’s stringers sit under a deck that leaks; corrosion fatigue grows a crack faster and has no threshold.

The criterion is empirical. Fisher’s boundary was fitted to tests on welded bridge details with holes of a certain range of sizes and finishes. A hole that is flame-cut rather than drilled, or drilled and not reamed, or that misses the tip, has a rougher root than the tests had, and the crack may restart from it. The tip has to be found — usually with dye penetrant, because a fatigue crack’s tip is finer than the eye can follow — and the hole centred a little beyond it.

Still open: the cope that was drilled before it cracked #

The crack started at the cope’s corner because the corner was sharp. A cope cut with a generous radius at its corner is a notch that was given its hole before any crack existed, and Fisher’s criterion read the other way says how large that radius has to be for no crack to start: a crack of zero length is replaced by the corner itself, with the corner’s concentration multiplying the nominal stress range. Whether a cope radius that a fabricator can cut cheaply — 15 or 20 mm, against the 10 mm or less a flame torch leaves — is enough to keep the corner below the threshold for the life of a stringer, and so to make the drilled repair unnecessary, is a question about a dimension that appears on no drawing and is decided in the shop.