The land that is already a crack

The root that has to stretch for it counted a partial-penetration butt weld made from one side twice. The plates carry their pull on their mid-thickness and the fused throat does not lie on it, so at half penetration of a 20 mm S355 plate the throat carries its force with an eccentricity half its own depth. Counted elastically — the throat full when its most loaded fibre reaches the limit — it keeps a quarter of what a weld made from both sides would carry. Counted plastically — every fibre at its limit, the face pushing back while the root pulls — it keeps 0.41.

That essay found that the plastic count needs the root to strain several times its yield strain while the compression block forms at the welded face, and then it named the difficulty. The root of a single-sided weld is not a corner of a rectangle. It is where the fused metal stops, and beyond it, from the root to the far face, the two plates sit against each other unjoined. Pulled across, the unfused land opens. It is a crack, its tip is the root, and the plastic count asks the most strain of exactly the fibre that sits at the tip.

Neither count knows that. The elastic count refuses to let the root yield at all, which is safe if a crack tip could be made to stop at its yield stress and is not, since an elastic crack tip has no finite stress. The plastic count lets the root yield without limit, which is safe if the metal there can tear a little and stop. The question this essay takes up is what a third count — one that treats the land as the crack it is — makes of the same weld.

The land is an edge crack, and its front is the whole weld #

Draw the joint plane, the face along which the two plates meet. Over a depth $a$ from the welded face it is fused; over the rest, $c = t - a$, it is not. In section that is a strip of width $t$ with a crack $c$ deep cut in from one edge: the single-edge-cracked strip, the most tabulated geometry in fracture mechanics.

The pull across the weld is carried by plates that are free to rotate, so the strip is loaded in tension on the plates’ mid-line with its ends free to turn, and its stress intensity at the crack tip is

$$K_I = \frac{N}{t}\sqrt{\pi c}\;F_I(\alpha), \qquad \alpha = \frac{c}{t},$$

where $N$ is the pull per millimetre of weld and $F_I$ is Tada’s factor for the strip, good to half a per cent at any depth. At half penetration $F_I = 2.83$. The eccentricity that cost the elastic count three quarters of the throat is already inside that factor: it is the reason the factor is nearly three, against 1.12 for a shallow crack in a wide plate.

So a resultant of $R$ newtons per square millimetre of throat, pulled straight across, puts $N = 10R$ on each millimetre of a 10 mm throat and a stress intensity of $(10R/20)\sqrt{10\pi} \times 2.83 = 7.9R\ \mathrm{N/mm^{3/2}}$ at the root. At the plastic count’s 183 N/mm² that is $46\ \mathrm{MPa\sqrt{m}}$; at the elastic count’s 110, 28.

Then there is what the crack’s front is. In a plate with a crack through its thickness, the front runs from face to face and is as long as the plate is thick. Here the front runs along the root, parallel to the weld, for the whole length of the weld. That matters because the toughness of a ferritic steel in its transition range is not one number: the toughness that belongs to the plate found that cleavage starts at the weakest spot the front passes through, so a longer front samples more weak spots and is less tough. A 300 mm weld has a crack front twelve times as long as the 25 mm specimen on which toughness is measured.

Two checks and the line between them #

A cracked member has two ways to fail, and the crack between the two checks found that the usual practice of checking each separately and taking the lower answer overstates the strength where the two are close. Fracture mechanics alone says the root tears when its stress intensity reaches the toughness. The plastic count alone says the throat collapses when every fibre is at its limit. Near either extreme one of them is right; between them, the crack tip’s plastic zone is large enough to matter and too small to have spread over the ligament.

Dugdale’s strip-yield model joins the two, and written as a failure assessment line it is one curve in two ratios: the stress intensity over the toughness, $K_r$, and the load over the plastic collapse load, $L_r$. Here the plastic collapse load is exactly the plastic count of the throat, because the throat is the ligament the crack leaves.

Both ratios grow in proportion to the load, so a joint loaded up from nothing moves out from the origin along a straight line, and its capacity is where that line meets the curve. The slope of the line is the stress intensity at the plastic collapse load divided by the toughness — $46\ \mathrm{MPa\sqrt{m}}$ over whatever the root has. A tough root’s ray is shallow and runs out along the bottom of the diagram to meet the line near its vertical end, at the plastic count. A brittle root’s ray is steep and meets the line near its flat top, where the answer is fracture mechanics alone.

The dotted line is the elastic count’s load, 0.60 of the plastic one. Where a ray crosses that line inside the curve, the elastic count is safe; where it crosses it above the curve, the root tears before the load the elastic count would have allowed.

Toughness picks the point between a quarter and 0.41 #

Read across the bottom of the diagram: the assessed share is a smooth curve that starts near the toughness-alone line, climbs through the elastic count at $30\ \mathrm{MPa\sqrt{m}}$, and bends over to meet the plastic count by about 100. Above $100\ \mathrm{MPa\sqrt{m}}$ the root is tough enough that the plastic count is the whole answer, and the strain the earlier essay worried about is strain the metal can give. Below 30 the root tears before the elastic count’s load, and the count that refuses to let the root yield is not the conservative one.

The dashed line is the simple practice, the lower of fracture mechanics and plastic collapse, and the gap between it and the solid line is what the interaction costs. It is largest where the dashed line has its corner, at $46\ \mathrm{MPa\sqrt{m}}$, where fracture mechanics alone says the root will see 46 at the plastic load and so can just reach it: the assessment gives 0.34 against 0.41, 18 per cent less. That is the same 0.81 of the simple check that Dugdale’s model gives for any crack at its transition length, translated into a weld.

The middle of the curve is where the master curve lives. A good weld metal at a comfortable temperature has a median toughness well above $100\ \mathrm{MPa\sqrt{m}}$ and a 5 per cent value somewhere in the middle of this axis; a weld metal close to its transition temperature has a 5 per cent value of 40 or 50. The weld that sits there keeps 0.30 to 0.36: much more than the elastic count, which was written for a root that could not yield at all, and a fifth less than the plastic count, which was written for a root that could yield without limit.

A long weld is a cold weld #

The toughness on that axis is the toughness of the crack front the weld actually has, and the master curve says how that depends on the front’s length. At a temperature $T$ and a reference temperature $T_0$ — the temperature at which the median toughness of a 25 mm specimen is $100\ \mathrm{MPa\sqrt{m}}$ — the scale of the Weibull distribution for a 25 mm front is $31 + 77e^{0.019(T - T_0)}$, and for a front of length $B$ it is $20 + (K_0 - 20)(25/B)^{1/4}$. A front twelve times as long has its excess over the floor cut by $12^{1/4} = 1.86$.

At −20 °C, for a weld metal whose reference temperature is −38 °C, the 5 per cent toughness of a 25 mm front is $77\ \mathrm{MPa\sqrt{m}}$, of a 300 mm front 51, and of a 3 m front 37. Those are three welds of the same metal, the same throat and the same plate, distinguished only by how long they run, and they keep 0.41, 0.36 and 0.29 of a double-sided throat. The short one is effectively at its plastic count. The long one is closer to the elastic count than to the plastic.

Put the other way round, a longer weld behaves like a shorter one at a lower temperature. The curves for 300 mm and 3 m are the 25 mm curve shifted to the right, by about 18 °C for the first and 40 °C for the second over most of the range. A weld specified with a toughness measured on a 25 mm specimen at the design temperature has had that margin spent before it is loaded, and nothing in the specification says so, because the specimen is the right size for a crack through a plate and the wrong size for a crack along a weld.

This is the same steel brittle in January, with the calendar replaced by a tape measure. It is also the reason the weld’s length, which every count of throat stress divides out, comes back into the answer. A capacity per millimetre of weld that depends on how many millimetres there are is not something a stress count can produce.

Drawn larger, the same weld tears sooner #

The elastic count of a single-sided throat depends only on the ratio of penetration to thickness: the root factor $1 + 6e/a$ is a proportion. So does the plastic count, whose share at half penetration is $\sqrt2 - 1$ at any size. A weld drawn twice as large, throat and plate together, keeps the same share by either count.

A crack does not scale that way. At the same stress, a crack twice as deep has a stress intensity $\sqrt2$ times as large, and the toughness it is measured against does not change with the drawing.

In a 10 mm plate the half-penetration root keeps 0.40, practically the plastic count. In 20 mm it keeps 0.36, in 40 mm 0.28, and from 56 mm it keeps less than the elastic count. The two flat lines are what a stress count says about every size; the falling curve is what the crack says. This is the bigger one is the weaker one in a joint, and its mechanism is the oldest one in the subject: the energy a crack releases as it grows scales with its size, and the energy the metal absorbs to let it grow does not.

There is a second size effect hiding in the first. A plate twice as thick is usually in a larger structure with longer welds, so its crack front is longer as well as its crack deeper, and the two compound. The figure holds the weld at 300 mm in every plate, which is generous to the thick ones.

It also says something about how the rule for these welds was found. Partial-penetration butt welds made from one side were in use in thin plates long before they were used in thick ones, and in a 10 mm plate the crack costs almost nothing; whatever was learned about them in thin plate was true there and stops being true as the plates grow.

How deep the land may be left #

The penetration decides two things at once: how eccentric the throat is, which both stress counts see, and how deep the crack is, which only the third sees.

The crack takes the largest share of the plastic gain in the middle of the range, around 0.7 of the plate, and the smallest at either end. At shallow penetration the throat is so eccentric and so small that its plastic capacity is low, and the stress intensity at that low load is modest even though the land is deep: the crack takes 15 per cent of the gain at 0.3. At deep penetration the land is short, the crack is small and the throat is nearly a full butt weld: 24 per cent at 0.9. Between the two the throat carries a large load across a land still several millimetres deep, and the crack takes 44 per cent of what yielding would have given back.

That is an uncomfortable place for the cost to peak, because 0.7 is where the root that has to stretch found the single-sided throat most useful: deep enough to outcarry a fillet once the load has turned along the weld. The plastic count’s best case is the crack’s worst.

Along the weld the land slides instead of opening #

Turn the load. The part across the weld still opens the land, now in proportion to $\cos\theta$. The part along it, a shear on the throat parallel to the weld, has no lever arm about the throat and puts no tension across the joint plane, so the land does not open under it. It slides. The faces of the land move past each other parallel to the root — the third way a crack can be loaded, tearing, in which the crack front is sheared along its own length.

The stress intensity for that mode, for the same edge-cracked strip, is $K_{III} = (V/t)\sqrt{\pi c}\,\sqrt{2\tan(\pi\alpha/2)/(\pi\alpha)}$, smaller than the opening mode’s at the same force because the strip’s bending does not amplify it. The two are combined here by their energy release rates, $K_{eff}^2 = K_I^2 + K_{III}^2/(1 - \nu)$, which is the rule the British assessment procedure uses and which treats a tearing crack as being exactly as dangerous, energy for energy, as an opening one.

Up to about 23° nothing changes but the scale: the throat’s capacity rises as the double-sided one does, for the reason the earlier essay found, and the crack takes the same share of it. Beyond that the counts spread apart. The light line, which counts only the opening, rejoins the plastic count as the load turns along the weld: with no pull across the land, nothing opens it. The solid line, which counts the sliding as well, does not. At 84° the root keeps 0.92 of a double-sided throat against 0.99 plastically, and straight along the weld, 0.93 against 1.

That last stretch is a reversal worth stating. Straight along the weld the eccentricity has gone, the elastic and plastic counts coincide, and both say the single-sided throat is as good as a double-sided one, since it carries the shear in a throat that faces it squarely. The land is still there, though, and still a crack, and the shear slides it. From 78° to straight along, the crack is the only thing that distinguishes a single-sided weld from a double-sided one, and the elastic count, which knows nothing of it, is no longer below the assessed capacity.

Whether the sliding mode should be counted as fully as the opening one is the softest assumption here. Cleavage is an opening event, and steels in the transition range are generally tougher against tearing than against opening, so the energy rule is cautious. With the sliding left out, the root’s capacity along the weld is the plastic count; with it counted fully, it is 7 per cent less. The truth for a given weld metal is somewhere between, and is a matter for a test rather than a calculation.

The tear by hand at half penetration #

Take the default case: a 20 mm S355 plate, a 10 mm throat, pulled straight across, a weld 300 mm long in weld metal of reference temperature −38 °C, at −20 °C. The double-sided throat’s capacity is the cap on normal stress, $0.9 \times 490 = 441$ N/mm².

The toughness first. At 18 °C above the reference temperature, the scale for a 25 mm front is $31 + 77e^{0.019 \times 18} = 139\ \mathrm{MPa\sqrt{m}}$. For a 300 mm front it is $20 + 119 \times (25/300)^{1/4} = 84$. Its 5 per cent value is $20 + 64 \times (-\ln 0.95)^{1/4} = 20 + 64 \times 0.476 = 51\ \mathrm{MPa\sqrt{m}}$, or $1{,}598\ \mathrm{N/mm^{3/2}}$.

The plastic count is $0.414 \times 441 = 183$ N/mm² of throat. At that load the stress intensity is $7.92 \times 183 = 1{,}447\ \mathrm{N/mm^{3/2}}$, 0.91 of the toughness — so fracture mechanics alone would pass the plastic count with 10 per cent to spare, and the simple check would stop there.

The assessment does not. Try 157 N/mm². Then $L_r = 157/183 = 0.86$, and the strip-yield line there is $0.86/\sqrt{(8/\pi^2)\ln\sec(0.43\pi)} = 0.86/\sqrt{0.811 \times 1.50} = 0.78$; the joint’s own $K_r = 7.92 \times 157/1{,}598 = 0.78$. The ray meets the line at 157 N/mm², a share of $157/441 = 0.36$ of the double-sided throat. The elastic count was 110, the plastic 183.

What the crack count rests on #

The plates are free to rotate. Tada’s factor for a strip with free ends carries the whole eccentric moment into the crack, which is what the elastic and plastic counts assumed too. Plates held against rotation by the structure round them would carry part of the moment themselves, and the stress intensity would fall faster than either stress count’s root stress does, because a clamped edge crack has a much smaller factor at depth. Every share here would then rise, the assessed one most.

The weld metal’s reference temperature is chosen, not found. It is set here at −38 °C, the plate’s value from the toughness that belongs to the plate. A weld metal’s reference temperature depends on its consumable and its heat input, and the metal at the root of a single-sided weld is the first pass, diluted by the parent plates and reheated by every pass above it. A root 20 °C colder in its transition than assumed moves every curve here the same distance, and sends the long weld below its elastic count at −20 °C.

There is no residual stress. A weld cools and shrinks against the plates it joins, and leaves a tension across its own root that can be as large as the yield stress. It adds to the stress intensity at the root without adding to the load, which in the assessment raises $K_r$ and leaves $L_r$ alone. For a weld that has not been stress-relieved, the assessed shares here are an upper limit.

The land is a sharp crack. A real unfused land has a root radius of a fraction of a millimetre and its faces are often slightly apart or partly fused, which blunts it a little. The assessment’s question is whether the metal at the tip can tear and stop, and a blunt tip helps it; but the flaw that sets the strength is the flaw that is there, and the land is there by design.

Still open: the stress the welding left behind #

Every count here started the joint with no stress in it. The welding did not: the root cooled last against fused metal above and cold plate around it, and the shrinkage it was refused is held as a tension across the land. In a crack assessment that residual stress is a secondary stress — it adds stress intensity at the tip but cannot cause plastic collapse, because it relaxes as soon as the ligament yields — and the stress that was there before the load found how large it can be in a rolled member before any weld is laid on it. Whether the root’s residual tension takes most of the margin the strip-yield line leaves above the elastic count, whether a proof load that yields the root relieves it for good, and whether a single-sided weld that has been loaded once is then tougher than one that never has, is what the welding’s own history asks of the crack it left.