Strongest across, and first to crack across

The first four essays on the weld group were about finding its worst point. The corner that is not the worst point showed that the point furthest from the centroid is not always it; the radius rule, and where it fails that the direction of the two stress components decides where the peak is; the weld that is stronger where it is pulled that a fillet weld’s capacity itself depends on direction, which moves the peak again. All four were static. Each of them has named the same case as the next to be worked out: “weld group fatigue, where the peak point rather than the average decides the category.”

What happens is that the direction argument does not disappear. It reverses.

The same two directions, rated the other way round #

A fillet weld’s static strength depends on the angle between its force and its length because of the plane it fails on. The weld that is stronger across than along worked it through: a force along the weld puts pure shear on the 45-degree throat, a force across it splits into equal normal and shear stresses there, and the directional check $\sqrt{\sigma_\perp^2 + 3(\tau_\perp^2 + \tau_\parallel^2)}$ counts shear three times over and normal stress once. So the weld pulled across carries $\sqrt{1.5} = 1.22$ times the weld pulled along.

EN 1993-1-9 reads the same stresses for fatigue and does something quite different with them. It combines the two throat stresses that a force across the weld produces into a normal range, $\Delta\sigma_{wf} = \sqrt{\Delta\sigma_\perp^2 + \Delta\tau_\perp^2}$, which for a force across the weld in the plane of the joint is simply that force over the throat. The force along the weld gives a shear range $\Delta\tau_{wf} = \Delta\tau_\parallel$. And the two are checked against different detail categories: a load-carrying fillet weld’s normal range against category 36, on a slope of three, and its shear range against category 80, on a slope of five, combined by the interaction

$$\left(\frac{\Delta\sigma_{wf}}{\Delta\sigma_R}\right)^3 + \left(\frac{\Delta\tau_{wf}}{\Delta\tau_R}\right)^5 \le 1$$

with each resistance read off its own curve at the design number of cycles.

The reason is where the crack starts. A load-carrying fillet weld cracks from its root or its toe, and the crack is opened by the stress normal to it. A force across the weld opens it directly; a force along the weld slides its faces past each other, which drives a crack far more slowly. The detail decides and the steel does not made the general point that a detail category is a statement about geometry, and this is geometry in its plainest form: the same weld, rated twice, because the notch at its root is oriented.

Plotted against the angle of the force, the two checks cross. Static capacity rises from along to across by 22 per cent; fatigue range falls by 55 per cent, to $36/80 = 0.45$ of the along value. At right angles to its length a fillet weld is at its strongest statically and at its weakest in fatigue, and the two disagree about it by a factor of 2.72.

Two criteria that weigh the same stresses oppositely #

It is worth being exact about why the static and fatigue ratings run in opposite directions, because the reason is not that one of them is careless.

The static check is a yield criterion applied to the throat. Von Mises counts a shear stress as $\sqrt{3}$ times as damaging as a normal stress of the same size, because shear is what makes metal flow, and the weld pulled across its length turns half of its force into normal stress on the throat. So the transverse weld is stronger: less of its force arrives as the component the criterion weighs heavily.

The fatigue check is a crack-growth criterion applied to the root. A crack grows under the stress that opens it, and a normal stress on the throat opens a crack lying in the throat, while a shear stress along the weld only slides the faces past each other. So the transverse weld is weaker: more of its force arrives as the component that opens cracks.

Both are right. They describe different failures of the same weld, and the directional enhancement in static design is paid for by exactly the stress component that fatigue is most sensitive to. A designer can use the 22 per cent where the load is static and must give it back, and more, where it is not.

A group whose two peaks are on different welds #

On a group, the direction of the force varies from point to point, and so the reversal moves the worst point. The simplest case is the plainest. Take the C-shaped group from the earlier essays — a 200 mm vertical run and two 80 mm returns, a 5 mm throat — and put the load through the centroid, so that every point carries the same force, $P/A$, with $A = 360$ mm of weld.

Statically, the worst point is on the vertical run, because there the force runs along the weld, the weak direction. The group’s static capacity is 526 kN.

In fatigue, every point on the run has a shear range and nothing else, checked against category 80; every point on the returns has a normal range, checked against 36. The allowable load range at two million cycles is therefore, at the run, $80 \times 5 \times 360 = 144$ kN, and at the returns $36 \times 5 \times 360 = 64.8$ kN. The fatigue peak is on the returns, where the static check has most in hand.

Around the group the two checks are nearly each other’s mirror image. The static use is highest along the run and 18 per cent lower on the returns; the fatigue use is highest on the returns and less than half as high along the run. A designer who found the static peak, as every earlier essay on this group taught, and checked fatigue there, would allow a load range of 144 kN on a group that takes 64.8. That is the first refutation, and the factor is the ratio of the two categories.

Drawn on the group, the reversal is a statement about which welds carry the connection’s life. The run, which static design is sized by, is the least used part of the group in fatigue. The returns, which static design treats as the reserve, are the fatigue-critical welds.

With an eccentric load, sometimes the same point #

Move the load away from the centroid and the torsion adds a force that rotates with position. The radius rule showed that this puts the static peak at the tip of a return for moderate eccentricities, where direct and torsional components add and the combined force runs at a steep angle to the return. There, both checks agree: the force is largely across the weld and large, and both put their peak at the tip.

At large eccentricity they part again. At 400 mm, the static peak is still at the tip of a return, where the force runs at 38 degrees to the weld. The fatigue peak has moved to the corner of the vertical run, where the torsional force is pointing straight across the run.

The overstatement is smaller here — 1.24 — because both points carry large forces with sizeable components across their welds, and the difference is only in the angle. But it is there, and it is on the unconservative side.

Swept across eccentricity, the two checks agree over a middle band, from about 20 to 180 mm, and disagree on either side of it. Near the centroid the disagreement is the full ratio of the categories. At the far end it is a quarter. Where they agree is where the rule taught for static design happens to work in fatigue as well, and nothing about the group tells a designer which band it is in without doing both checks everywhere.

A group that never agrees #

A box — the C closed by a second vertical run — is the shape used to weld a hollow section or a box stiffener. It is the opposite case.

For the box, the static peak and the fatigue peak are never the same point over the whole range. Two vertical runs share the torsion with two horizontal ones, the combined force is at a different angle at every corner, and the point where the force is most nearly along a weld is never the point where it is most nearly across one. A fatigue check at the static peak overstates the allowable load range by a factor between about 1.2 and 2.2 everywhere on the plot.

The longer the life, the further apart #

The two detail categories sit on curves of different slope. The normal-stress curve falls with a slope of three, the shear curve with a slope of five, so across a range of lives the normal-stress resistance falls faster. At 10⁵ cycles the ratio of the two resistances is 1.51; at two million it is the categories’ own 2.22; past five million the normal-stress curve’s slope changes to five as well, and the ratio settles at 2.54. The longer the life asked of a weld group, the more the across-the-weld direction dominates, and the further apart the static and fatigue peaks’ allowances become. A crane runway girder, a bridge detail or a machine support, designed for tens of millions of cycles, is on the right-hand side of the plot.

When fatigue decides at all #

The last number the calculations give is how small a load range has to be before fatigue stops mattering. Across the eccentricities drawn for the C group, the allowable load range at two million cycles is between 0.11 and 0.16 of the static capacity. So a group whose load varies by more than a tenth to a sixth of its static capacity, two million times, is designed by fatigue rather than strength. For a crane bracket or a vibrating machine mounting that is the ordinary case, not the exception. The load that never came near failing anything made the same point about members: a fatigue life is set by a range much smaller than anything a strength check would notice.

That changes what the group’s geometry is for. A static designer adds returns to a vertical run because a return, loaded across its length, is the stronger weld, and because returns increase the polar moment of the group. A fatigue designer sees those returns as the category 36 welds in a group that is otherwise category 80, and for a load through the centroid would do better with a longer vertical run and no returns at all: a single run of 360 mm along the load, all of it in shear, would allow 144 kN of range rather than the C’s 64.8. Its static capacity is the same 526 kN, since the C’s static check was governed by its run anyway: for a load through the centroid the returns bought nothing statically and cost a factor of 2.2 in fatigue. Under an eccentric load the returns earn their place, by the polar moment they add, and the choice between the two shapes depends on which check governs at the eccentricity the connection actually has.

A spectrum rather than one range #

The comparison so far is at one constant range. Real loads come as a spectrum, and the cycles that do not count showed that the damage from a spectrum is a cube-weighted average of its ranges, dominated by the heaviest few per cent of the cycles. The two categories do not share a weighting. The normal range, on its slope of three, is averaged as a cube; the shear range, on its slope of five, as a fifth power, which weights the heaviest cycles more still.

So under a spectrum the two peaks of a group are not only at different points but are sensitive to different parts of the load history. The returns, checked on the normal-stress curve, accumulate damage from the whole of the heavy end of the spectrum. The run, on the shear curve, accumulates it mostly from the few heaviest cycles, and many of the rest fall below the shear curve’s cut-off, which sits at a higher share of its category than the normal-stress curve’s does. A load history with a few very heavy cycles among many light ones therefore narrows the gap between the two points, since a power mean of five exceeds a power mean of three; it would take an extreme history to close a factor of 2.2. Doing the check at every point, with each point’s own components and each component’s own curve, is the only procedure that does not depend on the history’s shape.

What a designer does with this #

In practice it means three things. First, the group’s fatigue check has to be made round the whole group, point by point, and the static peak is no guide to where to look. Second, a group loaded near its centroid should be laid out with its long welds along the load where that is possible, because along-the-weld is category 80 and across is 36. Third, where returns are needed for an eccentric load, their tips and the corners where the torsion runs across a weld are the places to improve the detail: a fused root, a ground toe, or a weld size chosen for the normal range rather than the static force.

The same numbers by hand #

Everything here reduces to three numbers per point: the force per unit length, its angle to the weld, and the throat. With the load at the centroid of the C group, the force is $P/360$ kN per millimetre everywhere. On the run the angle is zero, so the whole force is a shear range on the throat, $\Delta\tau_{wf} = \Delta P/(360 \times 5)$ N/mm² per kN; setting it equal to 80 N/mm² gives 144 kN. On the returns the angle is 90 degrees, so it is all normal range, $\Delta\sigma_{wf}$; setting it equal to 36 gives 64.8 kN. The static check, by contrast, gives the run $5 \times 430/(0.85\sqrt{3}) = 1{,}460$ N/mm per millimetre and the returns 1.22 times that, so its peak is on the run at $1.46 \times 360 = 526$ kN.

Where the model stops #

The forces are elastic. The group’s force distribution is the elastic one, which is right for fatigue, where the stresses are well below yield, and conservative for static strength, where the group redistributes. The comparison here uses the elastic distribution for both, so the static peak is found in the same way the earlier essays found it.

The categories are EN 1993-1-9’s. Category 36 for a load-carrying fillet weld’s normal range is a lower bound over root and toe cracking together, and a weld whose root is fused or ground may be rated higher. The ratio of the categories is what reverses the direction argument, and it would take a normal-stress category above 80 to remove it.

The connected plates are rigid. A flexible plate concentrates the force near its stiffer parts, which changes the distribution around the group more than any refinement of the weld’s own behaviour does.

What the pictures cannot show #

That a fatigue crack in a weld group does not stop the connection at once. A crack at the tip of a return transfers its share to the run, whose shear range rises, and the group’s life after the first crack depends on how much the remaining welds can take. The interaction check is a first-crack criterion, and a group can survive its first crack by a long way if the rest of it is sized by a static check with margin to spare.

Still open: the plate that concentrates the weld’s load #

Everything above assumes the plates the weld joins are rigid. A plate welded across an unstiffened flange is not: the flange bends away from the load except near the web, and the weld next to the web carries far more than its share. EN 1993-1-8 handles it with an effective width, $t_w + 2s + 7kt_f$, and a requirement that the weld be sized to the full strength of the plate — which is a statement that the weld must be ductile enough for the flange to redistribute. How that effective width compares with the elastic distribution, and what it means for fatigue, where there is no redistribution to rely on, is the next question.