The shear that helps a butt weld

The throat that misses the load found that a partial-penetration butt weld’s throat, facing the load squarely, is not limited by the von Mises part of the directional method at all when it is pulled straight across. It is limited by the method’s second, separate condition — a cap of 0.9fu on the normal stress — in every grade. In S355 that makes it 441 N/mm² of throat against a fillet’s 385, 15 per cent stronger per millimetre.

That essay considered the pure pull and noted in passing that a butt throat can carry some shear alongside it before the von Mises condition takes over from the cap. It ended on the combined case: a throat carrying a transverse force and a longitudinal shear together, where the two conditions each govern over part of the range. This essay follows the combination round from straight across to straight along. The butt throat’s advantage over a fillet does not fall steadily from 15 per cent to nothing as the shear grows. It rises first.

Two conditions, drawn as a region #

The directional method checks a weld’s throat against

$$\sqrt{\sigma_\perp^2 + 3(\tau_\perp^2 + \tau_\parallel^2)} \le \frac{f_u}{\beta_w} \qquad\text{and}\qquad \sigma_\perp \le 0.9 f_u,$$

with $\sigma_\perp$ the normal stress on the throat, $\tau_\perp$ the shear across the weld in the throat’s plane, and $\tau_\parallel$ the shear along the weld. A load per millimetre of weld that has a part across the weld and a part along it puts different stresses on the two kinds of throat.

A butt throat faces the load. Its transverse part is all normal stress, $\sigma_\perp$; its longitudinal part is all $\tau_\parallel$. A fillet throat is inclined at 45°, so its transverse part splits equally into $\sigma_\perp$ and $\tau_\perp$, and the $\sqrt{3}$ on the shear terms makes it weaker across than a throat facing the load would be; its longitudinal part is $\tau_\parallel$ as for the butt.

Drawn this way the two conditions become one picture. The fillet’s region is an ellipse — the von Mises condition with the transverse stress split between $\sigma_\perp$ and $\tau_\perp$ — and the cap never reaches it, because a fillet never puts more than $1/\sqrt{2}$ of its transverse stress into $\sigma_\perp$. The butt’s region is a wider ellipse, since none of its transverse stress is weighted by three, cut off by a vertical line: the cap on $\sigma_\perp$, which does not care how much shear there is.

That straight edge is the whole of what follows. A region with a flat side is not a region whose largest radius lies on an axis.

The strongest direction is off square #

Walk a load round from straight across to straight along and read off how large a resultant each throat can carry.

The fillet behaves as expected: every degree of turn from across to along moves some of its load onto shear, which the method weighs by three, and its capacity falls from 385 to 314 N/mm² without a turning point.

The butt throat does not. Straight across it is at its cap and nowhere near its ellipse: in S355 the von Mises condition would allow 544 N/mm², and the cap stops it at 441. Turn the load a little and the normal stress on the throat is the resultant times $\cos\theta$, so the same 441 of normal stress now comes with a resultant of $441/\cos\theta$ — larger — and a shear the ellipse has room for. The resultant grows while the cap is the binding limit, and it keeps growing until the growing shear brings the point out to the ellipse. There the two conditions meet, and from there on the ellipse governs and the capacity falls.

The meeting point is where $\sigma_\perp = 0.9f_u$ and $\sqrt{\sigma_\perp^2 + 3\tau_\parallel^2} = f_u/\beta_w$ hold together, so

$$\tan\theta_s = \frac{\tau_\parallel}{\sigma_\perp} = \frac{1}{\sqrt{3}}\sqrt{\left(\frac{f_u/\beta_w}{0.9 f_u}\right)^2 - 1}.$$

In S355, with $f_u = 490$ and $\beta_w = 0.9$, the ratio in the bracket is $544/441 = 1.234$, $\tan\theta_s = 0.418$, and $\theta_s = 22.7°$. The resultant there is $441/\cos 22.7° = 478$ N/mm². A butt throat in S355 is strongest when the load on it is 23° off square, and is 8 per cent stronger there than when it is pulled straight across.

At the same angle the fillet carries 371 N/mm², and the ratio of the two is 1.29 — twice the 15 per cent the butt had over the fillet in pure tension. The ratio has a closed form at its peak, found by putting $\tan\theta_s$ into both ellipses:

$$\left.\frac{\text{butt}}{\text{fillet}}\right|_{\max} = \sqrt{\frac{2 + 3\tan^2\theta_s}{1 + 3\tan^2\theta_s}},$$

which is the ratio of the two ellipses on the ray through their meeting point, because at that point the butt is on its ellipse and the fillet is always on its own.

Every grade has the peak, in a different place #

The correlation factor $\beta_w$ decides where the peak falls. In S235 it is 0.8, the directional limit $f_u/\beta_w = 450$ is far above the cap of 324, and the ellipse is so far away that the cap governs until the shear is 0.56 of the pull, at 29°. Straight across the butt throat is barely stronger than a fillet in S235 — 2 per cent — and at its peak it is 23 per cent stronger. In S460, $\beta_w$ is 1.0, the directional limit of 540 is barely above the cap of 486, the switch comes at 15.6°, and the advantage there is 35 per cent.

The pure-pull comparison understates a butt weld’s advantage in every grade, and by most in the grades where it looks smallest. A designer choosing between a fillet and a partial-penetration weld in S235 on the strength of the pure-pull numbers would conclude that the extra preparation buys 2 per cent. For a load with even a modest shear component it buys ten times that.

The same shape wherever a smooth limit is cut off #

The butt throat’s region is a smooth criterion truncated by a separate limit on one stress, and that shape is common. A soil’s or a concrete’s strength envelope is usually drawn with a tension cut-off, a straight line across a friction curve, because the material cannot be pulled apart however it has been sheared; a material that carries no tension has its whole interaction diagram shaped by a limit on one side. Whenever a flat cut-off crosses a smooth boundary, the region’s largest radius does not lie on the axis the cut-off is perpendicular to. Where a limit on one component governs, a little of the other component is free, until the smooth part of the boundary is reached — and a ratio against a material without the cut-off peaks exactly there.

In the weld’s case the cap exists for a reason the rest of the criterion does not capture. $\beta_w$ was calibrated on fillet-weld tests, whose throats always carry shear with their normal stress, and a throat in pure tension, opening any flaw that lies across it, is the case the calibration least describes. So the cap is a rule written for the direction the tests did not cover — and one of its consequences is that the weld is rated stronger in a direction slightly off that one.

A web welded to an end plate #

The combination is not a curiosity. It is what the welds of a moment connection’s web carry.

A beam’s web, welded to an end plate from both sides, has to deliver to the plate the web’s share of the bending moment and the whole of the shear. The bending stress in the web runs from zero at the neutral axis to its largest at the flanges; the shear stress is nearly uniform over the depth. So along the web’s weld the load changes direction continuously — straight along the weld at the neutral axis, where there is only shear, turning toward straight across as the bending takes over near the flanges.

At the neutral axis the two welds need the same throat, 1.6 mm, because they carry pure shear and there is nothing to choose between them. Moving up the web, the butt pulls ahead faster than the pure-pull comparison would suggest, and at 200 mm above the axis, where the mix of shear and pull passes through the switch angle, it needs 28 per cent less throat than the fillet. At the flange the load is only 15.7° off square — the bending has nearly taken over — and the advantage is 1.21, still well above the 1.15 of the pure pull.

For a weld of one size along the whole web it is the flange end that governs, and there the fillet needs 4.9 mm and the butt 4.0.

In S235 the effect is starker. Pure pull says a butt throat is worth 2 per cent over a fillet in this grade. At the flange end of this web it is worth 15 per cent, and further down, 23. The bending at the flanges is lower in S235 while the shear is the same, so the load at the flange is further off square — 23° — and sits almost on the grade’s switch angle of 29°.

The throat is only half of the comparison. A fillet with a 4.9 mm throat has legs of 6.9 mm and a cross-section of $4.9^2 = 24$ mm² of weld metal for every millimetre of length. A partial-penetration weld with a 4.0 mm throat, made in a 45° bevel, has a cross-section of $4.0^2/2 = 8$ mm². At the flange end of the S355 web the butt weld puts down a third of the metal the fillets do. What it costs in exchange is the bevel, cut into the edge of the web before welding, and the control of penetration that a fillet does not need — and that is why fillets are the usual answer anyway, with the butt weld reserved for places where the metal, the heat or the distortion that comes with it matter.

Along the weld, one side is as good as two #

The essay on partial-penetration welds found a much larger penalty for making them from one side. The throat then runs from one face to a depth $a$ while the plates carry their force on their mid-thickness, so a transverse pull misses the throat’s centroid by $e = (t - a)/2$, and the moment $Ne$ puts $1 + 6e/a$ times the mean stress on the root: four times at half penetration. That essay’s single-sided weld kept a quarter of a double-sided weld’s capacity.

A longitudinal shear does not do this. It acts along the weld’s own axis, so it has no lever arm about it and puts no moment on the throat, and the single-sided throat carries it exactly as a double-sided one would.

So the single side’s penalty is a property of the direction of the load, not of the weld. Straight across at half penetration it is a factor of four; at 45°, two and a half; at 75°, one seventh; straight along, nothing. A single-sided partial-penetration weld is a poor transverse weld and a perfectly good longitudinal one.

That is how single-sided welds are used where they are used well. The corner welds of a steel box girder can only be made from outside the box, and they carry mainly the longitudinal shear flow that makes the flanges and webs work as one section; the transverse forces they see, from distortion of the box or from a load between diaphragms, are small beside it. The trough welds of an orthotropic deck are the counterexample: made from one side for the same reason, they carry the deck plate’s local bending straight across their throat, and that is where they crack.

Where the combination turns up #

The web weld is one case of a general pattern: a weld’s load changes direction along its length whenever it carries a force distributed one way and a shear distributed another.

The flange-to-web welds of a plate girder carry the shear flow that no bending diagram shows, $VQ/I$ per millimetre, along the weld and nothing across it. For them the butt throat has no advantage whatever — the load is at 90°, where the two throats are the same — and continuous fillets are the right weld. Put a crane wheel on the girder’s top flange and the same welds, unless the web is fitted to bear on the flange, carry the wheel into the web as well, spread over a length the flange’s own stiffness chooses. A 250 kN wheel spread over 400 mm is 625 N per millimetre across the weld; beside a support, the shear flow in a girder of ordinary proportions can be a third to a half of that. The load on the weld under the wheel is then between about 18° and 27° off square, around the direction in which the butt throat is strongest.

A weld group under an eccentric load varies its direction round the line from point to point, and the gain a fillet group gets from the transverse parts of that load was found to peak at a small eccentricity. A butt-welded group has its strongest direction off square rather than square, so its gain over the longitudinal value should peak at a different eccentricity — one that puts much of the line near 23° rather than near 0°. That is an expectation from the shape of the region, not a result computed here.

A moment connection’s end plate is the case drawn above. A moment crosses the gap as a couple, carried mainly by the flanges, and the web’s weld carries the shear and the web’s share of the couple together. The flange welds see a pull that is nearly square; the web weld sees every direction from square to along, which is why its size is decided by a mix rather than by either component alone.

The web weld by hand, at the flange #

At the flange the S355 web carries 355 N/mm² of bending and 100 of shear, so each millimetre of weld length, shared between two welds, receives a transverse force of $355 \times 10 = 3{,}550$ N and a longitudinal one of $100 \times 10 = 1{,}000$ N. The load is $\arctan(1{,}000/3{,}550) = 15.7°$ off square, and each weld carries half its resultant, $\tfrac12\sqrt{3{,}550^2 + 1{,}000^2} = 1{,}844$ N per millimetre.

As a butt throat: at 15.7° the cap still governs, and the resultant a throat can carry is $441/\cos 15.7° = 458$ N/mm², so the throat is $1{,}844/458 = 4.0$ mm. As a fillet throat: the transverse stress splits into $\sigma_\perp = \tau_\perp = 1{,}775/(a\sqrt{2})$ and the longitudinal is $\tau_\parallel = 500/a$, so the von Mises condition is $\sqrt{2 \times 1{,}775^2 + 3 \times 500^2}/a \le 544$, which gives $a = 4.9$ mm. The ratio of the two, 1.21, is the advantage at 15.7° on the grade curve.

The directional method, as written #

The method is taken at its word. Its two conditions, and the correlation factors of the four grades, are applied exactly as the design rules state them. Neither condition is a physical law: the ellipse is von Mises with the shear weighted to match tests, the cap is a separate rule, and the corner where they meet is where two approximations intersect rather than a place where the weld metal does anything special. What is robust is the shape — a smooth limit cut off on one stress — and the consequence that a resultant a little off the capped direction can be carried at a higher value.

The shears are taken to be uniform along the weld and the pull linear over the web’s depth. At the flanges of a real connection the web’s bending stress is not linear and the weld there shares its load with the flange welds, which the web weld’s calculation ignores. The single-sided throat is counted elastically, with the unfused root carrying its peak stress; the fully plastic count, set out in the partial-penetration essay, raises every curve but changes none of their shapes.

What the pictures cannot show #

That a weld is not the throat the method assumes. The directional method takes a throat of a nominal size and a stress uniform across it, and the weld that is made has a profile, a penetration that varies along its length and a root that is fused in some places and not in others. A partial-penetration weld’s advantage over a fillet depends on its penetration being what the drawing says, and a fillet’s does not: a fillet’s throat is visible and measurable from outside, and a partial-penetration weld’s is not, which is part of why inspection regimes treat the two so differently.

Nor can the pictures show fatigue, which is where the comparison between fillets and partial-penetration welds is usually decided. A fillet has a toe on each plate where its profile meets the surface, and a partial-penetration weld has an unfused root; which of the two cracks first under repeated load depends on stress ranges that the static criterion does not see.

Still open: the single-sided throat pulled and sheared at once #

Every single-sided weld here was counted elastically, with the moment of its eccentric pull carried by the throat alone. Under a combined load the throat’s root carries the pull’s bending peak and the shear at the same time, and a throat allowed to redistribute plastically would carry the two in proportions no elastic count can find. Whether the plastic interaction of an eccentric normal force and a longitudinal shear across one throat recovers the double-sided weld’s peak advantage at some intermediate direction, or whether the root’s notch forbids relying on any redistribution at all, is the question the single side leaves.