The weld that is stronger where it is pulled

The weld that is stronger across establishes a fact about one weld: a fillet loaded square across its axis carries √1.5 times what the same fillet carries along it, because the two load directions produce different combinations of normal and shear stress on the same throat.

The radius rule and where it fails establishes a fact about a group: an eccentric load makes the force per unit length vary round the weld, and the largest force is not always at the largest radius.

Put the two together and a third fact appears that neither contains: the capacity varies round the group as well as the force, because the direction of the force is different at every point of it.

Which free body produced the number #

The free body is a millimetre of weld, cut across its throat.

What crosses the throat is a force per unit length, in some direction in the plane of the connected parts, and the criterion the weld has to satisfy is the one a fillet always satisfies: on the throat plane, $\sqrt{\sigma_\perp^2 + 3\tau_\perp^2 + 3\tau_\parallel^2} \le f_u/\beta_w$.

Resolve a force $F$ per unit length at an angle $\theta$ to the weld’s own axis. The component along the axis is $F\cos\theta$ and it is pure shear on the throat, parallel to the weld. The component across is $F\sin\theta$, and on a throat at 45° it resolves into equal normal and shear parts, $F\sin\theta/\sqrt2$ each.

Substituting and collecting gives the whole of the direction dependence in one expression:

$$F_{Rd}(\theta) = \frac{a f_u}{\beta_w \sqrt{3 - \sin^2\theta}}$$

At $\theta = 0$ that is $af_u/\beta_w\sqrt3$ — the along-the-weld value. At $\theta = 90°$ it is $af_u/\beta_w\sqrt2$, which is $\sqrt{1.5}$ times as much, and is the rung below’s result recovered from the same algebra.

Everything between is available too, and on a group the whole range is in use at once.

Reading the profile #

The first figure repays a slower reading than a peak value, because the shape of the utilisation curve is a picture of the group’s geometry.

Walking the C-shaped group from the bottom of the lower return, along it to the corner, up the vertical, and out along the upper return, the force does what an eccentric load makes it do: small at the centroid’s own height, largest at the two far corners.

The capacity does something else entirely. Along the vertical welds, a mostly vertical force runs nearly along the weld, so the capacity is near its lower bound. On the returns, the same force is square across them and the capacity is at its upper bound. So the capacity curve is nearly a step function following the geometry, while the force curve is smooth and follows the radius.

The utilisation is the ratio of two curves with completely different shapes, and it has a feature neither of them has: a step at each corner, where the weld’s direction changes by ninety degrees and the capacity jumps by 22 per cent while the force does not change at all.

That step is the most interesting object on the figure and it is a genuine discontinuity in the model rather than an artefact. Two millimetres of weld either side of a corner, carrying the same force in the same direction, have capacities differing by a fifth — which is a statement about the model rather than about the steel, since the real weld turns the corner continuously and the throat there is a complicated shape nobody analyses. It is the same kind of idealisation as the shear strength nobody measured — a criterion doing work at a point where the geometry it assumes does not exist.

Why the criterion has this shape at all #

The direction factor is not a fitted curve and it is worth two paragraphs on where it comes from, because the same argument decides several other things in this collection.

A fillet weld’s throat is a plane at 45° to both connected parts, and it is the plane on which the weld is assumed to fail. A force in the plane of the parts, resolved onto that throat, produces three components: a normal stress across the throat, a shear across it, and a shear along it. The failure criterion combines them by von Mises — squares, with a factor of three on the shears — and that factor of three is the whole reason the direction matters.

A force along the weld produces only shear along the throat: all of it gets the factor of three, and the capacity is the smallest it can be. A force across produces equal normal and shear parts: half the square escapes the factor, and the capacity rises by √1.5.

So the direction factor is a consequence of the yield criterion rather than of anything about welding, and it would appear in exactly the same form for any material whose failure is governed by von Mises and any joint whose failure plane is fixed by geometry rather than by the load. The shear strength nobody measured is the same factor of three seen from the other side: a material’s shear strength is $f_y/\sqrt3$ for exactly this reason, and nobody measures it because the criterion supplies it.

Where the gain actually is #

The obvious expectation is that the directional method is worth √1.5 — 22 per cent — and it is not, for a reason that is worth following.

Three regions, and each has a mechanism.

At no eccentricity the gain is exactly nothing. A concentric vertical load on this group is carried mostly by the two long vertical welds, and it runs along them. The along-the-weld capacity is the right capacity, and the directional method returns it.

At a small eccentricity the gain is at its largest. The load is still nearly vertical everywhere, but the critical point has moved to a return weld — where a vertical force is square across the weld’s own horizontal axis, and the full √1.5 is available.

At a large eccentricity the gain settles at about seven per cent. Rotation dominates, the force at the critical corner is at about 38° to the weld, and $\sqrt{3 - \sin^2 38°}$ is only a little below $\sqrt3$.

So the gain is a property of the group’s shape and the load’s position rather than of the weld. That is the practically useful statement, and it means the directional method is worth doing on some connections and not on others — which is not what a rule that says “22 per cent” would suggest.

The peak that does not move #

There is a claim the first figure invites and it turns out to be false, which is worth recording because the reasoning that produces it is sound.

If the capacity varies round the group, then the most utilised point need not be the most loaded one — the critical location should sometimes move to a place with a smaller force and a much smaller capacity.

It nearly never does. The capacity spans a factor of 1.22 between its extremes, and on any group with an eccentric load the force per unit length spans several times that, so the utilisation peak sits exactly where the force peak does.

Which means the practical consequence of everything on this page is a number rather than a location: the simpler check finds the right point and gives it the wrong capacity. That is a much easier error to live with than the reverse, and it is why the directional method can be applied as a correction at the critical point rather than as a sweep round the group.

What the simplified method does instead #

Codes offer two methods for a fillet weld and the second one exists because of everything above.

The directional method is the criterion applied as written: resolve the force onto the throat, combine the three stresses, compare with $f_u/\beta_w$. It is exact within the model and it needs the direction of the force at the point being checked.

The simplified method takes the along-the-weld capacity and applies it to the resultant force whatever its direction. It is the $\theta = 0$ value used everywhere, so it is conservative everywhere — by nothing at $\theta = 0$ and by 22 per cent at $\theta = 90°$.

The choice between them is a straightforward trade of arithmetic against steel, and the figures above say what the trade is worth on a given connection: nothing on a concentrically loaded lap joint, a fifth on a bracket with a small eccentricity.

The simplified method is the right default, and the reason is not caution. A weld’s direction relative to the force is a property of the drawing rather than of the calculation, and drawings change — a bracket rotated, a load path revised, a return weld omitted on site. A capacity that does not depend on the direction survives all of those, and one that does has to be revisited every time.

What it is worth on a bracket #

Setting the arithmetic on a real connection gives the size of the decision, and the answer is that it is usually a bolt or a millimetre of leg.

A beam-to-column bracket with a 6 mm fillet in a C round a 200 mm deep plate, carrying 100 kN at 150 mm, is the group in the figures. The along-the-weld check gives 165 kN of capacity; the directional check gives 182. The difference is 17 kN — or, expressed the way it would actually be used, the difference between a 6 mm weld and a 5 mm one.

That is real and it is small, and the reason to know it is not usually to exploit it. It is to know how much a connection has in hand when something changes. A load that grows by ten per cent on a connection checked by the simplified method is very often still adequate by the directional one, and finding that out is a calculation rather than a redesign.

The place it is worth exploiting is the opposite one: a connection where the weld is at its limit and the plate cannot be made thicker, in a fabricated assembly where a larger fillet means a bigger heat input and more distortion. There the fifth is worth having, and the check is half an hour.

One practical note before the limitations. The directional method needs the angle between the force and the weld, and on a group that angle has to be computed at the critical point rather than assumed from the weld’s orientation. A vertical weld under a nominally vertical load is not at θ = 0 once the group is eccentric: on the C-shape at 150 mm the critical point is at 47 degrees, which is neither of the two values anybody would guess.

Where the model stops #

The elastic force distribution. Everything here uses the vector analysis of the rung on bolt groups, applied to a weld: the group rotates about its centroid and the force is proportional to the radius. A weld group has an instantaneous-centre method too, and it returns 10 to 30 per cent more again — so the two refinements compound, and neither is in the simplified check.

The throat is 45 degrees and equal-legged. An unequal-leg fillet has a throat at a different angle, and the resolution of the force onto it changes both components. The expression above assumes the standard fillet.

The parent metal is not checked. A weld group at full capacity is delivering that force into a plate, and the plate has to carry it — often through a Whitmore section or a block shear path that governs long before the weld does.

One load case at a time. A bracket carries several combinations, and the critical direction moves between them: a load nearer the vertical uses one part of the direction curve and a load with a horizontal component uses another. The directional method has to be applied to each combination, which is part of why it is skipped.

And nothing here yields. The criterion is a strength one applied at a point, with no redistribution along the weld. A ductile weld metal in a long weld does redistribute, which is the same argument a long bolted joint has and which the elastic distribution ignores in the conservative direction.

What the pictures cannot show #

The weld’s actual size, which is what all of this is a coefficient on. The capacity is proportional to the throat, and the throat delivered on site is a function of the process, the position, the fit-up and the gap — a 6 mm fillet specified on a drawing may be 5 mm in an overhead position with a 2 mm root gap, and the throat rather than the leg is what the capacity is proportional to.

Against that variability, the difference between 1,460 and 1,789 N/mm is a refinement being applied to a number known to perhaps 15 per cent. The directional method is a real gain and it is smaller than the uncertainty in the leg length, which is the most useful thing to know about it and the reason nobody argues very hard for it.

They also cannot show the residual stress, which in a fillet weld is at yield along the weld’s axis before any load is applied. The criterion is applied as though the throat starts unstressed; it does not, and the reason that does not matter is that the weld metal is ductile enough to redistribute a self-equilibrating stress field — the same argument that lets any residual stress be ignored at the ultimate limit state.

The assumption the figure rests on #

That the force on the throat is the force in the plane of the connected parts.

A fillet weld connecting two plates at right angles transmits force between them, and the resolution above assumes that force is delivered to the throat as a single vector in a known direction. In a real bracket the connected plate also bends, so there is a component out of the plane of the weld group — a peeling action on the throat, with a normal stress the two-dimensional resolution has no term for.

That component is what the codes exclude by requiring welds to be “not subject to out-of-plane bending” and what a designer excludes by making the connection stiff enough that the assumption holds. It is the same requirement an in-plane bolt group analysis makes, and it fails in the same way: the moment that was drawn in the plane of the connection is the one the analysis contains, and the other one is carried by a component nobody named.

What to carry away #

Three sentences.

A fillet weld has a capacity that depends on the direction of the force on it, from $af_u/\beta_w\sqrt3$ along to $\sqrt{1.5}$ times that across, and the expression between them is $\sqrt{3 - \sin^2\theta}$ in the denominator.

On a group the direction varies from point to point, so the capacity does — but the utilisation peak still sits where the force peak does, because the force varies more than the capacity can. The simpler check finds the right place and understates its capacity.

And what that understatement is worth depends on the arrangement rather than on the weld: nothing at all on a concentrically loaded joint, a fifth at a small eccentricity, seven per cent at a large one. A design rule quoted as a single percentage is describing one point of a curve, and knowing which point is most of what it is worth knowing about it.

The ladder from here #

Later rungs on this anchor: the instantaneous-centre method for welds, which is the plastic distribution and compounds with everything here. Unequal-leg fillets and deep-penetration welds, where the throat is at a different angle and the resolution changes. Partial-penetration butt welds under combined actions, where the criterion has a different form and the root is a crack. Welds under out-of-plane bending, which is the component the plane analysis omits. And the fatigue of the same connection, where none of the direction argument survives — a detail category is a stress range on a nominal section, and the weld’s own orientation appears in it as a category number rather than as a factor.