The weld that is stronger across than along

A fillet weld is a triangle of deposited metal in the corner between two plates. It is specified by a leg length, checked on a throat, and quoted in kilonewtons per millimetre — a number a detailer looks up and multiplies by a length.

The number depends on which way the load is going, and by more than it is comfortable to ignore.

The throat, and why it is at forty-five degrees #

Start with the geometry, because the whole result is in it.

A fillet weld deposited into a right-angled corner has two legs against the plates and a hypotenuse exposed to the air. For an equal-leg weld of leg $s$, the shortest path across the triangle — the throat — is the perpendicular from the corner to the hypotenuse, and it has length $a = s/\sqrt{2} = 0.707 s$.

That is the plane the weld fails on, and it fails there for the same reason a chain breaks at its thinnest link: it is the least material the crack has to cross. A 6 mm fillet weld has a 4.2 mm throat, which is the number every capacity is computed on, and the factor 0.707 is the reason weld sizes and weld capacities never look like round numbers together.

What matters for this essay is the throat’s orientation. It sits at 45° to both plates, which means that a force applied in any direction resolves onto it as some combination of a normal stress and two shear stresses, and the combination depends on the direction.

The three components, and the criterion that combines them #

Set up the components on the throat plane:

• $\sigma_\perp$ — normal stress on the throat, pulling it apart;
• $\tau_\perp$ — shear on the throat, across the weld’s length;
• $\tau_\parallel$ — shear on the throat, along the weld’s length.

The check is the directional method, which is a von Mises criterion written for these three:

$$\sqrt{\sigma_\perp^2 + 3\left(\tau_\perp^2 + \tau_\parallel^2\right)} \le \frac{f_u}{\beta_w}$$

The factor 3 on the shear terms is the von Mises factor and it is the same one that makes shear yield at $1/\sqrt{3}$ of direct yield — which appeared in block shear as the 0.6 on the shear planes and in torsion as the reason a shear stress is compared against $f_y/\sqrt{3}$. It is one criterion, met three times in three parts of the subject.

Working out the two limiting cases #

Now the result, which takes two short calculations and no data.

A weld loaded along its length. The force is parallel to the weld’s axis, so it produces only $\tau_\parallel$, of magnitude $F/(a\ell)$. There is no normal stress on the throat and no transverse shear. The criterion becomes

$$\sqrt{3}\,\frac{F}{a\ell} \le \frac{f_u}{\beta_w} \qquad\Longrightarrow\qquad F = \frac{a\ell f_u}{\sqrt{3}\,\beta_w}$$

A weld loaded across its length. The force is perpendicular to the weld’s axis and in the plane of the plates. Resolve it onto the throat, which is at 45° to that direction: it produces a normal stress and a transverse shear of equal magnitude, each $F/(\sqrt{2}\,a\ell)$. Now

$$\sqrt{\frac{F^2}{2(a\ell)^2} + 3\cdot\frac{F^2}{2(a\ell)^2}} = \sqrt{2}\,\frac{F}{a\ell} \le \frac{f_u}{\beta_w} \qquad\Longrightarrow\qquad F = \frac{a\ell f_u}{\sqrt{2}\,\beta_w}$$

Divide one by the other and everything cancels except

$$\frac{F_\perp}{F_\parallel} = \frac{\sqrt{3}}{\sqrt{2}} = 1.224745$$

Nothing was measured. The enhancement is a consequence of the throat being at 45° and of the criterion having a 3 on its shear terms, and it would be the same number for any material the criterion describes.

Why across is stronger, in words #

The formula is short enough to hide the mechanism, and the mechanism is worth having.

A weld loaded along its length is in pure shear on the throat. Pure shear is the state the von Mises criterion punishes hardest — it is the direction in stress space that reaches the yield surface soonest, which is exactly what the $\sqrt{3}$ means.

A weld loaded across its length is in a mixture of tension and shear. Tension is a less efficient way to reach the yield surface than shear is, so splitting the load between the two reaches further before the surface is met. The transverse weld is stronger not because it is doing something clever but because it is doing something less bad: half of its stress is in a component the material is comparatively resistant to.

That reframing generalises. Whenever a criterion weights components unequally, the strongest orientation is the one that puts the least of the load into the heavily weighted component — which is why an interaction curve bulges rather than being a straight line between its two intercepts.

The weld that is bigger than it needs to be #

There is a practical asymmetry in weld design that follows from the throat geometry and is worth stating because it inverts the usual economics.

The capacity is linear in the throat, so doubling the weld size doubles its strength. But the volume of deposited metal goes as the square of the leg, so doubling the size costs four times the consumable, four times the arc time and four times the heat put into the plate. A weld twice as big costs four times as much and is worth twice as much.

That is the opposite exchange rate from almost everything else on this site. Section depth buys strength faster than it costs material; bolt group spread buys capacity quadratically for free. Weld size buys linearly and costs quadratically, which is why the design instinct in welding is the opposite one: run it longer, not bigger.

A 6 mm weld over 400 mm and a 12 mm weld over 200 mm have the same capacity. The first uses half the metal and is preferred every time, provided there is 400 mm of edge to run it along. Weld design is therefore mostly a search for length, and connections are detailed to provide it — end returns, longer cleats, larger gusset plates.

Those two are worth drawing rather than asserting, because the arithmetic that makes them equal is the same $a\ell$ that makes the first figure’s curve what it is.

There is a second reason to prefer the long thin weld, and it belongs to the materials field. A large weld puts a great deal of heat into a small volume of plate, which leaves a larger region at yield-level residual stress when it cools, and a coarser microstructure in the heat-affected zone. The structural consequence lands on fatigue rather than on static strength — but it lands, and it lands on a check the weld capacity calculation does not touch.

A weld’s ends are not weld #

“Run it longer” has a floor under it, and the floor is a consequence of how a weld starts and stops.

An arc struck at the beginning of a run takes a moment to establish, so the first few millimetres are shallow and often contain porosity. Breaking the arc at the end leaves a crater — a depression where the pool solidified while shrinking, frequently with a crack in the middle of it. Neither end has the throat the drawing says.

Codes handle it by deducting a length equal to the throat at each end, so the effective length is $\ell - 2a$, and by refusing to count a weld shorter than the greater of 30 mm and $6a$ at all. Both matter more than they look. A 4 mm throat run over 40 mm loses 8 mm to its own ends — a fifth of the weld — and one run over 30 mm loses more than a quarter. The loss is a fixed length, so it is a fixed cost per run rather than per millimetre, and it falls hardest exactly where a detailer is trying to fit a weld into a short piece of edge.

The same curve drawn at the effective length is what a short run actually delivers, and the gap between it and the drawing is the whole of the deduction.

Which sharpens the economics of the previous section rather than contradicting them. Length is still the cheap variable, but runs are not: two 100 mm welds are worth less than one 200 mm weld, by two throats’ worth, and a connection detailed as a row of short intermittent welds has paid the end deduction once per segment. Continuous is better than intermittent for the same total length, and the difference is computable rather than a matter of good practice.

It is also why end returns are detailed. Taking the weld round the corner puts the crater on a face that is not carrying the peak stress, and moves the start and stop away from the end of the joint — which is where the eccentric weld group’s stress is largest, and the worst possible place to leave a solidification crack.

The check that is not on the weld at all #

A weld can be adequate and the connection still fail, because there are two things being joined and the weld sits between them.

The force that leaves the weld has to enter the plate on either side, and it enters over a strip as wide as the weld’s leg. That is a very concentrated introduction of load, and the plate has to spread it — which is the disturbed region problem again, with the weld as the point of application.

Two failures follow from it and neither is a weld check.

Lamellar tearing. A weld that pulls on a plate through its thickness is loading rolled steel in the one direction it is weakest, because rolling flattens inclusions into planes parallel to the surface. The weld is fine; the plate delaminates behind it. This is the direct descendant of the flaw that sets the strength, with the flaw population being a property of how the plate was made.

Parent metal shear. The material immediately behind a weld carries the whole of that weld’s force in shear over its own thickness. For a weld larger than about 0.7 times the plate thickness, the plate runs out before the weld does — which is where the detailing rule that a fillet weld need never exceed the thinner plate’s thickness comes from.

Both are reminders that a weld capacity is a capacity of the joint, not of the deposited metal, and that the deposited metal is often not the weakest thing in it.

The simple method, and what it costs #

Most weld design does not use the directional method. It uses the simple method: compute the resultant force per unit length whatever its direction, and compare it against the longitudinal capacity.

That is the same as assuming every weld is loaded along its length, which is always safe and is sometimes 22% wasteful. The trade is deliberate and it is usually the right one, because:

• most weld groups carry loads in more than one direction at once, so there is no single angle to enhance by;
• the enhancement requires knowing the direction accurately, and the direction depends on the analysis assumption that produced the force;
• the arithmetic is faster, and weld design is done at high volume by people who are not going to enjoy resolving three components at every point.

The simple method is worth reaching past in one situation specifically: a transverse weld carrying a large, well-defined, single-direction force — the end returns on a lap joint, a flange plate welded across a beam. There the direction is not in doubt, the force is large enough for 22% to be worth having, and the calculation is short because there is only one angle.

What happens at forty-five degrees #

The intermediate angles are worth a look, because the curve between the two limits is not a straight line and its shape says something about the criterion.

At 45° the enhancement is 1.0954. At 60° it is 1.154701, which is $2/\sqrt{3}$ exactly. Those recognisable constants at particular angles are the signature of a computed curve rather than an interpolated one, and they are a cheap way to check that a figure like the one above is being generated from the criterion rather than drawn between its endpoints.

The shape also shows how little of the enhancement is available at small angles. Half of the 22% has not arrived until about 50°, so a weld inclined at 30° to its load — which is a perfectly ordinary angle in a bracing connection — collects 4.4% and is not worth the arithmetic. The enhancement is concentrated at the transverse end, which is another reason the simple method loses so little in practice: the cases where the bonus is large are the cases where the geometry is simplest, and those are rare.

What the correlation factor is doing #

The $\beta_w$ in the denominator is worth a paragraph because it looks like a safety factor and is not.

It is a correlation factor that adjusts for the parent steel’s grade, and it goes the wrong way from intuition: 0.8 for S235, 0.85 for S275, 0.9 for S355, 1.0 for S460. A higher number is a lower capacity, so a weld on stronger steel is worth less per millimetre than the same weld on weaker steel.

The reason is that the capacity is written against the parent metal’s $f_u$, and weld metal does not track parent metal grade proportionally. Welding consumables come in a small number of strengths; a weld on S460 is likely to be undermatched relative to the plate it joins, and $\beta_w$ is the adjustment that keeps one formula applicable across grades.

It is a small thing, and it is a good example of a coefficient whose direction is meaningless until the quantity it normalises is named. Read as “the strength factor for the steel”, it is backwards. Read as “the correction for using the plate’s ultimate strength in a formula about the weld”, it is the right way round.

Two welds that are not this weld #

Everything above is about a fillet weld, and it is worth naming the two other kinds briefly, because their capacities are arrived at completely differently and confusing them is easy.

A full penetration butt weld joins two plates edge to edge with weld metal filling the whole thickness. Its capacity is not calculated. Provided the consumable matches the plate and the weld is sound, the joint is as strong as the parent metal in every direction, and the design check is on the plate rather than on the weld. That is why butt welds are specified where a connection has to develop a member’s full capacity, and why their cost is in preparation and inspection rather than in arithmetic.

A partial penetration butt weld is the awkward middle case: it has a throat, like a fillet, but the throat is normal to the plate rather than at 45°, so none of the directional results above apply to it. A tensile load across it produces $\sigma_\perp$ alone with no shear at all, which the criterion treats generously — and the reason partial penetration welds are nonetheless treated with suspicion in tension is not the criterion but the root, which is an unfused notch of exactly the kind fracture mechanics exists to worry about.

So the three weld types are three different design problems: the butt weld is a plate check, the fillet weld is the throat calculation above, and the partial penetration weld is a fatigue and fracture question wearing a strength calculation’s clothes.

Where this leaves the weld group #

Every result above is about one weld carrying a load in one direction. Real weld groups are lines that turn corners, carrying loads that miss their centroids, so the direction of the stress varies continuously round the group and so does the enhancement.

What to take from it #

A fillet weld’s capacity is direction-dependent by a factor of 1.2247, and the factor is computed. It comes from the throat sitting at 45° and from the von Mises criterion weighting shear three times as heavily as direct stress.

Across is stronger because tension is a less efficient route to yielding than shear is. The transverse weld gets to put half its stress into the component the material minds least.

The simple method throws that away on purpose, and usually correctly. A weld group loaded in several directions at once has no single angle to enhance by, and the 22% is only collectable where the direction is known and constant.

And the throat is 0.707 of the leg, which is where every unround number in weld design comes from. A 6 mm fillet has a 4.2 mm throat, and everything is computed on the smaller of the two.

Run it longer, not bigger. Capacity is linear in the throat and cost is quadratic in the leg, which is the reverse of the exchange rate every other part of this subject offers. A 6 mm weld over 400 mm and a 12 mm weld over 200 mm carry the same load; the first uses half the metal, puts half the heat into the plate, and leaves a smaller region at residual stress behind it.

And the weld is frequently not the weakest part of the joint it is in. Lamellar tearing behind it and parent metal shear beside it are failures of the plate, and neither appears in any of the arithmetic above. A weld capacity is a number about deposited metal; a joint capacity is a number about everything the force passes through, and this field’s recurring lesson is that those are not the same list.