The throat that misses the load

A fillet weld is stronger across its length than along it, by exactly the square root of three halves, because the directional method resolves the force onto a throat inclined at 45° and weighs the normal stress and the shear on it differently. That essay set aside two other kinds of weld. A full-penetration butt weld fills the whole thickness of the joint and is as strong as the plate, so it is checked as plate. A partial-penetration butt weld is the awkward case in between. It has a throat, like a fillet, but the throat is normal to the plates, and that essay noted that its root is an unfused notch and left it there.

This essay follows the partial-penetration weld through a single transverse pull, and finds two separate things wrong with the obvious calculation. The first makes the weld look weaker than a naive application of the directional method would suggest. The second, which applies only when the weld is made from one side, takes away most of what is left.

A throat that faces the load #

The directional method treats every weld the same way. It finds the plane of the throat, resolves the force into a normal stress $\sigma_\perp$ on that plane and two shear stresses $\tau_\perp$ and $\tau_\parallel$ in it, and checks

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

where $f_u$ is the weaker part’s ultimate strength and $\beta_w$ a correlation factor that rises with the grade. The first condition is a von Mises form with the shear weighted by three. The second is a separate limit on the normal stress alone.

A fillet weld loaded along its length has shear $\tau_\parallel$ only, and the first condition gives $f_u/(\sqrt{3}\beta_w)$. Loaded across its length, its throat is inclined at 45°, so the force splits into equal parts of $\sigma_\perp$ and $\tau_\perp$, and the first condition gives $f_u/(\sqrt{2}\beta_w)$ — which is where the factor of $\sqrt{3/2}$ between the two came from.

A butt weld’s throat is the plane of the joint itself, normal to the plates. A transverse pull puts $\sigma_\perp$ on it and nothing else. The shear terms vanish, and the first condition alone would allow $f_u/\beta_w$.

So the butt throat really is the strongest of the three per millimetre. It is not as strong as the criterion’s first condition suggests, because the second condition intervenes, and it is worth seeing that this is not an accident of one grade.

The limit that is not the criterion #

The design strength of a butt weld’s throat in tension is 0.9fu, in every grade, and the von Mises part of the rule never decides it. The reason is in the correlation factor. $\beta_w$ was calibrated on tests of fillet welds, whose throats always carry some shear with their normal stress, and for those the first condition is the binding one. A throat in pure tension is the case the calibration least describes, and it is also the case in which a flaw lying across the throat is opened most directly. The cap on $\sigma_\perp$ is what the rules put there instead.

The cap also says how much shear a butt throat can take before the orientation argument stops holding. With the normal stress at its cap, the first condition is still satisfied until $\sqrt{\sigma_\perp^2 + 3\tau^2}$ reaches $f_u/\beta_w$. In S355 that is $\sqrt{544^2 - 441^2}/\sqrt{3}$, a shear of 184 N/mm² alongside 441 of normal stress. A butt throat can carry shear up to about 42 per cent of its normal stress before the von Mises condition takes over from the cap, which is why the pure pull is a genuine case and not a curiosity: most transversely loaded butt welds are well inside that range.

That is a small difference in the number and a large one in the reasoning. A designer who thinks of the partial-penetration weld as a fillet weld with a better orientation will expect the orientation to buy the whole $\sqrt{2}$ over a transverse fillet. It buys 15 per cent in S355 and 8 per cent in S275.

Which free body produced the number #

Everything so far treated the throat as though the force passed through it evenly. For a weld made from both sides that is true. For a weld made from one side it is not, and the free body shows why.

Cut the joint along the weld and take one plate as the free body. The force it carries per millimetre of weld, $N$, acts along the middle of the plate’s thickness, at $t/2$ from either face, because that is where the plate carries it. The throat that has to receive that force runs from the welded face down to a depth $a$, so its centroid is at $a/2$ from that face. When $a$ is less than $t$ the two lines do not coincide, and the force passes the throat’s centroid at a distance

$$e = \frac{t - a}{2}$$

Moving a force to a new line of action adds a moment, here $N e$ per millimetre, and the throat has to carry that moment as well as the force. The stress across the throat is then what a section carrying an axial force and a bending moment always has, $N/a \pm 6Ne/a^2$. It is the same short-column stress that turns out to hold a fixed-ended member’s end moments, in a very different place.

The root takes the difference #

The sign of the moment decides where the extra stress goes. The force lies on the root side of the throat’s centroid, so the moment adds tension at the root — the unwelded end of the throat, deep in the joint — and removes it at the welded face.

For the weld in the first figure, $t = 20$ mm and $a = 10$, so $e = 5$ mm and $6e/a = 3$. The root carries four times the mean throat stress and the face minus two, which is compression. In general the root factor is $1 + 3(t/a - 1)$, and it grows quickly as the penetration falls: at a third of the thickness it is 7, at a quarter 10.

Below the root is the unfused land, the part of the joint the weld never reached: ten millimetres of steel in contact with steel and joined to nothing. A notch of that kind is what fracture mechanics measures strength against, and the essay that introduced partial-penetration welds treated the notch as the reason for suspicion. The eccentricity is what makes that suspicion concrete. It puts the highest stress in the throat at exactly the notch’s tip, on its opening side, under a single static load.

The double-sided weld in the lower half of the figure has the same 10 mm of throat, half at each face. Its throats are symmetric about the plates’ mid-thickness, their combined centroid lies on the line of the force, and every part of both carries the mean stress. Its unfused land is in the middle of the joint, where no bending stress reaches it.

How much of the throat a single side can use #

The eccentricity takes capacity away as well as moving stress, and how much depends on how the throat is allowed to fail.

Elastically, the throat’s capacity is reached when the root stress reaches the limit, so the concentric capacity is divided by the root factor: a quarter of it at half penetration, an eighth at 0.3. Fully plastic, the throat can redistribute until part of it is at the limit in tension and part in compression, and the capacity is $2d/a$ of the concentric value with $d = \sqrt{e^2 + a^2/4} - e$ — which at half penetration is $\sqrt{2} - 1$, 0.41.

Neither bound is comfortable to use. The plastic figure needs the weld metal to yield across its throat while the root, a notch under tension, does not crack first, and that is precisely the property an unfused root casts doubt on. The elastic figure is what the throat carries before its root sees more than the limit. Codes resolve it by not relying on either: single-sided partial-penetration welds, and single fillet welds, are not to be used where the load puts their root into tension by bending about the weld’s own axis.

Turned into a design, the difference is a matter of how much welding a joint needs. Suppose the two 20 mm plates have to carry 3,000 kN per metre across the joint. From both sides, the throat needed is $3000/441$, 6.8 mm in total — 3.4 mm at each face, about a third of the thickness. From one side, carrying the eccentricity elastically, the throat has to satisfy $441a/(1 + 3(20/a - 1)) \ge 3000$, and the smallest $a$ that does is 14.5 mm: nearly three-quarters of the plate’s thickness, and more than twice the total throat the double-sided weld needs for the same force. Most of the extra penetration is not carrying the force; it is carrying the moment the single side created, and shortening the lever arm of that moment by making the throat’s centroid come closer to the plate’s mid-thickness.

Both bounds converge on the double-sided value only as the penetration approaches the full thickness, because only there does the eccentricity vanish. At full depth the double-sided weld would carry 8,820 kN/m, more than the plate’s yield capacity, and the plate governs; at that point the weld is a full-penetration butt weld and belongs to the plate check anyway.

The rest of the joint decides #

The moment $Ne$ has to be carried by something, and the analysis so far gave all of it to the throat. That is right for two free plates pulled apart, which rotate about the weld as soon as they are loaded. It is not right for a weld inside a larger assembly.

A plate welded into a box section, restrained by the other walls of the box, cannot rotate about its weld, and the moment the eccentricity would have produced is carried as bending in the walls around it. A stiffened assembly does the same. The capacity of a single-sided weld is not a property of the weld but of how much rotation the joint around it allows, and the same weld detail is adequate in one place and a quarter adequate in another. A connection is not a point, and the partial-penetration weld is a case where the parts of the joint a drawing does not show decide the weld’s strength.

The practical difficulty is that restraint is rarely known with any precision, and a joint that is restrained under one load case may not be under another. Designing the single-sided weld as unrestrained is safe; designing it as restrained needs the restraint to be demonstrated, and a weld that depends on it has a strength that changes if anything around it is altered.

The root is a crack either way #

Static strength is not where partial-penetration welds usually cause trouble, and the eccentricity explains why they cause it where they do.

Under repeated load, the root stress is what drives a fatigue crack from the root, and it scales with the same factor. A single-sided weld at half penetration puts four times the mean stress range on its root, and since fatigue life goes as the cube of the stress range, a detail’s category for root failure in such a weld sits far down the table. The most familiar example is the weld that joins the troughs of a steel bridge deck to its plate, a partial-penetration weld that can only be made from outside the closed trough, whose root cracks grow unseen through the throat. The deck plate’s local bending loads the root directly, and the single side it is welded from is the reason it is exposed to that bending at all.

A rough fracture comparison makes the same point in the currency the root is actually judged in. A crack’s stress intensity grows with the stress across it and with the square root of its length. The single-sided weld’s unfused land is a notch 10 mm deep running in from one face, loaded at four times the mean throat stress. The double-sided weld’s land is a flaw 10 mm long buried in the middle of the joint — so half that length measured from its centre to either tip — loaded at the mean stress. Ignoring the geometry factors, which differ between an edge notch and a buried flaw and do not reverse the comparison, the single-sided root sees four times the stress on $\sqrt{2}$ times the effective length: roughly five to six times the stress intensity of the double-sided weld’s land, from the same 10 mm of throat and the same steel.

The eccentricity and the notch are two separate defects that happen to point the same way. The notch makes the root the weakest place in the joint; the eccentricity makes it the most highly stressed. A double-sided weld removes the second and hides the first in the middle of the joint, where the stress is at its mean.

What a designer does with it #

The options follow directly from the free body.

Make the weld from both sides where access allows, and put the same total throat symmetrically. That keeps the whole of the throat’s capacity and moves the unfused land to where the bending does not reach.

Take the penetration to full depth where the joint has to develop the plate, and check it as plate. That removes the land and the eccentricity together, at the cost of preparation and inspection.

Put the line of action through the throat. A tee joint loaded through a member whose centroid lies on the weld’s side, or plates offset so that their mid-thicknesses align with the throat, has no eccentricity to carry.

Or design for the eccentricity explicitly, with the elastic root stress, and only where the joint’s restraint is certain. The weld is then carrying a known moment and the root is a known concern, rather than a surprise.

Where the model stops #

A single transverse force. A real throat usually carries shear along the weld at the same time, and then the first condition, with its factor of three on the shear, comes back into play alongside the cap; the orientation argument above holds only for the pure pull.

A linear or fully plastic throat. The real throat is neither. Weld metal is ductile, the root notch is not, and residual stresses from welding are already at yield in the throat before any load arrives, so the actual stress distribution under the first loading is somewhere between the two bounds and depends on the weld.

Perfect alignment. Angular distortion from welding one side, and misalignment between the plates, both add eccentricity of their own, usually in the same direction as the eccentricity here because the weld shrinks on the welded side.

And the throat is taken as the depth of penetration. Some welding processes do not reliably fuse to the full prepared depth, and codes then reduce the design throat below the preparation, which moves the throat’s centroid and increases the eccentricity again.

Still open: the throat that carries shear as well as tension #

The pure pull is the simplest case, and the combined one is where the criterion’s two conditions compete. A partial-penetration weld carrying both a transverse force and a longitudinal shear, where the cap on σ⊥ and the von Mises form each govern over part of the range, and the orientation advantage over a fillet is spent at a rate the shear decides. The weld loaded out of its own plane, bending about its axis in the sense that closes the root and the sense that opens it, where the same throat has two capacities. The root crack itself, as a stress intensity at the tip of the unfused land, which turns the elastic root factor into the input of a fracture calculation. And the instantaneous-centre method for a weld group with partial-penetration welds in it, where each weld’s throat faces the load differently and the eccentricity of a single-sided one is added to the group’s.