The eccentricity at right angles to the drawing

Everything about the weld group so far has been planar: a load in the plane of the welds, an offset in that plane, and forces on the weld in that plane. The whole argument about which point of the group is the worst one lives inside that plane.

A bracket welded to a column flange stands proud of the flange. The beam it carries sits on the outstand, so its reaction acts at some distance in front of the plane the welds lie in — set by the bracket’s own thickness, the beam’s bearing position, and any packs that were needed to make something fit. That offset is a second eccentricity, at right angles to the first, and the planar calculation cannot see it at all.

Two things about that figure are the whole essay. The stress is linear through the centroid, which required no assumption. And the two components do not add.

Why the axis is where it is #

A moment about an axis lying in the plane of a fastener group has to be resisted by tension on one side of that axis and compression on the other. Where the axis sits depends entirely on how the compression is carried.

A bolt group cannot carry compression. A bolt in a hole pulls; it does not push, and the plates it holds together do the pushing, by bearing on one another. So the compression is carried by contact somewhere over the region below the axis, and where that contact is, is a modelling assumption.

A fillet weld carries compression across its throat exactly as readily as it carries tension. There is no contact problem, no bearing block, and no region of the group that has stopped working. The neutral axis is the centroid of the weld line, by the same argument that puts a beam’s neutral axis at the centroid of its section, and the plane-sections assumption underneath both is doing the same work in each.

That figure is worth reading slowly, because the two assumptions do not disagree in a simple direction.

Move the axis down to the bearing edge and every row is in tension, so the total tension is shared over six rows rather than three and each row carries less. But the compression resultant has moved down too, and the couple’s lever arm has shortened from 156 mm to 147 — so the total tension has gone up. One assumption is more onerous for a bolt and the other is more onerous for the connection, and which of them a check reports depends on what the check is asking about. The bolt group’s own essay takes the bearing case as the one that matters, and it is right to, because a bolt group’s governing check is a bolt.

None of this exists for the weld. The elastic section modulus of the line is $I/c$, it is 13,333 mm² for the group drawn, and it is not an approximation to anything.

Where the force lands on the throat #

The second half of the difference is about direction, and it turns the awkward-sounding case into the favourable one.

A fillet weld’s throat is the 45° plane through the root, and every force on the weld has to be resolved onto it. A force running along the weld’s own axis produces shear in the plane of the throat and nothing else. A force running across the weld — whether in the plane of the plate or normal to it — splits into equal normal and shear components on that 45° plane.

The arithmetic of that combination is where the strength comes from. A stress running along the weld enters the criterion multiplied by three; a stress running across it splits in half first, so it enters as $\sigma^2 + 3\tau^2$ where both are the same number — a total of four halves rather than three wholes.

$$\frac{\text{across}}{\text{along}} = \sqrt{\frac{3}{2}} = 1.2247$$

That is the same $\sqrt{1.5}$ a single weld gets for being loaded transversely, and here it applies to the entire group at once, because every point of a group in out-of-plane bending is being bent the same way. The in-plane case has no such luxury: there the direction of the force varies round the group, so the enhancement varies with it and the peak point is not the peak-force point.

So the out-of-plane eccentricity, which sounds like the awkward one because nobody drew it, is the direction in which a fillet weld is at its strongest and at its most predictable.

What the two components do to one another #

They are perpendicular, and they are not added as vectors.

In the planar problem the direct shear and the torsional shear both lie in the plane of the plate, so they are added as vectors and the resultant is what the weld sees. That is the arithmetic the corner-versus-peak argument is entirely about.

Here the bending force is normal to the plate and the vertical shear is along the weld. They are at right angles in space, they land on the throat as different stress components, and the criterion combines them quadratically with different weights. On the group above the bending contributes 265 of the 274.8 comparison stress and the shear contributes the rest — the shear is doing almost nothing, which is a useful thing to know before spending any effort on it.

That ratio is a property of the numbers rather than a general result, and it moves. Make the group deeper and the bending term falls as the square while the shear term falls only in proportion, so the shear becomes relatively more important as the connection is made more adequate.

Depth beats throat, by a square #

Which is the design conclusion, and it is the one a drawing invites getting wrong.

Take the 200 mm group to 300 mm and the utilisation falls from 54 per cent to 25. Take the throat from 6 mm to 8 instead — a third more weld metal, a third more heat into the plate, and a fillet at the size where a single pass stops being enough — and it falls to 41.

The lever arm is free and the weld metal is not. That is the same statement this collection makes about a truss’s depth and about a beam’s section, arriving in a connection detail where it is easy to forget, because a connection is drawn at a scale where 100 mm of extra plate looks like a change and 2 mm of extra fillet looks like nothing.

There is a second and larger version of the same move, and it is what a real bracket does.

Three times the section modulus for three-quarters more weld, and the reason is exactly the reason a flange is where it is on an I-beam: material at the extreme fibre contributes at the square of its distance, and material near the axis contributes almost nothing. Geometry beats material is the thread this site keeps returning to, and a weld group is one of the smallest objects it applies to.

What a drawing has to say for any of this to be checkable #

There is a practical difficulty that follows from the whole argument and it is worth stating before the refinements.

The section modulus above was computed from where the weld runs are. A drawing that specifies a fillet weld by its leg length and a symbol — which is what a drawing does — has told the fabricator everything and told the calculation almost nothing: the extent of the weld is a line on an elevation, its returns are a matter of shop practice, and whether the top return is 150 mm or 20 is often decided by whoever holds the torch.

That matters here more than it does in the planar case, because every millimetre at the extreme fibre is worth $y^2$. A return that stops 50 mm short costs 23 per cent of the section modulus of the flanged group below; the same 50 mm missing from the middle of a vertical run costs 1.6 per cent. The two omissions look identical on a site inspection.

The response, where it is taken seriously, is to specify the extent as well as the size — an explicit dimension on the return rather than a symbol — and to check the group with the returns discounted where it cannot be. That is a conservatism worth spending, and it is not the conservatism most connection checks spend, which is in the weld size.

What the in-plane calculation was doing meanwhile #

The two eccentricities do not interact in the arithmetic, and they very much interact in the connection.

The honest position is that a real bracket has both, that they produce stress components on the throat which are neither parallel nor perpendicular to one another in any convenient way, and that the general combination is a three-dimensional resolution at every point of the group. It is done by computer or it is not done at all.

What is done by hand, and what most brackets are checked by, is the two calculations separately with the results added on the assumption that the worst point of one is the worst point of the other. That is conservative when it is right and it is not always right, because the in-plane peak has already been shown to move with the eccentricity.

Two numbers that decide the whole check #

It is worth ending the arithmetic with what the group’s answer actually depends on, in order.

The depth, at the square. From 70 mm to 330 mm the utilisation runs from 430 per cent to 21 — a factor of twenty, from a dimension nobody costs.

The eccentricity, in proportion. The out-of-plane offset multiplies the moment directly. On the bracket above, 15 kNm at 100 kN is an offset of 150 mm; take it to 200 and the utilisation goes from 54 per cent to 71, and the 50 mm came from a thicker bracket plate or a shim.

The throat, in proportion, and only after those two. Six millimetres to eight is a fall from 54 per cent to 41, for a third more weld metal.

The shear, hardly at all. It contributes 10 of the 274.8 N/mm² comparison stress on the group drawn — under four per cent — and doubling it moves the utilisation from 54 per cent to 60.

That ordering is the useful thing to carry, because it is the reverse of the order in which the four are usually thought about. The shear is the number the connection is nominally designed for and it is the least of the four. The eccentricity is the one that is not on any drawing.

Two rigid things that are not rigid #

The plate is rigid. The whole of the linear stress distribution assumes the bracket rotates as a stiff body about the neutral axis, so the strain and hence the force at each point of the weld is proportional to distance. A thin bracket plate bends, its far edge lags, and the stress near the extreme fibre is lower and nearer the axis higher than drawn. This is the largest omission in the model — larger than any of the refinements above — and it is where a finite-element analysis changes the answer rather than confirming it.

The column flange is rigid too. A weld group on a thin flange is loading a plate that will bend away from it, which shifts the load toward the stiffer parts of the group — the ends near the web, or near a stiffener — and away from the middle. The same effect halves the effective length of a long joint and it is unmodelled here.

The returns are drawn as though the corner were a point. A weld turning through 90° has a starting and stopping crater at the corner, its throat is not a clean 45° there, and codes require the return to be at least twice the leg size before it is counted. The 150 mm returns above are comfortably past that; a 15 mm one would not be, and the arithmetic above would have credited it in full.

And no residual stress appears anywhere. A welded connection is in a state of self-equilibrated stress before anything is applied to it, at yield in places, and the elastic distribution here is superimposed on that rather than replacing it. What saves the calculation is that yielding at a point redistributes rather than fails, which is the same argument the whole elastic method leans on.

The throat is 0.707 of the leg, and an elevation has no thickness #

The figures draw the weld group as a line, and it is a solid of triangular section with a root at one edge and a face at the other. The throat is the shortest path through it and it is 0.707 of the leg, so a “6 mm fillet” on a drawing has 4.24 mm of resisting section, and every stress on these figures is computed on the smaller number. A reader looking at the elevation sees no thickness at all.

The other absence is the load itself. The out-of-plane eccentricity is a distance measured perpendicular to the page, so on an elevation of the bracket it is a point — and it is the dimension the whole calculation rests on. It is set by a bracket’s thickness, a bearing position and a packing shim, each of which is a length on a fabrication drawing rather than on a frame diagram, and each of which is a lever arm.

Why the elastic answer here is not conservative but exact #

Every stress here is elastic, and elastic means the group has not redistributed. That is a much safer assumption in bending than it is in the planar case, and the reason is worth carrying.

The in-plane calculation has a plastic alternative — the instantaneous centre — which gives a higher capacity and which the rung below argues should not be spent, because a fillet weld’s deformation capacity is a fraction of a millimetre. In out-of-plane bending there is no equivalent reserve to argue about at all: the extreme fibre yields, the stress block would have to become plastic over the depth of the group, and the deformation that requires is a rotation the weld cannot supply. The elastic answer here is not conservative by 10 to 30 per cent; it is the answer.

The same object, one field over #

There is a section-property calculation hiding in all of this and naming it makes the whole check easier to hold.

A weld group in out-of-plane bending is a thin-walled section in bending, and the arithmetic is the arithmetic of a section modulus. The runs are the web, the returns are the flanges, the throat is the wall thickness, and the section modulus of the line multiplied by the throat is the section modulus of the weld considered as a shape. Everything this collection knows about where to put material in a section applies here without translation.

Which is why the returns are worth three times what they cost, and why they are worth it for the same reason a flange is. It is also why the two failure modes are the ones a section has: the extreme fibre reaches its limit first, and there is no reserve past it because the material cannot redistribute.

The correspondence has one instructive break in it. A steel section’s flange is limited by local buckling — it has a slenderness beyond which it cannot reach its own yield — and a weld return has no such limit, because it is a solid triangle 6 mm across and nothing that shape buckles. So the weld group is the one thin-walled section on this site that can be made as efficient as its geometry allows without a compactness check arriving to spoil it.

The break runs the other way too, and it is the harsher of the two. A steel section can be checked after fabrication with a tape measure. A weld group’s section is its throat times its length, the throat is inside the weld, and neither dimension can be measured on the finished connection without cutting it up.

A calculation gets attention in proportion to how visible its inputs are #

Later rungs on this anchor: the two eccentricities resolved together at every point of the group, which is the three-dimensional problem and is a computer calculation. Welds sharing a connection with bolts, and why the sharing is not permitted. The connected plate’s own flexibility, which is the largest omission above. Weld group fatigue, where the peak point rather than the average decides the category and where the direction argument disappears entirely.

There is a history worth knowing in why this case is the neglected one. The in-plane weld group has tables, charts and a coefficient method going back to the 1930s, because it is the shape a bracket takes on a drawing. The out-of-plane eccentricity is a dimension nobody draws — it is the thickness of the thing the beam sits on — and it produces the larger stress on a great many real brackets. A calculation gets attention in proportion to how visible its inputs are, which is not the same as in proportion to how much it decides.