The corner that is not the worst point

A bracket is welded to a column rather than bolted to it. The load is 100 kN at 150 mm from the weld group’s centroid, which is the same problem the bolt group faced, with one difference: the fasteners are not points any more. The weld is a line, and the stress varies continuously along it.

The standard rule for finding the critical point is short and memorable. Check the point furthest from the centroid.

The calculation, which is the bolt group with an integral #

The mechanics is the same as for bolts and the bookkeeping is different in one respect: everything is per unit length of weld, and the group’s properties are integrals along the line rather than sums over points.

Treat the weld as a line of unit throat. Then:

• the area is the total length, $A = \int \mathrm{d}\ell$;
• the centroid is $\bar{x} = \frac{1}{A}\int x \,\mathrm{d}\ell$, and likewise for $y$;
• the polar second moment is $J = \int (x^2 + y^2)\,\mathrm{d}\ell$ about that centroid.

Shift the load to the centroid, adding a torque $T = Pe$. Then at any point on the line the stress per unit throat is the vector sum of

$$f_{\text{direct}} = \frac{P}{A} \qquad\text{and}\qquad f_{\text{torsion}} = \frac{T r}{J}$$

with the first in the load’s direction everywhere and the second perpendicular to the radius from the centroid.

The C-shaped group above has $A = 360$ mm, a centroid 17.8 mm from the back of the group, and $J = 2{,}494{,}198$ mm⁴. The peak stress is 0.885 kN per millimetre of throat, at the bottom corner.

Why the weld line has no thickness in the arithmetic #

The unit-throat trick above deserves a note, because it is one of the tidier pieces of bookkeeping in the subject and it is easy to use without noticing what it buys.

The weld’s actual throat appears nowhere in $A$, $\bar{x}$ or $J$. Every one of those is computed for the line at unit throat, which makes them pure geometry — properties of the shape of the group rather than of any weld deposited on it.

That is legitimate because the throat is constant round the group and therefore factors out of every integral. The stress at any point is then a force per unit length divided by the throat, and the throat can be chosen at the end:

$$a_{\text{required}} = \frac{f_{\text{peak}}}{f_w}$$

with $f_w$ the weld’s capacity per unit area. So the whole analysis is done once for the shape, and sizing the weld is a division at the end.

The payoff is that the group’s properties can be tabulated by shape, which is exactly what the older design handbooks did: a page of weld group configurations with $A$, $J$ and the critical point for each, ready to be scaled. Those tables are still in circulation, and the critical points in them are precisely the ones this essay is questioning — computed under the radius rule, for shapes chosen because they were symmetric.

The same factorisation does not survive an unequal-leg fillet or a group that changes size round its perimeter, and both occur: a bracket welded with a heavier run along its loaded edge is a common and sensible detail. There the throat is inside the integrals, $J$ is weighted by throat, and the centroid moves. It remains a two-line calculation and it is not the one in the tables.

The rule, and the shapes it is exact for #

It is worth being clear that the standard rule is not a bad rule. Three of the five shapes tested here put the peak exactly at the largest radius.

A plain vertical run. Peak 1.1995 at the top of the line, which is the furthest point. Correct.

A C-shape. Peak 0.885 at a bottom corner, which is one of the two furthest points. Correct.

A deep C — a long vertical with short returns. Peak at the corner, which is furthest. Correct.

What those three share is a symmetry about the axis of the load. When the group is symmetric about the vertical line through its centroid and the load is vertical, the direct and torsional components combine in the same way at the two extreme points, and the extreme radius is the extreme stress.

That is a real and common configuration. It is also exactly why the rule is dangerous.

The shape where it is wrong #

The asymmetry does it. With only one return, the centroid moves towards that return, and the geometry of the two extremes stops being a mirror image. At one of them the direct and torsional vectors are nearly aligned; at the other they are inclined. The larger radius no longer wins.

Fifteen per cent is not catastrophic and it is a real underestimate on a mechanism that is not especially ductile.

That last clause is the part that turns an arithmetic error into a design problem. A weld group does not redistribute the way a bolt group does: a bolt in bearing has millimetres of ovalling available before anything fractures, and a fillet weld at its throat capacity has a fraction of a millimetre. So the argument that saves the elastic bolt calculation — the worst fastener reaches its limit and the others take up the slack — is not available here in anything like the same measure.

Which means the peak is the answer rather than the beginning of one. There is no plastic reserve behind a 15.6% underestimate, and the whole group is only as good as the point the calculation was supposed to find.

The shape where the rule cannot even choose #

The sharper failure is not that the rule picks the wrong point. It is that the rule sometimes cannot pick at all.

That is the finding worth carrying out of this essay, because it says why the rule fails rather than that it does.

Only one of the two components depends on position. The direct shear $P/A$ is the same vector at every point on the group — same size, same direction, everywhere. The torsional shear $Tr/J$ has a magnitude that depends on radius and a direction that rotates with the point, always perpendicular to its own radius.

So at one end of the group the two vectors point roughly the same way and add. At the other end the torsional vector has swung round and they partly cancel. Two points at identical radius therefore have identical torsional magnitudes and different resultants, and no rule expressed in terms of radius can see the difference.

The box section shows the same effect more mildly: four points at maximum radius, spread of 1.297.

The comparison the bolt group gets and this one does not #

It is worth setting the weld group beside the bolt group at this point, because the two problems are the same and the available answers are not.

So the bolt group’s elastic calculation is conservative and known to be, with a route to the extra capacity when the ductility is there. The weld group’s calculation is not conservative in the same way; it is simply the answer, and the only question is whether the right point was checked.

That asymmetry is why this essay is about where the peak is rather than about how much of it can be redistributed. There is no second method waiting behind the first.

Why a sometimes-exact rule is worse than an always-approximate one #

There is a general point here that reaches past welding.

A rule that is always approximate gets used with a margin. Everybody knows it is approximate, the error is bounded, and the calculation carries that knowledge with it.

A rule that is exact for the common case and wrong for the uncommon one gets used without a margin, because everyone who has checked it has checked it on the common case and found it exact. Its failures sit in exactly the configurations nobody validated it against — and the configurations here are not exotic: an L-shape is what a return run along one edge only produces, which happens whenever a bracket meets an obstruction.

This site has met the shape before. The count that does not see it is a rule — $m + r = 2j$ — that is exactly right for every frame anybody draws to check it, and silently wrong for the arrangements that defeat it. Euler’s formula is exact for a perfectly straight column and describes no real one. In both cases the rule survived because the cases used to test it were the cases it was derived on.

The scarce thing is not accuracy; it is a test that could have failed.

Which is why the figures here mark every point at maximum radius rather than only the peak. The rule’s own prediction and the actual answer are both on the drawing, and a reader can see whether they coincide instead of being told that they do.

What the shapes cost, compared #

Set the four groups against each other on the number that matters and the design guidance falls out without any argument.

The C-shape carries 6% more weld than the L and has a peak 23% lower. That is not a small return, and it comes from where the extra weld went: the second return is at the largest radius the group has, so it contributes to $J$ far more than the same length placed anywhere nearer the centre.

Which is the second moment of area argument once more, in a fourth setting. Material far from the axis does nearly all the work, whether the material is a flange, a bolt or a run of weld — and the quantity is the same integral each time.

The box goes further and the return is smaller per millimetre: 56% more weld than the C for a 40% lower peak. Diminishing, because the fourth run is between the other three rather than outside them. There is nothing left at a larger radius to put weld at, and adding it nearer the centroid is buying $A$ rather than $J$.

So the design sequence is: get the group as far across as the geometry allows, close it where that is possible, and only then decide the throat. The throat is the last decision and the cheapest one to change.

What to do instead #

The honest method is to evaluate the resultant everywhere and take the maximum, which is what the figures above do at 360 sample points. That is trivial arithmetic for a computer and tedious by hand, which is presumably why the radius rule exists.

By hand, the workable compromise is to check every corner and every free end rather than the furthest point. Those are where the direction of the weld changes or stops, and the peak of a resultant that varies smoothly along a line is always at a corner, an end, or a stationary point in between — and for these groups there is no interior stationary point, because both components vary monotonically along each straight run.

That gives four checks instead of one for an L-shape, and it is exact.

The other practical answer is to make the group symmetric about the load, which is free at detailing time and removes the problem entirely rather than managing it. It is also better for the magnitude: symmetry puts the centroid where the two extremes are equivalent, and the peak is lower for the same total weld length.

That is a rare kind of design move and worth naming as one. Most decisions in this field trade a capacity against a cost — more weld, more plate, more bolts. This one trades nothing: it changes the arrangement so that the rule everyone will apply to it becomes correct, at no expense in material. A detail that makes the standard check exact is worth more than a detail that is marginally stronger and needs an unusual calculation to prove it, because the second one has to survive being checked by somebody who will use the rule.

The direction the peak points, which nobody records #

There is a loose end that connects this essay back to the last one and is almost always dropped.

The resultant at the peak has a direction as well as a magnitude, and a fillet weld’s capacity depends on the angle between the load and the weld’s own axis by a factor of up to 1.2247. So a full directional check needs, at every point, the angle between the resultant there and the run of weld there.

On the C-shape the peak is at a bottom corner where the weld runs horizontally and the resultant is inclined — so the transverse component is real and some enhancement is available. On the vertical run the peak is at the top, the weld runs vertically and the resultant is very nearly vertical too, so there is essentially none.

Nobody does this. The universal practice is to compute the resultant and compare it against the longitudinal capacity, which is the simple method and is always safe. The reason is not laziness: the angle varies continuously round the group, the enhancement is only worth collecting near the transverse limit, and the arithmetic of tracking it is out of proportion to the return.

It is worth knowing that the conservatism is there, though, because it is the reason a weld group that fails its check by ten per cent is not necessarily inadequate — and because it explains why weld groups so often check out in tests at loads well above their computed capacity, which is a discrepancy that otherwise invites the wrong explanation.

What to take from it #

The stress in a weld group is a vector sum in which only one term rotates. That single fact accounts for every discrepancy in this essay, and it is invisible in any formula written in terms of $r$.

The furthest-point rule is exact for a group symmetric about the load’s axis, and only for that. Measured here: exact for three shapes, 15.6% low for one, and unable to distinguish four points differing by 66% for another.

Check corners and free ends, not radii. Four checks by hand, exact for every group in this essay, and it costs nothing at all.

And a rule that is exact on the cases it was tried on is not a validated rule. It is a rule whose failures are still ahead, filed under the geometries nobody has needed yet.

There is no plastic reserve behind the answer here. A weld group cannot redistribute the way a bolt group can, so the difference between the peak and the point the rule found is the difference between the capacity and what was checked — not the start of an argument about ductility.