The radius rule, and where it fails

A weld group’s worst point is not always its corner, and the reason is that the check everybody makes is about a distance while the quantity being checked is a vector. Distance is scalar and monotonic; a vector sum is neither, and the whole of this page is the difference between them.

What the two components are #

The load is moved to the centroid and replaced by a force plus a couple, and each produces its own throat stress.

The direct component is the force divided by the total throat length — the same magnitude everywhere in the group, in the direction of the load.

The torsional component is $Tr/I_p$, with $r$ the distance from the centroid and $I_p$ the polar second moment of the weld line. It grows with the radius, and its direction is perpendicular to the radius.

The resultant is the vector sum, and the peak is where that sum is largest. Since the two components point in different directions at different places, the sum’s maximum need not be at the largest $r$.

The radius rule is a shortcut that assumes the direction does not matter, and its accuracy depends entirely on the shape.

It is worth writing the two components out, because the arithmetic is short and the shortcut’s failure is visible in it.

Put the centroid at the origin, the load $P$ at an eccentricity $e$, and the total weld length $L$ with polar second moment $I_p$ about the centroid. At a point $(x, y)$ on the weld line:

$$f_{\text{direct}} = \frac{P}{L}\ \text{(in the load's direction)}, \qquad f_{\text{torsion}} = \frac{Pe}{I_p}\,(-y,\ x)$$

The torsional term’s magnitude is $Per/I_p$ and its direction is perpendicular to the radius vector — which is where the $(-y, x)$ comes from and is the whole of what the radius rule discards.

The resultant’s magnitude is the norm of the sum, and a norm is not monotonic in either argument. Two points can have identical $r$ and different resultants because the angle between the two vectors differs; a point at smaller $r$ can beat one at larger $r$ if its angle is more favourable. Both happen on the shapes above.

The reason the rule survives is that it is exact for the shapes people mostly draw. Symmetry about the load’s line makes the two components orthogonal at the extreme point, and then the resultant is $\sqrt{f_d^2 + f_t^2}$, which does increase with $r$.

Which free body produced the number #

The free body is the weld group taken as a line rather than as an area, and that idealisation is what makes the calculation possible.

Cut the connected plate away and replace the weld by a distributed force per unit length along its line. Equilibrium requires that force to sum to the applied load and to have the right moment about the centroid, and the elastic assumption — that the plate rotates rigidly about the centroid and the weld’s force is proportional to the local displacement — closes the system.

The throat’s own thickness never enters. The polar second moment is computed on the line, and the answer comes out in force per unit length of weld rather than as a stress. Dividing by the throat at the end turns it into one.

That is why weld group properties are tabulated per unit throat: the whole calculation is a geometry of a line, and the throat is a scale factor applied afterwards. The throat and its direction are a separate question, answered at the point the radius rule found.

Where the rule is exact and where it is not #

Four shapes make the pattern clear.

The pattern is about symmetry.

A group symmetric about the load’s line of action has its peak at the maximum radius, because the two components are perpendicular there and the sum is a clean hypotenuse. The C-shape and the box are of this kind, and the rule is exact.

A group with no such symmetry has its peak elsewhere, at a point where the two components are more nearly aligned than they are at the extreme. The L-shape is the ordinary case, and the error is one-sided: the rule is always low, never high.

So the rule is unconservative exactly where a designer cannot see that it is, which is a poor arrangement and is why the safe practice is to check several points rather than one.

Where the peak moves to #

The eccentricity changes the balance between the two components and can move the peak.

Hold the shape and move the load further out instead, so that the torsional component grows against a direct component that does not.

In the limit of a very large eccentricity the group is in pure torsion, the direct component is negligible, and the peak is at the maximum radius by construction. The radius rule becomes exact as the eccentricity grows and is worst near the concentric case.

That is the opposite of what intuition suggests. A designer worried about an eccentric connection reaches for the radius rule at exactly the eccentricities where it is safest, and applies it without thought at the small ones where it is not.

What to check instead #

The practical answer is not to abandon the rule but to bound it, and three habits do that.

Check every corner and every free end. The peak on an asymmetric group is at a re-entrant corner or at the end of a run, never in the middle of a straight length, so the candidate set is small — usually four or six points — and computing the resultant at each is arithmetic rather than analysis.

Notice when the group has no axis of symmetry through the load. That is the whole of the diagnostic. An L, a Z, a single run with a return, a group with one weld omitted for access: each is a shape where the rule is low, and each is common because it is what fits round an obstruction.

And treat the error as one-sided. The rule never over-estimates the peak, so a group checked by it and passing comfortably is fine; one passing marginally needs the corners checked. That is a cheap rule and it puts the effort where it matters.

There is a fourth habit that is better than all three and is available on any machine: evaluate the resultant at every point of the line and take the maximum. The expression is two lines long, the line is a parameter, and the whole ambiguity disappears. That the profession still uses a rule about a radius says more about the age of the tables than about the difficulty of the calculation.

What the elastic method is leaving on the table #

Everything above is an elastic analysis, and there is a second method that gives a different answer.

The elastic method assumes the group rotates about its centroid. The instantaneous centre method assumes it rotates about the point that makes the fastener forces balance the applied load, with each fastener at its own load–slip capacity rather than in proportion to its distance.

Finding the point a mechanism turns about is the same construction, and the answer is always a higher capacity because the elastic assumption forces a distribution the group need not adopt.

For bolts the difference is used; the instantaneous centre method is in the codes and the tables are published. For welds it usually is not, and the reason is worth stating: a fillet weld’s load–deformation behaviour depends on the angle between the force and the weld axis, so the instantaneous centre calculation for a weld needs a different capacity at every point of the group. It is done, and it is done by a computer rather than from a table.

Why the two methods disagree by a varying amount #

The gap between the elastic answer and the instantaneous centre is not a constant, and understanding why says when it is worth chasing.

At zero eccentricity the load is concentric, every fastener carries the same share under both methods, and the two agree exactly.

At large eccentricity the group is in nearly pure torsion. The elastic distribution — force proportional to radius — is then close to the plastic one, because a fastener far from the centre both carries more elastically and is asked for more plastically, and the two converge again.

In between the gap opens, because the elastic method forces the direct and torsional components to be shared in one fixed proportion while the plastic one lets the group choose the rotation centre that suits it. The 13.87 per cent at 150 mm is near the widest part of that curve.

That gives a useful reading of when to bother. A connection with a small eccentricity relative to its group size has the most to gain, and it is also the connection a designer is least likely to think of as an eccentric one — a fin plate with the bolt line 60 mm from the weld, a cleat with the load 80 mm out. Those are the ones where an elastic check is most conservative and where the conservatism is most often simply accepted.

What the redistribution needs #

The instantaneous centre method is a plastic argument, and it needs what plastic arguments always need.

The group has to be able to reach the assumed state, which means the most heavily loaded fastener must deform enough for the others to catch up without losing its own capacity. For a bolt in bearing that is generous — several millimetres of hole elongation at nearly constant load. For a fillet weld it is a fraction of a millimetre, and the deformation capacity depends on the loading angle: a weld loaded transversely is stronger and less ductile than one loaded longitudinally.

So the same property that makes a transverse weld stronger makes the group’s redistribution less available, which is an unhelpful pairing and is why weld groups are more often designed elastically than bolt groups are.

The practical consequence is a rule of thumb worth carrying. An elastic weld group check is conservative by 10 to 30 per cent and the reserve should not be spent, because the mechanism that would deliver it is the one the weld is least able to provide.

The check that comes after #

Finding the peak point is half of the work, and what is done with it is the other half.

So the resultant found by the vector sum has to be resolved again, this time relative to the weld’s own axis, and compared against a capacity that depends on that angle. A fillet weld is stronger across than along by 22 per cent, and the peak point’s resultant is at some angle in between.

That produces a mild irony. The point with the largest resultant is not necessarily the point with the highest utilisation, because a slightly smaller resultant arriving at a more favourable angle can be further from failing. Strictly the check is on the ratio of resultant to directional capacity, point by point — which is one more reason the single-point radius rule is a simplification of a simplification.

The simplified method most codes offer resolves the issue by ignoring the direction entirely and using the along-the-axis capacity everywhere. That is conservative by up to 22 per cent, it removes the ambiguity, and it is what almost every real connection is designed by.

What to carry away #

The peak is a vector sum, not a radius. The direct and torsional components point in different directions, and the sum’s maximum is not always where either component’s is.

The radius rule is exact for a symmetric group and low for an asymmetric one, by around nine per cent on an L-shape — and it is unconservative in the case a designer cannot spot.

Equal radius does not mean equal stress. On a box group two points at the same distance differ by 1.23.

And the elastic method leaves capacity unclaimed. The instantaneous centre gives more, by an amount that varies with the eccentricity, and for welds the reserve is real and hard to justify spending.

The bolt group’s version of the same story #

The bolt group is worth setting beside the weld group, because the two calculations are identical and their design treatments have diverged.

A bolt group under an eccentric load has the same two components, the same vector sum and the same radius rule — with a sum over discrete fasteners in place of an integral over a line. Everything on this page applies to it unchanged.

What differs is that the bolt group’s plastic reserve has been measured. A bolt in bearing deforms several millimetres at nearly constant load, the redistribution is real, and the instantaneous centre method has been in the codes with published tables since the 1960s. A designer of bolt groups routinely claims 10 to 30 per cent that a designer of weld groups does not.

The comparison is instructive because the mechanics is the same and the difference is entirely in the ductility of one component. A method is available when the material will deliver the state it assumes, which is the same condition every plastic argument in this collection carries, and here it separates two connections that look alike on a drawing.

There is a practical corollary. A connection with both bolts and welds sharing one load cannot be designed by adding their capacities, because they reach them at different deformations — the weld is at its capacity while the bolts are still taking up their clearance. Codes forbid the sharing outright, and this is why.

Where the model stops #

The weld is treated as a line of zero width. Its properties are computed on the centreline of the throat, which is standard and is an approximation for a group whose members are short relative to their throat.

The connected parts are rigid. They are not, and a flexible plate does not rotate about the weld group’s centroid — which is a larger error than anything the radius rule makes, and one nobody quantifies.

The direction of the resultant is used only to find the peak. Whether the weld can carry that resultant depends on the angle it makes with the weld axis, and the directional strength is a separate check made afterwards at the point found here.

Only in-plane loading is covered. A group loaded out of its plane has a different set of components and a different rule, and most real connections have both.

The group is assumed to carry all of the load. In a real connection a fin plate, a seating cleat or a bolt group carries some of it, and the split between them is a stiffness question the weld group calculation never asks.

And the throat is assumed constant. A weld with a varying leg, a start or a stop within the group’s length has a different line and a different polar second moment from the one drawn.

One last observation, about what makes the error survivable. The radius rule is low by nine per cent on the shape drawn, and a fillet weld group is almost never designed to within nine per cent of anything — the leg size is a whole number of millimetres, the length is a round figure, and the run is usually made continuous for fabrication rather than sized. The rule’s error is smaller than the granularity of the thing it is sizing, which is the honest reason it has never caused trouble and is not a reason to keep using it now that evaluating the whole line costs nothing.

The group’s geometry decides more than the rule does. A weld is stronger across than along, so the direction of the stress at the critical point matters as much as its magnitude; a bolt group under the same eccentric load distributes it by a different rule with the same failure of intuition; and the shape that comes out of the shop is what the welding does to the member before any of this is checked.

The ladder from here #

Later rungs on this anchor: the instantaneous centre for welds, with the directional load–deformation relation that makes it a computer calculation. Out-of-plane eccentricity, where the group is in bending as well as torsion. The weld group with a longitudinal load, where the two components align and the rule is exact for a different reason. Welds in combination with bolts in one connection, and why sharing between them is not permitted. Weld group fatigue, where the peak point rather than the average decides the category. And the connected plate’s own flexibility, which is the largest omission in the model and is where a finite-element analysis changes the answer rather than confirming it.

The elastic weld group method is a direct transcription of the elastic bolt group method, which is itself an application of the torsion formula to a set of discrete points. What is unusual is how long the transcription has survived unexamined: a bolt group’s plastic reserve was measured and codified in the 1960s and a weld group’s has not been, on connections that are more numerous and whose reserve is smaller.