The angle that uses half of itself

An angle is a cheap and useful tension member and it has one awkward property: it is not symmetric, so bolting it through both legs requires two lines of bolts on two planes, and nobody does that. It is bolted through one leg.

The force then has to get from the bolts, which are in one leg, into the whole section, which includes the other leg. Over a long enough connection it does. Over a short one it does not, and the outstanding leg spends the connection at a lower stress than the connected one.

What the correction is #

The reduction is written as an efficiency factor on the net area:

$$A_{\text{eff}} = U A_n \qquad\text{with}\qquad U = 1 - \frac{\bar{x}}{L_c}$$

where $\bar{x}$ is the distance from the connected face to the centroid of the whole section, and $L_c$ is the length of the connection — first bolt to last, or the length of the weld.

Both quantities are lengths on a drawing. There is no material property anywhere in it, no yield stress, no modulus. This is geometry beating material in its purest form: two distances, measured, divided.

Why those two lengths #

The formula reads oddly until the mechanism is clear, and the mechanism is Saint-Venant’s principle wearing a different hat.

The force enters through the bolts, in one plane. Immediately at the first bolt, the stress in the section is concentrated in the connected leg and is nearly zero in the outstanding one. As the force travels along the member it spreads, by shear across the section, until the stress is uniform — which is exactly the settling that Saint-Venant’s principle describes, and it takes a similar distance.

Two things follow directly.

The spreading distance scales with how far the force has to travel across the section, and the measure of that is $\bar{x}$, the offset between where the force enters and where the section’s centre of resistance is. A section whose centroid is close to the connected face barely has to spread at all.

The spreading has $L_c$ to happen in. If $L_c$ is large compared with $\bar{x}$ the force arrives fully spread and $U \to 1$. If they are comparable, it does not.

So $U = 1 - \bar{x}/L_c$ is a ratio of the distance the force has to move across the section to the distance it has along the member to do it in. That is a real, if crude, model, and it is the reason a formula fitted to test data has the shape it has.

The numbers, which are worse than expected for short connections #

A two-bolt connection throws away a quarter of the member. That is a large number for a detail that would be drawn without hesitation, and it is the reason minimum bolt counts appear in codes for angles in tension.

It is worth noticing what the table is a table of. Every row is the same angle: same steel, same area, same net area, same holes. The only thing that varies down the column is how far apart the first and last bolts are — a dimension that belongs to the gusset plate rather than to the member, and that would ordinarily be settled by whoever was fitting the connection into the space available. The member’s capacity is being decided by a length nobody thought of as structural, which is the sentence this whole field turns on appearing for the seventh time.

Notice the shape of the curve. The gain from two bolts to three is 13 percentage points; from four to six it is 3.5. The return on connection length is steeply diminishing, which means the design decision is nearly always at the short end: getting from two bolts to three or four is worth a great deal, and going beyond that is not worth the connection length it costs.

Which leg to connect through #

The centroid offset $\bar{x}$ is measured from the connected face, so an unequal angle gives two different answers depending on which leg the bolts go through.

For the 100 × 75 × 10 angle: connected through the long leg, $\bar{x} = 19.8$ mm. Connected through the short leg, $\bar{x} = 32.3$ mm — because the centroid is now further from the plane the load enters on.

At three bolts that is the difference between $U = 0.868$ and $U = 0.785$. At two bolts it is 0.736 against 0.570, which is to say that the same angle with the same bolts loses 26% one way round and 43% the other.

Connect through the long leg. It is free, it is a detailing decision rather than a structural one, and it is worth more than a bolt.

The reason it is easy to get wrong is that the other choice often looks tidier. Bolting through the short leg leaves the long leg standing proud, which fits neatly against a chord or a gusset edge, and it is the arrangement that draws well. Nothing about the drawing suggests that the angle has just lost a further eight percentage points of its own area at three bolts, or seventeen at two — where the total loss reaches forty-three per cent.

The eccentricity that comes with it #

There is a second consequence of connecting through one leg, and it is separate from shear lag although it has the same cause.

The bolts are in one leg; the member’s centroid is 19.8 mm away. So the force in the member and the force in the connection are not collinear, and the difference is a moment of $P\bar{x}$ applied to the member at each end.

For a tension member that moment is usually shrugged off, and reasonably: the member straightens under load, the eccentricity reduces, and the tension stabilises it — the opposite of what happens to a compression member, where the same initial eccentricity is amplified rather than relieved.

For a compression member connected the same way, it is not shruggable at all. An angle strut bolted through one leg is loaded at an eccentricity it cannot straighten out of, and the design of single-angle struts is essentially the design of a beam-column — which is why their capacities in tables look so poor compared with their areas, and why the tables exist rather than a formula.

The same drawing therefore carries two different penalties depending on which way the load goes: a modest area reduction in tension, and a substantial capacity reduction in compression. Neither is visible on the drawing, and the drawing is often reused for both.

Where else the same lag appears #

The mechanism is not specific to angles, and recognising it elsewhere is the useful part.

A wide flange member connected by its flanges only. The web is the outstanding element, and the force has to spread into it from the flanges. Same formula, $\bar{x}$ measured from the flange face to the section’s centroid.

A welded connection along two edges only. No bolts, so $L_c$ is the weld length, and the same reduction applies. A transverse weld across the full width has $U = 1$, because the force enters everywhere at once and has nothing to spread.

A long bolted lap joint, where the same lag runs along the joint rather than across the section: the plates stretch between bolts, so the end bolts take more load than the middle ones. This is the effect the bearing essay meets as the long-joint reduction, and it is the same physics with the axes exchanged.

A wide flange in a box girder, where the flange is so wide relative to its span that the parts far from the webs never reach full stress. The correction there is called an effective width, and it is shear lag with a different name and the same cause.

That last one is worth flagging because it is the case where the effect is not a connection matter at all. Shear lag is a property of how load spreads across a section, and connections are simply where the spreading is most abrupt.

Two reductions that multiply #

For the angle above: $A_n$ is 86.7% of gross after one 22 mm hole is deducted from 1,650 mm², and $U = 0.868$, so the effective area is $0.867 \times 0.868 = 0.752$ of the gross section.

Taking the worse of the two would give 0.867, which is 15% optimistic — and 15% optimistic on a tension member is a great deal more serious than 15% optimistic on most things, because a tension member has no plastic reserve to speak of once its net section has ruptured.

The welded case, and the one detail that removes the problem #

Welding an angle to a gusset gives a designer something bolting does not: control over where the weld goes, and therefore over both variables at once.

Run the welds only along the connected leg and nothing has changed — $\bar{x}$ is the same, $L_c$ is the weld length, and $U$ is computed the same way.

Add a transverse weld across the end of the angle and the picture changes qualitatively. Now some of the force enters the section across its whole width simultaneously, with nothing to spread. The portion of the load that enters that way has $U = 1$, and the correction applies only to the rest.

Weld the outstanding leg as well, where the gusset geometry allows it, and the problem disappears altogether: the force enters both legs, there is nothing to spread into, and $\bar{x}$ has stopped being the relevant distance because the load is no longer entering on one plane.

That last option is the reason welded angle connections are so often detailed with a return onto the second leg where there is room, and it is a good example of the general shape of this field: the detail is not a consequence of the analysis, it is the variable the analysis is about. A bolted connection has to accept the eccentricity because bolts have to be reachable from one side. A welded one does not, and the choice is made on the drawing.

The check this competes with #

As always with a tension member, the effective area only matters if net rupture is the governing limit. The competition is with gross yielding, at $f_y A_g$, which knows nothing about holes or connections.

For a 1,650 mm² angle in S275: gross yield gives 454 kN. Net rupture with the effective area gives $430 \times 0.752 \times 1650 / 1000 = 534$ kN. Gross yielding governs, comfortably, and neither the holes nor the shear lag is the answer.

Change to a two-bolt connection and net rupture becomes $430 \times 0.867 \times 0.736 \times 1650/1000 = 453$ kN — against a gross yield capacity of 454. The two land within one kilonewton of each other, which is to say that at two bolts the connection has taken the member’s capacity down to exactly the point where it stops being a member calculation at all.

That crossover is the practical content of the whole essay. Shear lag rarely governs, and when it does it is because the connection was made short. Three bolts instead of two moves the answer back to gross yielding and costs 75 mm of gusset plate.

What a formula fitted to tests is allowed to be used for #

$U = 1 - \bar{x}/L_c$ is, like the $s^2/4g$ term in the last essay but one, a fit rather than a derivation — and the two sit close enough together in the same calculation to be worth comparing as pieces of evidence.

The difference between them is that this one has a mechanism attached. Saint-Venant’s principle really does say that a disturbance settles over a distance comparable to the scale of the disturbance, and $\bar{x}$ really is the scale of this disturbance. So the formula’s shape is derived even though its calibration is not, and that matters for how far it can be trusted outside the range it was fitted in.

A formula with a mechanism can be extrapolated cautiously: if $\bar{x}$ and $L_c$ are both larger by a factor of three, the physics is unchanged and the ratio is what matters. A formula without one cannot be extrapolated at all.

Two limits are worth stating anyway. $U$ has to be capped at 1, which the formula does not do on its own — a very long connection gives $\bar{x}/L_c$ near zero and that is right, but nothing stops somebody entering a negative $\bar{x}$. And $U$ is meaningless for $L_c$ below about $\bar{x}$, where the formula returns zero or a negative number: a single-bolt connection has $L_c = 0$ and $U = 0$, which is not a capacity of nothing but a statement that the model does not apply. Single-bolt connections are covered by their own rules, and the reason is exactly this.

What to take from it #

A member’s capacity depends on its connection, and not only through the holes. A short connection cannot reach the whole section, and the part it cannot reach is not carrying its share.

Both variables are lengths. The centroid’s distance from the connected face, and the connection’s own length. No material property appears in the correction at all.

The return on length is steeply diminishing. Two bolts to three is worth 13 percentage points; four to six is worth 3.5. The decision is always at the short end.

Connect through the long leg. It halves the loss, it costs nothing, and it is a drawing decision made by somebody who may not know it is a structural one.

And it usually does not govern, which is exactly why it is worth checking. Gross yielding wins comfortably at three bolts and loses by one kilonewton at two. A quantity that is irrelevant at one detail and decisive at the next, with nothing in between to signal the crossing, is the kind that gets left out of a check list and stays out until the day the connection was drawn short.